Selected Papers of J Robert Schrieffer: In Celebration of His 70th Birthday (World Scientific Series in 20th Century Physics, V. 30)

1184

BARDEEN,

COOPER,

energy with respect to the distribution function for excited particles, / , and ground pairs, h. I t turns o u t that hk is a function of temperature and therefore excited and ground pairs are not necessarily orthogonal t o each other a t different temperatures, although the excited states form a complete orthonormal set at each temperature. The most probable distribution of single particles also varies with temperature and thus the great majority of states contributing to the free energy a t different temperatures will be orthogonal in any event. The situation is similar to taking the lattice constant temperature dependent as a result of thermal expansion. This freedom in choosing k* allows us to work in that representation which minimizes the free energy a t the specified temperature. A typical excited state wave function can be written as the projection of

AND

SCHRIEFFER

and p refer t o excited electrons; below the Fermi surface, h>\ and j and p refer to holes. The diagonal element of » k „ follows immediately from (3.3), (3.4), and (3.5). Including the factor giving fractional number of configurations for which each decomposition applies, we have (* I » * , | * ) = (1/2) J k + M l - A k ) + (l-sk-/>„)/(„, t > 0 ; (*|l-»k.|*)=(l/2)*+Mk + (l-sk-pk)(l-kk),

(<0.

Upon using (3.7) and the fact t h a t 1—Ak(—t) = hi{t), the Bloch energy contribution to W,

(H T. 'k»k„+ T. U k | ( i - » k , ) l * ) , )t>kpa

(3.8)

k
becomes

*„c= n [ ( I - A O » + * * V ] n [ ( i - M ^ - * k(s) k'(5>)

W r D-ZI*k|C*k+2# k +2(l-* k -2#0* k (|€ k |)l (3.9) k

where we have carried out the spin sum. T o calculate the matrix elements of t h e pair interaction operator X! Vkt-bk^bk, we assume t h a t I7kk< onto the space with N pairs, where g, <$>, and S specify varies continuously with k and k' so t h a t Vw may be the states occupied by ground pairs, excited pairs, and considered to be the same for all transitions from states single particles respectively and c (k »)* denotes either in Ak to states in Ak'. L e t k and k ' represent two k"f or — k " | is included in the product. For any speci- specified wave vectors in Ak and Ak'. T o obtain nonfied k, this function can be decomposed into a portion vanishing matrix elements for &k<*6k, these states must with k occupied and a portion with k unoccupied. The be occupied b y either excited (—) or ground ( + ) pairs, decompositions for the three cases in which k is in the giving the four possibilities + + , -, H , 1sets 9,

and S are for k and k', respectively. For any one of these cases, a typical wave function may be decomposed into comGround: ponents in which the pair occupancy of k and k ' is ¥ = *„**!(• • l k - • - ) + ( l - f c k ) W - • -0 k - • •), (3.3) specified:

-**••: n e(k")**o, (3.2) k"(S)

Excited: * = ( l - A k ) V i ( - • - U - • - ) - A k V o ( - • -0 k - • •), (3.4) Single in kf: *=tfkt*#>o(---Ok---),

(3-5)

where the
* k , k ' = aiioi(" 'Ok- • - l k ) +aoo*>oo(---0 k ---0 k .---),

(3.10)

where the
where W is the energy calculated by a n ensemble average over the wave functions of the form (3.2) and lh{\-h)h'(l-h')J{{\- -p){\-s'-p') S 5 is the entropy. +pp'-(\-s-p)p'-p(l-s'-p')) To enumerate the systems in the ensemble we divide = Zh(l-h)h'(l-h')2i Ai-space into cells Ak containing 3l k pair states as before. X{(ls-2p)(l-s'-2p')}. (3.11) Let there be 5 k single particles and P k excited pairs in Ak, with the rest of the 3l k states being occupied by Introducing these matrix elements into the ensemble ground pairs. The probability that either kf or —kj average of the interaction Hamiltonian, we find is occupied by a single particle is J k = S k /9l k , while the probability for an excited pair in state k is p*.=P*/3l* Wj=~ E Vw[hy(1-*„)*„'(1-Ak<)]* k,k' and therefore the probability for a ground pair is (1 — Si—pi). Above the Fermi surface, h<% and s X{(l-ik-2/>k)(l-jk— 2pk,)}. (3.12)

THEORY

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1185

TABUS I. Coefficients for the decomposition of * according to Eq. (3.10). Wave function k k'

+ +

Fractional No. of cases

(l-s-p)(l-s'~p') pp' (l-s-p)p> p(l-s'-p')

+ +

[(l-AHl-A')]' [A(l-A')]» [(1-*)*'?

It should be noted that the Bloch energy (3.9) and the interaction energy (3.12) depend on (s+2p) or the total occupancy probability. The energy does not depend upon the relative probability for single-particle and excited-pair occupation. Thus one may use a distribution function / which gives the over-all probability of occupancy, where (3.13)

*k-2/k(l-/0,

[A(l-A'):1

II(1-*)(1-A')? -[(1-*)A']»

C(I-A)(I-A'):*

of the form *k-i[l-(«i/£k)l

(3.19)

and [Ak(l-*k):]*=W£t. (3-20) The energy £ k , a positive definite quantity, is denned as £ k = + (2)i,

(3.21)

where

^.= K E k { A k . ( l - M ] ! ( l - 2 W (3-22) (3.14) It will turn out that 2e0 is the magnitude of the energy which follows from the fact that sk is the probability gap in the single-particle density of states and therefore that either kf is occupied and — k | is empty or the the distribution of ground pairs is determined by the reverse and pk is the probability that both kf and — k^ magnitude of the gap at that temperature. When we minimize F with respect to / k , we find that are occupied. The free energy can be minimized directly with respect to Ak, pk, and s* without introducing / k 2 € k ( l - 2 A k ) + 4 £ k . F k k . [ M l - M A k ( l - A k ) ] * and one indeed finds that (3.13) and (3.14) hold. X ( l - 2 / k . ) + 2 W l n ( / k / ( l - / k ) ) = 0 , (3.23) Since the excited particles are specified independently for each system in the ensemble, the usual and using (3.19) through (3.22), we find that expression for the entropy in terms of J/ may be used J r*k2 to 2 i r S = 2 * r E k ' { / k ' l n / k . + ( l - / k . ) m ( W k < ) } . (3.15) -M/k/(l-/k))=0[—+—J=0£k, (3.24) and

Minimization of the Free Energy If the expressions (3.13) and (3.14) are introduced into (3.9) and (3.12), the free energy becomes

where fi = l/kT and £ k is a positive quantity. The solution for fk is A = — — = /(£*). e»*H-l

J?=2£|ck|C/k+(l-2/k)Ai,(l«k|)] k

- Z F k k'CA k (l-A k )A k -(l-A k -)] J k,k'

X{ ( 1 - 2 / 0 ( 1 - 2 / . - ) } - r . S .

(3.16)

When we minimize F with respect to Ak, we find that 2 e k - E F k k '[Ak'(l-A k 0;P(l-2/M k0 k'

(l-2A k )

x-CAk(l-A k )J

=0,

(3.17)

or CAk(l-Ak)]* _ ^ F k k . [ A k - ( l - A k . ) ] » ( l - 2 / k - ) — 2^ , l-2A k v 2 €k

(o.lo;

where the energy ek is measured relative to the Fermi energy and « k <0 for k 'kp. Assuming as before that the interaction can be replaced by a constant average matrix element —V, defined by (2.34) for |« k |
(3.25)

Thus the single particles and excited pairs describe a set of independent fermions with the modified dispersion law (3.21). For A>Aj?, / k specifies electron occupation while for ke0 for the electron and when «k—K>~~, then £k—>e0 for the hole or the corresponding electron energy—>—eB. Thus the new density of states has an energy gap of magnitude 2e0 centered about the Fermi energy. The modified density of states is given by dN(E) dE

dN(t) dt dt

dE

--N(py

E

(3.26)

(&-<•
which is singular at the edges of the gap, £ = e 0 . The total number of states is of course unaltered by the interaction. If the distribution functions (3.19) and (3.25) are introduced, the condition determining the energy gap

1186

BARDEEN,

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AND

SCHRIEFFER

Tc, the gap may be expressed as 10

«««»

eo=3.2*r c [i-(r/r«)]»,

~ " ~ ^ ^ \

oe

\.

CIS

\

Oi4

02 i

i

i

i

1

FIG. 1. Ratio of the energy gap for single-particle-like excitations to the gap at 7"=0°K vs temperature.

(3.22) becomes (dividing by to) 1

/•*»

dt

tanhC^(e8+e„2)*],

(eHto2)*

N(0)V~J0

(3.27)

where we have replaced the sum by an integral and used the fact that the distribution functions are symmetric in holes and electrons with respect to the Fermi energy. The transition temperature, Tc, is denned as the boundary of the region beyond which there is no real, positive eo which satisfies (3.27). Above Tc therefore, e0 = 0 and /(£*) becomes /(«k), so that the metal returns to the normal state. Below Te the solution of (3.27), to^O, minimizes the free energy and we have the superconducting phase. Thus (3.26) can be used to determine the critical temperature and we find 1

/•*" dt

= I N(0)V 0)K JJ„ 0

— tanhQ&e), €e

kTc=1.14hwexp

L

iV (0)7.1

(3. 28)

(3.31)

which has the form suggested by Buckingham.** It can be seen from the distribution functions that our theory goes over into the Bloch scheme above the transition temperature. As T—>TC, -Ek—»|«k|, and h* vanishes for k>kp and is unity for k kp and complete hole occupancy for kkF and excited holes for k
f dtlog(l+trl>>) *o = -i^N(0)(kTy.

(3.33)

With the aid of (3.25) the entropy in the superconducting state, (3.15) may be expressed as TS=4kT £ Dn(l+«-" £ k )+/9£k/k].

(3.34)

Replacing the sum by an integral and performing a partial integration, we find (3.29)

as long as kTc<£hji, which corresponds to the weakcoupling case discussed in Sec. II. The transition temperature is proportional to ha, which is consistent with the isotope effect. The small magnitude of Te compared to the Debye temperature is presumably due to the cancellation of the phonon interaction and the screened Coulomb interaction for transitions of importance in describing the superconducting state, and the resulting effect of the exponential. The transition temperature is a strong function of the electron concentration since the density of states enters exponentially. It should be possible to make estimates of the change in transition temperature with pressure, alloying, etc., from (3.29). A plot of the energy gap as a function of temperature is given in Fig. 1. The ratio of the energy gap at T=0°K to kT, is given by combining (2.36) and (3.28): (3.30) 2 £o /*r c =3.50. From the law of corresponding states, this ratio is predicted to be the same for all superconductors. Near

TS = 4N(0)f

de\-+E\f(£E),

(3.35)

where the upper limit has been extended to infinity because f(J3E) decreases rapidly for /3t>l. If (3.34) and (3.16) are combined with the distribution functions (3.19) and (3.25), the free energy becomes F,= -4N(G)f

dtEfiQE)

+mo)

l

d

V-l\-v'

(3 36)

-

which with the aid of (3.27) may be expressed as r2e2-Ho2l

r"

F.=-2N(O)J

^i

mm

-* ( o ) ( H[ i + G)7-4 (3-37) « M. J. Buckingham, Phys. Rev. 101, 1431 (1956).

THEORY

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The critical field is given by combining (3.31), (3.32), and (3.37): J

-=N(Q)(fay 8TT

i»

([<)]

)

1

1187

r/T c
(3.43)

H a # 0 [i -1.07 ( r / r „ ) 2 ] .

(3.44)

or

IT 2

N(0)(kT)'

(3

4-*f4vw

38)

A plot of the critical field as a function of (T/Tcy is given in Fig. 2. The curve agrees fairly well with the \—(T/Tcy law of the Gorter-Casimir two-fluid model,19 the maximum deviation being about four percent. There is good experimental support for a similar deviation in vanadium, thallium, indium, and tin; however, our deviation appears to be somewhat too large to fit the experimental results. The critical field at 7/=0 is ffo=[4T#(0)]*eo(0) = 1.75[4Ttf(0)]»*r,,,

(3.39)

where 2eo(0) is the energy gap at T=0 and the density of Bloch states N(0) is taken for a system of unit volume. A law of corresponding states follows from (3.39) and may be expressed as

This approximation corresponds to neglecting the freeenergy change of the superconducting state, the total effect coming from F„. The electronic specific heat is most readily obtained from the entropy, (3.34) : dS dS dfk C.,= T— = - 0 — = - 4 * 0 £ 0 £ k — , dT dp y>kr dp C„ =

(3.45)

0<2eo2l

r

£/*(i-A) L

2 dpi

(3.46)

The expression for C„ is simply interpreted as the specific heat due to electrons and holes with the modified spectrum (3.21) plus the change in condensation energy with temperature. At the transition temperature, the energy gap vanishes and the jump in specific heat associated with the second order transition is given by IW1

2 7r<;vw=Kftre/eo(o)] =o.i7o,

(3.40)

where the electronic specific heat in the normal state is given by C.„=-r 7/(ergs/°C cm'), (3.41) and -r-fjrW^)* 2 . (3.42) The Gorter-Casimir model gives the value of 0.159 for the ratio (3.40). The scatter of experimental data is too great to choose one value over the other at the present time. Near T=0, the gap is practically independent of temperature and large compared to kT, and hence for

(ca-cen)\Tc=2kp»j: / k ,(i-A„) dp — hc *>**•

I dp J

-kN(0W\ — \ , I dplTc

where

An=l/(« S 'K+1).

(3.47) (3.48)

The derivative dt^/dp can be obtained from the relation between to and T, (3.27). After some calculation we find 10.2 de
PcZ

Tc

and the jump in specific heat becomes Ce,—yTc yTc

(3.50)

= 1.52. Tc

The Gorter-Casimir model gives 2.00 and the Koppe theory9 gives 1.71 for this ratio. The experimental data in general range between our value and 2.00. The initial slope of the critical-field curve at the transition temperature is given by the thermodynamic relation

T^/dB^y

(T/-y* FIG. 2. Ratio of the critical field to its value at r=0°K vs {T/Tc)\ The upper curve is the \-(T/TtY law of the GorterCasimir theory and the lower curve is the law predicted by the theory in the weak-coupling limit. Experimental values generally lie between the two curves.

. -II 4v\dT / IT.

=(C.-Cn)

(3.51) Tc

With use of (3.47) this becomes l/dEey (— ) y \dT/

=l°-4, \TC

(3.52)

1188

BARDEEN,

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SCHRIEFFER

creation and destruction operators, U=T.Hj=

£

B*.v..cv..*c*„

(4.1)

where .BkTk',,' = I

^•.'*Hfl>i..dTj

(4.2)

is the matrix element for scattering of a single electron from k1, the specific heat can be expressed in unoccupied in a typical initial configuration. A similar factor occurs in the second-order perturbation theory the form expansion of U:

yTe

2iAwV

£ 7

r

S8.5e-'-" ''' ,

(3.54)

where K„ is the modified Bessel function of the second kind. The ratio C.,/(yTc) is plotted in Fig. 3 from (3.46) and compared with the P law and the experimental values for tin. The agreement is rather good except near Tc where our specific heat is somewhat too small. The logarithm of the same ratio is plotted in Fig. 4 to bring out the experimental deviation from the P law. The recent work of Goodman el al.m shows that the data for tin and vanadium fit the law: CM/{yTe) = aer™
k,ffrk'.a'

'*—12//(1-/).

(4.3)

e— c

If Hj is independent of spin, a' =
\^^»!^2,_E]BufzI\

(4.4)

(3.55)

with high accuracy for TC/T>IA, where o=9.10 and J=1.50. These values are in good agreement with our results in this region, (3.54). Thus we see that our theory predicts the thermodynamic properties of a superconductor quite accurately and in particular gives an exponential specific heat for r / Z \ . « l and explicitly exhibits a second-order phase transition in the absence of a magnetic field. IV. CALCULATION OF MATRIX ELEMENTS

There are many problems for which one would like to determine matrix elements of a single-particle scattering operator of the form f/=ZL#i> where Hj involves only the coordinates of particle j . In terms of

FIG. 4. A logarithmic plot of the ratio of the electronic specific heat to its value in the normal state at Tt vs TJT. The simple exponential fits the experimental data for tin and vanadium well for Tc/T> 1.4.

THEORY

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TABLE II. Matrix elements of single-particle scattering operator. Wave functions* Initial, ¥< Final, * /

(kt, (k't, -14), -k'i)

(k't, or excited (—) -k*), -k'+) k k' (a)

xo xo

00 XX

XX 00

OX

xo

OX

00 XX

XO XO

(b)

ox

ox ox

XX 00

00 XX

XX 00

(c)

(d) XX 00

00 XX

Ground ( + )

(kt,

ox

xo

+ — + — + — + — + + — + — + —

+ — — + + — — + + — — + + — — +

Energy difference Wi-Wf E-E E-E E+E -(E+E')

Probability of initial state

Matrix elements Ck-i*cyi or «_kt*c_k'i or c-vi*cui -c-ki*«k't

is(l-s'-p')

:(1-A)(l-A')]i

W W

(AA')»

-(AA')» -C(1-A)(1-A')J

-Ld-h)h'y

-:A(1-A')?

-CA(l-A')]*

-C(l-A)A']t -[(1-A)(1-A'):» -(AA')» [A'(1-A)D» [A(l-A'):»

hs(ls'-p')

wa-s-p)

(AAO*

is'P is'd-s-p) is'P

Z(l-h)(l-h')J [A(l-A')3* :(1-A)A']»

E+E' -{E+E) E-E E-E

its' iss' iss1

CA(i-A')3» -CA'(l-A)]* -(hh')i

t«'

C(1-A)A':» -CA(l-A')]* [(1-A)(1-A')]» -(AA')»

-(E+E) E+E' E'-E E-E

(l-s-p)(l-s'-p') pp' (l-s-p)p' Pd-s'-p")

[A(l-A')]« -[(l-A)A']i -(kk')i C(l-A)(l-A')]«

C(I-A)A':*

E'-E E-E' -(E+E') E+E'

ia-h)d-h')j -[A(l-A')]t £(l-i)(l-h')J -(AA')»

• For transitions which change spin, reverse designations of (k't, —k'i) in the initial and in the final states.

which the pair states k' in Vi and k in ¥7 are unoccupied, designated by X0 00 for ¥< and 00 X0 for * / . There is a nonvanishing matrix element of Ck'T*«kt between these components. Other components of the same wave functions have the pair states k' in SFj and k in ¥ / occupied, as is indicated by the designations X0 XX and XX X0, respectively. While the matrix element of ct't*£tt between these latter vanishes, that of c-ki*c-t't does not. Since both Ck-t*£kt and c_n*c_k't are included in the sum in (4.5), they will give coherent contributions and must be considered together. Matrix elements between these states of all other terms in'the'sum are zero. Nonvanishing matrix elements of <;k't*£kt are obtained Matrix elements of type (b) are for single-particle only when the single and excited-pair occupancy of &; occupancy of — k'j in *,• and of —k| in ¥/, while k in and *> is the same except for those designated by wave •*,• and k' in ¥ / may be occupied by either excited or vectors k and k'. Further, * i must contain a configura- ground pairs. With interchange of spin and of k and k', tion in which kt is occupied and k't unoccupied and they are similar to type (a). Type (c) represents single •*/ one in which k't is occupied and kt unoccupied. occupancy of kt and — k'l in >J\ and either excited or The various possible transitions along with the matrix ground pair occupancy of both k and k' in tyf. Transielements are listed in Table II. While the individual tions of ck't*Ckt are allowed for the component 00 XX matrix elements are complicated, fairly simple results of * / and of c_kj*£_w for the component XX 00. are obtained when a sum is made over all transitions in Again, these are coherent. Finally, type (d) represents which k is in a volume element Ak and k' in Ak', it excited or ground pair occupancy of * ; and single being assumed that Bkk- is a continuous function of k particle occupancy of — k|, k't in * / . and k'. The energy differences W — Wf listed in the third The first type of transition, (a), listed in Table II column are obtained by taking an energy 2E for an corresponds to single occupancy of kt in the initial and excited pair, E for single occupancy, and zero for a of k't in the final state. Pair occupancy of k' (i.e., the ground pair. We have used the notation £ = £ ( k ) ; pair k't, — k'^) in *< and of k in ¥ / may be either £ ' = £ ( k ' ) , etc. excited or ground, giving the four possible combinations In column 4 are given the probabilities of the initial listed in the second column. These have components in state designations, based on taking 55 for a specified The factor of two comes from the sum over spins in the initial configuration and the second form from the fact that | i W | 2 is symmetric in k and k'. The calculation of the corresponding factors for the superconducting case is complicated by the fact that any given state k
1190

BARDEEN,

COOPER,

single occupancy, p for an excited pair, and (1 — s—p) for a ground pair. For example, SFi in the top row corresponds to kf occupied and a ground pair in k', and the fraction of the states k in the volume element Ak and k' in Ak' which have this designation is

is(k)i\-s(W)-P(kn To calculate the matrix elements, it is convenient to decompose the wave functions into components corresponding to definite occupancy of excited and ground pairs as in (3.10). Thus, for the top row in which k' in *v and k in * / are both ground pairs, *v=Ckt*[fc'WOk,U0+(l-A')WOk,O k .)],

(4.6a)

* / =c lt .t*[A^ 1 o(li„0 1 ,0 + (l-A)Voo(O t ,0 1[ 0],

(4.6b)

AND

SCHRIEFFER

c_ki*ek't. They are the same as the corresponding matrix elements for parallel spin, except for a reversal of sign of the reverse spin transitions. For example, for the type (a) transition of the top row, the final state is now */=c_k.1*[AWlk,0k<)+(l-A)W0k,0k.);]-

The matrix element of c_k-i*ckt is E(l—A)(l—A')D' as before. To obtain matrix element of c_ki*ck-t, we now have, corresponding to (4.9) and (4.10), c_ k 't*6 k '6 k *voi = c k 't* k *^oi, C - W * C k ' t Ckt* ¥>01 = C k ' t & k V o i ,

giving + (hh')*. We have indicated the change in sign in the table by listing —c_ki*ck-t at the top of column 6. In a second-order perturbation theory calculation, one is interested in determining

where ViO=6k*6k'V01.

The matrix element of ek't*«kt is (*/ M*Ckt 1*0 = [(1 -A) ( 1 - A'Wck'tVoolck'fW = [(l-A)(l-/i')?. (4.7)

c_kt*c_k' *Ckt*
| ( * / | E Bkk.ek./6k.|*0l

E-

The matrix element of C-u*C-vi is given by (*,|<;_,a*c_wl*<) = (AA'),(ek't*&k**k<«'oi|c_kt*c_,,<*Ckt,Voi). Since Ck't*6k*6k'poi= — c_k'i&k*yoi,

(4.15)

(4.8) (4.9)

k.k'.r

(4.16)

W~Wf

where the sum is over all intermediate states, / . The initial state should be a typical one for a given temperature T. In general, one might have either £„„. = + £ _ „

(case I)

(4.17)

(4.10)

(case II) (4.18) B kk - = —Bthe matrix element is — (hh')*. The other matrix elements of types (a) and (b) may be calculated in a The latter applies to the magnetic interaction. To take similar manner. For types (c) and (d) we make use of the decompo- the coherence into account, one may take the spinindependent sum over k and k', which designate initial sition (3.10). For example, for type (d), and intermediate states: *i=aiiio, so that we find (¥/|c_n*^k.t|*i)=aM.

(4-13)

{[(i-*)(i-*0]»=F(**0,},Ci*(i-*'-^) +$ps'+iss'+P(ls'-p')l

(4-H)

Those for type (c) can be obtained by interchanging initial and final states and spin up and spin down. We have so far assumed a spin-independent interaction . One involving a spin flip may be treated by exactly similar methods. Initial and final states differ from the parallel spin case by interchange of spin designation of k' in the initial and in the final state. There is a coherence between the matrix elements for c^.fi*Cki and

«'Teo 2

EE'

/(!-/').

(4.20)

Table III lists the average matrix elements for the various values of W—W/. The second-order perturbation theory sum may be written - E \BW\*LM, (4.21) k.k'

31 THEORY

OF

S U P E R C O N D U C T I V I T Y

TABLE I I I . Mean square matrix elements for possible values of Wi — W/.

Wi-Wj


E-E'

The corresponding expressions for Wf— Wi=E-\-E - ( £ + £ ' ) are: 2ir

/

eo2 \

h*M>kF

\

EE'/

E'-E 6J

— Z fl k.k->kr

E+E'

where 6£'=F£02\//-/\

1/

2V

EE

1/

)\E-E')

M'=Frf\/l-/-/\

££' A £+£' /

2V

l/(l-2/)£-(l-2/')£'

)

;(\ ( ^ 4 \

/(l~2f)E'-(l-2f)E\

+•A ££' A (4.22)

The upper signs correspond to case I, the lower to case II. To determine the probability of a transition in which an energy quantum hv is absorbed, we have a sum of the form 2w — E ( |Bkk- | 2 (| CJvUk
(4.23)

For the matrix elements for which Wf—W,=£—£', we may interchange k and k' in the sum and combine them with those for which Wt—Wi=E'—E. This just gives either one multiplied by a factor of two: 6£'Teoz\

2ir

/

ft k.k'

V

EE'

(4.24)

One may interpret the factor of two as accounting for the sum over the two spin possibilities of the initial state. If |Bkk'| 2 is symmetric with respect to the Fermi surface, so that we may sum over + and — values of t and t', terms odd in e and e' drop out, and we find 2ir

— £ fl h.k'>ir

/

4|A*|»(l=F \

e0* \

) EE'/

Xf(l-f')S(E'-E-hv).

X(l-f)(l-nS(E+E'-hv),

(4.26)

2\Bkk.\*(l±-^~)ff'6(.E+E'+hy), V ££'/

(4.27)

respectively, where again we have dropped terms odd in e and t'. Hebel and Slichter34 have used (4.25) to estimate the temperature dependence of the relaxation time for nuclear spin resonance in the superconducting state from the corresponding value in the normal state. They are able to account for an observed initial decrease in relaxation time in Al as 'the temperature is lowered below Tc. The increased density of states in energy in the superconducting phase more than makes up for the decrease in number of excited electrons at temperatures not too far below Tc. For this problem, the lower sign ( + ) is appropriate. These expressions may also be used to determine transport properties, such as electrical conductivity in the microwave region and thermal conductivity. Note added in proof.—The marked effect of coherence on the matrix elements is verified experimentally by comparing absorption of ultrasonic waves, which follows case I, with nuclear spin relaxation or electromagnetic absorption, both of which follow case I I . For frequencies such that hv<£kTc, one expects for case I I an initial increase in absorption just below Tc, followed by a decrease to values below that of the normal state as the temperature is lowered, as is observed experimentally. On the other hand, for case I one expects the absorption to drop with an infinite slope at Tc, such as is found for ultrasonic waves. The expressions for the transition probabilities are simplified if we change our convention for the moment to give E the same sign as t, so that E= — (e 2 +« 0 ! )* below the Fermi surface. One may then write (4.26) and (4.27) in the same form as (4.25), with E and E' now taking on both positive and negative values. Considering both direct absorption and induced emission, the net rate of absorption of energy in the superconducting state is proportional to

«.«/(IT^,):/(WO-/'U-/);M^(£')<*E,

/

XfO.-f)6{Wf-Wt-hv).

and

2x

*(l- l^)(l-/HW')

-(£+£')

1191

(4.25)

(4.28)

where E'=E+hv and p(E)=N(0)E/(,E>-t
11°2

BARDEEN,

COOPER,

state is similar, except that e 0 =0. We thus find «./«„=2/(«„). (4.29) R. W. Morse and H. V. Bohm (to be published) have used (4.29) for analysis of data on ultrasonic attenuation in an indium specimen for which the electronic mean free path is large compared with the wavelength of the ultrasonic wave, so that one might expect the theory to apply. Values of to(T) estimated from the data by use of (4.29) are in excellent agreement with our theoretical values (Fig. 1). For case II, the integral may be expressed in the form: V a, hrJ {(^-.o'JC^+Ai-)'-.^:)*' ' The integral diverges at E—ta if hv is set equal to zero in the denominator. Numerical evaluation of the integral indicates that for hv~%kTc or less, (4.30) gives an increase in absorption just below T, as observed by Hebel and Slichter in nuclear magnetic resonance and by Tinkham and co-workers (private communication) for microwave absorption in thin superconducting films. In order to have absorption at r = 0 , hv must be greater than the energy gap, 2c0. We take B negative and E positive, and find for this case: o. _1_ /-«. CgfcS+AiO+toflfcE = ! ( ' ' an hpJ«r-K, {(.E*-«„«)[(.E-f-AiO< -e(,»])r The integral may be evaluated in terms of the complete elliptic integrals, £(y) and K(y) as follows:

where 7-(*r-2«o)/(*»+2«J. (4.33) This expression is in excellent agreement with data of Glover and Tinkham (reference 20, Fig. 6) on infrared absorption in thin films. V. ELECTRODYNAMIC PROPERTIES The electrodynamic properties of our model are determined using a perturbation treatment in which the first order change in the wave function is used to calculate the current as a functional of the field. For such properties as the Meissner effect this approach is quite rigorous since we are interested in the limit as A(r) approaches zero. I t is assumed that the medium is infinite and that the sources of the field may be introduced by inserting current sheets in the interior. This method has been applied previously to the calculation of the diamagnetic properties of an electron gas. 36 We first derive an expression, valid for arbitrary temperatures, relating the current density to the total field (the field due to the sources and to the induced currents). The fact that the system displays a Meissner effect is established by investigating the Fourier transform of the current density in the limit that q—K). I n this limit we obtain the equation, 1 limj(q) = i-0

«(q),

(5.1)

c\T

"• This method was first applied to the calculation of the diamagnetic properties of an electron gas by O. Klein, Arkiv Mat. Astron Fysik, A31, No. 12 (1944). Our treatment follows that of one of the authors as given in reference 7, pp. 303-321, where further references to the literature may be found.

AND

SCHRIEFFER

where AT is a function of temperature, increasing, in the free-electron approximation, from the London value A—m/ne* at T=0 to infinity a t the transition temperature. The limiting expression (5.1) is valid only for values of q smaller than those important for most penetration phenomena. In general we find the current density is a functional of the vector potential A which, with div A = 0, may be expressed in a form similar to t h a t proposed by Pippard (1.3): 3 j(r) =

rR[R-A(r')]./(#,7Vr'

I 4TCAT£OJ

.

(5.2)

R*

T h e kernel, J(R,T), is a relatively slowly varying function of temperature, and at T=0°K is not far different from Pippard's exp(—R/£o). To calculate penetration depths, it is more convenient to use the Fourier transform of (5.2), which may be expressed in the form j(Q)--(«/4r)*(j)a(q),

(5.3)

where K(q) is a scalar which approaches the constant value 4ir/(Arc 2 ) in the limit q—K). One m a y determine K(q) directly from the perturbation expansion of the wave function, or one may first calculate J(R,T) and then find the transform of (5.2). The latter procedure is followed in Appendix C, where an explicit expression for K(q) valid for q not too small is derived. In this section we shall give a direct derivation of t h e transform which can be used to investigate the limit q—*0, and then give the derivation of (5.2). A comparison of calculated and observed values of penetration depths is given a t the end of the section. In the absence of the electromagnetic field, the system at a given temperature is characterized by the complete orthonormal set of wave functions which we denote by with corresponding energies W,(T),W1
(5.4)

We choose for ^O(JT) a typical wave function of the type described in the previous sections, where the occupation of "single particles" and "excited pairs" is given by the 5 and p distributions, respectively, appropriate to the temperature T; the rest of the phase space is available for "ground pairs" whose distribution is specified by h which is also a function of T. T h e set of orthogonal states is obtained by varying s and p (in analogy with the normal metal) and not changing h. We thus are choosing a representative configuration of the most probable distribution and taking system averages with respect to this representative configuration. The electromagnetic interaction term for an electron of charge q= — e, e > 0 , is, in second quantized form,

THEORY

OF

SUPERCONDUCTIVITY

T o the lowest order in A(r), the expectation value of the current operator is then

—ieh

Br

=JA**»[- 2mc -(A-v+V-A)

e2

+

A 2 (r)W 2m
(5.5)

We choose a gauge in which V • A = 0 and in which A = 0 if the magnetic field is zero. We expand if/ and ^* in creation and annihilation operators 86 : 1 £2*k.„ (5.6) ^*(r)=-2:ck-„.*«,-*e-ik'', G»k'.,'

eh where

j,(r) = - - E E E (2k+q)k' 2m2Cfl w*0 k,q,r k',q'.»'

l

x(*,(r)ick+,./ck,,]*o(r))-

Wo-Wi

+complex conjugate ,

— E Ck+^T*Ck.aa(q)k, 0 k.q.
«(q)=

•(£)/

while for the diamagnetic part we have j i ) ( r ) = — (ne 2 /wc)A(r),

<2r A(r)<-'''-'.

(5-8)

Expanding ^ and ij/* as in (5.6), we get

e2 1

•a(-q)e-i"L(ekJek+,)> (5.9)

E Ck+,..V^-i,"A(r).

In the presence of the electromagnetic field the wave function for the system may be written *(A) = * o + * i + (terms of order A2- • •),

(5.10)

where the usual perturbation expression for * i is *i=E

t?h? (2*-)* j * f r ) " . 2. . . E ( 2 k + q ) k 2m c Q? k, q (5.15)

i I

E ck+,.Ak,(re- ' (2k+q), 2mii k.t.ir

3b(r) =

(5.14)

where n is the number of conduction electrons, of both spin directions, per unit volume. The spin sums and the calculation of the average matrix elements can b e carried out as indicated in Sec. IV, (4.16) to (4.22). We then obtain

ieh e2 3 ( r ) = — ( ^ * v ^ — H e r m . conj.) ^*A^ 2m ntc -3PW+3I>(')-

(5.13)

(5.7)

The current operator 3 ( r ) is

eh

(5.12)

where the last equality follows if the current in the field free state, jo(r) = (<£ol3( r )l*o), vanishes. 36 " If (5.7), (5.9), and (5.11) are combined with (5.12) and the condition q- o(q) = 0 is used, we obtain for the paramagnetic part of the current density,

(2T)»

mC

3PW =

j(r)-<*|3M|*)-=(#il3*(*)l*o) + (*o|3j>(r)l*i)+<*o|3j>(r)|*o),

• [o(q')r-i''(^0(r) I Ck.+1.. ,-%.„-1 *«(r»

where the e's satisfy the usual fermion anticommutation relations, (2.1) and (2.2), u, is a two-component spinor, and Q is the volume of the container. The interaction Hamiltonian becomes, when one neglects the term of higher order in A, BJ=

1193

*.)•

(5.11)

86 At this point we insert plane waves for the Bloch functions. It would be possible to carry through an analogous procedure formally with Bloch functions. The average matrix elements which enter cannot be evaluated explicitly, but can be expressed in terms of empirically determined parameters. The appropriate modifications of the free-electron expressions are as indicated in the introduction.

where i(«t,«k+q) is given b y (4.22) with t h e lower signs, corresponding to case I I . Setting tk=t and «*+, = e', the explicit expression for L is 1 / 1 - / - A / I(e,e') = - ( )( 1 2\ E+E' A

«+eo2\ ) EE J

««' ++ee„o2 >2 \ /J-f\/ + ( ) ( 1 + ) . (5 16) 2\£-£'/\ EE J 1

86 * Note added in proof.—We neglect the effects of the momentum dependent cutoff on the expression for the current density; as shown explicitly by P. W. Anderson (private communication) errors introduced are negligible in the weak coupling limit. In a general gauge, A=Ao-f grady, with div Ao=0, it would be necessary to include in the perturbation expansion collective excitations of the electrons, the simplest of which corresponds to a uniform displacement of all of the electrons in momentum space. Energies and matrix elements of collective excitations are nearly the same in normal and superconducting phases. We assume the collective excitations make a negligible contribution to the current form A0 (see J. Bardeen, Nuovo Cimento 5, 1766 (1957)).

1194

BARDEEN,

COOPER,

T o show how (5.16) reduces to the usual expression for a free-electron gas in the limit «<j—*0, we note t h a t E is denned so that it is intrinsically positive, while the Bloch energy e may be either positive or negative. As eo—*0, E—* | e |. If £ is negative, / « = / ( - l « l ) = W ( M ) = W(-E),

T h e M e i s s n e r Effect T o establish the existence of the Meissner effect, we investigate the Fourier transform of j(r) in the limit q—>0. If j(q) does not go to zero in this limit, and is opposite in sign to o(q), then the system will expel the field from its interior and behave like a perfect diamagnet. T h e Fourier transform of the paramagnetic part of the current density, jp(r), is given by fdr

jP(r)«-'«".

(5.18)

Referring to (5.15), this can be written as jp(q) — (2

— 1 I < * (2k+q)k- 0(q)i(e k ,e 1[+ ,) J (5.19)

f» cV2ir/ J

where in the limit that q—»0, ek=« k +q and i(tk,ei, + q ) becomes Bet>E limZ,(«k,Ck+0= ,. , . „ .2(5.20) (1+e"*) Thus we want to evaluate ,PF.

limJHq)=

( — )2/Hikkk-a(q)

(1+e^)2

(5.21)

Choosing a(q) as the polar axis, the angular integration can be done; then, using the relations n=kr*/3n2 and Sr=Wkri/2m, we get limij>(q)

mc\

AT/

«(q),

SCHRIEFFER

T h e London constant Ar can also be expressed in terms of derivatives of eo, as is done in Appendix C where the following result is derived A7" = A ( 1 +

\

(5-17)

To get a nonvanishing contribution from the first line, t and e' must have opposite signs, while for the second line they must have the same signs. We thus find that £ ( « , * ' ) - » ( / ' - / ) / ( « - « ' ) , as it should [see (4.4)].

jp(q) = ( - )

AND

(5.22)

) en dB /

The total induced current thus becomes: A we2 limj(q)=jP+J2,= «-°

, A r mo

«(«0 =

«(q).

(5.26)

Arc

I n the two limiting situations T—»0 and T—*TC, A/Ar can very easily be evaluated. As r—K), B becomes infinite, so that Ar becomes equal to A, the London value lim(A/Ar) = 1 - 2 f dy exp[-(y ? +/3 2 eo 2 ) i ] = l .

(5.27a)

This could have been seen immediately from (5.19) since in the T—*0 limit L(ti,u+q)=0 and the paramagnetic part of the current density is zero. This limit just gives the equation obtained by London assuming complete rigidity of the unperturbed system wave function in the presence of an electromagnetic field. When T—>TC, toP goes to zero and A / A r also goes to zero, since lim( — | = i -

2

e'dy f

=0.

(5.27b)

The small Landau diamagnetism would appear only in a higher order. We thus find that when the system is in the superconducting phase, the current density in the London limit has the form (5.1), with Ar varying from A to oo as T goes from 0 to Tc. I t is interesting to note that the Meissner effect occurs for any value of eo^O. If eo=0 (for example if we let V=0), then, for T>0, /S« 0 =0 and from (5.27b) we see that A / A r = 0 . The paramagnetic p a r t of the current density is then we2 limji>(q)=—tt(q), «-*° mc

(5.28)

and the total current in the -q—>0 limit becomes limj(q)=0.

where

(5.25)

(5.29)

q-»0

A • i

2/3 $F =

AT

6

kr

r" I **rfjte**(l+«**)-». •'o

(5.23)

Letting y= t/kT, and using the sharp maximum of the integrand a t « = 0 , we find

Ar

•'o

[l+exp(y!+^€oJ)*?"

(5.24)

This is also true when T"=0, though in this case one must be careful of the order in which the various limits are taken. Current Density We now evaluate the spacial distribution of the current density and exhibit it in a form similar to t h a t proposed by Pippard. T h e method we use follows t h a t

T H E O R Y OF

SUPERCONDUCTIVITY

of one of the authors' who carried out a similar calculation for an energy-gap model. Beginning with (5.15), setting q=k'—k, and using the fact that q- o(q)=0 and that we can write terms like keik,r as ke < k r = -iVe*-', (5.30) we can reduce all of the angular integrations to integrals of the form

rT

I sinfc#e± - ~ >=—sinkR, •>„ kR where R= | r—r' |. Then, using the relation sin/U? \

kR coskR— sinkR

r=

AJP

as cos(k+k')R is very rapidly oscillating in the region of interest. Now, making the change of variable dk= (dk/dS)dt, approximating the slowly varying terms k+k'c~2kr and dk/dS~\/fivti, where »o is the velocity of an electron at the Fermi surface, and using the rapid convergence of the integral to extend large finite limits to infinite ones, we obtain G.(R)-Gn(R)

l ik (r r,

(— ) -

1195

**.

(5.31)

= -ikr*S*t where and

— I xVO)—J(R,T),

V dSf p

(5.39)

AT

J(R,T)=I(R,0)-I(R,€o)

(5.40)

(5.32) /(*,*„)=—i— f ATWOV

fdtde'Lit.e') J

we finally obtain Xcos|(«-c')—1. j*«-

2»» «r

The current density then becomes

where G.(R)= f dkf

dk' kk'/(k,R)f(k',R)L(.€,e')

and J(k,R)=kR cosMJ-sinW?.

(5.34) (5.35)

These equations correspond to (21.4)-(21.6) of reference 7. The entire current density j(r) can now be written in?

(5.36) A(r). mc It is convenient to subtract and add G„(R) inside the integral (5.33), whereGn(R) isG,(.R) evaluated at eo=0. The integral evaluated with G„(R) gives the paramagnetic current contribution of a free electron gas and just cancels the term — (n#/mc)A(t), leaving the small Landau diamagnetic term in which we are not interested. We have left the interesting part of the current density: i(')-W»)

eW j(r)=- 2 2m cv'

(5.41)

(5.33)

dr'-

2

[G.(Jg)-G,(ig)]R[A(r')-R] i?6

(5.37)

An inspection of G,(R), (5.34), reveals that the major contribution to the integral comes from the region k, k'^stk? (i.e., very close to the Fermi surface). Since we are interested in values of R ~> penetration depth which is 2~10~' cm, in the region of the major contribution, kRzstk'R2>l. With this in mind, the product f(k,R)f(k'R) becomes /(kJQfik'rR^k'R1 coskR cosk'R =>ikk'R1icos(.k+k')R+cos(k-k')R-\ c*lkk'R'cos(k-k')R,

j(r)=

3 ireo(0) C 7(U,r)R[A(r')-R] — - dr' , 4«Ar »J>o J R

where Ar has been given by (5.25). The current density has now been written in a form in which it is easily comparable to Eq. (1.3), proposed by Pippard for a pure superconductor, where we identify l/£o with the microscopic quantities: 1 «o(0) -=——, (5.43) in nvo and where J(R,T) is to be compared with the exponential function, exp[—jR/fo]. We have denned J(R,T) so that it has the same integral as exp[—R/£o]:

I"

J(R,T)dR=*h.

(5.44)

As evaluated in Appendix C, J(R,T) is given by J(R,T)=

A r « 0 (r)r

€ 0 (r) tanh A to(0) L 2kT /2R l \l-2/(£')l ^Cd^J™A -^^\ (5. 45)

2to(T) C"

T J0 U»o / *'E' J from which (5.44) can be established. In the London limit, where A(r) varies so slowly compared to the coherence distance £0 that it may be taken out of the integral sign, we obtain

j(r)

-A(r), CAT

(5.38)

(5.42)

as has been shown previously.

(5.46)

1196

BARDEEN,

COOPER,

reflection and for random scattering, based on corresponding expressions derived b y R e u t e r a n d Sondheimer" for the anomalous skin effect. T h e penetration depth, denned b y

1.0 0.8

S* -"*.

\

AND SCHRIEFFER

0.6

A=

Q4

is given b y 02

0

2 r"

^!i>ft=^__

"*J(R,0) .5

f H(x)dx, B(G)J0

1.0

13

(5.51)

dq (5.52)

2.0 2.5 3.0

3.5

4.0 4.5

5.0

for specular reflection and b y FIG. 5. The kernel J(R,0) for the current density at 7" = 0°K vs R/(a, compared with the Pippard kernel, exp(—R/£t). (5.53) With this normalization of J(R,T) it turns out that most of the temperature variation of the current density is contained in AT, and that the integral does not produce effects that vary very much with the temperature. This will be made clear later in the calculation of the penetration depth as a function of the temperature. At T=0, J(R,0) has the simple form J(R

2 .o)«-1r

Ko(y)dy,

(5.47)

2

ln[l+ ? - X(?)] •'ft

for random scattering, where K(q) is denned b y (5.3). Limiting expressions have been given for f 0 A large or small compared with unity. T h e London limit corresponds to £o
iir«'2. r v 2€ 0 (0)J!/«w

\L(T)=

and when R=0 7(0,0) = 1.

(5.48)

Thus J(R,0) not only has the same integral as the exponential b u t also the same value at .R=0. A comparison of the two given in Fig. 5 shows that they are quite similar. We may express £0 in a form similar to that suggested by Faber and Pippard 2 6 :

(4*-/A r c 2 )i.

(5.54)

For a free-electron gas, this reduces to the London value (m
|o=a(W*7\,),

(5.49)

where a was adjusted empirically from observed penetration depths. From (5.43), we find fivo

1 kTc flVo ^"o £„= = =0.18 . weo(O) w e0(0) kTc kTc

(5.55)

Aoc — -

while the value for specular reflection is smaller b y a factor 8/9.

(5.50)

Our theoretical value of 0.18 is between the empirical estimates of 0.15 b y Faber and Pippard 26 and of 0.27 by Glover and Tinkham. 27 Thus a t the absolute zero our theory gives a current density very much like that proposed by Pippard. Since J(R,T) varies slowly with T, most of the temperature variation is contained in the constant Ar.

1.0

- -_^^ 4

> ^ '

as Q€

0.4

vy

\\\\

- fe-f-[ss — a?]*

0.2

Penetration Depths "03

A most important application of the equations we have derived for the current density is to the calculation of field penetration at a plane surface. T h e results depend to some extent on the boundary conditions for scattering of electrons a t the surface. Pippard 2 ' has given general solutions for the limiting cases for specular

04

OS

0.6 0.7 t . T/Te

0^

09

FIG. 6. The temperature variation of the penetration depth, \ „ in the infinite coherence distance limit, (foA)—*°°, compared with the empirical law, [\(0)/\(T)y=l—t>. " G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. (London) A195, 336 (1948).

THEORY

OF

1197

SUPERCONDUCTIVITY

TABLE IV. Penetration depth, X, at T - C K . "

(D

(2)

(3)

cm

cm

1<X{. cm

3.5 1.6

1.4 8.2

0.2S l.S

10«XL

Tin Aluminum

(4)

7.3 93.

(S)

C6>

(7)

C8>

Specular reflection \/\L 10"X

Random scattering \/\L 10«\

1.4 2.8

1.6 3.2

4.8 4.4

5.7 5.2

9) iObs.

10«X cm

5.1 4.9

• Values of Xz. and vo are those estimated by Faber and Pippard" from high-frequency skin resistance and normal electronic specific heat. The value of to is OAShvo/kTt., as in (5.43). Ratios in columns 5 and 7 are taken from Fig. 6. Observed values are from reference 25.

The temperature dependence of X«, can be obtained directly from (5.45). The second term vanishes when R=0, so that we have X.'(O)

«(T) tanh[|ff6 0 (r)]l»

"t

(5.56)

A plot of this quantity on a reduced temperature scale is given in Fig. 6. It is plotted in this way so that a comparison can be made with the empirical law:

x«(O)A,(0-i-<',

(5.57)

based on the Gorter-Casimir two-fluid model. It is seen that our theory is very close to the empirical law except for temperatures very close to Te. It is in this region that the approximation fo5s>X becomes invalid as a result of X increasing with T. The corrections are such as to reduce the theoretical values so as to bring them closer to the experimental values. To determine X(r) for intermediate cases, we plot in Fig. 7 the quantity \(T)/\L(T) as a function of £o/ \L(T). The calculations are based on use of the asymtotic forms for K(q) near T=0° and T=TC given in Appendix C, and numerical integrations of (5.52) and (5.53). The curves for the two limiting temperatures are quite close together, indicating that the effect of the variation of J(R,T) with temperature is rather small. This procedure is analogous to that of Pippard6 who gave a plot by means of which one can determine X from known values of £0 and Xt for the case J(R,0) = exp(-2?/{o). Using our theoretical expression (5.50) for £o and empirically estimated values of v<, and XL(0) for tin and aluminum,26 we have obtained £O/XL(0), and, using Fig. 7, have determined X(0)/AL(0) for these two metals (Table IV). The agreement between theory and experiment is reasonably good, the experimental values falling between the theoretical values calculated for random scattering and specular reflection. The theory we have presented applies to a pure metal. Pippard6 has shown experimentally that the existence of a finite mean free path, I, due to impurity scattering, has the effect of increasing the penetration depth, and has suggested that it may be taken into account by introducing an extra factor, exp( — R/l), into the kernel for the current density. One of the authors7 has shown why such a factor may be expected from a theory of the diamagnetic properties based on

an energy-gap model. Similar considerations apply to the theory developed in this section; however, effects of coherence on the scattering matrix elements introduce complications and the proper correction factor has not yet been worked out. To complete the electrodynamics we should give a relation corresponding to the second London equation, the one which gives the time-rate of change of current when an electric field is present. Such a theory would require a calculation, not yet completed, of transport properties with our excited-state wave functions. We expect to find something similar to a two-fluid model, in which thermally excited electrons correspond to the normal component of the fluid. For frequencies such that &>«€o, one may determine dj/cli for the superconducting component by taking the time derivative of the integral relation (5.2) and then setting dk/dt = —cE in the result (see reference 7).

£./X L m FIG. 7. The ratio \(T)/\L(T)

vs &/\L(T)

for the boundary

conditions of random scattering and specular reflection and for temperatures near r = 0 ° K and near T=Tc. The temperature variation of XL(.T) is given by XL(T) — (Aj«y4ir)t.

1198

BARDEEN, VI. CONCLUSION

COOPER,

AND

SCHRIEFPER

much more difficult to use in such calculations than the determinantal wave functions oi the Bloch theory. Although our calculations are based on a rather For calculation of boundary energies and related idealized model, they give a reasonably good account of the equilibrium properties of superconductors. When problems, one would like to introduce an order paramthe parameters of the theory are determined empirically, eter which can decrease continuously from an equiwe find that we get agreement with observed specific librium value for the superconducting phase to zero in The heats and penetration depths to within the order of the normal phase as the boundary is crossed. Ginsburg-Landau theory and its extensions7 appear to 10%. Only the critical temperature involves the supergive a good phenomenological description of such effects. conducting phase; the other two parameters required Perhaps the energy gap, 2*0, or, what is equivalent, (density of states and average velocity at the Fermi the coherence distance, fo, could be used for such a surface) are determined from the normal phase. This parameter. quantitative agreement, as well as the fact that we can Another problem, not yet solved, is the calculation account for the main features of superconductivity is of the paramagnetic susceptibility of the electrons in a convincing evidence that our model is essentially superconductor, such as is required to account for correct. Reif's data88 on the Knight shift in the nuclear paraThe basis for the theory is a net attractive interaction magnetic resonance of colloidal mercury. Our ground between electrons for transitions in which the energy state is for total spin S=0. It is possible that there is difference between the electron states involved is less no energy gap between this state and those for S^O. than the phonon energy, fas. For simplicity we have While a finite energy is required to turn over an assumed a constant matrix element, — V, for transitions individual spin, it might be possible to construct states within an average energy fun of the Fermi surface and analogous to those used in spin-wave theory in which have neglected the repulsive interaction outside this each virtual pair has a small net spin, and for which region. In more accurate calculations one should take the energy varies continuously with 5. The explanation an interaction region dependent on the initial states of of the observed electronic paramagnetism (about twothe electron and the transition involved, and also take thirds that of the normal metal) would then be similar into account any anisotropy in the Fermi surface and to that suggested by Reif himself. in the matrix elements. The fact that there is a law of In view of its success with equilibrium properties, corresponding states is empirical evidence that such it may be hoped that our theory will be able to account effects are not of great importance. Neglect of the for these and for other so far unsolved problems. repulsive part of the interaction is in the spirit of the The authors are indebted to many of their associates Bloch approximation for normal metals, and appears to for discussions which have helped to clarify the probbe well justified in first approximation. Our theory may lems involved. We should like to mention particularly be regarded as an extension of the Bloch theory to discussions with C. P. Slichter and L. C. Hebel on superconductors in which we introduce only those calculation of matrix elements, with D. Pines on the criterion for superconductivity, and with K.. A. interactions responsible for the transition. An improvement in the general formulation of the Brueckner on the exactness of the solution for the theory is desirable. We have used that of Bardeen and ground state. Pines in which screening of the Coulomb field is taken APPENDIX A. CORRECTIONS TO GROUND into account by the Bohm-Pines collective model, and STATE ENERGY the phonon interaction between electrons is determined We may estimate the accuracy of our superconducting only to second order. Diagonal or self-energy terms in the net interaction have been omitted with the assump- ground state energy, W0, measured relative to that of tion that they are included in the Bloch energies of the the normal state, by making a perturbation theory norma] state. When the phonon interaction is so large expansion, using the complete set of excited state as to give superconductivity, higher order terms than superconducting wave functions as the basis functions the second may well be important. One should really of the expansion. We first consider the reduced, Hrei, have used a renormalized interaction in which such which includes only pair transitions for pair momentum q=0, and then the effect of the neglected portion of higher order terms are taken into account as well as the Hamiltonian, H'=H—Hn&. possible. Very likely the assumption of two-particle To the second order, the ground state energy of Hre& interactions is a reasonably good one, so that the only effect would be a redefinition of the interaction constant V in terms of microscopic quantities. ! ! •W,= (*.|tf„«|*.)+£ +••• (Al) The discussion of the matrix elements in Sec. IV w*> Wo— Wi should be a good starting point for calculation of = WVr-WV2)H . transport properties in the superconducting phase. Our excited state many-particle wave functions are not » F. Reif, Phys. Rev. 106, 208 (1957).

THEORY

OF

SUPERCONDUCTIVITY

Since the smallest excitation energy is 2eo, the second order energy is overestimated b y setting j Pfo—W.-| = 2e0 and performing a closure sum. We obtain the inequality ^0<2)<[(*o|ff,ed2|*o)-(*o|ffred|*o)2]/(-2e0)

(A2)

1199

the strong coupling limit, it is possible that *^o is exact in the statistical limit for all values of the coupling constant, although no proof has been found at this time. We may estimate the effect of the neglected portion of the Hamiltonian, H'=H—Hna by a similar perturbation calculation. T o second order in W the correction to the superconducting ground state energy Wo (measured relative to the normal ground state) is

-2to \(i\H'\0)\>

where W„=(*o|#o

2

W' = Y.

2

|*o)-(*o[i7o[*o) ,

(A3)

wi=(*0|flofl'.+ff.ffo|*o) -2(*0|.ff(,|*o)(*o|ff.|*o), w«=(*o|fl.'|*o)-(*.|fl.|*o)

,

where (A4)

and # 0 = Z £k*k*6 k + Z k>kF

k|Mk*,

(A6)

K*f

(A7) k.k'

the sum in (A7) being carried out over the region |e*| and | tt' |
j

dth(t)£l-h(t)y,

(A8)

•'o

o

rfei I <^o

XA(62)[l-A(e2)]} [l-2A(,0]ei,

4

X f

i€2[l-2A(t2)]2.

(A10)

With the use of the relations (2.35), (2.36), and (2.37), terms linear in the volume in (A2) become -2E 0 WO ( 2 )

<8N(0) f

dte2

to'

—+—

= 0,

(All)

2E> 4E> and therefore WV /Wo vanishes in the limit of a large system. I t is likely t h a t (*o|ffred n | , * r o)-( 1 i r o|flred|*o) n also vanishes in this limit for n small compared with the total number of valance electrons in the system. For * o to be an exact eigenfunction of .ff red, in the statistical limit, it is required that the above condition hold for all n. Since this requirement can be shown to hold in 2)

W>

7 2 [3iV(0)eo]V3eo (—)~10-Wo, 2*o

(A14)

where the factor (3eo/£y) comes from the average reduction in phase space as a result of conservation of momentum of the pairs making transitions. These estimates indicate t h a t although the total energy associated with H' may be significant, the effect of H' on the condensation energy is very small. I t should be possible to obtain any required quantitative corrections to We by use of perturbation theory. APPENDIX B. CHANGE IN ZERO-POINT ENERGY OF LATTICE VIBRATIONS

*MCl-*(«)] to'

(A13)

The first term in (i|fl"'|0) may be identified with breaking up a pair in ko, the spin-down member going to — ko'l and breaking up a pair in k, the spin-up member going into k'f. T h e second term arises from k 0 ' and k' being occupied in the ground state and arriving at the same intermediate state by ko'f—>kot and — k'i—>— k | . The transitions involving particles with parallel spin have been neglected since their contribution is reduced b y exchange. Since the distribution of particle changes differs only over an energy region several eo wide, it is to be expected that most of W' will cancel between the normal and superconducting phases. To estimate the energy difference, we shall carry out the sums over a region 6« 0 wide. Inserting typical values in (A12), we find

(A9)

dtJ,(*i)U--k(*

«

-[A(k„')(l-A(k„))Hk')(l-A(k))]*}.

de2{A(«i)[l-*(0] 4

wc = 8lN(0)Vj\

(i\H'\o)=-v{Wko)(i~h(W))h(kxi-h(k'))y

(A5)

>

(A12) Wi-Wi

T h e contribution to the condensation energy from the change in zero-point energy of the lattice can be estimated on the basis of the Bardeen and Pines collective ion-electron treatment. 1 ' Their theory gives

Wzr^-Wzp™ M . 12{ (»k<«>- wk<">) «£• »k(*>

fr^w

- m.+.c))} , (Bl)

fk — ek+«—hu>

where is the average occupation number in the superconducting state and M*'"' that for the normal

1200

BARDEEN,

COOPER,

state. The sum over K ranges over the first zone and k + K is to be interpreted as the corresponding reduced wave vector so that Umklapp processes may be taken into account. If k + K is replaced by k and K by — K in the terms containing «k + », (Bl) reduces to

SCHRIEFFER

for positive and negative values of e and t', the integral may be written as the sum of two integrals with limits 0 and <» : 7(i5,r) = CAr/^6o(0)A](/1+/2),

=£

. (B2) 2

*,K

(fk-«k+„) -(foj«)

---2(\M.\2)

WZPM-WZPU

I •'o

/•Sz

N(0)h(t)de e-H'

NW)\

h=2

2

If the sum over K is carried out, it is seen that the quantity multiplying the difference in occupation numbers is almost independent of k. Since rek(,) —« k ( n ) is antisymmetric with respect to the Fermi surface and differs from zero only in a range of the order of several «o, it follows that the change in zero-point energy will be small compared to Wo. A rough estimate of the sum gives

-+-

I"'

n

°° P?(«)-'* ? («')] cosae cosae' iiede . [ G ( e ) - G ( c ' ) 3 « ' sinae

I •'o

(C5)

G( e ) = [ l - 2 / ( e ) > - 1 - [ l - 2 / ( £ ) ] ^ 1 .

(C6)

Unless the argument is given explicitly, e<, = eo(T). If integration over the region about e'= e is made b y principal parts, each of the two terms in Ii (and each of the two terms in i 2 ) will give equal contributions. Care must be taken to take the same limits for e and e'. T h u s we may write 6

N(G)k(t)df2i[N{8z)/Sz3

2


-[N(0)Jt/Sz)WW0,

(B3)

§z

where &z is the energy at the zone boundary and typical values have been inserted for the parameters. Thus, it appears that the lattice zero-point energy should have little effect on the condensation energy. APPENDIX C. EVALUATION OF THE KERNEL IN THE PIPPARD INTEGRAL According to (S.41), the kernel of the Pippard integral expression for the current density at any temperature T may be written J(R,T)=I(Rfl)-I(R,(o) Ay f f{

jrV0)AJ J I

- Z , ( M ' ) 1 eosa(t-t')
e-t'

I

—00

rb F(e) cosae

If

cosatdtdt' (C7)

Here 6>' indicates the principal part of the integral of e' past e'= 6 is to be taken. Since F(e)—*0 as e—*<*>, the value of the integral does not depend on how the upper limits of the integrals over e and c' are approached. Thus we may set b="*> for both, and integrate over e' first. This would not have been true if we had not included I(R,0), the normal state contribution, which is equivalent to the term proportional to A in the expression for the current density. T h e lower limit is more critical; we must take a the same for both e and t' and approach the limit a = 0 only in the final result. The integral over e' m a y be obtained from f" COSae'it Ja

f'-e'

2

p™ cosae'de' Jo

e'-e'

/•" dt'

2

(C8)

Jo e 2 -«' 2 '

In the second integral on the right we have assumed that a is sufficiently small so that cosa«' m a y be replaced by unity. The first integral on the right m a y be evaluated by contour integration and the second directly to give

(CI) where a=R/hv
(C4)

F(e) = [ l - 2 / ( 6 ) > - [ l - 2 / ( £ ) ] ( £ + e o 2 ^ 1 ) ,

«-*0,fr-MO

-2{\M^)

sinat'dtdt'

(C3)

and

7i=4(P' lim

~-2-

(C2)

where

WzpW-Wzp(n)

XI

AND

1

<"/"

2e

/t+a\

(C9)

\t—a/

Similarly, to evaluate Ii, we have • " e' svaat'dt'


Jn

(CIO)

THEORY

OF

SUPERCONDUCTIVITY

In this integral we may take a = 0 , since the integrand is not singular a t the origin. Combining I\ and It, we find r"

/,+/.= - 2

integrate under the sign in (C13): lim f

e-i"*sin(2«.R/*bo)AR=!We.

F(e)ln(

)-

\t—a'

«*

J

Thus, remembering that {o=faio/|}reo(0)], we find

e

sin2ae [*(«)-*?(«)] rfi.

0

(Cll)

/i+/S=irseo[l-2/(eo)]

-2W f

1 r" — I

firo

£

In the limit a—»0, the first integral gives a contribution only near e = 0 . We may therefore replace F(e) by F(0). The logarithmic integral may then be evaluated, and we find

J{R,T)dR «e*Ar r* 1 1 " 2/(eo)

AK


f"|l-2/(«o)

d Jo dE\ dE\ •'O

N(0)V

daJ0


J*o

tE

E

•'o

i

E

d

(C14)

(C15)

1 dfao) 0= - + 0 dp

eo /•*» d /1-2 —( 0Jo d £ \ E

r"

(y)dy.

(C20)

f(E)\de )-. IE

(C21)

AT

1

d(^t(j)

A

to

d0

(C22)

where Ko is a modified Bessel function39 which falls off exponentially for large values of its argument. If we now observe that Ji°°K0(y)dy=Tr/2, we obtain

2

•dx.

The integral converges sufficiently rapidly so that we may replace the upper limit by » . Comparing (CI9) and (C21), we find t h a t

= 2Ko(2ae0),

J(R,0)

»"l-2/((*N-yO*)

We now differentiate with respect to 0, remembering that y depends on. j3. I n the weak-coupling limit [eo«fto>, / ( / J # « ) ^ 0 ] , we find:

This can be p u t into a form convenient for evaluation by using cos2a« r~" co&Zat

(C19)

f(E)

(C13)

2 e 0 ( 0 ) / a/•° , sin2ae \ ( - - / dt). 7T \2« 0 (0) Jo tE /

r" sinlat sin2ae d r"

*'1-2

sin2ae

which is equivalent to (5.45) of the text. Since / = 0 when 2 , = 0, we can write

/l-2f(E)\de - JE IE

This integral m a y be expressed in terms of es and its temperature derivative by differentiating the defining integral for to with respect to fi=l/kT. By a change of variable, Pt=x, ffto—y, we may write 1

l-2/(£)

~ I

(C18)

According to (5.44) of the text, AT is defined so t h a t the expression on the left is equal to unity. If we integrate the right hand side b y parts, we find that

(C12)

ireo(0)A/, 'o

/(tf,0) =

t.

l-2/(E)sin2ae

Note that this expression vanishes in the two limits eo—>0 and a (or R)—>», as it should. The second term vanishes as R—*0. From (C2) and (CI2), we may write 2ATt
1 - 2f(E) ] it

to

AT

=

(CI7)

/t+a\de

•'o

J(R,T)

1201

(C16)

We next consider the integral of J(R,T) with respect to if. If we introduce a convergence factor, we may " See Erdelyi, Magnus, Oberhettinger, and Tricomi, Higher Transcendental Functions (McGraw Hill Book Company, Inc., New York, 1953), Vol. 2, pp. S and 19.

as stated in (5.25). Finally, we shall derive an approximate expression for the transform K(q) valid when ?fao>e 0 (0). First we express K{q) as an integral of J(R,T). The transform of j (r) may be written j(q) =

*(«)ft(q) 4ir

J

4ircAr?i

R[R-0(q)]e i '-i -J(,R,T)dr. R*

(C23)

1202

BARDEEN,

COOPER,

To carry out the integration over angles, we take the polar axis in the direction of q and set u—cosB. This gives K(q)=

I

(l—tii)ei>«»'J(R,r)dudR.

j

When (C13) is substituted for J(R,T), following integral : j eiRi»{\—u?) J-i

g(b) = I Jo

= (l-62)lnf

(C24)

qc^AhvaJo 'o^o I•

4 /«"l-2/(£) 4 / 1 - I de^.—I txJt E tiXN^V

fuc\ In— I i,/ 4

flftflo

(C2S)

l-2/(£)

g(b)-.

(C34)

The latter form follows from the expression for eo(0) in the weak-coupling limit. The remaining terms cancel when fco! is neglected:

We thus find

eo to

which can be determined from (C20), the defining integral for eo. For e > « i we may take / ( £ ) = • 0. T h u s we find

= — In . «i « ( 0 )

sin(iRq/e{)dudR

6 W r"\\-2J(u) K(q)=-

SCHRIEFFER

we require the

J+2b,

where ei= \qhvo and b=t/a-

AND

n

db b1

15

—=0.

(C26)

Jj. 136

£

Thus we find that for qhvo/eo(0) =

Expansions of g(b) for i small and b large are

(C35)

156" irqio^l,

16e0 l-2/(€.)ln(ir?£ 0 ) qc?Ahva I TpqhVo 37^60

fe
g(6)=4^6—ft V 3

3

b>---\ 15

(C27)

K(q)=-

/

(C36)

For q very small, K(q) approaches the constant value,

J>1: g(b) =
+

}

15b3

\3b

(C28)

Further, r"

db (C29)

We may change the variable of integration in (C26) from c to b, and then use different expansions for b < 1 and b> 1. One of the integrals required is 1

l-2/(£)r

1 6—b»

4|

£

I

+4

i

3

1

Z ( 9 ) = 4x/(Ac 2 ) = l A i . ! ( 0 ) , 3x

b'--

49XL2(0)£„

1 ' l-2/(£) 1 - + — + • 3b 1563

—.

(C30)

(C37)

(q-0);

(C38)

16

K(q) = -

b

( ? ->0)

where A T is the temperature-dependent London constant defined by (C22). For intermediate values of q, one may interpolate between this constant value for ir?fo
db

15

= ^/(ATci),

K(q)=\L~'(T)

/

ln(a-^o) , (large?). (C39)

1

Near Tc, one may determine KT from (C22) and (3.49) : A/A r =0.23/3 C W(7') = 2 ( 1 - / ) ,

(C40)

In terms of b,

£=6 1 (fr 2 +W) } ,

(C31)

w h e r e 0 c = l / k T e and l=T/Tc.

When t0(T) is small,

1 -2/(«0) = tanh(i/8e0)-*|^o(r).

where (C32)

6o= eoAi-

An approximate expression valid in the limit 4o<SCl may be obtained by neglecting bo in the integral for b> 1, and also in terms in b' and higher in the integral for 6 < 1 . The only integral in which V is not neglected in comparison with b* is the first term, 1

r' >ll--22//((££}) f t

J

£

44 f/•' « l - 2 / _( ,£v _) _,

db=— j ti^o «i*A>

<**,

£

Thus the limiting expressions for K (q) as T-^TC 1

are

2(1-0 (
K(q)-*L*(T)

(C42)

AL»(0)

18.3

3.75*#(?) =

(C41)

s

4?Xx, (r)f0

ln(ir?fo) , (large ? ) . (C43)

1 v'qio

(C33) The close similarity of the expressions for the two

THEORY

OF

SUPERCONDUCTIVITY

limiting cases is evident. The reason for the similarity is that J{R,T) varies slowly with T. At temperatures very close to Te, X becomes larger than {o, so that X(T') approaches Xi(7"). A comparison of penetration depths measured near Tc to those measured at low temperatures would then be expected to give X(0) X(0) Xx,(0) X(0) -lJl=_H__ll=_L!v5(i_/)».

x(D x t (0)x(r)

an electron of given spin direction. The first term is just the average density, while the second gives the effects of correlation. The integral of .PAM gives the number of electrons of opposite spin correlated with a given electron. This is
r

eo (0) 1 dk

£

C*.|«*(k 1 yV(kwOe(ki,0«(k*OI*«>

ki,kt,ks,k4

Xe«k«-ki).r"+i(k,-k,).r' >

(DS)

T»e 2 »

**+* ksinkrdk

,.(l.)'fs£L±( J®.

kr-i C^+rf(0)]* 2irW*j

E

(D«)

To evaluate this, we make use of the sharp maximum of the integral near *=0 and the fact that the sine function is rapidly oscillating in this region for values of r of interest, krK2>l. We set (as in Sec. V) dk fcsdfeH--

dS

}r

kF+e/{hvo),

(D7)

and

CD3)

sinl kr-\ Irc^sin/feyrcosf — r I, V. A»o/ \hvo /

7=

kr 1

sinfcf''

PA (0 = »[*»+-?* M l where

(D8)

give slowly varying functions their value at the Fermi surface, and drop terms antisymmetric in t. Then, setting x=e/to, we get

use the matrix elements obtained in Sec. IV, and set r=r'—r", we get at the absolute zero of temperature

I

1

cos(rx/r(o)dx

(D9)

(«M-D»

where a=fa«i/«o»l. The integral may be expressed as e„'(0)

^W—(—)

« 2 -H„ ! (0)

J

(TJ2)

where 'J'o is our ground-state wave function. Of particular interest is the correlation function for electrons of opposite spin. If we define P-t=PH+PiT,

o t)

dt

where we have given slowly varying functions their value at the Fermi surface, and have used (5.43) and nc=N(0)eo(0). Thus the number of electrons of given spin correlated to one of opposite spin is of the order of the ratio of the number of electrons in coherent pairs to the total number of electrons in the system. The range of the spatial correlation may be determined by investigating PA^). We must then evaluate

p„.,-(r',r") = ( ^ o | ^ . * ( r " ¥ , - * ( r ' ) ^ ( r ' ) ^ " ( r " ) l * o )

=

1

~4*,{0

\2w/

This can be written in second quantized form, following the notation of Sec. V, as

]"/

dS\r IF

3

• •ir.«*(r l - • -rn) (Dl)

T2

In

APPENDIX D. CORRELATION OF ELECTRONS OF OPPOSITE SPIN

X*(ri- • •rn)S..(!i-*r)B...(rj-r").

<2+
2

xL(o)

Insight into the coherent structure of our ground state wave function may be obtained by an investigation of the correlation function for electrons of opposite spin. For the normal metal (in the Bloch approximation), electrons of antiparallel spin are entirely uncorrelated while for electrons of parallel spin there is the exchange correlation. The correlation function p,'„«(r',r") for an w-electron system is defined as

l

dk-

a

(C44)

This is to be compared with the ratio 2(1 —/)* which the empirical law, (1 —Z4)*, gives as t—»1.

p.,...(r',r")= ZJ--jdrv

1203

f

fdkdb'le**'-"(R

EE'

cos(r/V£o)z -dx=Ko<

(D4) o

(*H-1)»

(-)

1 cos(ra:/V£o) The integration is over flt, the region of interaction dx, (D10) (|«| ), and » is the number of electrons of both spin directions per unit volume. The terms in the bracket give the number of electrons of opposite spin where K$ is a modified Bessel function.™ Since o^>l per unit volume one would find a distance r away from except for very small values of r, (xt+l)^^x in the

44 1204

BARDEEN,

COOPER, AND

last integral. Then, setting l=rx/(*£<,), we have

I=— — smkJKo(—)irV hv„

L

Virfo/

J

f

^-dt\.

«rA{. t

SCHRIEFFER

behavior of PA(T) is given by

(Dll)

J Px(f)

Asymptotically

-I

t

di-<.SmX t

x

'

~S>1 '

(T)12t

while Ko(x) falls off exponentially. Thus the asymptotic

9«*02(0) 2(foo)2

smtkrrsin*(——'\ \fca> irf0/

.

(D13)

(kpr)*

Since o=Ao)/«o»l, the range of correlation is determined by the Ko function which drops off rapidly when r/(ir{o)j>l. Thus correlation distance is the order of irfo^lG -4 cm.

VOLUME 1, NUMBER 11

PHYSICAL

REVIEW

LETTERS

DECEMBER 1,

1958

to the vector potential, A. Coulomb interactions between electrons are neglected. By carrying out a perturbation theoretic canonical transformation based on the unperturbed Hamiltonian, H0, Wentzel eliminates the coupling of the s y s tem to the vector potential through order g2. Thus he finds H=e~K HeK= H0+Hg + 0(g3A),

(2)

where K = KA+KgA+KggA,

(3)

and the generators are defined by [H0,KA]=-HA, lH0,KgA\

= -[Hg,KA],

[B0,KggAh-[Hg,KgA].

(A) (5) (6)

To the same order in g, the transformed c u r rent density operator i s given by

where JM(q) is the current density operator in the presence of the vector potential in the original representation. Wentzel then calculates the c u r rent density by taking the expectation value of (7) with respect to the Bogoliubov'-Valatin* r e p r e sentation of the BCS ground state appropriate to H'=H. Since Hg is not treated as a perturbation in this last step, a Meissner effect is obtained and it is found that

THEORY OF THE MEISSNER EFFECT D. Pines* and J. R. Schrieffert Ecole d'Ete de Physique Theorique , Les Houches (Haute Savoie), France (Received November 7, 1958)

Urn J x ( 5 ) = - £ - 2 f ! A x ( q ) ,

(8)

q*~o In order to obtain an explicitly gauge-invariant description of the Meissner effect in superconductors, Wentzel 1 has recently suggested an a p proach which differs in several essential respects from that developed by Bardeen, Cooper, and Schrieffer. 2 Wentzel finds in the long-wavelength limit a relation between the current density and the magnetic vector potential which differs from the London value given by the BCS theory. We wish to point out that this discrepancy is due to Wentzel's assumption that corrections to his a p proach possess a convergent expansion in powers of the phonon-electron coupling constant, g. It is our belief that this expansion does not exist. Wentzel considers the Hamlltonian H=H, + Hg+HA*H'-mA,

(1)

where the three terms describe the free electrons and phonons, the electron-phonon interaction of strength g, and the coupling of the electrons

where p=g2N{0)-*i, A/(0) being the density of states in energy of electrons of one spin at the Fermi surface and J. denotes the transverse components. 8 To understand the difficulties with this approach, we note that the exact expression for the current density including t e r m s linear in A may be written as 8

^vKuvfaAyifo,

(9)

where [H',SA)=-HA,

(10)

and * 0 is the ground-state eigenfunction of W: fl'*Q=Wa*a. (H) If SA were expanded in powers of g, one would obtain Wentzel's expression for the current den407

46 VOLUME 1, NUMBER 11

PHYSICAL

REV

sity. The exact expression for SA is

As Schafroth7 has shown, the Meissner effect is established by considering; the form of K^v(q) in the limit q2~0. We argue that for a superconductor, the excitation energies that appear in (12) have a radically different behavior from the free-electron excitation energies in this limit. Thus^ the free-electron energies, ^k+q " £ k =q-(2k+q)/2w, vanish in the limit of small
EW

LETTERS

DECEMBER 1,

1958

National Science Foundation Senior Post-Doctoral Fellow, on leave of absence from Princeton University, 1957-58; present address: The Institute for Advanced Study, Princeton, New Jersey. 'National Science Foundation Post-Doctoral Fellow, on leave of absence from the Department of Physics and The Institute for the Study of Metals, University of Chicago, Chicago, Illinois, 1957-58. 1 G. Wentzel, Phys. Rev. I l l , 1488 (1958). 2 Bardeen, Cooper, and Schrieffer, Phys. Rev. 108, 1175 (1957), hereafter referred to as BCS. S N. N. Bogoliubov, J. Exptl. Theoret. Phys. (U.S. S.R.) 34, 66 (1958). 4 J. Valatln, Nuovo cimento T_, 843 (1958). *We choose units H = c = 1. 'Note that/(„ „(q) should not be confused withK d e fined by (2). 7 M. R. Schafroth, Helv. Phys. Acta 24, 645(1951). B G. Rickayzen, Phys. Rev. i n . , 817 (1958). 9 D. Pines and J. R. Schrieffer (to be published).

1050

PHYSICAL

REVIEW

VOLUME

121,

NUMBER

4

FEBRUARY

15,

196

Excitons and Plasmons in Superconductors"' A. BASDASIS AND J. R. SCHRXEPFER

University of Illinois, Urbana, Illinois (Received October 13, 1960) The Anderson-Rickayzen equations of motion for a superconductor derived within the random-phase approximation (RPA) are used to investigate the collective excitations of superconductors. A spherical harmonic expansion is made of the two-body interaction potential V(k,k') and a spectrum of excitations whose energies lie within the energy gap 2A is obtained. These excitations may be characterized by the quantum numbers L and M involved in the potential expansion. For an L-state exciton to exist, the i-wave part of the potential must be attractive at the Fermi surface. Odd-Z excitons have unit spin and may be considered as spin waves. For j-state pairing in the superconducting ground state, the plasmon mode corresponds to the i = 0 exciton whose energy is strongly modified by the long-range Coulomb interaction. For a general potential several bound states may exist for given L and M. If the i-wave potential is stronger than the s-wave part of the potential, the system is unstable with respect to formation of /.-state excitons. In this case, the ground state is formed with Z-state pairing, special cases of which are the ^-state pairing considered by Fisher and the <J-state pairing proposed recently by several authors for the ground state of He3 and nuclear matter. Corrections to the Anderson-Rickayzen equations are discussed which lead to a new set of exciton states if the i-wave potential is repulsive. These excitons are interpreted as bound electron-hole pairs, as opposed to the particle-particle excitons present with an attractive £-wave potential.

I. INTRODUCTION

excite a pair of quasi-particles from the ground state. The quasi-particle excitations are fermions and no N the original theory of Bardeen, Cooper, and boson excitations appear other than the phonons. 1 Schrieffer an approximation to the ground-state This independent quasi-particle approximation has wave function of a superconductor was obtained by a been surprisingly successful in explaining the thermovariational calculation. Basic to the theory is Cooper's result 2 that if a net attraction exists between the par- dynamic properties as well as the acoustic and electroticles, the Fermi sea is unstable with respect to the magnetic absorption, the nuclear spin relaxation, and formation of bound pairs. The BCS ground-state wave the Meissner effect observed in the superconducting function is formed from a linear combination of normal state. The derivation of the last has been criticized state-like configurations in which particles are excited because it is not strictly gauge-invariant. This is to states of low energy above the Fermi surface. In all primarily due to the neglect of residual interactions of these normal configurations, the single-particle states between particles in states ~ k and k V k . These interare occupied in pairs (kf, — k-i), so that interactions actions give rise to a set of collective excitations (bosons) other than those between pairs of electrons of zero net and lead to a gauge-invariant description of the momentum and spin are neglected. The theory leads to Meissner effect. For the investigation of these collective excitations, the single quasi-particle excitation spectrum given by 3 4 £ k = (ei^+Ak 2 )*, where ek is the Bloch energy measured Anderson and Bogoliubov, Tolmachev, and Shirkov with respect to the Fermi level and Ak is the energy gap; have used a generalized time-dependent self-consistent that is, 2Ak represents the minimum energy required to field or random-phase approximation (RPA). Their work shows that in the superconducting state, the * This work was supported in part by the Office of Ordnance Research, U. S. Army. One of the authors (A. Bardasis) was aided plasmon frequency and the plasmon coordinate in the by a Fellowship from the Minnesota Mining and Manufacturing Company. 'P. W. Anderson, Phys. Rev. 112, 1900 (1958). 1 4 J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, N. N. Bogoliubov, V. V. Tolmachev, and D. V. Shirkov, A New Method in the Theory of Superconductivity (Consultants 1175 (1957). ! Bureau, Inc., New York, 1959). L . N. Cooper, Phys. Rev. 104, 1189 (1956).

I

EXCITONS

AND PLASMONS

long-wavelength limit are essentially the same as in t h e normal state. They have also suggested the existence of the exciton modes lying within the energy gap which we investigate in t h e main body of this paper. A thorough discussion of the generalized R P A has been given by Rickayzen, 6 who used it to derive the complex dielectric constant of a superconductor and the Meissner effect in a gauge-invariant manner. T h e BCS quasiparticle states \a) and |/3) do not satisfy the continuity equation; that is, (a\ V-j-|-p|/?)^0. When collective modes are included, the current and charge density operators j and p are decomposed into a sum of individual-particle operators and collective operators. A virtual cloud of plasmons surrounds each quasi-particle, producing a back-flow current which leads t o over-all charge conservation of the excitation. Therefore, t h e continuity equation is satisfied within the generalized RPA. This condition is sufficient to guarantee a gaugeinvariant form of the electromagnetic response kernel. Tsuneto 6 has applied Rickayzen's analysis to t h e problem of the surface impedance at finite frequency. While he finds t h a t a precursor absorption exists for frequencies below that of the gap, his results, when applied to lead and mercury, predict an absorption which results from exciton states in the gap which is an order of magnitude smaller than that observed b y Ginsberg, Richards, and Tinkham 7 in these materials. The origin of the observed peak is uncertain a t present. I n this paper we interpret the exciton mode in t h e superconductor as a bound pair of quasi-particles whose center-of-mass [ ( r i + r 2 ) / 2 ] propagates with momentum hq. The exciton spectrum is investigated through t h e generalized R P A equations of motion proposed b y Anderson in the form introduced by Rickayzen involving the quasi-particle operators y* of Bogoliubov 8 and Valatin 9 rather than c*, t h e usual electron operators. In these equations we make an expansion of the interaction potential F(k,k') in terms of spherical harmonics. I t is found that excitons may be characterized by t h e approximate quantum numbers L and M describing the symmetry of the states with respect to the relative coordinate rj — r 2 . The existence of an L-state exciton (corresponding to the p,d,f, • • • excitons) is dependent on VL being negative, where Vi is the Z-wave part of F ( k , k ' ) . The plasmon state corresponds to an e s t a t e exciton whose energy is greatly increased by the longrange Coulomb interaction. To obtain solutions to the Anderson-Rickayzen equations, we take matrix elements of the equations between a state with one collective excitation and the ground state which has been renormalized so as to include t h e zero-point motion of the collective modes. The results give two sets of solutions Axjf(q) and 5 e

G. Rickayzen, Phys. Rev. 115, 795 (1959). T. Tsuneto, Phys. Rev. 118, 1029 (1960). ' D. M. Ginsberg, P. L. Richards, and M. Tinkham, Phys. Rev. Letters 3, 337 (1959). 8 N". N. Bogoliubov, Nuovo cimento 7, 794 (1958). a J. G. Valatin, Nuovo cimento 7, 843 (1958).

IN

SUPERCONDUCTORS

1051

VLM(q) which correspond to what Anderson has termed odd and even solutions. We show t h a t the k^M(q) modes are unphysical and that the r.LAf(q) modes correspond to the exciton states. The quantum numbers L and M are found t o be exact in the limit of zero center-of-mass momentum hq. For larger q, states of different L are mixed, although the mixing is small for g r £o«l, where £o is the coherence length. T h e magnetic quantum number M, however, remains a good q u a n t u m number for all q if t h e potential has no crystalline anisotropy. The exciton energy for t h e y = 0 case is plotted a s a function of the L-wave coupling constant gL defined b y gL= — J Y ( 0 ) F L / 4 I T , where N(0) is t h e density of states in the normal phase a t the Fermi surface. For gL>go, the excitation energy proves to be imaginary a n d t h e implications of this with respect to the original B C S ground state are discussed. The M^O exciton m a y be considered as transverse collective excitations since they do n o t couple with a longitudinal field. In t h e general case if the ground state is formed from La, Mo pairs, the La, Mo exciton becomes the plasma oscillation. In Sec. I I we discuss t h e generalized R P A from a diagrammatic point of view. Solutions for the collective excitations are obtained in Sec. I I I . I n Sec. IV we consider corrections to the Anderson-Rickayzen equations which lead to a new type of exciton state closely related to exciton states occurring in insulators. II. EQUATIONS OF MOTION We consider a system of electrons interacting via a n effective two-body potential V, whose matrix elements in the Bloch state representation are given b y ( k / , k 2 ' i F | k 1 , k 2 ) = -HK(k 1 ,k 1 ')+F(k 2 ,k 2 ')} XSki+k 2 ,ki'+k 2 '.

(2.1)

This potential arises from both Coulomb and phonon interactions between electrons and will be discussed in detail below. The Hamiltonian is expressed in t h e Heisenberg representation in terms of t h e operators c^ and Ck» which create and annihilate electrons in Bloch states of momentum k and spin a. They satisfy the usual Fermi anticommutation relations. T h e singleparticle Bloch energies ty, measured relative t o t h e Fermi energy EF, are assumed to be of the form (h2k2/2m)—EF. T h e Hamiltonian of the system is given b y

ff=£ €*;*/«*+* Jfc.ff

£

^(M+q)

k.k'.q.cr.ir'

XCk+q,» Ck'—q'.cr' Ck'.o'Ck,*.

(2.2)

In the generalized R P A one studies the time evolution of bilinear operators of the form * k t ( q ) = Ck+qttc-k*t,

(2.3a)

6k+,(— q) = c_k-,*ctt,

(2.3b)

Pk.(q) = Ck+q»tCk5,

(2.3c)

1052

h+q

A.

.,1.

BARDASIS

AND

k+ql

» -k

i

FIG. 1. The two vertices occurring in the full equations of motion for b^Hq). In the linearized equations only vertices with certain values of p' and q' are retained.

which create excitations with a fixed total momentum hq. It is helpful to consider the full-time development of these operators as being built up from the infinitesimal change of the operators in a time interval it; for example, 5*kf (q,0 = i k f (q,/+«) - 6 k ' (q,/)

= (l/ih)lH,bS(q,t)J&l.

(2.4)

In the absence of the interaction V, the commutator reduces to (ek+,—ek)ikf(q,0 so that except for a phase factor, the operators are independent of time. We call any operator pj an eigenoperator if its time dependence is given simply by a phase factor. The equation of motion, [/W>MW, (2.5)

J.

R.

SCHRIEFFER

six, . . . , etc., single operators to form the pj's in this case. The question arises whether there is a consistent approximation in which the eigenoperators are represented as linear combinations of the bilinear operators bf, b, and p alone. Consider a typical term in the commutator arising from the interaction potential $V'(p 1 p+q)[Cp+,.t t £-,._,'» t c_,'*£pt I ik t (q)] = jF(P>P+q){Ck+*+g't t C-p'-g'* 1 C_p'4C+_ H 5 Pi k + , — Ck+qt t Cp+q't t C_k-q'* t CptJp',k}.

(2.6)

This expression is shown in diagrammatic form in Fig. 1. In the diagram, time is increasing from right to left with the incoming particles in states k + q ? and — hi entering from the right. The first term on the right-hand side of (2.6) is represented by Fig. 1 (a) in which the interaction, represented by a dashed line, scatters the spin-up incoming particle to k-f-q-f-q'f, creating a particle and a hole in states — p'— q'| and — p'l, respectively. In Fig. 1(b) the analogous process for the spin-down particle given by the second term in (2.6) is shown. If at time 1=0 a pair of single particles is excited, at time U there is a finite probability that a particle-hole pair has been created from the background of particles in the Fermi sea, with the incoming particles scattering to new states. In the next interval of time a similar process may occur involving any of the four excitations, and in general the "bare" incoming particles will create a complicated cascade of excitations leading to a decay of the initial state. In the generalized random phase approximation one keeps only those terms in the commutator which conserve the number of excitations allowing for both forward and backward propagation in time (see below). This procedure corresponds to a linearization of the equations of motion by replacing two singleparticle operators in each term by a c-number given by the expectation value of this pair of operators with respect to a fixed state. If this state is chosen to be the BCS ground state, denned by

for the operator guarantees that pj, when applied to an eigenstate j/J) of H, creates an eigenstate \a) of H with an excitation energy Kl„. From the Hermitian conjugate of (2.5) it follows that pa has the inverse l^)=n[«k+^kt(0)]|0), (2.7) effect of pj. That is, while pj adds energy to the k system, pa subtracts energy, so that pj and pa may be thought of as creation and annihilation operators of where |0) is the state with no particles present, conexcitations of the system. A knowledge of the eigen- servation of momentum and spin leads to nonzero operators and their eigenenergies allows one to calculate average values only for the operators 4^(0), *k(0), and dynamic properties of the system as well as the thermo- pi,,(0)=nk,. In terms of the parameters «k and t)k, these averages are dynamic functions. In certain cases the state pj\f>) may vanish iden<^o|Jkt(0)|^0) = {^o|Jk(0)|!Ao)*=Mk«;k, (2.8a) tically ; for example, if pj creates pairs of fermions in 2 2 g W»k.|vJk . ( - b) states already occupied in \0). Another example is if the operator pj scatters excitations already present in The parameters « k and vt axe given by the initial state, in which case pj vanishes when applied «k= + (l+ek/£k) i , (2.9a) to the ground state. Both cases will be dealt with in ! »k= + (l- (2.10) (2.4) is complicated by the presence of terms involving p and A satisfies k four single-particle operators (c and c s). Therefore, the bilinear operators b1, b, and p are no longer eigenAk=E K(k,k')—. (2.11) operators of H and one must include products of four, k' 2£ < k

EXCITONS

AND

PI.ASMONS

This prescription gives a unique linearization of the equations of motion since for q ^ 0 there is at most one pair of operators with zero total momentum and spin in each term. The terms retained within this approximation are shown in Fig. 2. (1) As shown in Fig. 2(a), the conventional particleparticle scattering vertex arises from the first term in (2.6) when p' = k. The factor of | in front of V is cancelled by the term in the interaction with spins opposite to those in (2.6). This cancellation of the factor of 5 occurs in each vertex. (2) Another possibility, shown in Figs. 2(b) and 2(c), is for the scattered incoming particle to enter a bound state with the other incoming particle, the outgoing excitations being the particle-hole pair created from the sea. This possibility is allowed for in the linearization by including the finite average (^o!b^(0)\^ 0 ), which may be regarded as the amplitude for the pair to enter the g=0 bound state, which is macroscopically occupied in !^o). Since a finite fraction of all the electrons occupy this bound state in the superconducting state (corresponding to the finite fraction of helium atoms occupying the k=0 state is superfluid He4), the small fluctuation ~iV* in the number of pairs N described by (2.7) leads to no difficulties in a large system. Notice that in Figs. 2(b) and 2(c), the incoming pair of particles is transformed into a particle-hole pair by the interaction. Therefore, 6kf(q) and pk»(q) are coupled in the equations of motion. (3) In addition, there is the possibility that the scattered incoming particle enters the bound state with the particle created from the sea, leaving the hole and the other incoming particles as the outgoing excitations, as shown in Figs. 2(d) and 2(e). Due to the presence of the bound state, the incoming spin-up particle in Fig. 2(d) is transformed into a hole in the state of opposite momentum and spin. In the next instant of time the inverse process may occur. It is clear that the equations of motion are simplified if one introduces "quasi-particle" operators ytJ which are the proper linear combinations of particle and hole creation operators to account for these processes. The appropriate transformation, introduced first by Bogoliubov and by Valatin, is Tt - , ; „t = (2.12a) 7ko'=«kCkt — »kC_k4, 7klt=«kC_k*t+I'kCkt.

(2.12b)

For mathematical simplicity we will follow Rickayzen by expressing the final linearized equations in terms of quasi-particle variables. (4) The exchange contributions to the single-particle lines are shown in Figs. 2(f) and 2(g). As is well known, they lead to an anomalously low density of states at the Fermi surface in the normal metal unless a screened interaction is introduced. This point is discussed below. The exchange self-energy vertex can be accounted for, along with process (3), by the quasi-particle transformation (2.12).

IN S U P E R C O N D U C T O R S

1053

* i — * •

^>.

d

s

i

<

FIG. 2. The vertices retained in the full linearized equation of motion for ik^(q). Vertices/, g, h, and i were neglected by Anderson and by Rickayzen. The particle-hole excitons are obtained only if the interactions shown in h and i are included. (5) Finally, the unscattered incoming particle may enter the bound state with the particle created from the sea, leaving the hole and scattered particle as the outgoing excitations, as shown in Figs. 2(h) and 2(i). As in process (2), the pair of incoming particles is transformed into a particle-hole pair by the interaction. In the limit q —> 0, process (2) is more important than (5) in forming the plasmon state. Since the momentum transfer is always hq in the former process, the large matrix element of the Coulomb interaction 4?re2/<72 dominates the latter vertex in which the momentum transfer hq' may assume any value. Anderson and Rickayzen have neglected processes (4) and (S), suggesting that their effect is primarily to renormalize the single-particle energies and the effective interaction. The terms occurring in the linearized equation of motion for pk»( 0, the exchange correction to the plasmon frequency vanishes. Since matrix elements of the equations of motion are taken with respect to RPA eigenstates, two pairs may be spontaneously created from the vacuum and may interact with the incoming excitations as in Fig. 4. This process may be viewed as a propagation of the excitations backward in time, familiar in the Green's function formulation of the problem.

51 1054

B A R DA SIS

AND

J . R.

SCHRIEFfER

easily seen that the vertices 2(b), 2(c), and 3(b) are automatically screened within the R P A through the presence of the polarization vertex [Fig. 3 ( b ) ] in the linearized equations. For example, when the vertex 2(b) is followed in time by a series of vertices 3(b), the effect is to replace the bare interaction line in 2(b) by the screened line shown in Fig. 5. Therefore in vertices 2(4), 2(c), and 3(i), the unscreened interaction VD must be used. The potential VD is given by

z> < :

ire1

VD(d)=

jr

f

+0"-(«/)•'

(2.13)

where kil is the energy of the excitation involved. Also, »,' is the bare electron-phonon interaction matrix element introduced by Bardeen and Pines 10 and u,* is the bare phonon frequency. I t is essential, however, to -*-r introduce the interaction screened by the dynamical dielectric constant in the remaining vertices since it is impossible to replace the bare interaction line by the screened line through an iteration of vertices occurring FIG. 3. The vertices retained in the full linearized equation of in the linearized equations. The screened potential is of motion for j>kt (<j)- Vertices a, e, and/were neglected by Anderson the form and Rickayzen. 47re2 K(k,k+q) = — -, (2.14a) In the generalized RPA for the superconducting state jMq.'Ok,,) the presence of the bound state gives rise to the vertices represented in Figs. 3(c), (d), (g), and (h), so that an incoming particle-hole pair can be transformed into where the dynamical dielectric constant is given by either a pair of particles or a pair of holes. Therefore, the w operators ftkf(q) and 4k+«(— q) are coupled by the "(q> k,q) = l+47raion(q,6>k,q)+47rae((q,a>k,i,) ~ 1 - (k,q=ek+q ek and k, is the electronic screening Fig. 2 except that all arrows are reversed and the wave number. In a more complete treatment involving momentum q is replaced by — q. coupled equations of motion for the electrons and the We turn now to the question of screening. Within the lattice, the energy ftwk.q would presumably be given in random-phase approximation to the normal state, the terms of the quasi-particle excitation energies. screened interaction line is represented in the limit of For simplicity, we neglect the vertices shown in Figs. small wave-vector q by a sum of diagrams of the form 2(h), 2(i), and 3(a). We also neglect the exchange selfshown in Fig. 5. Rickayzen has shown that the dielectric energy correction since it simply renormalizes the constant is essentially unaffected by the pairing corre- single-particle energies. With these approximations, one lations occurring in the superconducting state. I t is obtains the equations first given by Anderson:

^ _

[fl',Jk t (q)]=(«k+Ck+,)** t (q)+l / c(Q)p(q)(«k tl k+«k+( 1 !'k+q) + A kPkt(q) + A x + , p . M l ( q ) - ( l - t k ! - » t + 1 ! ) E k ' F(k,k')fr
(2.15a)

[^,Ak+,(-q)]=-(
2

- W ) i : K K(k,k')Jk'+„(-q),

(2.15b)

[ H , P k t ( q ) ] = ( £ k + , - « k ) p k t ( q ) + ( r k 2 - f k + , 2 ) ^ ( q ) p ( q ) + A kAk t (q)-Ak+,6k+,(-q) + « k » k Z k . V ( k , k ' ) 6 k < t ( q ) - « k + ^ k + , i ; k ' J/(k,k')&k' + ,(~q),

(2.15c)

C^,p_k-,i(q)]=(«k-ek+,)p-k-,*(q)-(i'k 2 -i'k + q 2 )V'j)(q)p(q)-AkAk + ,(q)+Ak + ,*k + (q) +«k+,"k+, L k . V'fJUOMCq)-«*«>* £ k ' 7 ( k , k ' ) 6 k ' + , ( - q ) .

(2.15d)

The density operator p(q) is given by p(q)=Ek.<.Ck + „ t Ck''.

As mentioned above, the equations can be considerably simplified by transforming to quasi-particle

k

variables. The Anderson-Rickayzen equations are then: C#i7k+,0 t WJ = (£k+,+£k)7k+,o t 7ki t + Pi>(q)>»(k,q)p(q) -5*(k,q).4k(q)+5H(k,q)Bk(q), (2.16a) >° j . Bardeen and D. Pines, Phys. Rev. 99,1140 (1955).

52 EXCITONS

AND

PLASMONS

[ff,7k+qi7ko] = - (£ k + ,+JE k )7k+,i7ko-yc(q)>»(k,q)p(q) -i/(k,q)^k(q)-in(k)q)Bk(q), t

t

[H,7k+,,„ 7k,J= ( £ k + , - £ 0 7 k + , . . 7 k , , .

IN

< ^ (2.16b) (2.16c)

The coherence factors are defined by /(k,q) = M k M k+ ,+» k xi k+ „

(2.17a)

>»(k,q) = «k«k+q+»k«k+q,

(2.17b)

n(k,q) = M k tt k+q -i> k t> k+q ,

(2.17c)

£>(k,q) = w k zi k + q -ti k « k + q ,

(2.17d)

FIG. 4. A typical diagram retained within the randomphase approximation to pk«(q) in the normal state.

(2.18a)

Sk(q) = E k ' F ( k , k ' ) [ 6 k - ( q ) t + J k . + , ( - q ) ] = Ek.F(k)k')[»(k',q)[7k.+,ot7k-it +7k'+,i7k'o]-w(k',q)(7k'+qii t 7k'o p(q) = Ek.»Pk.(q) = Ek'[»n(k',q)(7k' + ,
XZ*'»»(k',q)(7 k . + q o t 7k'i t +7k'+ q i7k q)£*
Those operators pj(q) are now considered which are linear combinations of the bilinear products of 7t's and 7t t 's appearing in the two equations of motion (3.1), and which create one elementary excitation of type a. Thus we desire M«t(q) = Ek[v»(k,q)7k + q o t 7ki t +X„(k,q)7 k + q ,7 k o], (3.2)

(2.18b)

with FMat(q)|0>=[M«(q)+Wo>„t(q)|0))

(2.18c)

From (2.16c) we see that half of the normal mode operators are of the form 7k+ q « t 7 k »', which has the eigenvalue £ k + q — £ k - These operators describe scattering of excitations already present in the initial state and vanish when applied to the ground state. Since we will always take matrix elements of the equations of motion between the ground state and an excited state, these quasi-particle conserving operators may be safely neglected. III. SOLUTIONS OF EQUATIONS OF MOTION For the analysis of the plasmon and exciton modes at temperature T=0 we begin with the AndersonRickayzen equations of motion (2.16) for the pair operators 7*+»ot7nt and 74+51710. I t must be kept in mind that the equations have been linearized with respect to the ground state involving s-state pairing between electrons of opposite spin and momentum, as our results depend critically upon this fact. The collective variables defined by (2.18) are substituted into the equations in order to obtain them in a form involving only the Bogoliubov-Valatin quasi-particle operators:

7k + q o*7ki t =E(iC/ff(l',q)M? t (q)+/s(k J q)w(- q ) ] ,

(3.4a)

t

7k+qi7ko=E#fe(k.q)w (q)+2/»(k,q)w(- q)]- (3.4b) Taking matrix elements of Eq. (3.4) between |0) and | l(q,a)) and using the orthonormality property of the excited states, we find

(l(q,a)|7k + q .7ko|0)=£„(k,q) e x p [ t U , ( q ) a (3.1a)

(3.3)

where 10) is not the original ground state of BCS, b u t the renormalized ground state with / ^ a ( q ) | 0 ) = 0 . T h e quantity M)„(q) represents the energy of the excitation created by the operator M«f(q). The elementary excitation M«f(q) may be any one of the three types involved in the theory: a pair of excited quasi-particles in scattering states, a plasmon, or an exciton. From Eq. (3.3) and the discussion of Sec. I I , we have [ff,M„ t (q)]|0)=ftn a (q)^„ t (q)|0). Since the commutator [H,nJ(q)~} is related to the time derivative of /*o t (q), the matrix element of M«+(q) between the ground state |0) and the state | l ( q , a ) } containing one excitation of energy ftSl0(q) must have the time dependence exp[jQ„(q)0- Now, Eq. (3.2), expresses M«f(q), within the RPA, as a linear combination of the bilinear products 7 k + q o t 7 k i t and 7 k + q i7 k o, so t h a t we may write the inverse transformations as

= /«(k,q)exppfi„(q)0,

r.#,7k+»ot7kit3 = K k (q)7k + q o t 7ki t +l / o(q)»t(k,q) X£k'»»(k',q)(7 k . + q o t 7 k 'i t +7k'+,i7k'o) + ^(k,q)rk-F(k,k')/(k',q)(7k'V7k'it - 7 k ' + q i 7 k ' o m n ( k , q ) £ k . K(k,k')«(k',q) X (7k'+ q o t 7k'i t +7k'+nl7k'o).

k+ql

[#,7k+ql7k(fl = - >-k(q)7k + qi7ko- F D (q)m(k ) q)

X (7k'+ q 0 t 7k'l t +7k'+ql7k'o).

^k(q) = -Ek'F(k,k')C«k-t(q)-ftk-+,(-q)] = - £ k < V(k,k')[7(k',q)(7k.+,oW

+-rViW,i)l

<&

k'+qt—*-^J ^^ .u _^T R '

kt

and the three collective variables are

-7k'+,i7k-o)+/»(k',q) X (7k'+qO t 7k'0-7k'l t 7k'+ql)],

1055

SUPERCONDUCTORS

(3.5a) (3.5b)

The solution for the exciton mode dispersion relation is dependent on taking matrix elements of the equations

A.

1056

f-

BARDASIS

AND

i FIG. 5. The random-phase „ I approximation to the screened \ p l interaction line.

J.

R.

SCHRIEFFER

harmonics. The coordinate system is chosen so that q lies along the polar axis with 6 and

K(k,k')=i;F i (*,*')F ! o(r)

* ^

of motion (3.1) between the states |0) and 11 (q,a)) and = EVl(k,k') £ Fim(0,¥,)*F1„(@,*), (3.7) using the relations (3.5a) and (3.5b) so that we obtain (-0 m—l a set of e-number equations. The resultant system of linear equations may then be solved for the normal where V,(k,k')=(W2l+l)*?i(k,k'). mode frequencies and the transformation coefficients / A further approximation is made in setting Vi{k,k') = Vi, and g. a nonzero constant, for (e|0„(q)-**(q)]/„(k,q) order of the Debye energy. We define the coupling = VD(q)m(k,q)Zw m(k',q)[/ a (k',q)+g„(k',q)] constant gi by (3.8) +^(k,q)i:».'K(k,k')/(k',q)C/<,(k' ) q)-^(k',q)] g i =_JV (0)^/4*. + M k , q ) Z * - V(k,k')n(k',q) The BCS coupling constant is related to g0 by X [ / „ ( k ' j q ) + ^ ( k ' , q ) ] , (3.6a) ga= -iV(0)Fo/4x=A r (0)FBcs>0. C*Q.(q)+»(q)]f«(k,q) It is convenient to introduce three new variables = - Fz.(q)m(k,q)i: . wCk'.qJC/.fk'.qJ+g.O'.q)] k

+i'(k,«)E*-K(k > k')/(k',q)[/.(k' I q)-f.(k' ) q): -i»(k,q)Ek- F(k,k>(k',q) X[/a(k',q)+s„(k',q)]. (3.6b) From (3.6) it is evident that an explicit form for F(k,k') must be chosen in order to proceed further. As emphasized in the foregoing, the BCS ground state about which the Anderson-Rickayzen equations have been linearized is one involving j-state pairing. Thus in the absence of crystalline anisotropy, the q —»0 solutions must transform according to the irreducible representations of the full rotation group, i.e., the spherical harmonics. Because of this fact, we expand the two-body potential F(k,k') in terms of spherical

A(m(q) = L k »(k,q)F i F i m (e, ? )* XC/(k,q)+«(k,q)],

(3.9a)

r(-»(q) = £,J(k,q)F ( F,„,(0,p)* XC/(k,q)-£(k,q)I 2 ( q ) = * V q ) £ k W(k,q)[/(k,q)+£(k,q)],

(3-9b) (3.9c)

where the subscript a has been dropped from both sides of the equations for simplicity. Equations (3.6) then express the transformation coefficients / and g in terms of the new variables A, T, and Z. By substituting these expressions into the defining relationships (3.9), we obtain the following coupled integral equations to determine the eigenfrequency fi(q):

Ai»(q)= VL £ * ( k , q ) — — — {2^(q)m(k,q)Z(q)YLM*(e,v,)+Vk(q)n(k,q)ZYLM*(e,lp) I,m k C*n(q)] 2 -"k(q) : ' XF,m(0,¥>)Ai„(q)+^Q(q)/(k,q) £ YLM*(e,
(3.10a)

l.m

r ^ ( q ) = VL £ *(k,q)——: — {2hQ(q)m(k,q)Z(q)Y*LM(0!V,)+hn(q)n(k,q) * lhU(q)J-Ml)

£

Y*hM{6,v)

l.m

XF 1 ,(«,p)A ta ( ! ) + »k(q)/(k,q) £ Y*LM(e,V)Ylm(6,¥>)rb.(q)), Z(q)=VD(q)

£ M(k,q)

(3.10b)

2, t (q>»(k 1 q)Z(q)+, t (q)ra(k,q) £ 7ta(fl,».)A«m(q) +M2(q)/(k,q) £ F,„(9,v,)r im (q)}.

(3.10c)

l.m

From these three equations it is immediately seen that one good quantum number for the description oi an

excitation is the magnetic quantum number M. In the sum over k, the angular integration requires m = M, as

EXCITOKS

AND

PLASMONS

IN

SUPERCONDUCTORS

1057

the only ^-dependent quantities involved are the spherical harmonics. Thus, M is a good quantum number regardless of the center-of-mass momentum ftq. (1)

g—>(fCase

In the case of zero center-of-mass momentum, Eqs. (3.10) give L as an additional good quantum number. This follows since neither the coherence factors nor the energy i»k(q) of the quasi-particle pair are dependent on the polar angle in this case. T h e angular part of the sum £ k then reduces to

/

^ 9• i5O)

YLM * (e, k=&L$M „

L

The sum 2Zk is converted into an integral by letting

• [ n / W ] fdk&dwk where the volume v of the normalization box is taken as unity. The radial integrals over * are all of the form

q(k,0)Kk,0)3

-kHk,

(3.11)

(2*) J where each of the quantities a, b, c, • • • is one of the coherence factors, the energy vk(0) of the independent quasi-particles, or the excitation energy Ml. T h e integration over the magnitude of k is replaced by an integration over the Bloch state energy <*, as measured from the Fermi surface, by setting k'dk = (m/W)l(2EF)Ut=

2rW(0)d(,

VJhm«rLM (3.13a)

VLI^)VLM

=

\imZ(q)2VLImlm<'SLo.

(3.13b)

5-.0

From these equations it is seen that the direct Coulomb interaction 'lire 2 /? 2 involved in Z(q) only appears for the Z . = A f = 0 state. It will be shown below that this slate has a solution corresponding to a plasma oscillation with the usual plasmon energy fcn,=ft(4jr»ey»»)»~

10 ev

(l-m,„>°)=0,

( A t * mode),

(3.14a)

(1 - VJ,,*)=0,

{TIM mode).

(3.14b)

Setting x= (M2/2A) ^ 1 in the integrals 7,„>° and IHfi and using the definition (3.8) of the coupling constant gL, Eqs. (3.13) become:

(

1 gL

1\ /arcsinaA )= - ( J (I-*2)4 go'

V

X

(Az.Mmode),

/

(3.15a)

(
= lim 2(q)2Fi/, m „ 0 6Lo, •FL/W!„0ALM+(1-

and lies far above the gap 2 A ~ 1 0 _ S ev. I n this section only the M 5^0 cases will be considered, in which the right-hand sides of Eqs. (2.13) become zero. Since the integrand of IHOU0 is odd about the Fermi surface within the constant density of states approximation, Inain" vanishes and there is no coupling between t h e A and T modes. T h e excitation energies for t h e L^0 modes with zero center-of-mass momentum are then determined by the conditions:

(3.12)

where we have made the approximation of a constant density of states. The approximation leads to an error of order huc/EF— 10~3. The integrals I„\,...° are only performed over the region —•hoic<( 0 case are written as

( 1 - VJ,^)Uu-

FIG. 6. The £-state exciton energy in the limit q —> 0 as a function of the £-wave coupling constant &,, where s state pairing in the ground state has been assumed. The solid curve is based on the Anderson-Rickayzen equations while the slightly higher dashed curve includes the effect of the vertices shown in Figs. 2(A) and 2(«) for |o"=0.25. For gL>gt the /.-state exciton energy is imaginary. If gi is the largest coupling constant, the linearization should be carried out with respect to L-state pairing in the ground state.

(1-*J)»

(YLM mode).

(3.15b)

Values of x= (hU/2A) are plotted as a function of the left-hand sides of these equations in Fig. 6. T h e plot shows that when gL becomes larger than go, the frequency Q of the TLM mode becomes imaginary, indicating that the system is unstable when described by a ground state formed with s-state pairing. Therefore, if gL is the largest coupling constant present, the ground state should be formed from pair functions having L-type symmetry. T h e pair spin function is singlet or triplet depending on whether L is even or odd, since the wave function describing the exciton state must be antisymmetric on the interchange of all coordinates of the quasi-particle pair involved.

1058

A. B A R D A S I S

A N D J . R. S C H R I E F F E R

The growth of the TIM modes for gL>ga also indicates that the ALM modes have no physical existence. As is seen in Fig. 6, a ALM exciton cannot exist unless gL> goHowever, when such a coupling strength is reached, the corresponding YLM exciton is unstable so that the system decays before the AL.V mode can come into existence. Figure 6 also indicates the 2£-fold M degeneracy of q=0 L-state excitons. I t should also be mentioned that a continuum of scattering state solutions is obtained from (3.14b) corresponding to the vanishing of the denominator of the integrand. One such state exists between two successive unperturbed levels, £ k + £ k + q . Although the energy of a scattering state solution is unaltered from its value in the absence of interactions, its wave function is strongly modified since each particle is surrounded b y a depletion of the same type of particle leading to the backflow picture mentioned above.

£k+q= «|H by

fc% q h'q* 1 , m 2m (3.15c)

where fi^hvoq, ;u=cos0, and »o is t h e velocity of a particle a t the Fermi surface. This leads to a n error of order q/kr
-

b...=

1

f

-kUk,

I

(2»)»J

(3.16)

(ftQ)'-» k (q)>

in powers of /3. This procedure is valid so long as /S<M2—2A. T h e integrals over k are then of the form Iab...=hi...<>+vIah.J+SIab..z+

•••,

(3.17)

with superscripts indicating the powers of 0 involved. Keeping terms through order fP and using the relations

(2) q Finite Case From Eq. (3.10) i t is seen that L is not strictly a good quantum number for the case of finite q since the coherence factors and vir(q) now have a polar angle dependence. Because of the complexity of this dependence, the sum J2* cannot be carried out exactly.

ALM(q) = VL§dJ,2^-\

We approximate

C O S 0 = M = (4ir/3)K10(ff)

and 2

COS 0=M ! = !(4ir/5)F 2 0 (e)+ ( 4 f / 3 ) F „ , the equations for A and Y (3.10) become 2/47T\*

hmn1Z(q)YL„*Y10+\((.4*)irrnfi+—I„AYo0+-(—) 4*-\» XE YLM*YlMAtM(q)+(~\

rLM(q) = VLjdu\2\((^)iTmJ+-^-hmJ\Yl)o+-(—\ /4jr\* + ( — ) / A W F I O Z YLM*YlmA1M{q)+\

/„*KJ

hau'YnEY^Y^T

#(q)

(3.18a)

hmJYj\Z{q)YLM*

(4*)* r / (4*-)* \ f (4»)»/„*+ I,t*\Y,

2/4*y

+-( — ) ^ F J E YLM*YlMTlM{q) J. (3.18b) With the relation r(2/1+i)(2i2+i)-|* f I du Fi3m3*F/2mj>F!imi= C(h,h,h; mi,m2,m3)C(li,l2,h; J L 47r(2/3-t-l) J where the C's are usual Clebsch-Gordan coefficients,11 Eqs. (3.18) become

0,0,0),

I /4*V / W \ f 2 / + 1 Hi 2( — ) Iymn1Z(q)SLlSi!0+ ( (4») */,„.»+ /„* ) £ C(01L;0MM) I \ 3/ \ 3 / 1 l4ir(2I+l)J 2/47r\* f 5(2/+l) 1* XC(01L; 000)ALM(q)+-( — ) /„„' 2 Z C(2lL; 0MM)C(2lL; 000)A i M (q) 3\5/ i L47r(2L+l)J

ALM(q)=VL\

/47r\* <4TT\*

_.. rr;3 ( 2 / + l ) 1*

C(UL; QMM)C(UL; 000)r,.*(q) + ( — Ihmn'E V3/ ( L4 11 Refer to M. E. Rose, Elementary Theory of Angular Momentum (John Wiley & Sons, Inc., New York, 1957). L4^(2L+1)J

1'

(3.19a)

EXCITONS

AND TLASMOXS

IN

SUPERCONDUCTORS

1059

2/4*-\» l' i Af(q) = K I , 2 f ( 4 T ) » / H , , „ » H

— / A D f a ' W n W f - Y — j /tnr« 2 «/. 2 8AfJ^(q)

! / (4T \ r 2/+1 "I* r 2 . . » + — / , i >W ) EE I I + ((4ir)»/.i*+ 3 V 3 // ii LL4ir(2I+l)J 2/4*V T 5(27+1) I*

+-(

—) W

3\ 5/

E

i Ux(2L+l)J

C(QlL;OMM)C(OlL;000)T,.M(q)

C(21L; 0MM)C(21L;

000)TLM(q)

/47r\« r 3(22+1) i» / W E C(l^;0MM)C(l^;000)A L M (q) . i LM2Z+DJ

+ (V—3 )/

(3.19b)

As in the q —* 0 case, the Coulomb field represented by by a quantum number L within the approximations of the presence of the Z{q) term does not couple into the the calculation, due to A 0 M and TOM vanishing identiequations of motion except for the longitudinal modes cally for M 5^0. Thus, we may speak of a p-, d-, • • • state M = 0. Discussion of this case is deferred and the exciton when the additional term in the potential has transverse cases M9^0 are now considered. For a given L= 1, 2, • • • type angular dependence. M^O, Eqs. (3.19) represent a set of 2N linear simulIf the potential contains 5- and />-wave potentials, taneous equations in ALM and FLM, where N is the number of terms present in the spherical harmonic F(k,k')=FoF0(,*(e,¥.)Foo(©,*) + v 1 r 1 , ± 1 *(0„ P )r 1 . ± 1 (0,*), (3.20) decomposition of the two-body interaction (3.7). I t follows that for a given set of VL'S the normal mode the dispersion relations obtained from (3.19) are found frequencies of the system may be obtained by setting to be the determinant of the coefficients of the ALA/'S and TLM'S equal to zero. Once the frequencies have been — = (/•„'"+-K,„' 2 ), [Ai.±i(q) modes], (3.21a) obtained, the Az. w 's, T I M ' S , and the transformation Vi coefficients / and g may be determined. For simplicity we consider the case for which all but — = ( / w ° + i / , i « 2 ) , [Ti. ± i(q) modes]. (3.21b) two of the VL'S vanish. I t is assumed that the two-body Vi potential consists of a term V0l corresponding to the BCS parameter and another, VL, representing the We discard the A mode since it does not exist if the angular dependence of the interaction. Since M has system is stable. T h e dispersion relation (3.21b) for the been taken as nonzero, it is seen that the simplified Vo r ] l ± i ( q ) mode, when rewritten in terms of explicit and VL potential allows the modes to be characterized expressions for the integrals 1,^ and Iyl£ becomes 2x arcsinx Vi

( ? {„) 2 = 30* 6 '

(!-*»)»

go' 2

!

x(9+2* )-)

2(3* —6)

(!-*»)»

where * = M 2 i , ± i / 2 A < l . This dispersion relation is plotted in Fig. 7 for two values of gi with go=0.25. From the figure, it is seen that the curve intersects the origin for gi = go. For a value £2
(3.22)

arcsinz arcsina+fx*-

(1-**)»J

for Az,jv(q) and r z j / ( q ) . Before discussing the M = 0 cases, it should be emphasized that the equations of motion (3.1) which are the basis of this paper are those linearized by Anderson about the B C S ground state based on j-state pairing of the electrons. As Anderson 3 1 2 has pointed out, it is the j-state exciton which corresponds to a plasmon excitation, due to Z ( q ) coupling into the equations of motion. The Z = 0 mode is considered in the q—>0 limit. Because of the singular nature of the direct interaction, it is not possible to set q=0 in the calculation, so that 12

K. Yosida, Progr. Theoret. Phys. (Kyoto) 21, 731 (1959).

1060

A.

BARDASIS

AND

J.

R.

SCHRIEFFER

q —> 0 case: (3.23)

(1 - Ko/,,*)r 0 o= lim Z(q)2Ko/j n i „ ff~*0

From the definitions (3.5) and (3.9) an expression for Z(q) is obtained:

M?)

Z(q) =

I ( — ) / r a „ 0 A 1 0 (q)

+ 1 (4a-)*7W+ .7

.8

.9

1.0

FIG. 7. The p-state exciton energy as a function of momentum q for £o=0.25 and g^O.24 or 0.25. The parameter f0 is the coherence length ~10-* cm. Notice that the exciton states are strongly bound only for g~'>£o. the limit q —> 0 must be taken. For our starting point, we consider Eq. (3.13b) for the r 0 0 ( q ) mode in the

(ha,moyVo+I„m><>(l - VJ,tfi) = 0.

I»a,JT10(q)\.

(3.24)

Since the i = 0 mode excitation energy is being considered, only the Too(q) term in (3.24) need be used in substituting for Z(q) into (3.23). Rearrangement of terms then gives: (h^+T^/3)

( 1 - VJ.fP)

(3.25)

(3.26)

these relations, (3.26) becomes: 2A

i \ ( —+/,i'|* Vo

/

r'"" 2iV(e)AV-E /Km,o= I _ _ _^ej J-hu, (M2) 2 -4£ 2

(3.27a)

N(e)A/E

Q r *»
l

J-**, (hay-IE?

X

The validity of (3.26) is shown by considering the explicit form of the integrals involved:

J-hu

jr0o(q)

l - l W Since VD((])~1/<12, Eq. (3.26) indicates t h a t in order for the limit to be finite, the terms in the numerator which are independent of q must vanish:

*»•

(5 /

2 / 44irV T\*

+•

hm*!>(hmmo+hmJ/3)Vo+ l = \im&vVj>(q)

ImiJ

»»« ftue

N(e)A*/E

*»„ hue

(hn)2-4E?

(l-Voh,>)dt

With the validity of (3.26) established, (3.25) reduces to %*VD(
(hny-^E?

= 0.

| (3.28)

To determine the existence of a plasma oscillation for the L—0 mode, (3.28) must have a solution for * = (ftfi/2A)»l. Under this condition the term VoI,i*> in the denominator is much less than unity and may be dropped. The integrals involved in (3.28) are evaluated for x»l so that, to order 1/V, (3.28) reduced to

! (hays

J—tiuc

+ * >(to) ( - i -4£? 2

2

£\(to) -4£

2

L)U

(/ttt)2 2

(/ttt) /J

-2A

fcftKo

,hn[(hny-4E?)i

2Ar

1

hQ\

Vo

—

1

+/.!*,

i= (3.27b)

l

where the BCS integral equation 1 for Vo has been used to obtain the first term on the right. With the use of

6x2

MqJjWCOJtf

(3.29)

Using VD (?) = 4w»/j» and ftfN (0) = (3/2jr») ( W 2 A ) 2 , where
58 EXCITONS

AND

PLASMONS

IN

1061

SUPERCONDUCTORS

(4) The 1 = 1, M = 0 Mode To complete the investigation of the collective states present when only the V0 and Vi terms are kept in the potential expansion (3.7), we must determine the dispersion relation for the r 10 (q) mode. Setting M=0 is (3.19b) we obtain two simultaneous equations involving r0o(q) and Tio(q). There is no mixing of these modes in the equations. The Too dispersion relation gives the plasma frequency as discussed above while the IWq) mode dispersion relation becomes l/T1=(/,i'°+t/„'2).

(3.30)

In Sec. I l l (2) we found the dispersion relation for the ri ± i(q) modes to be 1/Fl= (/,!*+K'l*)-

(3 21b1)

Thus the Tio(q) dispersion relation can be obtained by letting q —* qv3~ in (3.22), indicating that for a given wave vector q the excitation energy of the longitudinal Tio(q) mode is raised above that of the transverse ri±i(q) modes. IV. CORRECTIONS TO THE ANDERSON-RICKAYZEN EQUATIONS We consider here the terms in the linearized equations neglected by Anderson and Rickayzen. For simplicity we treat these terms only in the q—>0 case. In the equation for 6*T( the terms shown in Figs. 2(h) and 2(i) were neglected. They contribute the factor -«k»k£k
Zk- Ck'V
to the right-hand side of (3.6a) and the negative of this term to the right-hand side of (3.6b). Introducing the variable •Rk=£-V(k,k')(/k'+gkO,

FIG. 8. The energy of the instate particle-hole exciton as a function of the £-wave coupling constant gL with go=0.25. For ££>0 the particle-particle exciton described by Figs. 6 and 7 is bound while for gL<0 the particle-hole exction is bound. In the absence of the direct interaction VD, the i-state exciton is essentially a bound particle-particle (and hole-hole) pair. With the inclusion of long-range Coulomb interactions, the i-state exciton becomes a plasmon described as a particle-hole pair. C0U

P' e< i equations: 2Ek

FI.M=TLMVLY1

(4.4)

k' Ek

one finds the Mj^O exciton states satisfy the set of

k (AQ)2—4£k; Ak\

Ml

RLMVLT,(—)

k E\Ek/(hS,y-iE/ k/

RLM=TLMVL

(4.5)

hn

X)

r\£ kk7/ ( M J ) 2 - 4 £ k 2 2Ak2 — RLMVL

Z

k £ k [(ftQ) 2 -4£ k 2 ]

Setting the determinant of the coefficients equal to zero, one finds the dispersion relation

(4.1)

to the right-hand side of (3.1a) in the limit q—>0, while the negative of this factor is added to the righthand side of (3.1b). The exchange scattering vertex shown in Fig. 3(a) was neglected in the equation for Pt„(g). Its contribution,

2 i l k k'

S0-+a25

/ , / ) + (/»oi»«)1-0, (4.6)

(—+uA ( or

(g1

L

arcsinx \ / 1

1

x arcsin*\

z(l-*D*/V

g„

(1-z2)*/ (arcsin*)2 +

0,

(4.7)

1-3C2

for the energy of the YLM exciton. The modification of the q—»0 exciton energy given by (4.7) is shown in Fig. 6 for go=0.25 and is seen to be small. A new type of excitation follows from (4.7) for gL<0, that is, a repulsive rather than attractive L-wave interaction between electrons. The energy of this state is shown in Fig. 8 as a function of —gL for go=0.25. From the form of the coherence factors entering the dispersion relation it appears the new state should be interpreted as a bound electron-hole pair in close analogy with the exciton states occurring in insulators. This interpretation is consistent with the fact that the electron-hole interaction is attractive when the corresponding elec-

1062

A. B A R D A S 1 S AND J.

tron-elcctron interaction is repulsive. Thus the electronhole exciton arises solely from the terms neglected in the Anderson-Rickayzen equations.

R.

SCHRIEFFER

thermally-excited odd L excitons (spin waves) by magnetic-resonance techniques. Since the precursor infrared absorption observed in Pb and Hg by Ginsberg, Richards, and Tinkham may be due to creation of V. CONCLUSIONS excitons, it would be interesting to carry out an explicit While we have approximated the i t h spherical calculation of the absorption coefficient for a thin-film harmonic of the two-body interaction by a separable geometry in an attempt to reconcile the difference potential, Fz,(k,k')= — Vi for |ck|, |ek-|c and between Tsuneto's prediction and experiment. We are zero, otherwise, in general, if the potential is independent at present calculating the infrared absorption due to of crystallographic orientation, the numbers L and M hole-particle excitons. remain good quantum numbers for the excitations in the limit q—>0. For a nonseparable potential, i.e., if ACKNOWLEDGMENT Vi(k,k') is not of the form ^t(k)^t*(k'), there may be The authors wish to thank the University Institute more than one exciton state for a given L and M. While the excitons should give a negligible contribution to the for Theoretical Physics, Copenhagen, Denmark, for its specific heat, it may prove possible to observe the kind hospitality extended during the summer of 1960.

60 PHYSICAL REVIEW

VOLUME 8, NUMBER 5

LETTERS

MARCH 1,

1962

CALCULATION OF THE QUASIPARTICLE RECOMBINATION TIME IN A SUPERCONDUCTOR* J. R. Schrieffer and D. M. Ginsbergt Department of Physics, University of Illinois, Urbana, Illinois (Received January 19, 1962) The previous Letter 1 gives an experimental upper limit for the average time required for quasiparticle recombination in a superconductor. Burstein, Langenberg, and Taylor 2 have calculated the contribution to the recombination rate from photon emission, and obtain a result which is much too small to account for the experimental results. 1 It is therefore of interest to calculate the recombination rate due to phonon emission. Using the BCS theory 3 in the limit in which kgT is small compared with the gap parameter, A, the recombination rate r j for a quasi-particle in the state k is given by

I> = ^ k

E Mk,k',X)l

2

*k',A

'2u>q,X

IMk,k<)lV(£rE.,-*^),

(1)

/

ir [ e x p ( W ) + i r '

and y(k, k', A) = phonon-electron matrix element. Here t-r is the Bloch single-particle energy in the normal state measured relative to the F e r m i energy £f» and OJJ ^ is the frequency of phonons of wave vector q and polarization A. Due to the complexities of the band structure in lead and uncertainties of the phonon-electron matrix elements we choose a spherical band model with deformation potential phonon-electron matrix elements, M k , k ' , A) I2 =C*
m * A'2

C

A

*

V^'+V^

7TP

m (2) (3)

(5)

where p is the mass density and C. is the Bloch interaction constant. Also, we set OJ; ^ = S^q and we work in a box of unit volume. Replacing the summation in Eq. (1) by an integration with r e spect to q, k' and the azimuthal angle, we find

where q = k - k ' + K (K = reciprocal lattice vector), ... k k' k k ' \ <(k,k')| 2 = U l = 511 + , £= -£„ k k<

(4)

5

Kk

^

F A

2

/»«

SA^ •I/ - « dH,f»,> k>'k'

(6)

where kp is the Fermi momentum. For states of interest E^ =* A^ so that the coherence factor H is approximately unity. Due to the complicated band structure of lead 207

61 VOLUME 8, NUMBER 5

PHYSICAL

REVIEW

it is reasonable to assume that the Bloch constants for longitudinal and transverse phonons are comparable. Thus we set C^ = C, where C is chosen to fit the high-temperature resistivity p: P

V

9 T T 3 / Hk

16

VV^/VV,

C)

Here, na is the number of free electrons per atom, kj) is the Debye wave number,M is the atomic mass, and Sj^ and Sy are the longitudinal and transverse sound velocities. We use the values for lead: A = 1.34xl0" 3 ev, 1 M=3.46xl0" 2 2 g, S L = 2.39 xlO 5 cm/sec," S r = 1.27 x10 s cm/sec, 4 na = 1.24,5 feD = 0.810x10" cm" 1 , fejr = 1.06x10 s cm" 1 , m*=m, p/feglT = 5.78x10"' esu. One obtains the reasonable value C / £ p = 1.16. With the value 3.44 xlO" 20 erg for the integral in Eq. (6) appropriate to the temperature 1.44°K used in the experiment, 1 we

LETTERS

MARCH 1, 1962

find for the recombination time 1/r = 0.43 x 10" sec. This result is an order of magnitude smaller than the experimental upper limit at this temperature. We wish to express our appreciation to Professor John Bardeen and Professor David Pines for several helpful discussions. •Supported in part by the National Science Foundation, the A. P . Sloan Foundation, and the Air Force Office of Scientific Research. tjv 'A P . Sloan Fellow. >D M. Ginsberg (to be published). 2 E. Burstein, D. N, Langenberg, and B. N. Taylor, Phys. Rev. Letters 6 92 (1961). 3J JJ.. Bardeen, L. N, Cooper, and J. R. Schrieffer, Phys . Rev._108, 1175 (1957). «W, P. Mason and H. E. B'dmmel, J. Acoust, Soc. A m . 28, 930 (1956). 5 R . G. Chambers, Proc. Roy. Soc. (London) A215, 481 (1952).

Nuclear Physics 35 (1962) 863—369; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE EFFECT OF COLLECTIVE EXCITATIONS ON THE SINGLE PARTICLE SPECTRA OF NUCLEI J. R. SCHRIEFFER Department of Physics, University of Illinois, Urbana, Illinois t Received 30 October 1961 Abstract: In deriving the collective excitation spectra of nuclei it is often assumed that the collective modes have negligible effect on the quasi-particle spectrum. An expression is derived for the selfenergy of the quasi-particle states including pairing interactions as well as virtual excitation of collective modes, from which the validity of the above assumption can be investigated.

1. Introduction In the so-called "pairing" approach to the nuclear many body problem 1) one concentrates on correlations between pairs of particles occupying single particle shell model states cpJ>m and
,&J..)|0>,

(2-1)

J,m>Q

ul+vjm = 1,

(2.2)

where b]m = c]mCj„m creates a pair in the valence shell and |0> is the ground state of t Work supported by the Office of Naval Research. 363

J, R. 3CH3UEFFER

364

the doubly closed shell nucleus. More generally, one could include a product of proton operators analogous to that for the neutrons and allow the products to run over all j and m. The Hamiltonian describing the valence nucleons is taken to be H-ixN = lejC]mcJm+i

£

(Js.^inWjJcW^c,,.

(2-3)

Due to conservation of angular momentum, we have m^ + m2 = m3+m4.. The matrix element {J3JA\V\J1J1) is assumed to include core polarization effects. One can determine the coefficients Uj and Vj and the ground state energy by minimizing (i^0, [H—fiN]\l/0) with respect to Uj and v} subject to the constraint uf + v/ = 1. The chemical potential \i is determined by requiring that N0 = (i/r0, Nij/0) be the number of valence nucleons, where N is the number operator. This is the procedure used by Bardeen, Cooper and Schrieffer in the theory of superconductivity *). We shall use an equivalent procedure introduced by Bogoliubov 5 ) and Valatin 6 ) in which one performs a transformation to the quasi-particle operators y / denned by ?! = UJC) + VJC.J,

jj = UjCj + Vjdj,

(2.4)

where v_j = —Vj. The operators y/ and y} satisfy Fermi commutation relations since the transformation is canonical. It follows from (2.1) and (2.4) that yj[//0 = 0,

for all J outside the core.

(2.5)

The Hamiltonian in the y representation is of the form H= U+Ho + H. + H,, where H is ordered by moving all creation operators to the left: U = constant = £ 2e,uj + ± £ {Jr~Ji\V\Jv-J,yj^j^j^j2, J

(2.6a)

JuJz

Ko-lEjylyj,

(2.6b)

j

^ i = I«/{y)y-j+y-/?/}.

(2.6c)

J

H2 = (terms involving four quasi-particle operators).

(2.6d)

We have set Ej = lj — \i. The Bogoliubov-Valatin procedure consists of setting OLJ = 0 and one finds

uj = i 1+

{ l)'

U2/ = where

*( 1- ?) '

(17a)

(2 7b)

'

THE EFFECT OF COLLECTIVE EXCITATIONS

Aj = - £ Vjj.uj.vj. = - £ VjjJf J'>a

J'>o

365

(2.9)

2Ej.

Here, V„. = (/', - / ' | K | / , - / ) - ( - / ' , / ' | K | / , - / ) and J ' > 0 means rri > 0. The energy Ej is required to create a quasi-particle so that the excitation energy of the nucleus in the pairing approximation is Wm^YdEJ,

(2.10)

j

where the sum extends over all states containing excitations and the excited state wavefunction is

i>A«c> = n ^ o > .

(2.H)

the product extending over the same values of J. Notice, in this approximation the entire excitation spectrum is formed by single-particle-Uke Fermion excitations and no collective boson states are included. 3. Greens' Function Approach To obtain the collective states and to treat their interaction with the single particle states, we introduce a single-particle Green's function, defined by G(J, t) = ,

(3.1)

where the exact ground state wavefunction !F0 and the operators y_,(r) and y / ( 0 ) are expressed in the Heisenberg representation based on H—fiN. The operator T signifies the chronological product: T{yj(x)yj{Q)} = ( _ y t ( 0 ) ? X T ) )

t < 0

(3.2)

Also, let G0 be defined as Go = <^olT{yXt)rJ(0)}^o>,

(3.3)

where the operators y and yt have their time dependence given by H0 and tj/0 is the ground state eigenfunction of H0 = £ y Ejyjfyj. The Fourier transform of G 0 (/, T) is easily found to be /•OO

h(J, e) =

G 0 (/, T)e'"dT

i B-Ej

,

(// = 0t).

(3.4)

+ iq

The pole of G0 is at the quasi-particle energy Es. More generally, the single particle excitation energy and damping rate are given by the real and imaginary parts of the pole of G(J, B), respectively. Thus, to find the effect of the collective states on the single particle spectrum, we must find the effect of these states on G(J, e). We first derive the collective modes by using the Green's functions.

J. R. SCHRIEFFER

366

4. Collective States The residual interaction of primary interest in the collective modes is that part of H2 which conserves the number of quasi-particles: (4.1) Ji'ji'ji'j*

The matrix element W involves a product of the parameters u and v and the matrix element (. . . \V\ . . ,>; it is readily obtained by ordering the Hamiltonian in the y representation. Suppose that we start with a two quasi-particle state in the H0 representation. The interaction V2 will modify this state by scattering the particles to new states, conserving angular momentum. The eigenstates of the two quasi-particle system in the presence of V2 are conveniently obtained by considering a t matrix in the space-time formalism. Let t be defined by the integral equation '3,4; 1.2W

= W3A.ia-i

f"

^

£ W^]5i6G0(5, s5 + e)G0(6, -e 5 K 6 ; 1 , 2 (e).

(4.2)

• / _ „ , 27t 5.6

We have replaced Jk by k in the above equation. The integral equation is shown in diagrammatic form in fig. 1.

+ Fig. 1. Integral equation for the t matrix, whose poles in the complex e plane give the energies of the two quasi-particle states and the collective states.

It is possible to show that the poles of / in the complex e plane give the energies and damping rates of the two quasi-particle states. If, for a particular angular momentum L of the two particle state, the relevant portion of the matrix element W3i 4. it 2 can be approximated by a separable potential: fw,2 =

^ i > n ,

(4-3)

the integral equation may be solved exactly and one finds: r

, . _ XL(a\ta>\2 l + Xri^e)

(4.4)

where L(e) = iT

p. £ |o)f6|2G0(5, s5 +e)G0(6, -es).

(4.5)

THE EFFECT OF COLLECTIVE EXCITATIONS

367

Inserting the expression for G0(J, e) we find, after performing the e5 integral,

L(e) = - £ la&l 2

1

5,s e—E5 — E6 + irj Thus, the poles of t are at energies e which satisfy \+XL<j>L(e) = 0,

(4.6)

(4.7)

or

r - Z KA2 At

j,y

„ V

.•

(4.8)

s—Ej — Ej' + iri

The right hand side has poles at the unperturbed energies EJ+EJ. ^ 2A, where A is the minimum value of E. In fig. 2, —L(&) is plotted as a function of e.

Fig. 2. A graphical solution of eq. (4.8) giving the energies of the two quasi-particle states and the collective mode of angular momentum L.

If kL < 0, a collective state of angular momentum L is split off from the bottom of the two quasi-particle spectrum. For L — 2, one obtains the familiar quadrupole vibration of a spherical nucleus.

Fig. 3. Vacuum fluctuation diagrams of this type are neglected in the theory but play a negligible role for weakly bound collective states.

For strongly bound collective states, vacuum fluctuation diagrams of the form shown in fig. 3 are important.

67 368

J. R. SCHRIEFFER

We assume that XL is sufficiently small that these diagrams do not play a strong role, an assumption this is reasonably well satisfied so long as one is near a doubly closed shell structure. 5. Modified Single-Particle Spectrum The effect of the collective states on the single quasi-particle energies is taken into account in the Green's function scheme by computing the self-energy I{J, e) with

Fig. 4. The diagram gives the self energy part 27. The wiggly line is represented in fig. 5.

the aid of the usual Feynman rules. The integral equation for I is given infig.4, where the wiggly line represents the sum of diagrams offig.5.

Fig. 5. The sum of these diagrams gives the interaction line entering fig. 4.

If we denote by M li23 4 the matrix element in the Hamiltonian which multiplies ?4 ?3 t ) , 2 t ') , i> w e n a v e f ° r t n e contribution of the L state collective mode to I : t

£Ci,O = (-02 I

|G0(2,£l-e)

„ ALo>34co5 L

| G 0 ( 4 , e 4 + e)G0(3, - e 4 ) de

-G0(5, s 5 +s)G 0 (6, - e s ) d 8 4 d e j M l i 2 3 4 M * 2

J (2TI) 3

l+XL (s)

(5-2J

Now, carrying out the e5 integral we have f"

rip

G0(5, e5 + e )G 0 (6, - 8 5 ) ^ •'-00

=

, e—£5 — £

2JI

6

(5.3)

+ »J

so that

^ ! . 0 = I f ^(^."i-ajp-—^ ^ 0 0 ] ( « - £ , - £ « + &,) 2,3...6j_00

[1+Aj/L'

1

Ml.234^1,256 W ^ S f i d f i .

(5.4)

(fi-£3-^4+''7)

By closing the e contour in the upper half plane we find. Z(J1,s1)=

V

^

*-L<**\<

Muii*

(5<5)

THE EFFECT OF COLLECTIVE EXCITATIONS

369

or y %A + XL(t> {&y-E2) J3,J,

4/l.£l) = I

L

Mlt23A(034 s1-E2-E3-EA

(5.6)

With the aid of the relation G~\JU

8l)

= Go V i . O + tfCi.Si) = - i ( e 1 - £ 1 ) + ' ^ i , « i ) = 0.

(5-7)

where EJ. is the energy of quasi-p article in state J±, we find the shift in energy of a quasiparticle in this state due to the interaction with the L-state collective mode is approximately given by I{Jy, Ej). It is clear from the form of (5.6) that the collective mode will couple strongly to the single particle spectrum if there exist two single particle states with energies E1 and E2 such that E, « E2±EL, where EL is the energy of the L t h collective excitation. In this case the factor A

is large, giving a large shift in the energy of state Jx. For the specific assumptions regarding shell model energies and the effective matrix elements V, one can estimate the shift of the single particle spectrum due to coupling to a given collective state with the aid of (5.6). References 1) B. R. Mottelson, The many-body problem (John Wiley & Sons, Inc., New York, 1959) p. 283 2) J. Bardcen and J. R. Schrieffer, in Progress in low temperature physics, ed. by C. J. Gorter (North Holland Publishing Company, Amsterdam, 1961) Vol. Ill, p. 170. 3) L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Med. Dan. Vid. Selsk. 32, No. 9 (1960) 4) J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175 5) N. N. Bogoliubov, V. V. Tolmachev and D. V. Shirkov, A new method in the theory of superconductivity (Consultants Bureau, Inc., New York, 1959). 6) J. G. Valatin, Nuovo Cim. 7 (1958) 843

VOLUME 10, NUMBBR 8

PHYSICAL

REVIEW

LETTERS

15 APKU 1963

EFFECTIVE TUNNELING DENSITY OF STATES LN SUPERCONDUCTORS* J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins University of Pennsylvania, Philadelphia, Pennsylvania (Received 15 March 1963) Recent tunneling experiments 1 ' 2 involving superconducting metals exhibit structure in the J - V characteristic which has been interpreted in terms of electron-phonon processes. In the preceding Letter 3 Rowell, Anderson, and Thomas present the results of improved experiments which more clearly resolve this structure. Below we summarize the results of a theoretical determination of the tunneling characteristic which is in good agreement with these experiments. To include dynamic interactions between phonons and electrons in a consistent manner, it is nec336

essary to extend the conventional expression for the tunneling current. 4 ' 5 We take the point of view of Bardeen and of Cohen, Falicov, and Phillips who characterize the tunneling process by an effective one-body Hamiltonian H

T= T

,s kk£

T \.kk t,»cb* ks

bt k's

-H.c.}.

(1)

Here c ^ s a and cbsb6t destroy and create electrons in Bloch states of momentum k, energy t^ m e a s ured relative to the chemical potential ix, and

70 PHYSICAL

VOLUME 10, NUMBER 8

REVIEW

spin orientation s in metals a and b, respectively. At zero temperature the transition probability per unit time for an electron to tunnel from a to b is given to lowest order in T by ifi -1) a b

~

2»S i S rkk>^ i b c nanbkk's

X
a

ks

Re

N(0) °-V).

(2)

Here \na) and Iw^) represent exact energy eigenstates of metals a and 6 in the absence of both H-j- and the applied potential V (measured in electron volts). The excitation energy Wna is given by the difference in energy of the states \na) and i0 ); Wn" is defined in a similar manner. For V« (i the dominant contribution in (2) comes from states k and k' near the Fermi surface. In this case we can approximate T^k' by its average value in this region and reduce (2) to the form = constf%fu;N_ 6(w)N_ fl(V-u.). (3) a—b *o 7+ i Here Nj ± '(w) is the effective tunneling density of states for metal i defined by w

NT±i(u!)=Ni(0)]~°JekP.±<M,u),

(4)

where

{n u

i kJ '°,>h(,

p.~fe,
(5a)

(5b)

and /v(C) is the density of Bloch states in energy at the Fermi surface. By definition p*(*, u>) are the spectral weight functions for the one-electron Green's function G(k,w), and hence it follows from (4) that NlO) f°° N„ (w)=?——I rf£kImG(*,±u).

15 APRIL 1963

Bloch energy and A(fc, w) = 4>(fe, u.)/2(/fe, w) is the complex energy gap parameter. By using the fact that Z and A are essentially independent of k, (6) and (7) may be combined to give

'V)

\cui ° | 0 )h(W b + W k's a ' n n

P. (fe,w)=S

LETTERS

I ul

(8)

[u, 2 -A 2 (w)] V 2

For A independent of u>, this expression agrees with the conventional BCS density of states di.U dE = E/(Ez- A2)V2 for IE|> A and zero otherwise. Note, however, if A varies with w, the effective density of states appropriate to the tunneling process Nj.{u>)/N(0) differs from the standard quasiparticle form [dEU^/dt^]'1, where E(ck) is the real part of the pole of G on the second sheet. This difference, essential in understanding the structure in the / - V characteristic, is due to the fact that cksa I 0a) and cksb^\ 0b) a r e not quasiparticle eigenstates of metals a and b a s they a r e within the simple BCS approximation. Thus, one cannot use the standard quasi-particle densityof-states expression which is appropriate, for example, in calculating the electronic specific heat. Equation (8) is easily generalized to finite temperature. We calculate A (ID) and Z(w) by including both electron-phonon and Coulomb interactions. The phonons are characterized by a frequency distribution F(u)g) and a r e coupled to the electrons by an interaction strength a(u>J. The screened Coulomb interaction is replaced by a pseudopotential U defined to include interactions between electrons outside a band of energies I u> I < wc, which is large compared with the Debye energy. The integral equation determining the complex gap parameter is then A(u

1 •2(w)

r

dm'Re

A'2)'

•

fc

ddj a 2(o) )

XF (to )[D (oi'+w)+£) (OJ'-U))]J

X. q

q

q

(9)

(6)

where Dg(x) = (x +wq-iO+)~\ A 0 =A(A 0 ) is the gap parameter at the edge of the energy gap, and x Thus, a knowledge of C(k, u>) suffices to determine labels the phonon polarization. Notice that Z does the tunneling current under the above conditions. not enter the integral in (9); it is reduced to quad(In the above derivation a spherical Fermi surrature, once A(UJ) is known, by face with an effective mass has been assumed.) To determine G it is convenient to use the forV A•<w) malism of Nambu6 and write

H-z M* -£W R e ^ ^ ^ p ,

G(£,u>) =

ui +e(M, w) u> 2 -e 2 (*,u))-A 2 (/fe,iJ+tO +

where e(£, w) = t.)l/Z(k, ui) is the renormalized

(7)

X-FAW

X q

)[D

q

(U/+U,)-.D

q

(a,'-

w)].(10)

Equations (9) and (10) have been solved by an

337

71 VOLUME 10, NUMBER 8

PHYSICAL

REVIEW

on-line computer facility 7 for a simplified model devised to represent the phonon spectrum of P b . The distribution of longitudinal and t r a n s v e r s e phonons w a s approximated by the sum of two Lorentzians, centered at frequencies ui, = 8 . 5 x 10~ 3 eV and u*/ = 4 . 4 x 10" 3 eV and having halfwidths < * 2 * = 0 . 5 x l 0 " 3 eV and u>,' = 0 . 7 5 x l 0 - 3 eV. The coupling parameter ax(uq) was taken to be a constant independent of both q and the p o l a r i z a tion A; this i s a reasonable approximation since the dominant part of the phonon interaction inv o l v e s umklapp p r o c e s s e s . The Coulomb pseudopotential was adjusted to U = 0 . 1 1 , the value we believe appropriate to lead. The cutoff u>c w a s taken equal to 7u>j + A 0 . While the s p h e r i c a l F e r m i surface approximation is a fairly good one for P b , 8 we need not r e l y on this fact in comparing with the experimental r e s u l t s 3 since these w e r e performed with dirty superconductors.

dl (V) dl (V) s In dV

1963

Njv) f

Re

N(0)

dV

A 2 (K)]*

(11)

The experimental data of Rowell, Anderson, and Thomas, a l s o shown in F i g . 2, a r e in remarkably good agreement with the theoretical curve, considering the simplicity of the model we u s e d . We would like to e m p h a s i z e that the e l e c t r o n phonon coupling i s s o strong for lead that the q u a s i - p a r t i c l e picture i s m e a n i n g l e s s over much of the energy spectrum. N e v e r t h e l e s s , the

1-10

i oe

1

At e n e r g i e s « / + A 0 and tvj + A 0 , the s t r o n g in-

1.12

1

N 7 .(^)/W(0) = l + [ A 1 2 ( u ) ) - A 2 2 ( u J ) ] / ( 2 w 2 ) .

I.Oi-

15 APSIL

c r e a s e of the phonon e m i s s i o n rate produces a rapid i n c r e a s e of A 2 , accompanied by a d e c r e a s e of Al. Both changes produce a sharp drop of Ny(u>) near these phonon e m i s s i o n t h r e s h o l d s . The experimental verification of the structure in Nj<(tx/) can be obtained from the I-V charact e r i s t i c s for tunneling between a normal metal (say Al) and lead in the superconducting and in the normal s t a t e s , a s d e s c r i b e d in the preceding L e t t e r . It follows from (2) that the ratio of the differential conductances in the two s t a t e s i s

1

In Fig. 1, Al and A 2 , the real and imaginary parts of A, a r e plotted a s a function of e n e r g y . The effective density of states (8) i s plotted in Fig. 2 along with the result of the s i m p l e BCS model. The g r o s s structure of Nf(w) can be understood by expanding NT(u>) to first order in A 2 ; thus

LETTERS

1.06

3

,04

V

\

** 11

\ 1

Z 1.02

I***"*.

1.00

V^"

0 96 0 86 0 «4

I

1

I

'


FIG. 2. The effective tunneling density of states NTfa)/N(0) vs (u -A 0 )/w,' (solid) and the density of states of the simplified BCS model u/(w* - A,,2)"2 (short dash). The ratio of the differential conductance of Pb in the superconducting to that in the normal state, dl (u>)/du> s dl (ui)/du' FIG. 1. Plot of the real (solid) and imaginary (dashed) parts of A(u)/wjt vs (u> - A 0 ) / u ; / . Here w1t = 4.4xl(T s eV and A, = 1.34* 10"3 eV. 338

is plotted (long dash) as a function of (a; -A„)/u>,' for T = 1.3°K. These data were obtained from the tunneling experiments reported by Rowell, Anderson, and Thomas.

'

72 VOLUME 10, NUMBER 8

PHYSICAL

REVIEW

Green's function approach we used above i s sufficiently powerful and simple to allow u s to treat this problem in detail without making the quasiparticle approximation. We a r e indebted to Thompson-Ramo-Wooldridge, Inc. for making the on-line computing facility available to us free of charge and to George Boyd for aid with the computations. We a r e a l s o g r a t e ful to J. M. Rowell, P . W. Anderson, and D . E . Thomas for s e v e r a l stimulating d i s c u s s i o n s r e garding the model used to represent the phonon spectrum of Pb and for prepublication u s e of their tunneling c u r v e s . *A contribution of the Laboratory for Research on the

LETTERS

15 APRIL 1963

Structure of Matter, University of Pennsylvania, covering research sponsored by the Advanced Research Projects Agency. 'i. Giaever, H. R. Hart, J r . , and K. Megerle, Phys. Rev. ^26' 9 4 1 < 19 62). 2 J. M. Rowell, A. G. Chynoweth, and J. C. Phillips, Phys. Rev. Letters 9, 59 (1962). 3 J. M. Rowell, P . W. Anderson and D. E . Thomas, preceding Letter [Phys. Rev. Letters M>, 334 (1963)J. 4 J. Bardeen, Phys. Rev. Letters 6, 57 (1961). 5 M. H. Cohen, L. M. Falicov, and J. C. Phillips, Phys. Rev. Letters 8, 316 (1962). 6 Y. Nambu, Phys. Rev. J_17, 648 (1960). T C. J. Culler and B . C . Fried, Proceedings of Pacific Computer Conference, 1963 (unpublished). 8 A. V. Gold, Phil. Trans. Roy. Soc. London A251, 85 (1958).

339

263

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148,

NUMBER

I

5

AUGUST

I96IS

Strong-Coupling Superconductivity. I* D. J. SCALAPJNOf, J. R. SCHJUEFFER, AND J. W. WlLKlNsJ Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania (Received 16 February 1966) The pairing theory of superconductivity is extended to treat systems having strong electron-phonon coupling. In this regime the Landau quasiparticle approximation is invalid. In the theory we treat phonon and Coulomb interactions on the same basis and carry out the analysis using the nonzero-temperature Green's functions of the Nambu formalism. The generalized energy-gap equation thus obtained is solved (at 7" = 0°K) for a model which closely represents lead and the complex energy-gap parameter A(u)) is plotted as a function of energy for several choices of phonon and Coulomb interaction strengths. An expression for the single-particle tunneling density of states is derived, which, when combined with A(u>), gives excellent agreement with experiment, if the phonon interaction strength is chosen to give the observed energy gap A0 at zero temperature. The tunneling experiments therefore give a detailed justification of the phonon mechanism of superconductivity and of the validity of the strong-coupling theory. In addition, by combining theory and the tunneling experiments, much can be learned about the electron-phon interaction and the phonon density of states. The theory is accurate to terms of order the square root of the electron-ion mass ratio, ~10~*-10""'. I. INTRODUCTION N the original BCS theory of superconductivity, 1 a central role was played by the concepts provided by Landau's theory of a Fermi liquid. 2 In Landau's theory,

I

* This work was supported in part by the National Science Foundation and by the Laboratory for Research on the Structure of Matter, University of Pennsylvania, covering research sponsored by the Advanced Research Projects Agency. Part of the work reported here was performed as a portion of a thesis of one of us (J.W.VV.) in partial fulfillment of the requirement for a Ph.D. degree in Physics, University of Illinois, 1963. t Alfred P. Sloan Foundation Fellow. J Present address: Department of Physics, Cornell University, Ithaca, New York. 1 J. Bardeen, L. N. Cooper, and J. R. SchrieSer, Phys. Rev. 108, 1175 (1957). 2 L. D. Landau, Zh. Eksperim. i Teor. Fiz. 30, 1058 (1956).

the excited states <J>.v of the Fermi liquid are placed in one-to-one correspondence with the excited states of a free Fermi gas. T h a t is, the excited states * # are labelled by the occupation numbers vk, of the "quasiparticle" states of momentum k and spin component s([ or 1) in analogy with single particle occupation numbers n*, of the free Fermi gas. Presumably the Landau configurations $t/ contain most of the manybody correlations occurring in the superconducting energy eigenfunctions *» except for those correlations which are specific to the superconducting phase, i.e., the pairing correlations. Since the states $N form a complete set, a state * ,

74 264

SCALAPINO, S C H R I E F F E R ,

AND

WILKINS

148

energy t k . If this is the case, as it is for strong-coupling superconductors, the Landau theory cannot be used as (1.1) a basis for treating superconductivity in these metals. JV'S, these states are re- Therefore one can treat the weak-coupling superconstricted-to a small subset by the pairing condition of ductors from the point of view of the Landau theory, the BCS theory. In simple cases, this condition corre- as in the BCS approach. In addition to this problem of quasiparticle damping, sponds to the requirement of correlated occupancy of time reversed states of the Landau theory (e.g., kf, — kl one is faced with the problem of treating the retarded nature of the phonon interaction between electrons. for a translationally invariant system). coupling The above simplifications reduce the configuration This is a nontrivial problem even for weak 4 interaction problem to a soluable form if one assumes the superconductors. As Eliashberg showed, even in the validity of the Landau theory and assumes a simple two- weak-coupling limit the correct form of the retarded equation differs body interaction between Landau's quasiparticles. This interaction which enters the energy-gap 6 6 simple but powerful scheme has led to a detailed under- from those given by7 Frbhlich, by Bardeen and Pines, 1 and by Bogoliubov. standing of the properties of most superconductors. Viewed as a field-theoretical problem, it might appear There is, however, a class of superconductors such as Pb, Hg, etc., which cannot be treated by such a that one has little hope of handling this strongly coupled procedure. The difficulty can be seen in the following fermion-boson system in an accurate manner. The manner. In constructing a superconducting energy reason that one can give an essentially exact treatment follows from an important discovery of eigenfunction ¥., the configurations $# which give the of the problem 8 largest contribution to the pairing correlation energy Migdal in his treatment of the coupled electron-phonon are typically those in which quasiparticles are excited system in normal metals. He showed that in normal self-energy to above the Fermi surface by an energy of order the metals one can calculate the one-electron 1 average phonon energy p.e., ^huo, where o>z> is the an accuracy of order 2(m/M) ** (m^ electronic mass, Debye frequency]. The average phonon energy enters M=ionic mass) ~10~ by what amounts to secondhere because the attractive interaction which causes order self-consistent perturbation theory. This resuperconductivity is due to virtual exchange of phonons, markable result does not depend upon the strength of on the and this interaction is strongest when energy is nearly the electron-phonon coupling but rather depends 1 conserved during the emission or absorption of the existence of the small parameter (m/M) '* in the problem. The generalization of Migdal's result to virtual phonon. Now the Landau theory works best when quasi- superconducting metals was given by Eliashberg* and particles are excited to states k in the immediate by Nambu.' In this paper we generalize the Eliashberg-Nambu vicinity of the Fermi surface kp. In this case an effective mass approximation for the quasiparticle energy «* can scheme by taking account of the Coulomb as well as 10 be used. £Note: ek—0 for | k | = kj; that is, quasiparticle the phonon interactions between electrons. Equations energies are measured relative to the Fermi surface/] More important is the fact that the lifetime T* of these • G. M. Eliashberg, Zh. Eksperim. i Teor. Fiz. 38, 966 (1960) low-lying excited states is so long that the corresponding [English transl.: Soviet Phys.—JETP 11, 696 (I960)]. • H. Frbhlich, Phys. Rev. 79, 845 (19S0). level width Tk= h/lr* can be neglected in comparison • T. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955). 7 with €*. This is the approximation of the Landau theory. N. N. Bogoliubov, Zh. Eksperim. i Teor. Fiz. 34, 58 (1958) [English transl.: Soviet Phys.—JETP 7, 41 (1958)J. As one goes to higher energy quasiparticle states, the •A. B. Migdal, Zh. Eksperim. i Teor. Fiz. 34, 1438 (1958) level widths of the states increases due to the increased [English transl.: Soviet Phys.—JETP 7, 996 (1958)]. rate for a quasiparticle to decay by emitting a phonon. »10 Y. Nambu, Phys. Rev. 117, 648 (1960). Using a similar "realistic" interaction, P. Morel and P. W. An important question is then whether the lifetime of Anderson have given an approximate solution for the pairing part a quasiparticle of energy a^hwD is so short that the of the self-energy [Phys. Rev. 125, 1263 (1962)3- A numerical level width of these states is of order the excitation solution for the pairing self-energy was reported by G. J. Culler, of the super phase can be represented as a superposition

B. C. Fried, R. W. Huff, and J. R. Schrierler, Phys. Rev. Letters ' J- Bardeen and J. R. Schrieffer, Progress in Low Temperature8, 399 (1962). In both these calculations the "normal" part of the Physics (North-Holland Publishing Company, Amsterdam, 1961), self-energy which is important for strong-coupling superconductors was neglected. Vol. IH.

75 148

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determining the electronic self-energy as a function of temperature are derived in Sec. II. In Sec. I l l an expression for the effective tunneling density of states is determined. This density of states depends upon the complex, frequency-dependent gap part of the electronic self-energy. In Sec. IV the parameters of the model used for Pb are discussed and the solutions of the self-energy equations presented. Using these solutions, the effective tunneling density of states is calculated and compared to the experimental tunneling data.11 The results show that tunneling measurements provide a delicate probe of the structure of the electronic self-energy and reflect the properties of the underlying effective electronelectron interaction. II. SELF-ENERGY EQUATIONS A. Structure of the Equations To treat strong coupling superconductors, we use the formalism of Nambu.* In this scheme one introduces a two-component electron field operator *p=(Cpt)

(2.1a)

I

265

scheme which is formally extended periodically throughout q space to allow umklapp processes to be handled automatically. Since we are ultimately interested in deriving expressions for the thermodynamic and transport properties of superconductors, we concentrate on calculating the Green's functions which determine these quantities. The one-particle thermodynamic Green's functions are defined in the Nambu scheme to be C(P,T) = - <£/r{*p(T)* p t(0)}),

A(q,r) = - (T{ * , x ( r W ( 0 ) } > ,

(2.3a)

(2.3b)

where the average is taken in the grand canonical ensemble (^)=Tr(e-* H ^)/Tr«-s». The operators in (2.3) evolve with the "imaginary time" ir according to *„(T) = e»'*p(0)^*',

(2.3c)

*,x(T) = e*V,x(0)
(2.3d)

The symbol T represents the conventional T-ordered product and the operator U in (2.3a) is given by

U=l+W+R, whose components cvt and c_p»' destroy an electron in a Bloch state of crystal momentum p and spin orien- where i?t converts a given state in an iV-particle system tation |, and create an electron in the time reversed into the corresponding state in the N+2 particle state —pi, respectively. The bare-phonon field operator system; thus for the ground states ¥>qA=«qX+<J-,>t

M\0,N)=\0,N+2),

(2.1b)

(2.4a)

tf|fyV>=|0,A--2), (2.4b) is a linear combination of a destruction and a creation operator for bare phonons of mode X and wave vector etc. Notice that G is a 2X2 matrix, whose diagonal q and — q, respectively. The Hamiltonian of the system components Gn and Gu are the conventional Green's can be expressed in terms of * p and ^,* as functions for up-spin electrons and down-spin holes, respectively, while Gn and Gn are Gorkov's12 F and F* B=£ t „ v > * p + £ n ^ x W functions which describe the pairing condensation. Due to the periodicity of G and D with respect to r, these t + E gpp'XPp_p'X*p' T8'J'p functions can be represented by the Fourier series pp'x

+1

£

Gto,T) = l/fi £

(*p,tr3*p1)

«-*-G(R»0,

(2.5a)

PlWPlPl

X(* P ,tT3* pl )+const. (2.2) Here e, is the Bloch energy measured relative to the Fermi energy Er, and n , T> and TJ are the Pauli matrices. We work in units with h=l. The quantities £2, g and Vt represent the bare phonon frequencies, the bare electron-phonon coupling, and the bare Coulomb interaction between electrons respectively. Translational invariance of Vc restricts pi-f-pj—p»—p< to be zero or a reciprocal lattice vector K. We work in a box of unit volume and impose periodic boundary conditions. The electrons are described in an extended zone scheme and the phonons described in a reduced zone

A ( q , r ) = l//9 t^e-'-'D^ivJ,

(2.5b)

where io„=(2n-T-l)ir/j9,

Vn=2n*/I3,

(2.5c)

n being an integer. The one-electron Green's function for the noninteracting system is easily seen to be given by Go(p,tu,) = [ia)„-£pTa3-1.

(2.6)

The electronic self-energy 2(p,tu„) (a 2X2 matrix) is

11

We have previously reported some of these results, J. R. Schrieffer, D. J. Scalapino, and 1. W. Wilkins, Phys. Rev. Letters 10, 336 (1963).

" L P. Gorkov, Zh. Eksperim. i Teor. Fiz. 34, 735 (1958) [English transl.: Soviet Phys.—JETP 7, 505 (1958)].

266

SCALAPINO,

SCHRIEFFER,

then defined by Dyson's equation [G(p,ia,„)]-'= [ G . ( P > w . ) ] - ' - S ( p , « . ) .

(2.7)

An important feature of the Nambu formalism is t h a t the familiar Feynman-Dyson perturbation series rules (and their finite-temperature generalization) hold in calculating the G and D. Our procedure is to set up a n integral equation for S(p,io)„) which treats the electron-phonon interaction accurately to order ( m / M ) 1 ' 2 ~ s / » F ~ ' W . £ i ' , where m/M is the electron-ion mass ratio, s/vp is the ratio of speed of sound and the Fermi velocity, and O>D/EF is the ratio of the Debye energy and the Fermi-energy. T h a t such an integral equation can be found in closed form was shown for normal metals by Migdal 8 and for superconductors by Eliashberg. 4 In their analysis the theory was worked o u t a t zero temperature and the Coulomb interaction Vc was neglected. Thus they took fl and g to be the appropriately screened quantities as in the Frohlich model of the coupled electron-phonon system. Since the Coulomb interaction plays an important role in a consistent theory of superconductivity, we work with the Hamiltonian (2.2) rather than that used by Migdal and Eliashberg and carry out the analysis a t finite temperature. In setting up an integral equation for 2 it is important to note that we are mainly interested in physical excitations of energy ~ W D < K £ F . Higher energy states are not thermodynamically populated a t superconducting temperatures. In addition, electron tunneling, electromagnetic absorption studies, etc. yield interesting information about the superconducting state primarily in this low-energy domain. Thus we are interested in the structure of 2(p,ioj») for p~pr and | w „ | « £ i r . In this range the Coulomb interaction leads to important screening and renormalization effects, however it does not lead to interesting variations of 2 in a region ~ u D about the Fermi surface as is evident on dimensional grounds. Thus, for our purposes the Coulomb interaction serves mainly to renormalize the bare electron and phonon-energy spectra and screen the electronphonon interaction, as assumed in the Frohlich model. In addition there remains a short range (screened) Coulomb repulsion which opposes superconductivity. As we will see, this short range (almost instantaneous) interaction must be handled in a manner different from that used for the (strongly retarded) phonon interaction between electrons.

O • A p-q p-q FIG. 1. Electron self-energy diagrams for the screened Coulomb (dashed line) and dressed phonon (wavy line) exchange by the self-consistently dressed electron propagator (solid line).

AND

148

WILK1NS

Our basic approximation for 2(p,io>„) is shown schematically in Fig. 1. T h e solid line represents G as given by Dyson's equation (2.7) in terms of this selfconsistently determined self-energy. In the first diagram of this figure, the dashed line represents the electronically screened Coulomb interaction. If the Bloch functions were approximated by plane waves, the screened Coulomb interaction would be given by K(q,tVJ=.[K.(?)A(?,*i..)]; Ke(9)s(W/
Vm

= COn-U,n'

(2.8)

where ic(q,ivm) is t h e electronic dielectric function. In the second diagram of Fig. 1, the wavy line represents the phonon propagator D\(q,ivm) and the right and left dots represent the electronically screened electron-phonon coupling functions, ^p.p-q,x(*J'm) and 9P-q,p.x(i"m), respectively. For a plane-wave approximation to the Bloch function, Q would be a function of the momentum and energy transfer (fi,ivm) alone and one would have g,K(2V m )=[gp, p _,,>.A(g,«i/„)].

(2.9)

In general, § will depend separately upon the initial and final states, p and p— q, of the scattered electron if crystalline anisotropy effects are important. Fortunately, g(ivm) always enters as a factor multiplying D(ivm). Since D drops to zero as l/vm2 for | and dynamical (as opposed to static) electronic screening enters only for »> m ~.E F »«z>, one can safely replace Qw-iiivm) by its static limit &,,'x(O)=0 pp .x. Furthermore, since the (longitudinal) dielectric function is essentially identical in the normal and superconducting phases, <7P,-x can be considered to be a fixed parameter determined in the normal state. 13 While the long wavelength transverse dielectric function is very different in the two-phases this need not concern us since: (a) for phase space reasons only short wavelength phonons contribute appreciably to the pairing correlations; (b) shear deformation, umklapp and collision drag interactions dominate the coupling for phonons of interest to us. These interactions, however, should not be affected by the Meissner currents which modify the long wavelength transverse dielectric function in the superconducting phase. Therefore j?pp.x is considered to be a fixed parameter which we attempt to determine from experiment. Unfortunately, first principles estimates of g are not fully reliable a t present. 14 In our approximation, phonon corrections to the electron-phonon vertex as shown in Fig. 2(a) have been neglected since they lead to corrections ~ ( m / M ) 1 / 2 as discussed by Migdal 8 and Eliashberg. 4 T h e essential point is that because of the rapid decrease of D(ivm) " R. E. Prange, Phys. Rev. 129, 2495 (1963). " Using a pseudopotential adjusted to fit high-temperature resistivity measurements, D. J. Scalapino, Y. Wada, and J. C. Swihart, Phys. Rev. Letters 14, 502 (1964), have calculated an effective electron-phonon coupling which is in good agreement with the results reported here.

148

STRONG-COUPLING

SUPERCONDUCTIVITY.

for [ xm | > WD, only vertices in which the energy transfer |w„—<>v| is of order wx> or less contribute appreciably to 2. For the low lying excitations of interest to us this restriction requires that |u>„| and |w„>| separately be q», q„_i, •••, qs, qi will be an appreciable contribution result, i.e., phonon line crossing graphs like that of Fig. 2(a) can be neglected.

(o)

i < «< * < »

£ t i <* « f

FIG. 2. Vertex corrections to the electron self-energy. The Coulomb corrections to the electron-phonon vertex (other than screening, which has already been included) are not so simple. The lowest corrections shown in Fig. 2(b), lead to a significant change in the effective electron-phonon coupling. Fortunately, these processes lead to essentially constant scale factors multiplying grr^ of Fig 1, as Rice has shown.16 Using this fact we will lump these vertex corrections in with g to be determined from experiment. Notice that we must not include phonon corrections of the electronCoulomb vertex if we include the corrections shown in Fig. 2(b), since this would double count graphs. Finally, there remains the Coulomb corrections to the electronCoulomb vertex. These again lead to scale factors on the screened Coulomb interaction of Fig. 1. Since phonons are not involved here, these corrections will not give interesting energy variations of 2 for |«„|
J-

< Le/

P

« V + •* P

P+K

267

I

< %
P

< ^<

pfK

P

- * i — « — t - t - i — < — 1 < +-«fc—«—-J-*-'—«—U-

+ < '

< \<J, piK.'

< \ii p+K

<—W-

FIG. 3. Umklapp corrections to the irreducible self-energy. Here K and K' are reciprocal lattice vectors. vector K. Therefore, if 2 is denned by (2.7) (Dyson's equation) we should, strictly speaking, include selfenergy graphs of the form shown in Fig. 3, where the momentum of the electron line connecting the various "irreducible" self-energy parts is not equal to the external electron's momentum. Since we are interested in electronic states p near the Fermi surface, a state p + K where K is a reciprocal lattice vector, will in general be far from the Fermi surface (unless p happens to be very near a zone boundary). Therefore, the state p + K will have high excitation energy and it will in general lead to a small effect in determining the excitation spectrum except for states very near zone boundaries, which are of no special importance to us. Therefore, we neglect diagrams of the type shown in Fig. 3 in calculating 2. In view of the above discussion, the integral equation determining 2(p,tu„) is directly obtained by writing down the contributions corresponding to the two diagrams of Fig. 1. 1 2(P,«»»)=

£ r 3 G(p'>„OT 3 /3P'»'

X{L l0p P 'x| 2 ZMp~p'>„-;av) + V(P-P')},

(2.10)

where for simplicity we have taken the screened Coulomb interaction to be a function of the momentum transfer alone. The phonon Green's function has a spectral representation of the form Dx(.q,ivm)=f

ixBx(q,>0{[l/(»,-„-„)] - D / (».+»)]).

(2.11a)

Here, the spectral weight function is given by £x(q,*)= ( 1 - « " E e-">\ (j\ * v | ; } | 2 »./ X * ( » - E y + E O / T «-»*) ,

(2.11b)

SCALAPINO,

268

Z

SCHRIEFFER,

AND

148

WILKINS

The contribution from the circle at infinity can be shown to vanish. This expression for 2 (p,iu>n) can be analytically continued with respect to iw„ to the real axis from the upper half-plane by replacing t&>„ by u+i&. In this form 2 is a function of the continuous (real) variables p and u. Note that (2.14) actually represents four coupled integral equations which determine the four components of the 2X2 matrix 2. It is convenient to express these components as the coefficients of the Pauli matrix representation of 2 :

plow

ST,. C

vf^ y^

2(p,a>) = (l-Z(p,u)))wl-|-*(p>«)T1+X(p)a,)T,) Fic. 4. Contours for changing the summation in the self-energy equation to an integration.

where (2.11c)

#|n)=£„|n>.

By substituting (2.11a) into (2.10) and transforming the H-summation to an integral along the contour c shown in Fig. 4, one finds

(2.15)

where we have chosen phases so that the coefficient of T2 is zero. It follows from (2.14) that Z, $ and X are even functions of o>. By combining (2.15) with Dyson's equation (2.7) one finds the analytically continued one-electron Green's function is given by u2(p,«)l-r-i(p,u)T8-f-^(p,w)Ti G(p,o>) =

<*0fo»)-*M-?tofi>)

, (2.16a) Imu»fJ

where 5(P,W.)=

e(p,w) = e f + X ( p , u ) .

£ / dzrfi{p',z, «)T 2m p' J „

Thus, the calculation of G is reduced to solving three coupled equations for the functions Z, * and X which determine the electron self-energy. The function A(p,u)=*(p,
XJE f d,Bx(p-p',,)[ I » Jo 1

Lia>„—J 1

X l+e~" +-iu„—z+y l+e he"f J -iK(p-p')tanh(0z/2)

(2.16b)

.

(2.12)

B. Reduction of the Self-Energy Equations

To obtain explicit solutions of the integral equations one is forced to use a computer. Fortunately a number of simplifications can be made which greatly reduce the labor involved in carrying out the computation. (1) For G(p, a+iS)-G(p, a-iS) = 2i ImG(p, o>-W«), (2.13) most purposes, X, which arises from the Coulomb which follows from the spectral representation of G, one interaction, can be included as a_simple scale change of (p which is the same in the normal and superconobtains ducting phases. Furthermore, x is a slowly varying 1 r" function of u for u out the integral over ep- so that one is left with two coupled equations determining Z and A, which for , |gf,-x|'^(p-p ,y) • (2.14) an isotropic (dirty) superconductor are one-dimensional equations involving the frequency variable u. In a clean By deforming the contour c to c' (see Fig. 4) and using the relation

-+-

]

148

STRONG-COUPLING

SUPERCONDUCTIVITY.

superconductor, the anisotropic phonon density of states, electron-phonon matrix elements and Fermi surface lead to an anisotropic self-energy." Here we treat the impure case in which these crystalline anisotropy effects are washed out by impurity scattering, and we wish to determine the spherically averaged self-energy

I

269

FIG. 5. Coordinate system for carrying out the momentum integral which occurs in 2" b .

2(p ;,«) = / rffl^(p,Ci))/4T. For the case in which the anisotropy due to the interaction is washed out by impurity scattering, Markowitz and Kadanoff" have shown that the spherically averaged self-energy is obtained if the effective electronelectron interaction \8n-\[*Bx(p— p', v) and ^(p,p') as well as 2(p',u>') which appears in G(p',w'). Eq. (2.14), are replaced by their spherical averages. The primary purpose of this section is to reduced (2.14) to onedimensional form. We begin with the phonon-interaction terms of (2.14):

i 2(p,u)" 1 's

r

£ / do' I m [ r « G ( £ > ' ) r , ] T P'» J_«,

/-^ISPP'AI*/

J

AT

r

^.(p-p'.x)

JO

i

i

i

i

1

Lai——u'+v+U u—u'+v+iS l+e*"'-l l+efl

p'» J o

angle

J (2T)> (2*ypF J where we have denned mdif=p'(

•/

:

Bx(p-p', ») Igpp'xl11-e*' 4x

(2.17)

(2.18b)

and sine d6^q(dq/pr).

(2.18c)

We emphasize that q is not restricted to the first Brillouin zone since we have formally extended the reduced-zone phonon spectrum periodically throughout q space. Thus u,x means the frequency of the phonon of mode X (acoustic or optic, longitudinal or transverse) and wave vector corresponding to q reduced to the first Brillouin zone. Since the dominant contribution to the if integral comes from the region | e,< | ~WB we extend the limits of integration of this variable to infinity and find

/ XT8;

(2.18a)

d«Vrj(7(/>»r>= { - ir\u'Z(u') - (
where

GAZ'K)-^'))1'2},

(2.19a)

2

(2.19b)

2

i!

Im(co' Z (o,')-* (a)'))>0.

The 0)' integrand of these terms decreases as 1/V for On combining (2.19) and (2.16) one obtains \o>'\'^>at> [when the contribution from &>' and —u' are m added]. Thus, because of the form of G, it follows that 2(pp,w)<>)'* (2x)'Pr if we are interested in excitations of energy a>~wj><5CE/?, the major contribution to the p' integral comes from r U'Z(W')1-*(D. For this reason we can replace S I/2 J-. L(CO'»Z'(U,')-« (CO')) J7 4* p' by Pr in Z and $, occurring in the integrand. The same situation holds for the last term of (2.14). Within fr f r 1 this approximation p enters the right-hand side of X qdq dV2ZB^q,v)\6,M x Jo Jo L
xf^Rer"^ -^']/-

l+r-f

" Starting from our results for the isotropic case, the effects of the anisotropic phonon density of states and Fermi surface on the Pb gap have been calculated by A. J. Bennett (to be published). " D . Markowitz and L. P. Kadanoff, Phys. Rev. 131, 563 (1963).

Jo

u-u'+v+iS

Jo

l+eO-'J

4ir/>,

* •-oL(u>. i! ^(a..)-^(a.,)) 1 ' il J

Xf%.^, J 4*

(2. 1-e"'

SCALAPINO, SCHRIEFFER,

270 where

w, = u + ( - l ) V + « ' S .

(2.20b)

The restriction q<2pF reflects the fact that the largest momentum transferred to a phonon occurs when an electron of momentum ~ P f scatters to a state of momentum ~ — p?. Equation (2.20) can be simplified by transforming the negative frequency part of the u' integral to the positive frequency interval /-

2(*F,o))ph= / rfu'Re

r W'Z(U')1+*(«O')TI 1

XQC^(afi>')f(-a')TK±^u,-o)')f(u')) fr£ <-oL(u!,2Z2(w,)—4>H">.))' J f'r qdq dQq Bx(.q,S)

Jo

2

XT.

l^xl -

9 W

(2.21)

x Jo 2pF* 4T e""-l Here, Imw' = + S and AT(0) is a single-particle density of states at the Fermi surface obtained from the effective mass denned by Eq. (2.18b). N(0)=mpF/2*2

(2.22)

As we noted earlier, Eq. (2.18b) is to an excellent approximation the same in the normal and superconducting phases so that N(Q) is just the density of states at the Fermi surface of the normal metal excluding the phonon renormalization of the electronic mass. The phonon interaction kernels K±fh are given by

^Llff.xI'SxfeOr——L e i ) + u -+r

x

The upper signs in (2.21) are to be used with the n matrix component and the lower signs are associated with 1 component. The Fermi function is denoted by

/(«)-!/(«»•+1).

(2.24)

ph

This completes the reduction of 2 to one-dimensional form. The Coulomb part of the electronic self-energy is given by (2.14):

J (2T) A

b\o>')+iS/ XV(p,p')tanb(fr>'/2).

(2.26)

In Appendix A it is shown that the upper limit of the o>' integration in this equation can be reduced to a cutoff uc (which is conveniently chosen to be of order 10 bio) providing V{p,p'} is replaced by an energyindependent pseudopotential Ue. One is then free to perform the p' integration in (2.26) by exploiting the rapid decrease of G{p',<ji') for large | cp-1, (so long as |o>'| <(P)2#c(pr) dco'Rej

2

(ZW -tf2(
du' • Im[VjG(y ,«')r J It

XV(p,p')\anh(fr>'/2),

(2.27)

The complete equation for 2 is given by combining (2.21) and (2.27).

v+iS (2.23)

3

*(*V)

Xtf,tanh(/3u'/2).

u'—u-j-y—iiJ

2*W

J (2r)3

Xlm

••-N(0)j

Jo 2pF*J 4*

Jo

rJ,

f tPp'

*(
r"r qdq f •dQ,

xf

148

where as previously discussed only the j-wave part of the Coulomb interaction V(p,p') = fV(p~p')dQ^.r./4ir enters in determining the s-wave part of the self-energy. Within our static screening approximation, the Coulomb contribution to Z(p,w) vanishes, since the 1 component of G(p',u') is an odd function of oi'. The Coulomb contribution to X{p,w) can be neglected if the single particle energy tp includes the static Coulomb correction to the effective mass in the normal state (which is altered by the transition to the superconducting state by a negligibly small term of order A / £ F ~ 1 0 - 4 ) . A small chemical potential shift ~ A 2 / £ F between the normal and super states is also neglected within this approximation. Thus we are concerned only with 1 f°°

O>,Z(U,)1—<J>{CO.)TI

-iwN{0)

AND W I L K I N S

(2.25)

III. THE SINGLE-PARTICLE TUNNELING CURRENT One of the initial motivations for obtaining solutions of the gap equation for a more realistic model of a metal was the structure observed in the I-V characteristics of Pb tunnel junctions.18'" This structure occurred at bias voltages of order typical phonon energies and indicated that the structure of the interaction responsible for superconductivity was experimentally observable. In this section we calculate the single-particle current flowing between a superconductor and a normal metal separated by a thin insulating barrier as a function of the applied bias voltage. 18 1. Giaever, H. R. Hart, Jr., and K. Megerle, Phys. Rev. 126, 941 (1962). 11 J. M. Rowell, A. G. Chynoweth, and J. C. Phillips, Phys. Rev. Letters 9, 59 (1962).

STRONG-COUPLING

148

SUPERCONDUCTIVITY.

For the purpose of determining the single particle I-V characteristics, the tunnel junction can be described by the Hamiltonian H=H,+H,+HT.

[2mtf(*)-iM»;W«. The first term in HT gives rise, when a bias voltage is applied, to a current flow from I to r. The transition probability per unit time for an electron to tunnel from / to r is given by w^i=2*(Y.\(F\ F

kikrl

Here c\.,S creates an electron in the Bloch state (ks) in metal (/) and ckr, destrovs an electron (k^s) in metal As Bardeen has shown, the tunneling matrix element can be written in terms of the expectation value of the current density operator in the oxide barrier. Since the density of electrons drops to a small value in this region, an independent-particle approximation is presumed to be valid in evaluating 2\,k,. Using Bardeen's expression, Harrison22 has evaluated the tunneling matrix element within the WKB approximation and finds ]rkk'|2=5fc.ii*!r expj —2/" k,{x)dx

j

4nV'Vr)-

where F and / refer to the final and initial states, respectively, and the angular brackets to an ensemble average over the initial states. In Appendix B it is shown that this expression can be reduced to Aut Wrr-i=

rx / du>NrT(o>)(l—f(u>)) XAV(u-P0/(«-K).

(3.6)

Here A n is the area of the barrier, / the effective square of tunneling matrix element £see Eq. (B8)J, and /(o))= (e s "-|-l) _1 is the Fermi factor. Most important, NTW) the effective tunneling density of states, is given by -Vr ( » ) = / " dftA(k,»),

/ 2 V /2 e'^w'smikiX+yt),

if one used a free-electron model, we should only consider positive values of k^ with a mesh of irjL. Since one generally performs averages over smooth, symmetric functions of kit one can alternatively include positive and negative values of &i if a mesh of 2ir/L is used for ky. The delta function in (3.3) involving the components of k parallel to the barrier reflects the fact that the transmission is specular. In the exponential, xi and xT

where A (k,u) is the spectral weight function 1 ^ ( M = -|ImG n (A,«)|.

(3.8)

IT

Here Gn is the one-one component of the Nambu Green's function. We observe that expression for iv^i (outside of numerical factors) is just what one might write intuitively—viz., the current that flows is proportional to the product of number of electrons capable of tunneling and the number of states available to be tunneled into. However, it is not that simple. For example, at r = 0 ° K , the single-particle tunneling current arises solely from a process in which one electron from a superfluid pair in I tunnels through the barrier to a single-particle state in r. The remaining electron of the pair fills a single-particle state in I. The Fermi factors and density of states perfectly represent this case. For temperatures greater than r = 0°K, the other term in HT gives rise to a current flowing from r to / which is proportional to

20

J. Bardeen, Phys. Rev. Letters 6, 57 (1961). » M. H. Cohen, L. M. Falicov, and J. C. Phillips, Phys. Rev. Letters 8, 316 (1962). » W. A. Harrison, Phys. Rev. 123, 85 (1961).

(3.7)

J —tc

(3.4)

where L is the length of the metal in the direction perpendicular to the oxide, kL is the component of the wave vector perpendicular to the oxide and £** is the energy associated with motion in this direction. Since the single-particle basis functions for the metal would be of the form «>* = ( — )

L 7Wtr.t,*„|/)1'&(EF-E,)), *>*" (3.5)

(3.3)

Here p < l r ) are the one-dimensional density of states in metals (l,r) for motion in the direction perpendicular to the barrier interface: p^L/TdkJdZn,

271

refer to the classical turning points, for a given energy of the tunneling particle, and a given barrier potential U(x) in the oxide layer. Here k^(x) is given by

(3.1)

Here Hi and HT are the full many-body Hamil tomans for the superconductor (I) and normal metal (r), respectively, and BT is the effective tunneling interaction discussed by Bardeen20 and Cohen, Falicov and Phillips21 HT= E (.Tkli,ckl.ickr.+H.c). (3.2)

I

/ J —go

dwNT'(<J)f(w)NT>(o>-V)V-J(o,-V)).

SCALAPINO, SCHRIEFFER,

272

AND

W1LKINS

148

where Eh= (t 2 +A 2 (£ t )) I/!! and hence

Thus, the tunneling current density j is given by

WBCS 0»)=d(k/dEk | BJ_„ = [u— AdA (oi)/doi/ »"=

(w„_|—W|«) ,

(ctf>-A2(w))"2].

-Ail el

j=

/••*

/

)NT'(u-V) X(/(«-V)-/(«)).

(3.9)

For normal metals, one can show that the tunneling density of states is just the single-particle density of states in the absence of electron-phonon interactions, since

JdaAM-l+cM—)

1

(3.10)

in this case. Here m/M is the electron- to ion-mass ratio. Thus, electron-phonon interactions cancel out to order (m/M)llt in the tunneling density of states for normal metals.28 Therefore, the normal-superconductor tunnel current density is given by 2

]/" £ = e//16ir Jo

da NT'(<») ,

(3.11)

and dj/dV=lel/l(y**ytr'(w)

j

u.r.

(3.12)

(3.16)

This expression is not correct even when the energy variation of the gap is small and of course fails completely for those energies where the gap varies rapidly with energy and/or has a large imaginary part. IV. ZERO-TEMPERATURE SOLUTIONS OF THE SELF-ENERGY EQUATIONS AND CALCULATION OF THE TUNNELING DENSITY OF STATES Taking the zero-temperature limit of the self-energy equations (2.21) and (2.27) one finds that the energy gap A.(a>)=0(a>)/Z(a>) satisfies the integral equation 1

r"

A(«) =

/

doi'

z(u);0

Re

A(«')

l(a>'2-A2(
(4.1)

and the renormalization parameter Z(co) is given by [1-Z(«)>= f Jo

du'Ke

(w' 2 -A 2 (w'))" !

#_(«». (4.2)

Thus, the differential conductance is proportional to the tunneling density of states of the superconductor. The spectral weight function for a strong-coupling superconductor is given by

As discussed in Sec. II, the frequency integral over the phonon kernel in the gap equation can be cut off at u«~ 10&>D because of the rapid convergence of that part of the integrand. With the introduction of the Z(k,w)w+tk Coulomb pseudopotential Ue, the entire integral can A(k,v) = - I m [ (3.13) be cutoff at uc. From a computational point of view, *l \(z»(*,»v-€t»-*»(*., " ) / the choice of A(&>) instead of (u) is very useful since Z Since Z and

,u') in the form NT(w)=Re

.1

(3.14)

(u»-A» («))"»J

L u '++ w -+f i ' + i 5

x Ja

This result should be contrasted with the naive extension of the simple BCS model to allow for the energy dependence of the gap. There the spectral weight function would be written at 2"=0°K as A Bcs(*,a>) = 1(1+e»/£i)«(o-£ t ) +1(1-•*/£*)«(«+£*),

A-±(«,o>') = L f d,cn?{i>)Fx(*{

(3.15)

" This does not exclude band effects of the type reported by L. Esaki and P. J. Stiles, Phys. Rev. Letters 14, 902 (1965). Also, implicit in our neglect of vertex corrections is the assumption that the phase velocity of the phonons is small compared to that of the electrons at the Fermi surface. In semimetals and degenerately doped semiconductors this need not be the case and structure can be observed. Further, we assume the tunneling matrix elements varies on the scale of the Fermi momentum. As W. L. McMillan has pointed out, certain tunneling anomalies observed in normal metals by J. M. Rowell can be accounted for by variation of Tu; (private communication).

1 O)' — u + V

« ] • ( 4 - 3)

Here Fx(v) is the phonon density of states for the X mode, d3q f
=

hf.

HrpF* J

9dq\$*\*Bx(q,v).

(4.5)

Jo

This represents an average of the electron (X mode)

148

STRONG-COUPLING

SUPERCONDUCTIVITY.

phonon matrix elements over the allowed momentum transfers. The properties of the physical system which are important in determining the electron self-energy in a region of order o>c about the Fermi surface are contained in a\2(v), F>,(v) and N(0) Uc- Once these are given the structure of Z(u) and A(u>) can be determined by solving Eqs. (4.1) and (4.2). Data from inelastic neutron scattering24 can be used to determine the phonon density of state 5ZxFx(c); and, as we will see below, the I-V characteristics of superconducting tunnel junctions provide information on X\ax*(<<>)-Fx(<»>). For Pb we estimated that the phonon density of states would have peaks near 4.4 and 8.5 (meV) of width 0.75 and 0.5 meV, respectively. In our original calculations" these peaks were represented by Lorentzians which were chosen because the integrals giving K± could then be carried out analytically. However, the choice of a Lorentzian gave rise to a small but finite nonphysical phonon density of states at negative frequencies. The effects of this were minor, but could be observed in the failure of the imaginary part of A(o>) to vanish properly as u approached the gap edge A0. To avoid this difficulty, it is convenient to represent the phonon density of states by cut-off Lorentzians'6:

Fx(«)I) — 0 > l X | < 0 > 3 X

10;

|ft>-«j x |>o)| x

It is perhaps of interest to note that these estimates of F\(o>) were made before a full understanding of the electron tunneling data and its implication about Hxax(ai)F\() were known; and they were regarded as only a rough guess. Recent work by Rowell and McMillan2' in which the tunneling data were used to extract c?(u)F(a), and calculations of F(ui) by Bennett18 using the phonon dispersion relations obtained from the neutron scattering data24 show that this estimate is in fact surprisingly realistic. To complete the specifications, the behavior of the electron-phonon coupling strengths ax2(o>) and the value of the Coulomb pseudopotential Uc must be specified. Since the peaks in the phonon density of states are relatively narrow, the frequency-dependent phonon-coupling strengths were approximated by their values at the peaks wi* [i.e., <*((<<)) =a<(wi') and ai(u) =ai(«i ! )]. The strength of the longitudinal electronphonon coupling ar^Mi')—a1 was adjusted so that the calculated value of the gap at the gap edge Ao=A(A0) agreed with the experimental value of 1.34 meV for ratios a»2/ai2 of 1 and 0.5. We estimated that Ut=0.11, and computations were carried out for Uc values of 0.11 and 0. Before discussing the solutions of the gap equation and the resulting effective tunneling density of states for this model of Pb, it is useful to briefly consider the nature of the results for a simpler model in which the phonon density of states has just one peak, see Fig. 7(a). In the absence of Uc, t i e solution of the gap

(4.6)

Here A x normalizes Fx(«) to unity and OJJ* is taken as 2o>jx. Calling the lower peak the transverse peak and denoting its values by \=t: a>i'= 4.4 meV and u>j'=0.75 meV. In the same way the upper peak will be designated as the longitudinal phonon peak with X=2 and ui'=8.5 meV, u s '=0.5 meV. A plot of the resulting phonon density of states F(p)~2Ft(u)+F,(u)

273

I

(4.7)

A "•'. 41 ° •3|<

1.0

in which two effective transverse polarization modes have been taken and each F\(oi) is normalized to unity is shown in Fig. 6.

— l\\ \

-I-'-'

F|«l

Z

1

2.0

(n>

\ /»-&» ,7c>

\

1.0

'

' **

FIG. 6. Model for the lead phonon density of states F(u). Here wi'=4.4 meV (milli-electron volts), «»'=0.75 meV, wi'=8.5 meV, and w,' = 0.5 meV. »B. N. Brockbouse, T. Arase, G. Caglioti, K. R. Rao, and A. D. B. Woods, Phys. Rev. 128, 1099 (1962). " D. J. Scalapino, Y. Wada, and J. C. Swihart, Phys. Rev. Letters 14, 502 (1964). J. C. Swihart, D. J. Scalapino, and Y. Wada, ibid. 14, 106 (1965).

FIG. 7. Single phonon peak model as an illustration of the manner in which structure in the phonon density of states is reflected in the gap and the effective tunneling density of states. The phonon density of states F(u) is plotted in 7(a); the real (solid) and imaginary (dashed) parts of the gap A(u) in 7(b); and the normalized tunneling density of states NTM/N(Q) (solid) compared with the BCS form (dashed) in 7(c). " W. L. McMillan and J. M. Rowell, Phys. Rev. Letters 14, 108 (1965).

274

SCALAPINO,

SCHR1EFFER,

Eq. (4.1) has the form shown in Fig. 7(b). Here for our present discussion we have simplified the form of these solutions by neglecting the weak structure at nuo+Ao which is associated with the nonlinear nature of the gap equation. As the frequency approaches wo+Ao, the real part of the gap increases and reaches a maximum at &)0+Ao. I t then decreases, becomes negative and finally goes to zero. The imaginary part of the gap exhibits a peak slightly beyond wo+Ao. This is a direct reflection of the structure of the effective electron-electron interaction. At frequencies w below uo+Ao the bulk of the phonons which can be exchanged have frequencies greater than u and the effective electron-electron interaction is attractive. Physically, charge fluctuations a t &) in the neighborhood of uo+Ao. In Fig. 7(c), the ratio of the effective tunneling density of states in the superconducting and normal states is plotted for this single-phonon peak model. T o see how the structure in A modifies this ratio it is useful to expand NT(O>)/N(0) in powers of A/w. Ar r (u)/Ar(0) = l + H ( A i ( o ) ) / o ) ) 2 - (A2(o>)/a>)2).

(4.8)

As Ai increases above Ao the effective tunneling density

AND

WILKINS

148

of states increases above the BCS value (the dashed curve). This is the situation just below o)o+Ao. However as the phonons at the peak in the phonon-density of states can be resonantly transferred, the imaginary part of the gap rises and the effective tunneling density of states decreases. Moreover, since just above coo+Ao, &i is decreasing while Aj is increasing to its peak value, this decrease in N.(w)/N(0) is sharp and the curve drops below the BCS value and, in fact, can drop below unity. I n Figs. 8 and 9 results of a numerical solution of Eqs. (4.1) and (4.2) for the gap A(o>) and renormalization parameter Z(u) are given. These solutions are for a ratio of coupling constants aiVaj 2 of 1 and a Coulomb pseudopotential N(0)Uc of 0.11. T h e value of a? necessary in order t h a t Ai(A 0 ) equal the experimentally measured 27 lead gap of 1.34 meV was 1.2. T h e peaks in Ai (solid line Fig. 8) which occur for (a>—Ao)/ o>i' values near 1 and 2 reflect the two peaks in t h e phonon density of states. Beyond the second peak, the real part of the gap decreases and becomes negative since the bulk of all the phonons occur at lower frequencies. In the present case, the real p a r t of A remains negative, asymptotically approaching a value proportional to Uc- The additional structure a t «o>i' +mo)i'+Ao (n and m integers) is associated with multiphonon processes and arises mathematically the nonlinear nature of the gap equation. In Fig. 9, a plot of the real and imaginary parts of the renormalization parameter Z are given. T h e structure of the effective electron-electron interaction and the underlying form of the phonon density of states is clearly reflected in Z. Asymptotically it can be seen from Eq. (4.2) t h a t Z\ must approach unity from above. In some of the early numerical work this asymptotic behavior was violated because the integration in the Z equation was cutoff at o>c. This unsatisfactory feature can be simply eliminated by adding on to the numerical

FIG. 9. Plot of the real (solid) and imaginary (dashed) parts of the renormalization parameter Z(u) versus (a—Ao)/m' for the model parameters given in Fig. 8.

FIG. 8. Plot of the real (solid) and imaginary (dashed) parts of A(CD)/O>I' versus (m—4o)/ui' for the model of the Pb phonon density of states Eq. (4.7). Here o>i' = 4.4 meV, A„=1.34 meV, a,» = t.2,a,Var , = 1.0, and N(0) £/,=0.11.

" J. M. Rowell, P. W. Anderson, and D. E. Thomas, Phys. Rev. Letters 10, 334 (1963).

148

STRONG-COUPLING

SUPERCONDUCTIVITY.

result obtained for the Z from (4.2) the remainder R(a)

f du'Ref

V - (<",<•>')

A

J

_a, 2 f

2

.z

We

r-(a>+a> ( « + « !1 <+a) < + «<:«))+(a> » + ( 2w<)j ' n

L^-aii'-w^-r-^'^J

+

In

^W(o)Ue -0.11

\

_y A yjl

\

J

/

1 1

1 \ 1 1 \\3T—

2

\

[ | 4 --A,

if \) 5, \

f f6 / T ;

y

-.4

r(w+^i ! +« c ) 4 +(« 2 ') 2 H

a?

1 i

1 11

2

-.2

u I

l\

1 \ I 1 f!^ / \ 1 1

1.0

" ,e ~ .6 -

V(a)'2-A2(co'))"V

J„.

275

I

-.e

.

(4.9)

2a,2 L(o,-o )1 '-o )c ) 2 +(a J2 ') 2 JI The approximation is excellent since {&{wc/o>cY<£.lFrom the plots of A2/a>i' and Z 2 it is clear that in the vicinity of the phonon peaks the quasiparticle approximation fails completely for lead. The widths of the spectral weight function for w~a>ix are comparable with their positions and in addition multiple peaks associated with phonon admixtures are present. This shows the breakdown of the quasiparticle approximation referred to in the Introduction. It does not affect our calculations since we have not used this concept (or approximation) in our work. In Fig. 10, the ratio of effective tunneling density of states NT(w)/N(0) which we calculated from A is plotted as the solid curve. The short-dashed curve is the BCS constant gap prediction and the dash-dot curve is experimental tunneling data obtained by plotting the ratio of the differential conductance dl/dV in the superstate to that in the normal state as a function of the bias voltage (o>— V). Just as for the single peaked model (Fig. 7) previously discussed, the characteristic knees near (oi— Ao)/
FIG. 10. The effective tunneling density of states NT(a)/ Ar(0) = Re(u>/(u>s-A,(u))"») versus (u>-A 0 )/«i' (solid) obtained from A of Fig. 8. The ratio of the differential conductance of Pb in the superconducting to that in the normal state (Ref. 25) is plotted (dash-dot) as a function of (w—Ao)A>>'. The prediction of the simplified BCS model a/(uf~ Ao8)1" is shown as the short dashed curve.

-.8 -1.0

FIG. 11. Plot of the real (solid) and imaginary (dashed) parts of A(a)/m' versus (a — Ao)/W for the model of the Pb phonon density of states Eq. (4.7). Here «i»=4.4 meV, Ao = 1.34 meV, a? = 1.6, aW^O.Sa.nANiO) C/«=0.11.

upon the parameters of the model the gap equation was solved for the case in which the ratio of the relative electron-phonon coupling a^/ar* was reduced to 0.5. Since the size of the coupling a? was set by fitting Ai(Ao) to the experimentally determined value of 1.34 meV, the effect of reducing the at'/a? ratio is to place more weight in the longitudinal peak at OJI'. This is clearly visible in the behavior of the gap, Fig. 11. The second peak in Ai associated with &>i! is now considerably larger than the first peak associated with the transverse phonons at coi'. This behavior is reflected in the associated tunneling density of states Fig. 12 by the large knee near a> = a>i!+Ao. The strong oscillation between 2 and 3 is also a higher order manifestation of the increased strength of the longitudinal coupling. The experimental tunneling data (dash-dot) Fig. 10

FIG. 12. The effective tunneling density of states NT(U)I TVfOJ^Retw/fy-A'M)" 8 ) versus (u-A 0 )/
276

SCALAP1N0,

SCHRIEFFER,

AND

WILKINS

148

V. CONCLUSION A conclusion to this paper seems particularly in order since the bulk of this work was completed several years ago and many of the conclusions have in fact \ N(o>U = 0.0 been verified in great detail. The first and perhaps the most basic conclusion is that experimental data support the Eliashberg4 form of the interaction kernel in the gap equation. They do no support the form suggested by the work of Frohlich6 or Bardeen and Pines.6 These latter two interactions agree with Eliashberg's only in the static limit; however, the structure in the gap function arises from the dynamic behavior of the interaction. This work provides a good illustration of the usefulness of Green's-function methods in solid-state problems. Here these methods provide a framework within FIG. 13. Plot of the real (solid) and imaginary (dashed) parts which the retarded nature of the electron-phonon of A(u)/W versus (=0.5, and tf(0)£/.=0. approximation (because of the large damping at energies associated with the peaks in the phonon density of does not exhibit this additional structure so that it states) can be simply dealt with. For example, because is possible to distinguish quantitatively between these of the separation of frequency and momentum variables, it is clear that the effective tunneling density of states two different forms for Hx ax2 (<»)Fx (<•>)• 2 2 12 The effect of removing the Coulomb pseudopotential is W(0) Reu/O — A (ai)] ' rather than involving derivatives of A (<•>), as a naive quasiparticle approxiis shown in the behavior of A plotted in Fig. 13. The ratio of the electron-phonon coupling constants a}ja? mation would give. A second conclusion is that tunneling data provide is again taken to be 0.5, and the structure of A in Fig. 13 should be compared with A in Fig. 11. Without a direct probe of the electron-ion system capable of 2 the pseudopotential, the size of the electron-phonon measuring 23xax («)^'x(w) and the Coulomb pseudocoupling which is necessary to obtain Ai (Ac) = 1.34 potential Uc- The sensitivity of the results for different meV is reduced. This decreases the variations in A and choices of these parameters supported the choice of therefore ultimately reduces the structure in N,(w)/ Fx(u) given by Eq. (4.7) with a?/a?=l and <7e=0.11. N(0). This is shown in Fig. 14 where the tunneling The calculated value of 0^= 1.2 is an important measdensity of states for aj 2 /W=0.5 a n d Uc+0 is plotted. ure of the effective electron-phonon strength in Pb. A comparison of this case with the a/'/a? =0.5, Ut=0.11 Subsequently to the above work, a number of applicase, Fig. 12, clearly shows that Uc enhances the struc- cations of the theory have been made. One of the first ture in the effective tunneling density of states. of these was that not only is the gross structure of the phonon density-of-states peaks observable in the I-V characteristics, but the Van Hove critical points are reflected as log and square-root singularities in
FIG. 14. The effective tunneling density of states iVrM/ iV(0) = ReW(w'-A,(w))«») versus (a>-Ao)/un' for A obtained from Fig. 13. Comparison with Fig. 12 shows that V, enhances the structure in the effective tunneling density of states.

» D. J. Scalapino and P. W. Anderson, Phys. Rev. 133, A921 (1964). D. J. Scalapino, Rev. Mod. Phys. 36, 205 (1964). • M. Fibich, Phys. Rev. Letters 14, 561(E) (1965): 14, 621(E) (1965).

STRONG-COUPLING

14a

SUPERCONDUCTIVITY.

the damping of the log singularity in the ratio of normal to superconducting nuclear spin-lattice relaxation time at Tc. In addition, the agreement between theory and experiment for the temperature dependence of the phonon-limited electronic thermal conductivity of strong-coupling materials is greatly improved by the above type theory, as Ambegoakar and Tewordt have shown.80 Thus, it appears that the theory of strong-coupling superconductors can account in detail for most of the discrepancies between the weak-coupling theory and experiments on strong-coupling superconductors.

In the reduction we have used the relation 'da' r'da' JK. r

Il

1

6(E —ucc) e(Epp—a 2£„.

(A3) '

where

e(x)

1,

x>0,

0,

CO.

If we view the momentum indices p and p' as matrix indices and the summation over the index p' as d3p'

/•(2^'

The authors have benefited greatly from discussions with many people, particularly P. W. Anderson, L. P. Kadanoff, W. L. McMillan J. M. Rowell, J. C. Swihart, and Y. Wada. APPENDIX A: COULOMB PSETJDOPOTENTIAL Vc

then (A2) takes the matrix form (1+G)0«=W, where the matrix elements of fi are given by «W = [« (E>- ««)/2£,.] V (/>-/>'),

c(p)=~\ ( 'da'+f / Iml V

(A5)

f'da'

XtanhOSa)72).

l r cUt

(A4)

and

We derive here the Coulomb pseudopotential Uc which reduces the range of energy integration from 0-> « in (2.26) to 0—>a>e~10wB in (2.27). We write (2.26) in the form

J (2TCY (2TY

\

"Aw*-£„'+*«/

ACKNOWLEDGMENTS

d>p'

277

I

dJ]

(A6)

Since V is a repulsive potential, 1+fl is a nonsingular operator and (A4) can be written as #(f>')

'= (1+Q)-1 VF= VCF,

\2?(p',u>')u'*-tS-r(j,'>')/ XV(j>,p')tanh(J3a'/2).

(A7)

where the Coulomb pseudopotential Uc satisfies (Al)

(1+0)17.-7;

(A8)

c

Now Z(p,u) —> 1 and {p,a) —> 4> (p) for u»aiz), since or in component form, one has the integral equation the phonon-electron interaction is ineffective except for 3 p" f dd3p" energies OI>O>D- Furthermore /3u)«»l for temperatures Uc(p,p')=V(p-p> P") at which the material is superconducting, so that tanh(0o//2)~l for a'>ue. Therefore the integral from uc to » can be simplified by replacing Z(p',u') and 0(£„.-«e) tanh(0w/2) by unity, and *(/>>') by 4>'{p'), so that X-Uc(,p",p'), (A9) 2£„ (Al) reduces to

'(P)+

r d>p'
6J{EV -««) IE, da'

J ((2*)» 2T)'

JO

T

*'(P')

determining Uc- Finally, by taking components of (A7) we obtain (AlO) J

(2ir)'

It is clear from the form of (A6) that Fp. decreases extremely rapidly for i?>we. For this reason the major *(fV) contribution to the integral in (AlO) comes from states Xlml p' near the Fermi surface, |p'—p F \«.p F - Since the P's»V pseudopotential U has appreciable variation only on XtanhC8w'/2). (A2) the scale of p , asc is easily seen from (A9), we can r " V. Anbegoakar and L. Tewordt, Phys. Rev. 134, 805 (1964); replace Uc(p,p') by Uc(p,pr) in (AlO). The />' inteV. Ambegoakar and J. Woo, ibid. 139, A1818 (1965). gration can then be carried out with the aid of (2.18)

278

SCALAPINO,

SCHRIEFFER,

du'Rel

io

\(«*-A»(«'))"*/ X E/c(*>,/>,) tanh03u>'/2).

WILKINS

148

essential experimental condition is that the two metals be in thermal equilibrium at some fixed temperature. Hence we may take j
and (2.19) and one finds
AND

(All)

|<»,(#+1)| £ If we are only interested in (p) for momenta p~pF, p can be set equal to pF and one obtains (2.27), where can be written as

7We,,'t| W r (tf))|i

(B2)

c

UC=UC(PF,PF).

(A12)

£|ZW|»|<>»r|«k't|« r )|', (B3) The integral equation (A9) which determines Uc has a direct interpretation. It simply accounts for Coulomb since the operator Ckr* selects out that subset of exscatterings outside the energy band ±OJ<: about the citations which are characterized by the wave vector k. Fermi surface which have been excluded from the Then, explicitly introducing Harrison's22 expression numerical integration by the introduction of <>>c in Eq. for 7\k' we can write (All). Physically, because the Coulomb interaction is repulsive, the correlations induced by the multiple scatterings taken into account in (A9) reduce the Wf*-i=-SI / duexpi— 2 / ki(x)dx) ir*liy_«, \ J xt / probability that the two electrons are within the range of the screened Coulomb interaction. This has the effect of making Uc smaller than the corresponding X[Z £ P«.,|<«rk't|«,>|» i-wave average of the plane-wave matrix element V(ptp'). This reduction can be explicitly determined if we approximate V(p,p') by a factorizable potential X«(£, T—Emr—it-a) (A13) V{P,P')=V„ | e,| and | t , <&>m, = 0, otherwise. Pm\(n,\c„.'\md\* Here Vc is the average of the Coulomb interaction over L*/ pi (,) m "" the Fermi surface and um is of order the Fermi energy EF- Using this, the solution of (A9) is XS(.En,-Eml+v-V+w) (B4) U.<= Vc/ll+N(0)V< l n ( £ , / « . ) ] (AH)

]•

if the density of Bloch states is taken constant for where P =e" «-'< "-" '> and we have carried out the m I «»|<«»i- Using values appropriate to Pb we find spin sum. Note that ki and kj have only positive values. N(Q)Uc=0.ll. The sum over k\\ can be done first. In principle, the expression in the square brackets depends on k\\ but in APPENDIX B: REDUCTION OF THE practice such dependence is very weak. Furthermore, TUNNELING RATE EXPRESSION since the exponential factor decreases rapidly with increasing k\\, only electrons moving perpendicular to We derive here the reduction of the general expression the barrier contribute significantly to the tunneling for the tunneling rate (3.5) to the form (3.6). We current. If we consider a wave function which is mainly 7 define \m(N)) as an exact eigenstate of the A -particle in the right-hand metal with an exponentially demetal in the absence of both HT and any applied voltage creasing tail in the barrier, then N V. Em is the energy associated with the state \m(N)). Then the delta function conserving energy can be W{x) written -U(x)+ c*ii KM (BS) 2m w 1 w w lv 8(£F-£,) = «(£„r + +£„, -'-r-£Br -£„1 ) where U(x) is the barrier potential and ,

—I

A

tki\+n = k\i2/2m.

dui(E„r— Em,—Mr— ") X6(Enl-Em,+,il-V+u),

£

(Bl)

where in the last expression we have dropped the encumbering superscripts. The letters m and n will be used to denote excited states containing JV and A^fcl particles, respectively. It should be noted that an

The use of the bare electron mass is consistent with our treatment of the barrier as a potential step (with rounded shoulders). In addition to the possibility of structure in the barrier we are also neglecting image force corrections to the barrier potential and any asymmetry in that potential due to bias voltage or to

STRONG-COUPLING

148

SUPERCONDUCTIVITY.

fabrication procedures. While such effects are certainly present we do not believe their inclusion is essential to an understanding of the I-V characteristics of the metal-insulator—metal tunneling processes for very small voltages. The sum over k\\ can then be written as f'iT

/ 8ir»

rk
dip I

2

dx

X(2»«(£/(*)+£„-£„)+V)" } ,

(Bll)

and A(k,w) is the spectral (B12)

(B6)

where A\\ is the area of the barrier. We assume that U(x) changes rapidly (but not so rapidly that the WK.B approximation is invalid) in the vicinity of xr and xi so that for most of the barrier region U(x) = UmKL. We define the metal-barrier work function tp— (J max 04. A precise specification of the cutoff k? of the £u2-integral is immaterial since only the region around £u~0 is important. For small applied voltages and at low temperatures only states near the chemical potential will contribute to the tunneling current. So in the case where F«*5, the integral (B6) over k\\ is well represented by A\\t/4*, (B7) where /= l/d((2m^) 1 «4-l/2d) exp(-2d(2m«0 1/2 ), (B8) and d**xr—xi= barrier thickness. Now we consider the expressions in the square brackets in (B4) and their relation to the one-particle Green's functions. An alternative way of writing the one-one component of the Green's function defined by Eq. (2.3a) in Sec. II is

While a portion of the temperature dependence has been accounted for by the introduction of the Fermi factors, A (k,u) will also have temperature dependence. It is convenient to define the effective tunneling density of states by NT(<*)=°N(P)

E p m |(n| Cl t|»«>( 2 «(£:»-£ m - M -co) (BIO)

de„A(kfiO,

(B13)

XS(£„ r -£ m r —n—u) -Jff,(«)(l-/(»)).

(B14)

In a similar manner one can reduce the second square bracket in (B4) and obtain the desired reduction of Eq. (3.5) to (3.6), 1VT*-1!

no

f

since then our final expression for the transition rate Wr*-i (B15) has formally the same structure as that for tunneling between two systems of noninteracting electrons where NT(O>) would be just iV(0) near the Fermi surface. Finally, consider the kt sums. In the continuum approximation it is more convenient to sum over both signs of ki (and reduce pj. by a factor of two). Then since etl is not limited by any essential restrictions from ki\ we can replace t t l by ck- Thus, the square bracket involving the sum over ki in (B4) becomes

(B9)

Then by inserting a complete set of states one can easily show that

=A(k,o)(l-/(»)),

=.4(M/(«0> -1

1 i4(ft,u) = — ImCn(*,u+t8). 2

Gu(.k,T)=-Zp»(«i\Tc{TW(0)\m).

279

and

where /(&>) = (1+e*") weight function

H"

<Jfc|| exp|— 2 /

I

167T 2

[

)(l-/(u)) XNT'(<->-V)f(a-V).

(B15)

90 VOLUME 17, NUMBER 8

PHYSICAL

REVIEW

LETTERS

22 AUGUST 1966

EFFECT OF FERROMAGNETIC SPIN CORRELATIONS ON SUPERCONDUCTIVITY* N. F. Berk and J. R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania (Received 24 June 1966) If the superconducting transition temperatures of the transition metals are fit by the BCS r e lation 1 -9Dexp[-

^"MW

(1)

where 9 p is the Debye temperature and N{0) is the density of states at the Fermi surface, then the pairing interaction VgQg is found to be roughly constant, except for metals at the beginning and end of each series, where it r a p idly decreases. 2 Traditionally, one writes V

-V -V *. (2) v BCS ph C ' ' where V-^ represents the phonon interaction and VQ* is the Coulomb pseudopotential. 3 While at present one cannot rule out the possibility that Vph'^C* i s small or negative for metals near the ends of these series, the fact that their paramagnetic susceptibility is anomalously large 4 suggests that there are strong ferromagnetic exchange forces which enhance the spin susceptibility and presumably also suppress superconductivity for singlet spin pairing, as first discussed by Doniach5 on the basis of a phenomenological exchange interaction. In this note we show from first principles that ferromagnetic spin correlations which arise from strong Coulomb interactions between the valence electrons lead to an enhanced singletstate repulsion. 6 For realistic values of s y s tem parameters, this spin-induced repulsion can be many times larger than the conventional Coulomb pseudopotential. For example, in systems like Pd, whose static spin susceptibility is strongly exchange enhanced, this r e pulsion is sufficient to suppress superconductivity totally.

Physically, the enhanced singlet-state repulsion a r i s e s because of ferromagnetic spin polarizations induced by the interacting electrons. If the exchange interaction is nearly strong enough to produce ferromagnetic alignment, a given up-spin electron, A, will be surrounded primarily by electrons of up spin. Since the phonon interaction is very short range in space, a down-spin electron, B, attempting to lower its energy by taking advantage of the phonon attraction produced by A, must first pass through a large region of unfavorably oriented spins. Thus, there is an effective exchange potential b a r r i e r separating electrons of opposite spin, and if the space and time persistence of the induced spin-polarization cloud is sufficiently large, the integrated exchange repulsion can dominate the phonon attraction. Mathematically, the effects of ferromagnetic spin correlations are incorporated into the singlet-state pairing interaction by noting that the paramagnetic susceptibility function X(Q, W) develops a pole in the limit that q and w - 0 at the ferromagnetic instability. This behavior is due to the singularity in the amplitude for particle-hole scattering, as is conveniently seen in the random phase approximation (RPA) form for x shown graphically in Fig. 1(a). In the static, long-wavelength limit 7 x

RPA(0'°) = xP/[1_iV(0)P'c]'

(3)

if one assumes that the screened Coulomb interaction is zero ranged, a reasonable approximation in the narrow d-band metals because of the screening due to the s electrons. Here, Xp is the Pauli susceptibility. In RPA, \(Q, 0) diverges when N(0)VQ = 1, the Hartree-Fock 433

91 PHYSICAL

VOLUME 17, NUMBER 8

REVIEW

x ? r - ^ ' [ 0 CD <0>+---] p'.t

-P',*

P,»

-P,

*

- ^+0 + S3 + FIG. 1. (a) Graphs which give the susceptibility in RPA. (b) Graphs which give the singlet-state repulsion in the presence of strong particle-hole correlation. The thin dashed line is VQ and the heavy dashed line is VQ (Eq. (8)]. (c) Self-energy graphs which contribute to Eq. (6). criterion for ferromagnetism. For general q and uo, it has been shown that 7 Xp"(<7, ">) )

^>"

=

(4)

l-N(0)Vcu(g,uy

where u(q, w) - 1, as q and w - 0, with w/qv-p « 1 (vf= Fermi velocity). The particle-hole correlations which lead

LETTERS

22 AUGUST 1966

to the singular behavior of x a r e included in the singlet-state pairing interaction by summing the corresponding graphs shown in Fig. 1(b). The behavior of the particle-hole t matrix t(q, w) for small q and u> thus accounts for the dynamic effects of the induced spin polarizations on the singlet-state repulsion and leads to a singularity in this repulsion at the ferromagnetic instability. While short-range electron-electron correlation effects, important in rf-band metals, are neglected within the RPA, these effects can be included through the Landau theory of a Fermi liquid. 8 In the Landau theory x(0, 0) is given by replacing VQ in (3) by VQ, the average over the Fermi surface of f^-f^, where fss,(k,k') is the Landau function. We assume that for the small values of q and w of interest near the ferromagnetic instability, the important variation of \(,q, ui) in (4) is through the variation of u(q, ui) rather than VQ. Thus, in treating the spin fluctuations, one can p r e sumably use the RPA by introducing an effective short-range interaction VQ. The superconducting transition temperature Tc is found by solving the linearized gap equation obtained within the Nambu formalism, 9 which treats the strong particle-hole c o r r e l a tions both in the normal and superconducting phases. The linearized gap equation is

(u)=^JUc^{neA(w')}[K(w,w')-V*}t*nh(w>/2kT),

(5)

where [l-Z(w)]a>=W(0)X°°da)'if_(w,w'),

(6)

and K (w,(u') • Ik p = (2k2)~1 qdql dv\£\g\i\lmD(q,v)\*lmt(q,i>)][(u'+w+v+i6)-1±(tjJ'-<j)+v+i6)-1}. r J0 Ja q*. *Here we assume that kTc is small compared to both the Debye energy and the important spinpolarization frequencies. The function t(q, u) in (7) is the particle-hole t matrix, N(0)Vc2u(q,u) t(q,u) = Vn +

C

l-N(0)Vcu(q,u)'

and VQ* in (5) is the pseudopotential which accounts for the effect of VQ. Equation (6) for 434

(8)

(7)

the normal-state self-energy comes from the sum of graphs shown in Fig. 1(c). The first term treats the phonons exactly to order (m/ M)1'2 (where m is the electronic mass and M, the ionic mass), and the sum of the remaining graphs accounts for the dynamic effects of the spin polarizations. Since these graphs are slowly varying functions of three-momentum near the Fermi surface, we are able to write the

92 VOLUME 17, NUMBER 8

PHYSICAL

REVIEW

LETTERS

22 AUGUST

1966

that s p i n c o r r e l a t i o n s a l o n e l e a d t o rw* = 4»n. This may o v e r e s t i m a t e the m a s s r e n o r m a l i z a tion b e c a u s e of o u r n e g l e c t of t h e d e t a i l e d e n e r g y dependence of t h e d e n s i t y of s t a t e s in P d . However, it i n d i c a t e s t h a t the l a r g e s p e c i f i c heat of P d is not due e n t i r e l y to phonon e n h a n c e m e n t . A s in the c a s e of e l e c t r o n - p h o n o n i n t e r a c t i o n s , 1 0 the above m a s s r e n o r m a l i z a t i o n d o e s not influence the s p i n s u s c e p t i b i l i t y , s i n c e t h e spin-fluctuation c o n t r i b u t i o n to t h e s e l f - e n e r g y is i n s e n s i t i v e to a s m a l l uniform s p i n p o l a r i z a tion. FIG. 2. Strength of the spin-polarization contribution to the pairing interaction for U,OJ' = 0 as a function of W(0)V(-. (bottom scale) and susceptibility enhancement (top scale). s e l f - e n e r g y in the f o r m shown in (6) and to t r e a t the phonon and < - m a t r i x p a r t s of t h e i n t e r a c tion on equal footing in d e r i v i n g the gap e q u a tion (5). P a r t i c l e - h o l e s y m m e t r y i s a s s u m e d throughout. Equation (5) w a s solved for an E i n s t e i n m o d el of the phonon s p e c t r u m , with an a d j u s t a b l e e l e c t r o n - p h o n o n coupling. F o r s i m p l i c i t y , the Coulomb p s e u d o p o t e n t i a l w a s s e t equal to z e r o . N u m e r i c a l l y , we find for N(Q)VQ>0.5 that the c r i t e r i o n for s u p e r c o n d u c t i v i t y is a p p r o x i m a t e l y N(0)Vph=2N(0)g2/w0>N{0)Ks(0, 0), w h e r e o)0 i s the phonon frequency and Ks(fl, 0) i s the s t a t i c limit of the ^ - m a t r i x contribution to K+, (7) (the s u b s c r i p t s s t a n d s for t h e " s p i n " p a r t of the i n t e r a c t i o n ) . Note that c o r r e l a t i o n effects s i m i l a r to t h o s e which give the p s e u d o potential reduction of the Coulomb i n t e r a c t i o n do not a p p e a r to r e d u c e the s p i n - i n d u c e d r e p u l sion. Evidently t h i s i s due to the s t r o n g p e a k ing of Ks at z e r o frequency for N(0)VQ> 0.5. N(0)Ks(0, 0) i s shown in F i g . 2 a s a function of N(0)VQ. Susceptibility m e a s u r e m e n t s i n d i cate that N ( 0 ) ? c a 0 . 9 for P d . 4 A c c o r d i n g to the c u r v e in F i g . 2, P d will not be a s u p e r c o n d u c t o r for any r e a s o n a b l e phonon s t r e n g t h . We note that the p a r t i c l e - h o l e ^ - m a t r i x cont r i b u t i o n to the s e l f - e n e r g y [ F i g . 1(c)] l e a d s to a l a r g e effective m a s s n e a r the f e r r o m a g n e t i c i n s t a b i l i t y . F o r N(0)VQ =0.9 we e s t i m a t e

T h e s u s c e p t i b i l i t y of liquid He 3 i s a l s o s t r o n g ly exchange enhanced. Since the c o n v e n t i o n a l p a i r i n g i n t e r a c t i o n is thought to b e m o s t a t t r a c tive in the I = 2 s t a t e , 1 1 s i n g l e t s p i n p a i r i n g i s a p p r o p r i a t e . On t h e b a s i s of the a b o v e d i s c u s s i o n , it a p p e a r s that c r i t i c a l spin f l u c t u a t i o n s play a r o l e in s u p p r e s s i n g s u p e r f l u i d i t y in H e 3 . We wish to thank M. A. J e n s e n a n d S. D o n i a c h for helpful d i s c u s s i o n s . *This work was supported by the National Science Foundation and the Advanced Research Projects AgencyJ J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. H)8, 1175 (1957). 2 M. A. Jensen, thesis, University of California, La Jolla, 1965 (impublished); K. Andres and M. A. Jensen, to be published. 3 P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962). 4 D. N. Budworth, F . E. Hoare, and J . Preston, Proc. Roy. Soc. (London) A257, 250 (1960); W. E. Gardner and J. Penfold, Phil. Mag. U., 549 (1965). B S. Doniach, in Proceedings of the Manchester Many Body Conference, September, 1964 (unpublished). 6 A preliminary account of this work was described by N. F. Berk and J. R. Schrieffer, Bull. Am. Phys. Soc. .11, 78 (1966). ' P . A. Wolff, Phys. Rev. 120, 814 (1960); T. Izuyama, D. J. Kim, and R. Kubo, J. Phys. Soc. (Japan) 1_8, 1025 (1963). 8 P. Nozieres, The Theory of Interacting Fermi Systems (W. A. Benjamin, Inc., New York, 1963). 9 J. R. Schrieffer, Theory of Superconductivity (W. A. Benjamin, Inc., New York, 1964). 10 D. Simkin, thesis, University of Illinois, 1963 (unpublished). "V. J. Emery and A. M. Sessler, Phys. Rev. 119, 43 (1960).

435

93 Volume 24A. number 11

PHYSICS L E T T E R S

22 May 1967

ON T H E D E S C R I P T I O N O F N E A R L Y F E R R O M A G N E T I C FERMION SYSTEMS J.R.SCHRIEFFER* The Niels Bohr Institute, University of Copenhagen. Denmark and N. F.BERK Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania Received 4 May 1967 The relation between the Fermi liquid and spin fluctuation descriptions of a nearly ferromagnetic system of fermiors is discussed. It is argued that the spin fluctuation theory provides a more suitable framework for describing such systems. Recent work [1-3] has demonstrated the importance of spin fluctuations (paramagnons) in fermion system which are nearly ferromagnetic. We discuss here the relation between the Fermi liquid [4] and spin fluctuation descriptions of such systems. Consider the magnetic susceptibility x, which in the Fermi liquid theory is given in terms of * John Simon Guggenheim Fellow. Permanent address: Dept. of Physics, University of Pennsylvania, Philadelphia, Penna. 604

the Landau f u n c t i o nt-ss' / ^ ' by

x/x 0 -

m /m

*


(^

m*/m = l + (m*/m)N{0) ( ( / ^ , +/££,) cosBkk).

(2)

•(m*/m)N(0)
m

Here, Xo, "' andAr(0) are the free gas p a r a m eters and the averages are taken over the Fermi surface; eq. (2) relies on the Galilean invariance

94

of the interactions. In the Fermi liquid theory, the exchange enhancement factor is defined as FL oi ec l- (!)• In contrast, near the ferromagnetic instability, the spin fluctuation theory gives X

1-A'(0)/ 0 where A'(0)/o is a smoothly varying quantity which approaches unity at the ferromagnetic instability. By comparing (1) and (3), it appears that the spin fluctuation theory has overlooked m*/m corrections to \. We argue that near the ferromagnetic instability, the dominant contributions to m*/m (i.e. from paramagnons) cancel out in a consistent calculation of x, and eq. (3) is in fact the preferable form. This distinction is particularly important near the ferromagnetic instability where, according to the spin fluctuation theory, m */m tends to infinity. For liquid 3 He, m*/m « 3 and x/Xo "* 9> so a>gp ~ 9 while a>FL ~ 3 - T n i s modified definition of the exchange enhancement factor suggests that paramagnons are more important in ^He than the Fermi liquid theiry suggests. To see this, we note the dynamic susceptibility is Xo

X(g)

=2AifGU> + l)G(p)AU>,9) M (4)

a (2ny where G is the exact one -particle Green function and A is the spin flip vertex function, with q = = (q,q0), etc. In terms of the irreducible particle hole interaction /, A satisfies [4]

MP, <7) = 1 +i fHp,p';q)G{p'+q)

22 May 19G7

PHYSICS L E T T E R S

Volume 24A, number 11

G(P') A(p',q)

dV (2w)4

(5)

flip terms in the Hamiltonian). It follows that A varies with \ p | on the same scale. As in the elcctron-phonon problem, near the ferromagnetic instability paramagnons lead to the self-energy S(/>) varying rapidly with p0 on the scale of the Fermi energy but the \p\ variation is on the scale of A'p. By redefining ep and / 0 to absorb this weak \p\ dependence, one has as q —» 0, A

<*'?)

=

<6)

l-.V(0)/ o i,(g)'

where u(q) is the generalized Lindhard function [7] «(*> = 7r ^ / G (P' + Q) C{p<) ±£ A (0)

.

(7)

As q — 0, q0/ \ q \ — 0, the limit appropriate to eqs. (1) and (3), one finds [2] t u{q) =

G(p'+q)-G(p') MO) / tp'+q-tp'

d4/>' (2>r)4

(8)

;

_i

,-d(«/,')

A^J^7 A r M d ^'

1,

where(«/,) i s t n e bare occupation number a v e r aged in the physical ground state. We have a s sumed that near the ferromagnetic instability, (np) differs from the Fermi function only over a region having N{ep) ~ N(0), where ep is measured relative to the Fermi surface. This is the case for the spin fluctuation smearing. From eqs. (4), (6) and (8) we obtain eq. (3). We see from eqs. (7) and (8) that the quasiparticle and "incoherent" parts of G add in a coherent way in \ near the ferromagnetic instability essentially cancelling the renormalization effects characteristic of the Fermi liquid theory. A similar effect occurs in calculating S and other properties of the system. For this reason we suggest that the spin fluctuation theory provides a better frame work for describing these s y s tems, than does the Fermi liquid theory.

The Fermi liquid theory proceeds by replacing G(p+q) G(p) in eqs. (4) and (5) by the product of the quasi-particle parts G1P, the incoherent parts G m c = G -G IP being included by redefining / and A as the quantities appropriate to quasi-particles. We argue that near the ferromagnetic instability, low energy paramagnons make the quasi-particle / ill behaved and one should treat G IP and G^nc on an equal footing as below [2] *. We expand / in Legendre polynomials Pl{cos9pp') and only the 1 = 0 term I0(p,p') enters eq. (4) as q — 0. / 0 varies with \p\ on the scale of £ p (since paramagnons can not enter the c r o s sed channel, p -p', due to the absence of spin-

The authors are indebted to the members of the Niels Bohr Institute for their kind hospitality and to Dr. G. E. Brown for stimulating conversations.

* For a discussion combining the Fermi liquid and spin fluctuation approaches to account for temperature dependent renormalizations, see ref. 6.

t The derivation (4-8) has been criticized by Engelsberg [3] on the basis of stationary state perturbation theory. Explicit calculation shows his objections to our derivation are incorrect [9]. 605

95 Volume 24A. number 11

PHYSICS L E T T E R S

References 1 X. F.Berk and J.R.Schrieffer, Bull. Am. Phys.Soc. 11 (19CG) 76; Phys. Rev. Letters 17 (1966) 433; P r o c . 10th Int. Conf. on Low temperature physics, Moscow. 1966. to be published. 2. X.F.Berk. Ph. D. thesis. Univ. of Penn.. 1966 (unpublished). 3. S.Doniach and S.Engelsberg, Phys. Rev. Letters 17 (1966) 750; S.Doniach. S.Engelsberg and M . J . R i c e , Proc. 10th Int. Conf. on Low temperatur physics, Moscow, 1966. to be published.

** *

606

22 May 1967

4. P.Nozieres. The theory of interacting fcrmion s y s tems, (W.A.Benjamin. Inc.. 1964). 5. A.B.Migdal. Soviet Physics J E T P 7 (1958) 996; J.R.Schrieffer. Theory of superconductivity (W.A. Benjamin Inc.. 1964). 6. W. Brenig and H.J.Mikeska, Phys. Letters 24 (1967) 332. 7. J.Lindhard, Mat. fys. Medd. Dan. Vid. Selsk. 28 No. 8 (1954). 8. S.Engelsberg. to be published. 9. J.R.Schrieffer, to be published.

96

The sign of the interference current component in superconducting tunnel junctions A. B. Zorin Physics Department, Moscow State University, Moscow 117234, USSR

I. O. Kulik Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR. Khar'kov 310086, USSR

K. K. Likharev Physics Department, Moscow State University, Moscow 117234, USSR

J. R. Schrieffer Physics Department, University of Pennsylvania, Philadelphia, Pennsylvania 19104 USA

(Submitted May 7, 1979) Fiz. Nizk. Temp. 5, 1138-1157 (October 1979) The role of a finite-width Riedel peak in the interference conductivity of a Josephson tunnel junction is treated phenomenologically. It is shown that allowance for the finite width of the peak can lead to a change in sign of this conductivity in the region co
The theory of the Josephson effect in tunnel junctions 1 " 3 is now practically the only complete theory of nonstationary, nonlinear effects in superconductors. This theory explains quite accurately the results of many experiments with tunnel junctions. Relatively recently, however, experiments 4 were performed in which the interference current component through the tunnel junction was measured; according to these experiments the sign of this component is opposed 5 ' 6 to that predicted by standard theory. 1 ' 2 In this connection It is necessary to note that standard tunneling theory does not take into account processes leading to finite values of the Riedel peak width of the Josephson current component' and the discontinuity in the quasiparticle current component, while these a r e finite in experiments (see, for example, Refs. 8,9). It Is shown In the present paper that account of the finite peak width can bring about significant corrections to the magnitude of the dissipative components of the tunnel current, up to sign reversal of the interference current component. On the other hand, transport of Cooper pairs between 537

Sov. J. Low Temp. Phvs. 5(10), October 1979

the electrodes of a tunnel junction leads to some change in the charge, and, as a result, to a change in the chemical potential. The "superconducting" charge density is ps = —n(A/27')/V(0) etij. where N(0) is the density of states at the Fermi surface, and u, = -1 (ft ^ + 2eV) is the so-called invariant potential, the difference between the quasiparticle chemical potential (eV) and that of Cooperpairs (-V^hdy/dt). In the equilibrium state it is usually assumed thatM s = 0, but in the general case this may not be the case. Account of chemical potential oscillations can also lead to a change In the interference current component. In the first section of the present paper we derive a general expression for the current through a tunnel junction, as well as necessary dispersion relations relating the components of this current. The second section is devoted to considering the properties of the current components in the classical case: We write down the full expressions, their limiting values, and we trace out the temperature dependence of the quasiparticle and Interference conductivities in the cases of both an asymmetric

0360-0335/79/10 0537-10 $02.20

©1980 American Institute of Physics

537

97 (&i * A2) and asymmetric (A, =Aj) junction. In the third section we consider phenomenologically the finite relative half-width 6 of the Biedel peak, and present figures that illustrate the effect of 6 on the conductivity values at various frequencies and temperatures. In the fourth section we consider specific mechanisms capable of causing finite values of 6: inelastic relaxation processes during tunneling, inelastic relaxation processes in superconductors, anisotropies, and large- and small-scale inhomogeneities in superconductors. In the fifth section we compare the results obtained with experimental data. The problem of the sign of the interference component of the tunnel current is discussed in the concluding section.

type, valid for the response of a linear system:

1. GENERAL EXPRESSIONS FOE THE TUNNELING CURRENT

where the principal values of the integrals a r e taken, and the quantities Opf q and p p > q a r e defined by the expressions o„ (CD) = elm ./„ (co)/co; p„ (to) = e [Re J„ (0) + Re Jp (ujj/to; < 1 0 ) a, (to) = e Im / , (co)/co; p, (co) = e [Re J, (0) — Re J, (cu)]/to. (11)

The general expression for the current density J through the tunnel junction can be represented in the form (see, for example, Ref. 2) +.+r J (0 = Im \ da> \ dco' {W (to) W (0)') «*<•+-'+*••"J, (to' + to„) + W (u>) W* (01') e"—-'>'J, (to' + to„)}.

(1)

The function W(a>) is determined by the time dependence of the phase difference at the junction tp(t) (fi = 1): exp [19 (0/2] = exp [le J V (<*) df) = J W (to) e<(-.4~x da>,

<2)

.where V(t) is the voltage across the junction, and UQ is half the frequency of Josephson current generation av = 2 The functions J Pl q(w) appearing in expression (1) are, on the other hand, characteristic of the Josephson junction itself, and their form depends on temperature and on the values of the energy gap parameters A 1)2 of the superconductors forming the Junction. Mathematically, they are complex functions of the r e a l argument OJ, and possess the property •/„.,(-«>) = / „ * . , N .

4

<>

The physical meaning of the quantities J p „ becomes clearer if it is taken into account that, according to Eq. (2), in the special case of a constant voltage at the junction [V(t) = V] the Fourier-transform Wfcj) equals 5( u ), and Eq. (l) transforms to the well-known expression written down by Josephson himself 10 : / (/) = Re Jp (o)0) sin

(5)

It is seen from this that the quantity ReJp (a) 0) is the amplitude of the ordinary Josephson supercurrent.Im Jptoj) is the amplitude of the dissipative component of this current, due to interference of pairs and quasiparticles (see, for example, Ref. 5), andlmjq(w(|) provides the magnitude of the quasiparticle current. The fourth current component, due toRe J , though not appearing in the special case under consideration In expression (5), is of the same order of magnitude, and in the general case it can significantly affect processes in the Junction. We note that although the functions J p q characterize the nonlinear response of a tunnel junction at an external field, their real and imaginary parts a r e nevertheless related by dispersion relations of the Kramers-Kronig 538

Sov. J. Low Temp. Phys. 5(10), October 1979

o.(
,,

(6)

*<»<

0

(7) 2

f "pW

j

,

2 ?(D'p„(u') da'; o« (") = — -

2

Mm)

f "<<""> j

(8) '

— r » J = a=ra*».

O)

In the specific case in which the superconductors obey the classical BCS theory me validity of these relations can be proved directly, starting from the specific expressions •for J p > q . u They a r e proved for the general case in the Appendix. 2. CLASSICAL CASE As already mentioned above, the functions J («) appearing in expression (1) for a tunneling current characterize the Josephson junction Itself. They a r e expressed in terms of the unperturbed Green's functions of each superconductor as follows 1 ; "

J

"

'

\ Ira F? (u,) Ira F§ (01,)

>V> =2^7-1 J * . . * - ( « . £ + « . £ ) !(Dj -f- tl)

T l ^

,

1

I

off,

j

/(iKi,

3

0)\ImC

(12)

— 0) -f- IT]

,,

(o)1)lniC?(a>1)

(13) where i) = +0, o is the conductivity of a unit area of the junction in the normal state, and F ^ 2 M and Gj 2(u) a r e energy-integrated Green's functions, obtained by analytically continuing the corresponding temperature functions from discrete points of the imaginary semiaxis lti)n(ajn > Oi where aj n a r e the Matsubara frequencies) onto onto the real axis. For superconductors obeying the BCS theory the functions F R and G R are (see, for example, Ref. 1) '(<•>)=

.

"A

=

;G*(to) =

"M

(14)

where A = A(T) is the energy gap of tne superconductor. Substituting these expressions into Eqs. (12), (13) makes it possible to calculate the functions J p „itc) (for T * 0, however, only numerically) for arbitrary o . Curves of these functions can be found in Eefs. 2, 11, 12, 13; for the case A! = A 2 = A and temperature T = 0.3T C they are shown on Fig. 1 by curves 1. For what follows it Is important to write down the basic properties of the functions Jp q, whereupon for definiteness we henceforth assume Ai fe A2 and consider nonnegative u, while for w < 0 we use Eq. (4). For tij — Ai + A2 the real parts of the functions Jpfq have logarithmic singularities (a singularity i n R e J p is commonly called a Riedel peak'): Zorin etal.

538

98 T C2 /T C i Is shown in Fig. 2. For T — 0 the ratio <rp/cq becomes independent of temperature: a

i7 _ | / 2 n

A A

i »

(23;

,-i,/r.

(24]

£ - & < i . r«A,
(25)

Upon approaching the smaller of the critical temperatures (T c2 ) the quantity Oq tends to a finite quantity, the tunnel junction conductivity between a superconductor and a normal metal:

^r'S*- 1 ^*'^")' u/ZA

FIG. 1. Tunnel current amplitude component! Jptq as affected by finite relative Rtedel peak width: 1) 2<5 = 0; 2) 0.03; 3) 0.3. The functions are nonnalized to the quantity J, • o2A/t, and the relative temperature i s T / T c = 0.3. Re J„ (co) = Re / , (co) + const = - (/,/*) In [| to - (A, + A,) 1/8 (A, + 4,)),

(15)

o)^t A, -f A4,

where J c is the critical current density, equal to j

=a

(26)

CJ/BA

The imaginary parts in this case have finite discontinuities : lm./(,(A1 + As + 0)-Ini./,(A, + A , - 0 ) = -IIray,(A 1 + A1 + 0)-Imy,(A 1 + A , - 0 ) ] = yt.

(1?)

If the electrodes are different (At > Aj) and T * 0, then tor a — At—A2, on the other hand, the real parts of the functions J Pj q acquire finite values: Re ./„ (A, - A, + 0) - Re y, (A! - A2 - 0) [Re J, (A, - A,_+0) - Re / , (A, - As - 0) ] \m2T

Here K,(x) is the modified Bessel function of the second kind. For Aj — A2 the logarithmic singularity at to = Aj—A2 decreases in magnitude and approaches the point u = 0. As a result, In the case of identical electrodes (At = A2 = L the conductivities o Pl q have logarithmic singularities at zero:

in

2TJ'

while logarithmic singularities appear in the imaginary parts ta / , (co) = Im / , (co) = - a ^ 5 5 ( th £ _ Xln[|co — ^ - A J I ^ A . + A,)], co =* A, — A,.

th

"'• 'la

=

f &*+&ln (=•) + F>-«• c o « 2 A , T.

The t e m p e r a t u r e dependence of F p T —0

q

i s shown in Fig. 3 ; fo

f,-r«-«"[ln4-£-T]; F,_*.e-vr[|„ 4 + | r _

T

]

(28)

1

(29)

(30)

(7 = 0.577 i s the Euler constant); for T « T c (18)

= a£!^iMth^-th^ t

(27)

a 6 )

i£H5('th^' + th^.

4

and, consequently, the conductivity ratio satisfies (a^r-ra-O.

f p ( D - , 447"l n 8T4 8 '. F , ( r ) - * l + £4 l Inn ^

(31)

£) &9)

In the low-frequency region the real parts of J p qhave finite values: Re/„(0) = Rey,(co)_Re/,(0) = yf. (20) and the imaginary parts are proportional to u, so that the conductivities have finite values: 2f

a,A.

,~/T

(21) T/Ta

The temperature dependence of the quantities c_ _ for several values of the ratio of energy gaps A2(T = OyA^T = 0) =

FIG. 2. Temperature dependence of the interference and quasiparticle conductivities in the low-frequency region for several values of the ratios of superconductor critical temperatures: 1) T C! /Tci = °-5: 2) 0.8; 3) 0.95. The dashed line "aiming" at the point Oq/ o = 1, T / T c = 1 corresponds to the case when one of the electrodes is in the normal state. Broadening mechanisms are neglected (6 = 0).

539

Zorin et at.

o

T

.}V^-Ail/\>--AU'"/r+^

rfco.

Sov. J. Low Temp. Phys. 5(10). October 1979

(22)

539

99 For the case of identical superconductors the same procedure can also be applied to Eq. (28): In (r/u)-* In (77[co2 + (26A)2]"2). (35) We verify below that account of specific mechanisms of the finite peak width leads precisely to this expression.

$2 m

m

FIG. 3. Plots of the functions Fp.q appearing in the conductivity expressions (29), (30) intiiecase of a symmetric junction ( ^ = Aj = A). Equation (28) shows that for low frequencies the conductivity Q , related to the Interference current component, equals the conductivity o„ related to the quasiparticle component, i.e., ap/a,-*\ for to-*-0. (32) S must be noted, however, that due to the weak (logarithmic) u-dependence of the first t e r m in Eq. (28) even at experimentally utilized very low frequencies (w/T ~ 10" 3 10"2) it is not very important. Therefore the quantities Op and Oq coincide only at low temperatures, where F_ = Fq, but In this region they a r e exponentially small In comparison with the normal Junction conductivity c. On the contrary, at temperatures near the critical temperature, even at sufficiently low frequencies the inequality exp(—T/A) < <"/A, is satisfied, so that

i _ 4 l „ 5 * ^ . 0 fa

T-+Tc.

(33)

The temperature-dependence of the ratio op/oqfor u = 0.04A (T = 0) is illustrated in Fig. 4b (the curve c o r r e sponding to the parameter 6 = 0). 3. PHENOMENOLOGICALACCOUNTOFA FINITE RIEDEL PEAK WIDTH As already noted, the current amplitudes J (u) have logarithmic singularities (15), (19), (28) and acquire finite discontinuities (17), (18). We introduce the finite relative halfwidth 6 of the logarithmic singularities, including the Riedel peak, replacing the logarithms in Eqs. (15), (19): In [| no —(A, ± A2) |/8(A, + A , ) - * In U[u —(A, ± A2)|2 + [6.(A 1 + A ! )H'/ 2 /8(A 1 + A 2 )].

(34)

Since the functions J p „ must satisfy the dispersion relations (6)-(9), along with (34), (35) it is necessary to perform replacements which do not lead to violation of the dispersion relations. Such changes a r e the spreading of the sharp discontinuities (17), (18) into smooth ones. For this we turn attention to the fact that due to their linearity the dispersion relations are also valid for corrections to the functions appearing in them. These corrections, a r i s ing by introducing finite widths of logarithmic peaks, for example, into the real parts of the functions, a r e , by relations (10), (11), P*>.« = ±-41n([(co —a) J -f e 1 ]" 2 /^ —a)), A = const, a = A, + A,; e = « • (A, + A,)

(36)

According to relations (7), (9), to the functions (36) correspond °>., = ± A {-£ sign (co — a) — arctg [(co — a)/e]| t

(37)

which imply smoothing of discontinuities in the conductivities o_ q, and by (10), (11) also in the imaginary parts of the functions J p q (17) according to the rule sign (co — a)-^-|arctg[(co — a)lt\.

(38)

The same blurring of discontinuities, as easily shown, occurs also in the real parts of the functions J p „ In the case of nonidentical superconductors for w = A ! - A 2 (18). The changes described in the functions J_ „ under the action of a finite 6 in the case of a symmetric junction are shown on Fig. 1. In the first approximation in 6 in the real parts of the functions J_ „ they reduce to blurring of logarithmic singularities localized near singular points, and in the imaginary parts of Jp > q to corresponding smearing of discontinuities, while the "tails" ofthis smearing change significantly the original values of the functions in practically the whole frequency interval (see Fig. lc, d). For ai < 2 A the quantity ImJp(o)) acquires negative increments, andlm.Jq(u) positive ones. At sufficiently low temperatures, when the original values of the amplitude of the interference component, the functions Im J , a r e exponentially small (but positive), corrections due to sufficiently large values of <5 can change the sign of this function to the

FIG. 4. Effect of finite peak width on the temperature dependence of the conductivity ratio op/Oq of asymmetric junction (Ai = A2 = A) forvano 3 1 a) CJ = 0; b) u 0.02 [2A(0H for 1) * = 0; 2) 10" ; 3) W ; 4) ".03; 5) 0.1: 6) 0.2.

540

Sov. J. Low Temp. Phys. S( 10). October 1979

Zorin et al.

540

100

as-

r/T„

FIG. 5. Effect of finite peak width on the temperature dependence of tiie conductivity ratio °p/Oq of a nomymmetric junction (Aj > Aj) in the low frequency region Ho « A,^) for a) T ^ A ^ = 0.8 and b) 0.5. The values & are the tame a* in Fig. 4.

T/Ta

opposite one even at relatively low frequencies (see insertion in Fig. lc). These considerations also fully apply to the functions o P f q(u), related t o l m J p> q by expressions (10), (II). Thus, with increasing d the ratio op/oq can become negative, and even approach - 1 . We first write down the analytic expressions for the conductivities o p q for an asymmetric junction: (39) o„ = Op + CpS,

(40) o, = a", + ojfi, 6 (A, + A J ,
(41) 2eJc "(A. + 4JHere the upper subscript "0" denotes the original conductivity values, given by Eqs. (21), (22), and the second terms In Eqs. (39), (40) describe the corrections due to smeared discontinuities. The dependence of the ratio Op/oq on temperature for various 6 and T c 2 / r C i * s shown on Fig. 5. As seen, for 6 values varying from 10" 3 to 0.2 the change of sign in the ratio op/oq occurs at (0.2-0.4)T c ] , and the lower the temperature the smaller the 6 value for which this ratio approaches the level (—1). At higher temperatures the conductivity ratio, though positive, has a lower value than in the case 6 = 0. o. = — 0 . = — -

In the case of a symmetric junction, to calculate op q(u) in the region of small w it is necessary, besides accounting for the "tails" due to spreading of discontinuities, to take into account the logarithmic singularities of the functions CL _ at zero (35). As a result the expressions for op q of (28) transform to the following:

«1"' «*" + D!

In (77[
FIG. 6. The ratio Op/c^ as a function of 4 for a) T = 0.15 T c and b) T • 0.3TC at 1) W/$A = 0; 2) 0.01; 3) 0.03; 4) 0.3; 5) 0.1. 541

Sov. J. Low Temp. Phys. 5(10), October 1979

(«*" + D;

In (77[u« + (26A)']'/2) + F, (T) + -* 8,
where, according to Eq. (41), (44)

The temperature dependence of the ratio o-p/o-a, calculated by Eqs. (42)-(44) for several fixed values of 6 a t u = 0 and 6j = 0.02 [2A(0)], is shown in Fig. 4. Figure 6 shows th( 6-dependence of op/oq for several 6 values a t two t e m peratures : T = 0.15T C and 0.3T C . As seen from Fig. 6, the ratio changessign for T = 0.15T C at 6 ~ 10" 4 -10" 3 , and for T = 0.3T C at 6 ~ 10" : -5 • 10~2, and the values <jp/oq K (-1) a r e reached at 8 ~ 10~2 and 6 ~ 0.1, respectively. 4. SPECIFIC MECHANISMS BROADENING THE SINGULARITIES 4.1. Inelastic Relaxation Processes Superconductors

in

As already noted above, the functions characterizing the junction, and more precisely its superconducting electrodes, a r e Jp>q(iij). The electrode proeerties a r e determined by the Green's functions F£ 2 , Gi>2 of each of the superconductors appearing in expressions (12), (13); therefore, inelastic relaxation processes must be Included in the functions F ^ 2 . Gt,2 at the same time that J p i q as before must be calculated by Eqs. (12), (13). As the well-known, 1 '' 15 account of inelastic relaxation processes 1 ' in superconductors leads in the first approximation to the consequence that the energy gap of the supercondctor A acquires a nonvanishlng imaginary part A"(a>, p). When this imaginary part is small: (45) |A"|/A<£l its effect is easily estimated. The complete Green's functions of such a superconductor (not Integrated over the energy) differ from the Green's functions of an ordinary superconductor only in that the ordinary energy gap A In them is now replaced by a complex gap A(OJ, p) + iA"(o>, p), depending on momentum and frequency. Taking into a c count the weak dependence of A" on momentum p near the Fermi surface, 1 5 integration of the Green's function over this variable is easily performed and leads to the same result (14) for the functions F H and G R , with the only difference that A in these equations is now replaced by the complex function A: f *(<•>)

=

nA (to)

•.; G*(o.) =

~

V~A (a)' — A(w) = A(o)) + /A'(o)).

Zorinetal.

= V A' (B) — Kf

l 7

(46)

541

101 To calculate the functions J p > q M over the whole frequency interval it is necessary to choose a dependence of the real and imaginary parts of the energy gap on u (for example, selected from direct measurements of the eleetron-phonon interaction in specific superconducting metals 17 ). However, to calculate these functions in the regions to » \(&l)± A,(A,)= Ai:f A2one can get away without this information. Indeed, applying to the expression for J P j q (12), (13) the equation

r.._- o = r..^_M 6 ( ,_ a ) J

x — a + iO

J

i—a

m

v

and the dispersion relations relating the r e a l and imaginary parts of the functions F E and G R (see, for example, Hefs. 1, 18):

(48)

(38) , 2 ' with the relative half-width in them being , _ I A," (A.) I + I A,' (Aa) [ (55) 4 , + 4, Along with calculations of the functions J p n(u) a t the singular points, it is possible to estimate directly corrections to the conductivities o-_ q(w) for A" * 0 in the region of small u . The calculation in this case, however, gives a result differing from that obtained in the preceding section. Restricting ourselves, for simplicity, to the case T = 0, and estimating by Eqs. (50), (52) the conductivities °p q ^ ' f o r w « Ai 2, we note that the integrands in these expressions a r e nonvanishing only,in the small Interval [0, w]. Each of the functions Im Ffi2, Im G R 2 for these values of the variable of integration is a small quantity of order \&',i(«>)y\\,t, not mentioning their dependence on the argument due to the function A'fo). Thus, the corrections to the conductivities c p „, which coincide with the quantities Op q since the original values vanish, are already nonlinear and a r e at least quadratic in a small parameter. U _

we arrive at the single integrals

We thus reach a paradoxical result: While the logarithmic singularities of the current components are indeed broadened by the phenomenologlcally introduced Eq. (34), obtained from the dispersion relations, there is no change + Reff (u, + w) Im F? () Im F? (»,); resolved by noting that for a concrete broadening mecha~" (50) nism the change in the current components having at any frequency a logarithmic singularity can occur not only in the region of that singularity, as is assumed in a phenomR e }g ((0) . _i, j . *f ). imposed over a significant frequency range, can partially or completely cancel the contributions calculated above of (52) a strong change in the function near a singularity. In derivingthese relations we used properties of the Thus, in analyzing specific mechanisms and in what functions J p i q (4), as well as the facts that the functions R R 8 8 follows one must consider and estimate changes in the Re F and Im G a r e even while Im F and Re G are odd, functions not only near singular points, but also in the which are their general properties. Substituting in these whole frequency interval. equations the expressions for the functions FR I n e l a s t i c R e l a x a t i o n P r o c e s s e s in and G R from 4 . 2 . Tunnelinf Eq. (46), we verify that for the asymptotic expressions obtained under the assumption of small imaginary parts of Other possible mechanisms leading to the generation the energy gaps it is sufficient to choose for the compoof a finite width Riedel peak a r e inelastic relaxation procnents at singular points integrals which do not extend to esses in tunneling. We attempt to account for them as infinity, but extend over small intervals. Thus, for example, follows. for obtaining the expressionfor the broadenedRiedel peak in The expression for the tunneling current (1) can be expression (49) it is sufficient to take the integral of the written in a form similar to the Duhamel Integral with first term over a small neighborhood including the point Integration over time t' from t' = —«° to the current o>i = — Ai, and of the second term over a neighborhood of moment of time t' = t [see, for example, expression (6) of the point ux = — A2. Each of these integrals reduces to an Ref. 11. In this case the quantity n = +0, reflecting the integral of the type (53 causality principle in Eqs. (12), (13), appears in this V 1 1 > expression In the form of an exponential convergence facJ_ Vx + l* Yx + l + tp tor exp[-i7(t-t')]. If it is now assumed that the quantity V , + u)

542

Sov. J. Low Temp. Phys. 5(10), October 1979

Zorin et i l .

542

102 To illustrate the application of the method we provide an estimate of the Riedel peak at T = 0. In this case the main contribution to the double integral Re /„ (u) =

(56) A(

l

l

l o.

= ; L±-ReJ

,

A,

- • , / -

. ''"./--T— V tt\ —. ("i .+ - «))• V £ — {<*! +my.

\dalda.

w

l + Wj — W + IT]

u = A, + A, + |, U | « A , . j , Is provided by integrating over a small neighborhood of the points CJI - Aj, oj2 ~ ^2 in the plane of the variables OJI, o>2t i.e., Re J, (<•>)- —

Re ]

dio 2

do,

We discuss the physical nature of the change in sign of the interference conductivity. With this a i m we write the expression for J p > q in a form different from (12), (13) (see Bef. 2)

A,

4i

i7=f ] ' ojj —

scribed the conductivity ratio op/erq at low temperatures can be lowered from the value + 1 at T] = + 0 to values from — Vj to 0, depending on the ratio of Aj to A 2 . Assuming that the upper limit in (60), (61) is finite, I.e., the inelastic mechanism discussed Is effective In some energy region E near the Fermi surface, the conductivity r a t i o Increases The dependence of lop/fcql on E/A Is provided In Fig. 7b in the symmetric case.

A{ | ' w* — A*

(57) u, + CO, — A, — Aj — | + it)'

X {(1— n„ — /!*)(:e

» + e* + <" + "1+ ;

J

J]

1, . ' l l « e i . e 2 «A|, 3 . Choosing q « £2 "^Ai^ and integrating expression (57) first with respect to wi, and then with respect to wu we a r r i v e at the expression Re/„M (Jfin) InI(|= -f- •f)/16«r;]. (58) Other components can also be estimated by the same method. Thus, in the case of taking into account inelastic processes in tunneling the broadening of singularities is described by Eqs. (34), (35), and (38), where now

Despite the complication in the calculations, one can also estimate here corrections to the conductivity oy, q in the region of small u. Restricting ourselves again to the case T = 0 only, and linearizing expressions (12), (13) in u and IJ, we arrive now at the following expressions for cj^q and, due to the vanishing of the original quantities at T= 0, also for the corrections:

;

4=-M-..f

do.

.(60)

i A

I^M; — A; Va\ — A.* (w, + u,)'

i' = !' = !,, I'd*, L
"

\ j dlBdlk

X (I —n„ — nk)

*c + ek-j-u—

;+

— ("p — ih)

(". ")l. * = 1 (I

111 /

u> — in "*" *p — e» + <•> + ' 1 J ' ±

|„. »/e„.»), «». * = VA* + £j,

(59)

Ai + A,

°

J,(*)=~

I

i

= .

(61)

" V u? - A; V - u | - A ; (», + «,)•

Although here too there a r e linear corrections of different signs to the conductivities a , they a r e not equal In absolute value. For A4 = A2 the ratio of absolute values is of the order of %,.while for appreciably different energy gaps it tends to zero (Fig. 7). Due to the mechanism de-

The quasiparticle distribution functions at the electrodes Dp, nk vanish for T = 0. At high temperatures, whei these quantities are finite, the conductivities xrp,
b

03 1

-Ik*" 1

1

'%

0.1

S.S

a 543

FIG. 7. Ratio of the correction! to the conductivities Opq as a function of the energy gap ratio for low frequencies (u « a^j) at a) T_= 0, as well as of the ratio of the energy width of theinelastic region (E) to the gap width A (b).

Sov. J. Low Temp. Phys. 5(10). October 1979

3

JO

£/A

Zorin et al.

543

103 4 . 3 . I n h o m o g e n e l t y and A n l s o t r o p y Electrodes

of

Other mechanisms providing broadening of Riedel singularities a r e inhomogeneities and anisotropics of the electrodes forming the junction. Account of inhomogeneities is quite important if the electrodes, as is usually the case, are thin polycrystalline films, since the superconductor parameters (T 0 , A) always differ in different c r y s tals. The method of taking into account inhomogeneities depends essentially on their characteristic size D. If this size, as is usually the case experimentally, is much larger than the film width d and the coherence length {, the whole Josephson junction can be assumed to consist of a large number of independent junctions of parallel enclosures, while the parameters of these elementary junctions a r e randomly distributed. In this case the total current through the junction can be found, retaining the unperturbed equations for the current and averaging them over the ensemble of elementary Junctions. If the parameters of neighboring junctions do not differ strongly, i.e., me dispersions of the quantities A1>2 are small, the shape of the Riedel peak i s , obviously, determined by the expression In ] a — (A,> — (A2> + 11 w © dl.

Re/„.,(oi)<

(62)

where w(£) is the probability distribution density of the sum of energy gaps near their average value (Aj) + (A2). Equation (62) shows that the peak shape is determined by the shape of the distribution w({). For example. Fig. 8 shows the peak shape for a normal (Gaussian) distribution w. For comparison we also show the dependence described by Eq. (34). It is seen that the main difference in the change of peak shape is that for averages of type (62) the new points do not all lie under the original curve, but are located in such a manner that the average value of the change in current vanishes: -„,, (co) da ~ f da j dl [In ; a — (A,) — (A2> + 11 J Re/.. -(A.+O

-lnjffl — —
(63)

Here we used only the fact that the integral of w(|) over the localization region is equal, by definition, to unity, independently of the specific shape of the function w. If now the dispersion relations (6), (8) a r e used to find the changes

FIG. 8. The logarithmic singularity and Its broadening according to Eq. (62) (solid line) and (34) (dashed line): Sw is the parameter of Gaussian distribution w, 6 is the parameter in relationship (34), and 6 = 1.04Sv.

-2 - 7

544

0

7

Z

Sov. J. Low Temp. Phys. 5(10), October 1979

in the imaginary parts of the current components, due to equality (63) we verify that these changes a r e localized in a narrow neighborhood of the singular point to = Aj + A 2 and have no "tails," being drawn to the point OJ = 0. Thus account of weak large-scale inhomogeneities, though p r o viding broadening of the Riedel peak, does not lead to corrections to the quantities a (oj) in the region of small arguments. ' The same result is also valid when the small anisotropy of the electrode material is taken into account, since the averaging in Eq. (62) can also be performed over contributions to the current due to different directions of the electron quasimomentum. As to small-scale inhomogeneities, i.e., inhomogeneities of size D « d, { (but much larger than atomic dimensional), their treatment is by far more complicated. In particular, Eqs. (12), (13) may not be applied to them. Ihis case must be analyzed not by the tunneling Hamlltonian method, obtained specificially from the results of Refs. 1-3, but by finding directly the electrode Green's functions (see, for example, Refs. 3, 22).3> Nevertheless, it appears that account of precisely these inhomogeneities must give results near those of the phenomenological analysis given in Sec. 3. Indeed, while the large-scale inhomogeneities a r e the spatial analog of time fluctuations with long correlation times, the small-scale inhomogeneities a r e the analog of a 6-correlated random process. It is well-known from the theory of random proce s s e s In radiotechnological systems that a decrease in correlation time of a random process leads to a transition from a Gaussian shape of broadened resonance features (with a vanishing average of function values under the action of fluctuations) to a Lorentzian shape (with a nonvanishing average). 24 It can be expected, therefore, that transition from large-scale to small-scale inhomogeneities generates a transition from the peak shape (62) to (34) with a corresponding appearance of a nonvanishing average of ReJ P j qOver the peak, and, consequently, to the appearance of corrections to the quantities OJ, q. 5. COMPARISON WITH EXPERIMENT Experimental measurements of values of a p n at frequencies much lower than gap frequencies a r e described in Refs. 4, 25, 26. We compare the results of our phenomenological analysis with the results of these experiments. In experiments 1 performed with lead junctions at a temperature T « 0.6T e and a t a frequency of 9 GHz (corresponding to the value o>/2A ~ 10 -'), ratio values oyo,~ —0.9 ± 0.2. were obtained. Substituting in our results [Eqs. (42), (43), Figs. 4, 6] the maximum values of Riedel peak widths observed for lead (see, for example, the data ofSoerensen's measurements of dl/dV, given in Ref. 13, whose processing by Eq. (38) leads to these values), the indicated values of apjbq at the given frequency can be observed at temperatures not above (0.4-0.5) T c . We note that the experimental 6 values may contain parts giving peak broadening, but not providing significant corrections to the conductivities Op>q at small u. These terms may be due to the mechanisms considered in Sec. 4, Zorinet at.

544

104 go

that the effective 6 values providing corrections to oh can be smaller than those mentioned.

q

A strong temperature dependence of op/oq was noted experimentally, 25 ' 28 where for temperatures lowered below the critical this quantity initially increased from zero to approximately (+1), and then started decreasing and reached a value near (-1). Precisely such a dependence follows from our results (see Figs. 4, 5) if it is assumed that <5 does not vary too strongly with temperature. The negative value of the interference conductivity also follows from the results of Ref. 27. It must be noted, however, that the temperature values for which the maximum values op/on a +1 were experimentally reached 25 > 26 are significantly closer to the c r i t i cal one UT C -T)/T C « (1-5) • 10"2) than expected by our calculations starting from reasonable values for 6 (T » 0.9 T c ). B; is fairly difficult to identify the reasons for this discrepancy. We note, however, that the method of handling experimental results adopted in Ref. 25 allows, in our opinion, significant variation of the results due to the way of accounting for small additional factors. One such factor can be additional dissipation mechanisms of UHF-power (for example, ordinary losses in the electrodes), not taken into account by the authors. As to the experiments of Ref. 26, strong suppression of the critical current of an extended Josephson junction by a magnetic field was used in them to tune the plasma resonance (the total critical current was in this case only a few percent of the magnitude of the current in a vanishing field). For such suppression practically the whole Josephson tunnel current is compensated along the junction by the phase difference, and the residual observed current can be strongly subjected to an effect of small parasitic factors. An Indirect indication that factors not appearing in the theory strongly affect the results of experiments 25 ' 26 is the fact that the peak position in the temperature dependence of cp/cTq in them depends on the density of the junction c r i ical current,, while In all theories of the tunnel effect the current through the junction is a small quantity, and has no effect on the tunneling process. Other examples can be changes In the capacity of the junction, for which a p e r suasive explanation is difficult to find. 6. CONCLUSION Thus, using dispersion relations of cur rent components in a superconducting tunnel junction, we have shown that account of the finite width of a Riedel peak can lead to a significant change in the conductivities a p q, while theratlo of the interference conductivity a0 to the quasiparticle conductivity On can approach the value (-1). However, the estimates of this effect for a number of specific broadening mechanisms have shown that not all of them contribute to changes in the conductivities. The reason is the fact that the real parts of the current components under the action of some broadening mechanisms can undergo small changes even far from the Riedel singularity, whose contribution to the dispersion relations compensates for the change In these functions near singularities. It is possible that the main broadening mechanisms contributing to changes In c p q a r e small-scale inhomogenelties In the material of the tunnel contact electrodes. The known experiments, though qualitatively explained 545

Sov. J. Low Temp. Phys. 5(10), October 1979

by the phenomenologlcal theory developed h e r e , do not enable one to draw definitive conclusions that me negative signs of the interference conductivities observed in them a r e uniquely related to the effects considered here. F o r definitive clarification of the situation it Is required to p e r form further measurements of the quantity o-p/oq in a wide temperature range with tunnel junctions with small values of the critical current density. In any c a s e , experimental values of Op/oq differing from those obtained by the standard theory cannot provide serious criticism of the latter, since small values of ob,q (particularly at two temperatures) a r e greatly subject to the effects of various factors. The present work is a result of studies performed by the authors a t Moscow State University (A. B. Zorln, K. K. Likharev) and a t the Aspen Physics Institute, Colorado, USA (L O. Kulik, J. R. Schrieffer). The authors a r e grateful to J. M. Rowell, W. L. McMillan, R. C. Dynes, N. F. Pedersen, O. H. Soerensen, J . Mygind, S. Rudner, and T. Claeson for supplying publications of Refs. 17, 25, 26, and, besides, to the authors of the two latter publications as well as M. Yu. Kupriyanov and D. N. Langenberg for useful discussions. APPENDIX To find the relation between the real and Imaginary parts of the functions J_ _ consider the case of a small sinusoidal voltage at the junction V(/) = ReV,exp(—fa,/), «Wo> t «l. (A-D In this c a s e the function W(OJ) is in the first approximation in Vt W(a) = exp (i where q , V — cos ) -J- oq (a), Y" = cos q>„pp (),

(A.5)

where the quantities op q, pntq a r e determined by expressions (10), (11). The real and imaginary parts of Y must be related by the ordinary Kramers ~Kronig relations (see, for example, Ref. 28). Bearing in mind that these r e l a tions must be satisfied for any value of
545

quasiparticle current components; V is the voltage; o> is the frequency;

We note that an approximation was suggested 0 for the Riedel peak broadening, described by an equation deviating from (34) as follows (Aj = Aj); In (| 10 — 241/164) — In [ { | (0 — 24 | + 48A)/16A], 6 = | 4 ' (A) |/A. According to this (incorrect) equation the peak height is

R« ^ - V * ! " (4/8) which differs by In 2 from that obtained from Eq. (34;: Re .T„ = V n l n (8/8), It appears that certain deviations in estimating the quantity 6 by the Riedel peak width from the discontinuity in the quasiparticle current, noted in Ref. 20, Is related to this. s T h e quasiparticle component of the tunnel current between electrodes with small-scale inhomogeneities was calculated in Ref. 23. In this case equations for homogeneous superconductors were used, in which the Green's functions of each electrode were replaced by their statistical average. It seems to us that the validity of this approach is by no means justified. 'A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz._51,1535 (1966) [Sov. Phys. JETP 24,1035 (1967)]. : N . R. Werthamer, Phys. Rev. 147, 255 (1966). h. O. Kulik and I. K. Yanson, The Josepbson Effect in Superconductive Tunneling Structures, IPST, Jerusalem (1972). *N. F. Pedersen, T . F. Finnegan, and D . N. Langenberg, Phys. Rev. B6, 4151 (1972).

546

Sov. J. Low Temp. Phys. 5(10), October 1979

°D. N . Langenberg, Rev. Phys. Appl. 9, 35 (1974). • u . K. Poulsen, Rev. Phys. Appl. 9, 4T (1974). 'E. Riedel, Z . Naturfotsch. 19, 1634 (1964). *C. A. Hamilton and S. Shapiro, Phys. Rev. Lett. _26_, 426 (1971): see also S. A. Buckner, T . F. Finnegan, and D . N. Langenberg, Phys. Rev. Lett. 28, 150 (1972). *LGiaever, H . R . Hart, and K. Megerle, Phys. Rev. 126. 941 (1962). 10 B. D . Josephson, Adv. Phys. 14, 419 (1965). U R. E. HaniJ, Phys. Rev. B10, 84 (1974). ^ E. Harris, Phys. Rev. B l l , 3329 (1975). " U . K. Poulsen, In: Rent. Phys. Lab., D . T . H., No. 121 (1973). "l. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Lett. 10, 336 (1963). % . J. Scalapino, J. R. Schrieffer, and J. W. Wilkins, Rhys. Rev. 148, 263 (1966). " l . O. Kullk, Zh. Eksp. Teor. Fiz. 50, 799 (1966) [Sov. Phys. JETP 23, 529 (1966)]. n J . M. Rowell, W. L. McMillan, and R. C. Dynes, Preprint (1978). U A . A. Abrfkosov, L. P. GorTcov, and I. E. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics, Pergamon Press (1965). ° D . J. Scalapino and T . M. Wu, Phys. Rev. Lea. 17, 315 (1966). 2t S. A. Buckner and D . N. Langenberg, J. Low. T e m p . Phys. j!2_, 569 (1976). " A . E. Gorbonosovand L O. Kulik, Zh. Eksp. Teor. Fiz. j>5, 876 (1968) [Sov. Phys. JETP 28, 455 (1969)]. a A . I. Larkin, Yu. N. Ovchinnikov, and M. A. Fedorov, Zh. Eksp. Teor. . F I z . J l , 683 (1966) [Sov. Phys. JETP j!4, 4 5 2 (1967)]. ° A . I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. Jjl, 2147 (1971) [Sov. Phys. JETP £ 4 , 1144 (1972)]. *R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York (1963). ° N . F. Pedersen, O. H. Soerensen, and J. Mygind, Phys. Rev. B18, 3220 (1978). * S . Rudner and T . Claeson, J. Phys. C6,604 (1978). n O . P. Balkashin and I. K. Yanson, Fiz. Nizk. T e m p . 2, 289 (1976) [Sov. J. Low T e m p . Phys. 2, 143 (1976)]. Z8 L. D . Landau and E. M. Lifshirz, Electrodynamics of Continuous Media, Pergamon Press (1960).

Translated by Nathan Jacob!

0360-0335/79/10 0546-03 S02.20

©1980 American Institute of Physics

IL NUOVO CIMENTO

VOL. 100 A, N. 2

Agosto 1988

Rotating Superfluidity in Nuclei. G. F. BERTSCH(*), R. A. BROGLIA(**) and R. SCHRIEFFER(***)

Villa Monastero, Varenna sul Logo di Como - Como (ricevuto il 18 Aprile 1988)

Summary. — The limits of angular momentum in rotating nuclei may permit superfluidity with nonzero angular momentum, but the excitation energy appears too high to have observable consequences. PACS 21.10 - General and average properties of nuclei; properties of nuclear energy levels.

Nuclei in their ground state can be viewed, in general, as a condensate of pairs of nucleons coupled to angular momentum / equal to zero ('). Evidence for the existence of multipole (nonzero J) pairing has also been found in a variety of nuclear properties (2). Because the parity of two-particle states around the Fermi energy is positive, essentially only even multipolarities are allowed. A successful description of the variety of pairing correlations is obtained in terms of the Hamiltonian (1)

H = Hsp + Hp,

sum of a single-particle term (2)

flr1T

= 2 e » c Cffl».

(*) Permanent address: Michigan State University, East Lansing, MI 48824. (**) Permanent address: Dipartimento di Fisica, Universita di Milano and INFN Sez. Milano, 20133 Milano, Italy and The Niels Bohr Institute, University of Copenhagen, Dermont. (***) Permanent address: Institute for Theoretical Physics, University of California, Santa Barbara, California, CA 93106. 0) A. BOHR, B. R. MOTTELSON and D. PINES: Pkys. Rev., 110, 936 (1958). (2) R. A. BROGLIA, D. R. BES and B. S. NlLSSON: Phys. Lett. B, 50, 213 (1974). 19 - II Nxumo Cimento A.

283

G. P. BERTSCH, R. A. BROGLIA and R. SCHRIEFFER

284

and of a multipole pairing interaction (3)

H P - - 2 Gj 2 {[< a U [a* a j , } 0 .

As usual [a^al^j denotes coupling of the pair operators to total angular momentum J, and similarly for { } 0 . Restricting the single-particle states v< to the valence orbitals, the pairing coupling constant can be written as (4)

G,~G~^MeV,

where A is the mass number of the nucleus. Empirically, the d-state pairing energy of a single pair is about half that of a monopole pair. The reduction is due to the decrease in phase space for valence pairs with higher / . This is shown schematically in fig. 1. The effect is even more marked for J = 4. Consequently, near the ground state, we do not expect metastability for J&O. .

a)

b)

Fig. 1 - Phase space for particles in paired wave functions. The available momenta for valence particles in a finite Fermi system are shown in a). All momenta are allowed for a particle in a pair with total momentum zero. When the pair momentum in nonzero, the valence phase space is reduced as indicated in b). This situation may be rather different for rapidly rotating nuclei. In this case, large values of the angular momentum can be built by using a coupling scheme where both valence and core particles couple pairwise to angular momentum / . Because of the strong modifications suffered by the single-particle states, twoparticle states with negative parity are now readily available close to the Fermi energy. The lowest multipolarity different from zero to which pairs of particles can couple is J = 1. Under these circumstances, Galilean invariance allows one to redefine the phase space where dipole pairing acts, so that the resulting phase space is nearly the same as for / = 0 pairing.

ROTATING SUPERFLUIDITY IN NUCLEI

285

In this scheme, the lowest rotating superfluid state has angular momentum I = A/2. The pairing energy will be En = (Al + A$l2d - 2 . 5 MeV, where An and Av are the protons and neutrons pairing gaps (~ 1 MeV). The quantity d is the distance between single-particle levels, which, for medium-heavy nuclei is 0.4 MeV (cf. p. 653 of ref. (»)). Metastability arises because transitions with / - » I - AI require breaking of pairs, which may be energetically forbidden. For this scheme to work, two important questions have to be answered: 1) Does the nucleus allow spins as high as I ~ A/27 2) How does the energy cost to form a vortex compare with the energy gain of pairing? It has been calculated that a cold drop of Fermi liquid with A ~ 150 can accommodate up to about 80 units of angular momentum (4). Experimental studies of discrete lines in the quadrupole gamma-decay spectrum of strongly rotating nuclei have identified, in this mass region, rotational bands which carry up to 60 units of angular momentum (B). In what follows we give a simple answer to the second question, making use of a schematic model. We assume the system to be a cylinder of height H and radius R, and constant density po which rotates around the symmetry axis. Because of the J =£ 0 superfluidity, a vortex forms with a cylindrical hole along the axis of rotation. The velocity of the vortex can be written vt = glr, where g = h/2m for / = 1 vorticity. The energy of the vortex consists of a rotational part and a part associated with the surface energy of the role. The rotational energy is estimated as (5)

£™rt« = fpoJ d Z / ^ 2 T r d r = - ^ - L 2 l n ^ ,

where a is the radius of the cylinder, M is the mass of the nucleus, and L is the angular momentum, (6)

L = po J dz J 0

vt2?trdrmp0Hjdi2g.

a

We estimate from the surface tension a the energy cost to make the vortex (7)

E]Bt = 27tH(KT.

(*) A. BOHR and B. R. MOTTELSON: Nuclear Structure, Vol. II (Benjamin, Reading, 1975). (4) S. COHEN, F. PLASIL and W. SWIATECKI: Ann. Phys., 22, 406 (1963). (') P. TWIN, B. M. NYAKO, A. H. NELSON, J. SIMPSON, A. M. BENTLEY, H. W. CRANMER-GORDON, P. D. FORSYTH, D. HOWE, A. R. MOKTHAR, J. D. MORRISON, J. F. SHARPEY-SCHAEFPER and G. SLETTEN: Phya. Rev. Lett., 57, 811 (1986).

286

G. F. BEBTSCH, R. A. BROGLIA and R. SCHRIEFPER

Minimizing the total energy with respect to a, we find that a = 0.7 fm. To obtain the excitation energy of the state, we compare with the energy of rigid rotation, which is ErigjA^LVMR*. For a nucleus of mass A~ 150, rotating with angular momentum 75, the excitation energy is then A^ = - | ^ r ( l n ^ - - l N | + 27rHcrti«70MeV,

(8)

where the values
Discussions with B. R. Mottelson are gratefully acknowledged. The hospitality of the Societa Italiana di Fisica and of Ente Villa Monastero are appreciated.

•

RIASSUNTO

I limiti per il momento angolare di nuclei ruotanti sembrerebbe permettere l'esistenza di superfluidita con momento angolare diverso da zero, anche se I'energia di eccitazione e troppo elevata per avere conseguenze sperimentali.

Pe3MMe He noJiyMeHo.

110

John Bardeen, 1908-1991 J.R. Schrieffer Department of Physics, University of California, Santa Barbara, CA 93106. USA

Presented at the M2S-HTSC

Conference, Kanazawa, Japan, July

1991

When one speaks with friends of John Bardeen they frequently use words such as brilliant, profound, practical, quiet, devoted, family, friends, golf, humor, wise, determined, generous and a man for all seasons. He was awarded the highest honors an individual can be granted - he was the first scholar to receive two Nobel prizes in the same discipline. His colleagues awarded him every major prize in his field. He enjoyed a deeply devoted family with children who became leaders in their own fields, carrying on the tradition of their family. His scientific legacy includes the invention of the transistor, the mechanism of superconductivity and the shaping of four decades of scientists who carry on exploring the mysteries of nature in his footsteps. John Bardeen was born on May 23, 1908 in Madison, Wisconsin, son of Charles Russell Bardeen, founder of the Medical School at the University of Wisconsin. His mother, Althea Harmer, studied oriental art at the Pratt Institute and practiced interior design. John studied electrical engineering at the University of Wisconsin, where he received his Bachelors and Masters degrees. Following three years of geophysics research at the Gulf Research Laboratories, he continued his graduate studies at Princeton, where he received his Doctorate in 1936 for a pioneering study of electronic correlation effects at metal surfaces, carried out under the guidance of Eugene P. Wigner. He was a Harvard Junior Fellow from 1935-1938, where he interacted with John H. Van Vleck. After appointments at the University of Minnesota and the Naval Ordinance Laboratory in Washington, DC, he joined the newly formed research group in solid state physics at the Bell Telephone Laboratories in Murray Hill, New Jersey, in 1945. It was there that he carried out his fundamental studies on transport effects in semiconductors which led to the discovery of the transistor, for which he was to share the Nobel Prize with Walter Brattain and William Shockley in 1956. John Bardeen became Professor of Physics and Electrical Engineering at the University of Illinois, Urbana, in 1951 and played a key role in building the pre-eminent academic solid state research group in the US, as well as setting the agenda for this field and educating students and researchers throughout the world for four decades. While he became Professor Emeritus in 1975, he remained highly active in research, education and golf until his death on January 30, 1991. While many people feel they know John Bardeen quite well, as one begins to share those wonderful personal stories about him it becomes clear that each of us know but a small fraction of his true dimensions as a human being and a scholar. I first met John in the fall of 1953, when I became his graduate student in Urbana, and my stories begin then. Hopefully, one day there will be an opportunity for others to set down some of their stories from other times and occasions. My first recollection of seeing John Bardeen was on the day I arrived in Urbana that September. XI

Ill xii

John Bardeen, 1908-1991

I went to Professor Bardeen's office to introduce myself but no one was there. I went back to the Physics Department office, passing a man on the stairs, and I was told by the secretary that Professor Bardeen just went up to his office. So, up I went again, but still no professor. My father, who had accompanied me that day, said: "Why don't you ask this man we keep passing if he has seen Professor Bardeen". Needless to say, the man was John Bardeen. The point of the story, of course, is that John was a wonderfully humble appearing man and never called attention to himself - he didn't have to since he was continually surrounded by people seeks his wisdom and help, of which he gave freely. Most remarkable was his ability to quickly see to the core of a complex physical situation or a complex mathematical derivation. Soon after we completed our work on the theory of superconductivity, John received a 100-page manuscript from two distinguished theorists developing an elaborated mathematical version of the pairing theory. They applied their method to the calculation of the thermal conductivity in the superconducting phase. John looked at the last page of the manuscript where a long expression for the result appeared. Slowly shaking his head he quietly said: "They forgot the diffusion term". Days later we, with John's help, discovered an innocuous factorization approximation had thrown out the baby with the bathwater. Again and again it appeared that John simply thought differently than others. It was as if he communicated directly with nature in her own language, possibly the reason why so many people had difficulty in understanding John when he used this language with mortals. John had a joyous sense of humor. At a Physics Department picnic a small boy asked him to read a sign. His response was overheard as 'Wo elephants allowed here". He had an infectuous laugh and, as my wife Anne quickly discovered, this was activated by getting him to discuss his favorite topics - golf, basketball and his children, particularly Betsy. He was an exceptionally generous man. Having worked very hard on the theory of superconductivity since the late 1930's and particularly in view of the great excitement in having finally solved this half-century-old challenge, he insisted that Leon Cooper and I present the first announcement of the work at a hastily organized post-deadline session of the American Physical Society meeting in March, 1957, which Leon reluctantly agreed to do as I was travelling at the time. This remarkably generous act was his way of ensuring that the younger people were given proper credit for their contributions. This was followed some years later, I am told, by John nominating Leon Cooper and myself for the Nobel Prize in Physics since he was concerned that this earlier transistor prize might rule out the three of us sharing an award for superconductivity. John was indeed a principled and generous man. John's power of concentration is legendary. In the early spring of 1957, after the first announcement of the pairing theory had been submitted for publication, there remained a fundamental difficulty with the mathematical description in that the second-order phase transition was not properly included. One evening, the Bardeens were hosting a distinguished metallurgist and, as Jane Bardeen relates the story, John was rather distant, deep in thought, and apparently not following the conversation very closely. Jane would probe John about his views on the topic under discussion, to little avail. The slight embarrassment of that evening paid off as I learned when the phone rang at my house about 7 am and John said, "Hello Bob, when the excited pairs are orthogonalized to the ground state pairs the second order transition comes out OK". Who could sleep after that? The practical side of John is also well known. He was deeply interested in the applied side of research as well as the fundamentals. His contributions to Xerox, General Electric and Sony, the latter having recently endowed a Chair in his honour at Urbana, are well documented. He also served his country as a member of the President's Science Advisory Committee from which he resigned as a matter of conscience at the announcement of the so-called Star Wars program. He generously donated his share of the superconductivity Nobel Prize to enable the Fritz London Award to con-

112 John Bardeen, 1908-1991

tinue to honor outstanding work in low-temperature physics, a field to which he contributed so much. Finally, John and Jane have been true friends and inspirations not only to students and colleagues but also to those who only casually met them. Their unassuming nature, warmth, good humor and high ideals make each of us stretch to be a bit bigger than we are. And, finally, we will never look at elephants the same again.

113 PHYSICAL REVIEW B

VOLUME 55, NUMBER 18

1 MAY 1997-11

Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors M. I. Salkola Department of Physics, Stanford University, Stanford, California 94305 A. V. Balatsky Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 J. R. Schrieffer NHMFL and Department of Physics, Florida State University, Tallahassee, Florida 32310 (Received 16 October 1996) The consequences of localized, classical magnetic moments in superconductors are explored and dieir effect on the spectral properties of the intragap bound states is studied. Above a critical moment, a localized quasiparticle excitation in an s-wave superconductor is spontaneously created near a magnetic impurity, inducing a zero-temperature quantum transition. In this transition, the spin quantum number of the ground state changes from zero to ;, while the total charge remains the same. In contrast, the spin-unpolarized ground state of a rf-wave superconductor is found to be stable for any value of the magnetic moment when the normal-state energy spectrum possesses particle-hole symmetry. The effect of impurity scattering on the quasiparticle states is interpreted in the spirit of relevant symmetries of the clean superconductor. The results obtained by the non-self-consistent (T matrix) and the self-consistent mean-field approximations are compared and qualitative agreement between the two schemes is found in die regime where the coherence length is longer than the Fermi lengm. [S0163-1829(97)l 1217-6]

I. INTRODUCTION

In this paper, we revisit the problem of a localized magnetic moment interacting with a superconductor. A remarkable aspect of this interaction is the first-order zerotemperature transition which takes place in an s-wave superconductor as a function of the "magnetic moment," w = JS/2, where S is the local impurity spin and J is the exchange coupling. In this transition, the spin quantum number 5 of the electronic ground state of the superconductor

|*o)w changes from zero for a subcritical moment w<wc to 5 for w>wc. The total spin becomes S± \ depending on the sign of the exchange interaction J. The first to point out the phase transition was Sakurai12 who showed that the transition corresponds to a level crossing between two ground states as a function of the exchange coupling J. The level crossing occurs in a singlet superconductor between states where the impurity spin is either unscreened or partially screened. We note here that the crossing between two ground-state energies does not occur at the point where the excitation level becomes a zero-energy excitation, since the free energy of the superconductor is determined by taking into account continuum states as well as any localized intragap states in the self-consistent solution. We address the above problem at zero temperature by using the mean-field approximation both within the T matrix formulation and the self-consistent approach which takes into account a local gap-function relaxation. We also consider a local Coulomb interaction U which breaks particlehole symmetry leading to an asymmetric spectral density for the impurity-induced quasiparticle states. Figure 1 illustrates the local effect of a magnetic moment on the low-energy spectral density in an s-wave superconductor. Since we limit our considerations to a classical spin, 5 > 1 , the impurity moment cannot be screened completely by the quasiparticles. Our main results are as follows, (i) We show that the gross features of the impurity-induced quasiparticle states in s- and af-wave superconductors can be qualitatively understood within the non-self-consistent T matrix formalism, (ii) For an s -wave superconductor, we find that, above the critical moment w c , the order parameter changes sign at the impurity site and that it vanishes as w^°o. 13 In the self-consistent mean-field approximation, we find that order-parameter re-

0163-1829/97/55(18)/12648(14)/$10.00

12 648

One of the most interesting phenomena in superconductors is their response to defects. They can be applied as experimental probes in studying the properties of the superconducting state. The simplest example of pair-breaking defects is magnetic impurities in singlet superconductors, where magnetic scattering disrupts pairing in the singlet channel. Pair-breaking defects are known to lead to the formation of bound quasiparticle states in conventional nodeless superconductors1-3 and virtual bound states in gapless superconductors.4-7 These states are localized in the neighborhood of the defects and can have a very anisotropic structure, depending on the form of the energy gap. 47 Quasiparticle states, localized near impurities, carry spin and are expected to modify the ground-state properties of the superconductors as well. Because experimental techniques are becoming increasingly capable of analyzing the effect of impurities on the superconductivity, it is also interesting to explore theoretically the implications of impurity scattering together with the symmetries present in the problem. Indeed, the progression from tunnel-junction spectroscopy8 to scanning-tunneling microscopy911 provides a strong evidence of inhomogeneous quasiparticle states induced by defects in superconductors.

55

© 1997 The American Physical Society

114 SPECTRAL PROPERTIES OF QUASIPARTICLE .

55 1.5 i

.

1

1

.

i

r

'

12 649

i

U»K

i.o

? -3.0

-8.0

-1.0

0.0

1.0

2.0

3.0

u/A0 FIG. 1. The spin-unpolarized spectral density A(r=0,cu) in an .r-wave superconductor due to a magnetic moment located at r=0 as predicted by the T matrix approach. The continuum contribution is given in units of NF and the height of the intragap bound-state peaks, located at Cl0= ±O.43A0, denotes uieir integrated spectral weights in units of NFA0. Here, NF is the density of states at the Fermi energy in the normal state and A0 is superconducting energy gap. The magnitude of the magnetic moment is TTNFW = 0.7 and the local Coulomb interaction is TTNFU=0A. laxation shifts wc downwards and that the energy of the impurity-induced bound state does not reach zero before a first-order transition between the two ground states occurs, (iii) In contrast, a rf-wave superconductor has no quantum transition for any value of the magnetic moment when its quasiparticle spectrum in the normal state has particle-hole symmetry. The absence of the transition follows from the behavior of the impurity-induced quasiparticle states which are pinned at the chemical potential for an arbitrarily large magnetic moment. However, if particle-hole symmetry is broken or if the pairing state acquires a small s -wave component, the transition is again possible for a large enough moment. The rf-wave order parameter does not appear to change sign at the impurity site. The impurity moment induces two virtual-bound states which have fourfold symmetry and extend along the nodal directions of the energy gap. (iv) Finally, a mapping to an effective theory is derived which allows the formation of the impurity band at finite impurity concentrations to be explored. The Kondo effect in an s -wave superconductor interacting with a quantum impurity spin was considered in a recent work15 which found a similar type of a quantum transition from the spin-doublet state (unscreened impurity spin) to the spin-singlet state (screened impurity spin). For the antiferromagnetic coupling (J>0), the two states cross at 7V/Ao = 0.3, whereas, for the ferromagnetic coupling (7<0), the coupling constant flows to the weak coupling limit and the level crossing is absent. The situation resembles the antiferromagnetic Kondo problem in that the moment is screened. However, the crossover is replaced here by a sharp transition.15 The physical picture of the quantum transition follows from the behavior of the impurity-induced bound state. This transition is a consequence of the instability of the spinunpolarized ground state, because, for a large enough w, the energy of the impurity-induced quasiparticle excitation would fall below the chemical potential. For a weak coupling w<wc, the ground state of the superconductor is a paired state of time-reversed single-particle states in the presence of

FIG. 2. A schematic evolution of die spectral density in a superconductor as a function of a local Coulomb interaction U. The contribution due to the impurity-induced resonance state is depicted by the shadowed peaks. For |t/|~A 0 , the resonance has nearly equal weights in the electron and the hole components, whereas, for |C/|S>A0, it is mosdy either electronlike or holelike. die impurity scattering,16 |<£o)»

|<J>o), where

i* 0 >=n («,+f,^,^ u )io),

(i)

and the bound state at the energy H 0 is an intragap excited state, | * - U ) „ , < W c ~ | * - u > s 7 t - i | l * o > , where

I* -u >=*i

, n (",+^^T^,I)IO>.

(2)

The quasiparticle operators17 are y^u^^ — Vii/SL^ and yt.,j = «;^l,j + i>,\])wo)> does not appear in the superconducting energy gap. The processes of adding an electron with spin down to state i= — \ and removing an electron with spin up from state i = 1 in the ground state |$0}w<» lead to the same excited state: iAt_1j|*o) = "il*-ii> and - Thus, in a tunneling experiment, the bound state is responsible through these processes for a finite differential conductance proportional to u] and vf at energies &>= ±fl 0 . Figure 2 elucidates schematically the mixing of particle and hole degrees of freedom in a quasiparticle state using a potential scatterer as an example. Because the particle and hole components are part of the same state, diey appear symmetrically relative to the chemical potential. As w increases, ft0 approaches the chemical potential and, at the critical coupling wc, it becomes a zero-energy state. At this point, the ground state of the superconductor becomes unstable. The new ground state of the superconductor becomes a paired condensate for all but the impurity bound state with a finite single-particle am-

115 12 650

M. I. SALKOLA, A. V. BALATSKY, AND J. R. SCHRIEFFER

plitude at the impurity-bound state. Thus, it has the form \®o)w>w ~ | * - i | ) - This unpaired ground state is a result of the competition between the pairing-condensation energy and the magnetic interaction. For a strong magnetic interaction, the gain in the magnetic energy, due to the induced spin, dominates the condensation energy. The (variational) forms (1) and (2) allow a varying number of doubly excited quasiparticles of the pure superconductor created by the operators yj| yiyj from |4>0). These lead to a local suppression of the gap function A(r) and, in particular, for w>wc, to a negative value of A(r) at the impurity site (see below and Ref. 14). The spin density is spread over a volume of linear dimension of the order of the coherence length. The plan of the paper is as follows. In Sec. n, the general formalism is introduced. Sections III and IV explore the consequences of a magnetic moment in s- and d-wave superconductors. Both uniform and nonuniform order parameters are considered within the T matrix and self-consistent mean-field approximations. In Sec. V, the generalization to many impurities is outlined. Section VI contains a discussion of the number of electronic degrees of freedom associated with the impurity-induced quasiparticle states. In Appendix A, the interplay between impurity scattering and pairing symmetry is further elaborated from the quasiparticle perspective. In Appendix B, the T matrix is generalized to finite impurity concentrations. II. FORMALISM Our starting point in describing a localized magnetic moment in a superconductor is the lattice formulation of electrons hopping between nearest-neighbor sites and interacting via an effective two-particle interaction. The pairing interaction is assumed to be weak or at most moderate so that the mean-field approximation gives a qualitatively reliable description of the superconducting ground state and the lowenergy excitations. Specifically, consider a two-dimensional lattice model where a localized magnetic moment is created by a classical spin S at r = 0 . The model is defined by the effective Hamiltonian H=H0+Himp, where H0 describes a BCS superconductor18 and H^ is the contribution due to the magnetic moment. In the mean-field approximation,

^t(R+r)f3*(R)-At2

H0=-jW^ 4



2

A(R,r)^ t (R+r)riN['(R),

VHR)h*(«)

R

(3)

Rr

where (Rr) denotes nearest-neighbor sites separated by r, W is the half bandwidth (on a square lattice), fi is the chemical potential, and A(R,r) is the superconducting gap function. The operator , P(r) = [^ T (r)t/'|(r)] 7 ' is a twocomponent Gor'kov-Nambu spinor, ra (a= 1,2,3) are the Pauli matrices for particle-hole degrees of freedom, and r 0 is the unit matrix. Given that the pairing of electrons occurs in the spin-singlet state, the superconducting order parameter (amplitude) can be expressed in the form

55

l ? (R.r) = r - 2 ( i f 2 ) J W R + r ) i ( R ) > .

(4)

i. (TV

The relation between the order parameter and the gap function is given by the equation A(R,r) = -i>(r)F(R,r).

(5)

Here, i>(r) is the strength of the two-particle interaction, which is assumed to be instantaneous in time. Thus, the energy cutoff in the gap equation is set by the bandwidth. In a translationally invariant system, the BCS Hamiltonian reduces to Hn

k

•Akri)*k-

(6)

where A k = 2 r A ( R , r ) e " , k r and ^k=(ifrk^if'f-kl)T. The fermion operators in real and momentum space are related by the unitary transformation (^0.(r)=Af~1/22k^ko.e'k r , where N is the number of sites in the system. For a square lattice with the nearest-neighbor hopping, the single-particle energy relative to the chemical potential in the normal state is ek=-iWicosk^+co&kya)—^; a is the lattice spacing. At half filling, fi = 0, the Hamiltonian is symmetric under the particle-hole transformation generated by the operator TX: ^ k - ^ j ^ T i ^ Q - , , , where Q=(ir/a,7r/a). The nature of the superconducting condensate can be established by studying the order parameter F which incorporates the pairing correlations and gives an estimate for the size of the Cooper pairs.18 (i) An on-site attraction favors .s-wave pairing characterized by F(R,r=0)=£0 and A k = A0. As an example, consider a uniform superconductor close to half filling (/i~0) so that the Fermi surface is square. Let n be the normal to the two sheets of the Fermi surface which are most nearly normal to r and r x = n r > 0 . Thus, r x S 5 l l l ' where r|| = (e 3 Xn)-r is defined. For r±>a, F(R,r) =

«\i

Ko(rj_'t,±)cos(.kF±r1)6r,0,

(7)

where K0 is the modified Bessel function of the zeroth order, £sl = a(W/A0\l2) is the coherence length, and kF±~v/ayl2. In the direction perpendicular to the Fermi surface, the extent of the pairing correlations is determined by the coherence length £,± , whereas, in the parallel direction, their range is vanishingly short, f,||~0. (ii) For a strong repulsive on-site interaction and a nearest-neighbor attraction, pairing with d-wave symmetry may result:19 F(R,r=0) = 0 and Ak=2Ad(cositxa-cosfcva). Close to half filling and for r±>a, one can again estimate20 (ri/fl) n*,r)=£-[i 4ir\fJ[(r|/B)2+(r±/fx)^F8in(*^^)'

(8)

where fx = a(W/4Arf) is the characteristic coherence length perpendicular to the Fermi surface, A 0 =4A rf is the maximum energy gap at the Fermi surface, and T}=sgn(rxry). This result is most useful when there is a clear separation between the length scales a and f± . For a given r± , its maximum value and the position scale as a£Llr\ and

116 55

SPECTRAL PROPERTIES OF QUASIPARTICLE ...

r\\=a(r±/ij± >/2). The existence of nodes in the energy gap has resulted in the power-law decay of the pairing correlations in all directions at large distances instead of the exponential one as found in the j-wave case. F has dx2-y2 symmetry as expected. The interaction between the conduction electrons in the superconductor and the impurity spin is given by the Hamiltonian Hiap=JS-a(0) + Vn(0),

= ^S^V*k.,

(9)

12 651

A. Uniform order parameter First, consider a superconductor at half filling with a spatially uniform j-wave gap function A(r) = A 0 . As shown by Yu,1 Shiba,2 and Rusinov,3 the magnetic impurity gives rise to a bound state inside the superconducting energy gap. Their solution suggests that there exists a singular point w = wc(U) that separates two distinct ground states of the system. The nature of the bound state can be found by computing the spectral density per site 1 Aa(r,w) = --ImG^{r,r;
(12)

7T

For |o)||| is not too close to A 0 . Note that, for that the difference between these two approaches is imporA0<W, N{0) equals to the normal-state density of states tant only close to the impurity site. The assumption that the averaged over the energy interval ± A0 around ya=0. Thus, spatial variation of the superconducting order parameter can it is convenient to define the density of states per site in die be neglected leads to a considerable simplification of the normal state at the Fermi energy as NF=N(0). We can improblem, because the impurity interaction (9) is limited to mediately solve the above equations for the bound-state enone site. Scattering of quasiparticles from the impurity mospectral densities, (i) For w<wc{U) ment is described by a T matrix, T(T,r';a)) = T(a>)8r<)ST<0, ergy and 2 the = \I(TTNF)~ +U2, the bound-state energy n 0 = fl< , where which is local and whose Fourier transform therefore is independent of wave vectors. In the Gor'kov-Nambu presentation, the single-particle Green's function is G(r,r';r-f') O < = ? A 0 ^ = , +C" , (13) = -;'(r , !'(r,f) , I' t (r',f')}. In the presence of an impurity, °V(c2+ + l)(c2_ + l ) c±=cw+cu, and ^=sgn(c+c_). Here, the dimensionless G(r,r';to) = G <0) (r-r',ft)) + G ( 0 ) (r,w)f(w)G ( 0 ) (-r',w), parameters are CW = (WITTNF)I(W2 — U2) and c„ 2 2 (10) = (U/TTNF)/(W —U ). A straightforward calculation shows that Z ( i +) (0) = Z+ SZ, Z(T_)(0) = Z - SZ, and where both G <0) (r,w)=N~ 1 2 k G <0) (k,a>)e' k ' r and f((0) = 0, with Z=(NFA0)2-rr\cJ[(cl + c2u) matrices in two-dimensional Gor'kov-Nambu space. The + (c2-c2)2]/[(c2+l)(c2+l)]3/2 and SZ= single-particle Green's function of a clean system is (WFA0)sgn(*y)4Tr|cJc2/[(c + +l)(c 2 _ + l)] 3 / 2 . Hence, the (0) + _1 G (k,w) = (ft)f o -e k f3 + A k f,+/O ) . The Tmatrix can bound state is a local quasiparticle state with spin down;22 be obtained from the Lippmann-Schwinger equation in the i.e., the spin projection of the particle component is antiparform of allel and that of the hole component is parallel to the impurity moment w. Basic features of the low-energy spectral f(o)) = [V' 1 -G < 0 ) (r=0,a>)]- 1 . (11) density are illustrated in Fig. 1. As w increases, the minimum energy to make a quasiparticle excitation from the condensate decreases, until, for w>wc(U), it would be negative. At These equations allow a complete solution of the problem as this point, the superconducting condensate becomes thermolong as order-parameter relaxation can be neglected. dynamically unstable against spontaneous creation of a local quasiparticle excitation with spin down. As a result, the suIII. MAGNETIC MOMENT IN AN j-WAVE perconductor goes through a quantum transition from the SUPERCONDUCTOR spin-unpolarized to the spin-polarized state. Note that the average number of electrons remains the same. While, for Conventional pairing of s-wave symmetry is described by £/=0, (n(r)) is a uniform function, (^,.(r)) becomes more the gap function A(r)sA(r,0). It may be modeled by a peaked at r = 0 as w increases. (For U=£0, also (n(r)) would system with an attractive on-site interaction v(r) = v(0)Sr0, be nonuniform.) However, the true nature of the transition is u(0)<0.

117 12 652

M. I. SALKOLA, A. V. BALATSKY, AND J. R. SCHRIEFFER

55

1.0

(a) -

O.B 0.6 0.4 0.2

t-

0

o

o

-^m 0

-

0

0

''• -^»

o

''• ""•__

•^9

0.0 1

4.0 3.0

o.

8.0 1.0 0.0

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(b):

-

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~W

- * • " « ' " " " "

"a-

zr 0--TJ--CT—

z? - - ^ - .

^^-^•; 2\ -)

1

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(o)

II

(d) o II

— ' » * <• * *

FIG. 3. (a) The bound-state energy Cl0, (b) the spectral weight Z'/'CO) (in units of NFA0), (c) the spin polarization (i z (r=0)) ((w(r=0)) = 1), and (d) the gap function A(r=0) at the impurity site r=0 as a function of w in the s-wave superconductor for f/=0. Lines denote the T matrix results for the uniform order parameter and symbols the self-consistent mean-field results on a square lattice at halffilling.The quantities of the impurity-induced intragap quasiparticle state belonging to the branch w<wc are denoted by solid lines and solid symbols, whereas those ones belonging to the branch w>wc are marked by dashed lines and open symbols. Note thatZ<(r_)(0)=Z<_+(r)(0) because of particle-hole symmetry. revealed by considering the total spin polarization (sz) = I,r{sz(r)). At zero temperature we obtain (sz) = 0, for w<wc(U) and (sz)=~\, for w>wc(U). Because the ground state has qualitatively changed, so have the elementary excitations and, in particular, the nature of the impurityinduced bound state, (ii) For w>wc(U), the energy of the bound state is given by flo = ^ > > with H> = —il< . Moreover, it is now associated with a quasiparticle with spin par/ allel to the impurity moment: Z•^• ( + ) >(0) = Z-SZ, Z p ( 0 ) = Z+SZ, and z\+\0) = Z\~ ) (0) = 0. The bound state in both cases accounts for one electronic degree of freedom. Figure 3 summarizes the above results for U=0 and Fig. 4 describes the effect of nonzero U. The Coulomb interaction effectively increases the critical coupling wc(U). Thus, for a large enough U, the unscreened state is thermodynamically stable. It is also clear that the on-site Coulomb interaction breaks up the particle-hole symmetry. For example, given that the impurity moment can be described classically and the envelop of the experimentally observed intragap spectral weight of the electronlike state Z (+) (r) is larger than that of the holelike state Z ( - , (r), then either (a) £/>0 and w<wc(U) or (b) £/<0 and w>w c (i/). Interestingly, one is tempted to interpret the recent

FIG. 4. (a) The bound-state energy Cl0 and (b) the spectral weight Z'^O) (in units of NFA0) of the impurity-induced intragap quasiparticle state as a function of U in the s -wave superconductor for 7TNFW = 1.4. Lines denote the T matrix results for the uniform order parameter and symbols the self-consistent mean-field results on a square lattice at half filling. The quantities belonging to the branch w<wc(U) are denoted by the solid lines and solid symbols, whereas those ones belonging to the branch w>wc(U) are marked by dashed lines and open symbols. Note that Z^'CO^Z'J^CO), because particle-hole symmetry is broken by the on-site Coulomb interaction U. We emphasize that while the spectral weights of the intragap peaks are asymmetric their energies relative to the chemical potential are not. experiment11 on magnetic impurities as if the intragap spectral density decays on a length scale determined not by the coherence length but by a much shorter scale of the order of the Fermi length fF. This result appears paradoxical because a simple scaling estimate suggests that if the extent of the bound state were 0{/F), the energy of this state should be hvF//F~€F in a one-band picture and therefore could not be localized within the energy gap. One possible explanation of the experimental result is the power-law prefactor of the exponentially decaying bound state. For example, consider a uniform gap function A(r) = A0 and an isotropic en7
118 55

SPECTRAL PROPERTIES OF QUASIPARTICLE .

12 653

3.0n

*/,

V«

'«

'o

FIG. 5. The spectral weight Z\+\r) and Z\~\T) (in units of NFA0) of the impurity-induced intragap quasiparticle state in the j-wave superconductorfor(a) fi = 0, irNFU=0Awd (b) ji = -W/2, U=0. These results are computed self-consistently on a square lattice with the lattice spacing a, irNFw = 0.1, and A0/W=0.05. The bound-state energies are (a) il0=0.39A0 and (b) fl0=0A6A0. local attractive potential for quasiparticles which produces two bound states in the superconducting energy gap at the energy

n 0 =A 0 Vi _ « 2 .

(14)

where a=c&/fl. In addition, we have defined P2=\ (l-c2)2+cl (/3>0), c2 = cl-c2u-cl, CW = (WITTNF)I (w2-U2-SA2),

CU = (U/TTNF)/(W2-U2-SA2),

and c 4

= (SAITTNF)I{W2U2- SA2). In contrast to the bound state induced by the magnetic moment, this state is twofold degenerate by virtue of time-reversal symmetry. Their spinquantum numbers are ± \. Note that the bound states become essentially indistinguishable from the quasiparticle continuum (i.e., a2\. Positive w breaks the degeneracy by lowering the energy of the spin-down state and increasing the energy of the spin-up state:

n n /A n

l-C 2/8

•vr

(15)

For w>SA, the spectral density measured at the impurity site r = 0 is much larger for the former state at the energy fflSAj than for the latter one whose spectral density is either

transformed into the quasiparticle continuum or to the antibound state above the energy W. Next, compare the non-self-consistent T matrix predictions with the results obtained when the gap function is allowed to relax in the neighborhood of the impurity moment. Our numerical approach is based on the model defined by Eqs. (3) and (9) on a square lattice with the effective on-site electron-electron attraction of magnitude y(0) such that A 0 /W=0.05. The model, together with the mean-field equation A(r)= -y(0)F(r,0) is solved self-consistently (e.g., in the Bogoliubov-de Gennes scheme).23 As expected, the overall agreement between it and the T matrix calculation is good; see Figs. 3 and 4. However, the additional variational degree of freedom shifts wc to a lower value, and fl0 never becomes exactly zero. For w>wc, A(r=0) is overscreened to a negative value and, ultimately, A(r=0)—>0, as w—»». The nonzero total spin polarization residing in the neighborhood of the impurity has lead to a nearly complete pairbreaking effect at this site. The charge density remains uniform everywhere in the system for arbitrary w and U=Q. For w~wc(U), the results that assume a uniform gap function everywhere deviate most clearly from those ones allowing the gap function relax locally, because SA/W and SA/w have their largest values for the same values of w. At fi=0, the Fermi surface is a square producing a very anisotropic Fermi velocity. This anisotropy causes the

119 M. I. SALKOLA, A. V. BALATSKY, AND J. R. SCHRIEFFER

12 654

55

3

5

(a)

(b)

FIG. 6. The self-consistent j-wave gap function A(r) around a magnetic moment for (a) /u. = 0, TTNFU=0A, and (b) (t,= — W/2, (7=0, obtained on a square lattice with die lattice spacing a. The other parameters are TTNFW = 0.7 and Ao/W=0.05. impurity-induced bound state to have a shape which extends along the diagonal directions; see Fig. 5(a). A similar spatial form has also been observed in the context of relaxed quasiparticle excitations.25 Notably, the decay of the spectral density due to the bound state is governed by £n=hvJAa, where n is a normal to the Fermi surface at a given point and 1 vn=h~ (nVkek)k=k is die Fermi velocity at the same point. For a flat Fermi surface Z'-aL)(r)<xSr0e~2ri /x ^, where rj and r± are the components of r parallel and perpendicular to the sections of the Fermi surface for which r±3» |r|||. Here, \ x = f,x Vl -(O0/A0)2. This is in contrast to the isotropic r -i g -2r/X|) behavior found in two dimensions for a circular Fermi surface. The anisotropy in the Fermi velocity yields a similar structure in the gap function around the magnetic impurity.26 The slower relaxation rate along the diagonal directions is clearly observable in Fig. 6(a). Moving away from half filling, the anisotropy in the Fermi velocity decreases and the spatial form of the impurity-induced bound state and the gap function in the neighborhood of the impurity become increasingly isotropic, as shown by Figs. 5(b) and 6(b). Moreover, for fj.^0, the envelopes of the spectral weights ZJ +) (r) and z\~\r) differ, because the single-particle band structure of the normal state is no longer particle-hole symmetric. The sensitivity to the energy spectrum away from the vicinity of the chemical potential is a consequence of strong impurity scattering whose spatial extent is less than die lattice spacing and which efficiently mixes quasiparticle states with a wide wave-vector range around kF. A similar conclusion is reached in Ref. 14 where the effect of the band structure is discussed. The deformation of the order parameter in the vicinity of the magnetic moment creates an attractive potential with a finite range. This feature may lead to additional bound states which are no longer invariant under all symmetry operations of die square lattice (i.e., to higher "angular momentum" states) but whose binding energies are very small. Because these states have a node at the impurity site, tiiey do not couple to the impurity moment and they are at least twofold degenerate due to the spin degeneracy. In general, we find a twofold degenerate state slightly below A0 that transforms like quadrupole (x2—y2). However, away from half filling

and for w<wc, there is some evidence for the lowest energy, nonsymmetric quasiparticle state that is fourfold degenerate and transforms like dipole. IV. MAGNETIC MOMENT IN A d-WAVE SUPERCONDUCTOR Invariance of various perturbations under particle-hole transformation was significant in classifying the properties of the impurity-induced quasiparticle states in s-wave superconductors. In a similar fashion, we may describe the same properties in rf-wave superconductors under particle-hole transformation. In rf-wave superconductors, it is useful to introduce still another transformation, namely ' 'charge conjugation:" tyk—>Vk=T2yQ~k- It identifies the cases when the spin-unpolarized ground state of the superconductor interacting with a magnetic moment is unstable against spontaneous creation of a local spin polarization. As defined here, an .s-wave superconductor is not invariant under charge conjugation. A. Invariance under charge conjugation First, we consider a quasi-two-dimensional system whose energy spectrum is particle-hole symmetric in the normal state and whose superconducting energy gap has dxz-yi symmetry at the Fermi energy: Ak=A0cos2 (r=0,a>)f2=G (0, (r=0,w), it is immediately clear that G(0)(r=0,) = G0(
1 -Im

IT

G0(
(16)

where v± = w±U, |«^A0, one obtains the formula5'6 G0(a>) =-Aff(2
120 55

SPECTRAL PROPERTIES OF QUASIPARTICLE . . .

magnetic moment induces two quasiparticle states at energies fi± = ir/2c±A0/log(8/7r|c±[) with inverse lifetimes r ± = ir/2|fl ± |/log(8/ir|c ± |). For £/=0, the two states are degenerate: 0 + = fl_. Their spin quantum numbers are sz = -\. With increasing U, the degeneracy is lifted as fl + decreases and ft_ increases. For IKw, they appear as resonances in the spectral densities A t (r=0,o)) and Ai(r=0,&)) at u>= — fi+ (holelike) and w=il^ (electronlike), respectively. For U~w, there is no solution for fland no resonance structure develops in A|(r=0,o>). For U>w, fl_ is negative and approaches zero from below. Thus, for each spin orientation, the spectral density A°° with U fixed. In the neighborhood of the impurity, the virtual-bound state acquires the characteristic form of the rf-wave energy gap with tails extending along the diagonal directions. The spatial form of this state can be obtained by computing the spectral density Ao.(r,G>)=A0(a>) + (5A„.(r,co), where A0(&))= -Tr^1ImG0(o)) is the uniform spectral density of a clean superconductor and <5A„.(r,&)) is the impurity-induced contribution. For |GJ|\/A0. The impurity-induced contribution is M a (r,w) = --Im[G(+0)(r,a>)r<+
+ G)G{0\-T,w)l

(17)

where r(±")((o)= - 1/[G±(OJ)-(TVZI]. We use the notation in which G(a0)(r,w) = |Trr a G (0) (r,co) and G0(w) = GjJ0)(r=0,w) ( a = 0 , 3). In addition, we have defined G(°\r,w) = G(o\r,w)±Gf\r,(o) and G±(io) = G) = G 0 (w). For (7 = 0, Tif(a)) = Tif\co), and the spectral density is particle-hole symmetric: <X4j(r, -aj) = <SAj(r,G>). While the overall spatial dependence of the virtual-bound states generated by purely magnetic (U=0) and nonmagnetic (w = 0) scattering is the same, the resonance factors T^ lead to distinct states. The specific form of the spectral density is determined by the Green's function G(0)(r,a>). Equation (17) produces a qualitatively correct description of the quasiparticle states as long as the energy gap is not comparable to the bandwidth, because then the relaxation of the gap function in the vicinity of the defect can be neglected as a small correction. Nonetheless, the local suppression of the gap function A(r=0) could easily be included. For &>= + ft± , the spectral density is most strongly enhanced in the horizontal and vertical di-

12 655

rections (the extrema directions of the d-wave energy gap) at distances r<£0 ( = hvF/A0) from the impurity. Further away from the impurity ( r ~ f 0 ) , the situation is reversed: the spectral density is enhanced in the neighborhood of the diagonal directions at the resonances. That the Green's function Gm(r=0,a>) is proportional to r 0 has curious consequences which set rf-wave pairing apart from s -wave pairing. In the j-wave superconductor, lowenergy quasiparticles have both particle and hole character due to the nonzero pairing field F(R,r=0). In contrast, rf-wave symmetry forces the pairing field to vanish locally, F(R,r=0) = 0: the particle and hole degrees of freedom are effectively decoupled at the impurity site. Strong magnetic scattering leads to two virtual bound states whose components at the impurity site are either electronlike with spin down or holelike with spin up; the other components vanish because their orbital character is the same as that of the order parameter (i.e., they have a node at the impurity site). A nonzero U lifts the degeneracy without mixing the states. Note that, for an impurity whose nonmagnetic scattering strength (U>0) is larger than the magnetic one, two virtualbound states of hole character with quantum numbers sz= ± \ are obtained at the impurity site.6 In conclusion, a rf-wave superconductor appears locally as a simple metal with a pseudogap: at short distances, the particle and hole excitations are decoupled and the energies of the virtual-bound states do not cross the chemical potential due to the level repulsion. Another treatment in terms of elementary quasiparticle excitations is outlined in Appendix A. B. Broken charge-conjugation symmetry First, assume a mixed 5- and =fl crosses the chemical potential for cw(), and the quantum transition is shifted to higher values of the magnetic moment until, for As = 0, the transition is completely suppressed. Second, if the particle-hole symmetry is broken in the normal state, the Green's function acquires a nonzero r3 component. For a large enough magnetic moment, it will again induce a transition where the spin-quantum number is changed from zero to 3. In general, both of the above features may be present simultaneously. In such a case, one must replace F± by

121 M. I. SALKOLA, A. V. BALATSKY, AND J. R. SCHRIEFFER

12 656

55

j o-p^^§SSP? 0.9-

J D.e-J 2s0.6"

*^_~ 3?^-10

% ^"^TT" 0 ^ 10

y/a

(a)

(b)

FIG. 7. The self-consistent d-wave gap function (Ref. 28) A(r) around a magnetic moment (irNFw = 10 and C/=0) for (a) /t=0 and (b) fi—--W/2 obtained on a square lattice with the lattice spacing a and the coherence length f± = 10a (_4kdIW=0.\). The minimum at r=0 is not shown. Ad is the spatially uniformrf-wavegap function in the clean system. Note that the maximum energy gap at the Fermi surface is (a) A0=4Ad and (b) A0 = 2Arf. cw±ylgi + (cu+g3)2, where g a =(77^ F )- , G ( a 0 ) (r=0,n) (a = 1,3) and w>|l/|. Similarly, c^ is replaced by cw for which e"_=0. As a consequence, the critical coupling becomes a function of the chemical potential and the s-wave component: c* = <:_(/*,, A,).

(f-wave pairing, explaining the difference between Figs. 5(a) and 8(a). The spatial variation of the virtual-bound states is accurately given by Eq. (17) even though it assumes a uniform gap function as one can verify in this case. For rL>a and £/=0, one can estimate SA„(r,a~0)«sm2(kF±ri)/[(ril/a)2+

C. Nonuniform order parameter The Fermi-velocity distribution affects the precise form of the rf-wave gap function27 in the neighborhood of the magnetic moment; see Fig. 7. As in the s-wave superconductor, the sensitivity to the Fermi-surface geometry originates from the mode composition of the gap function: the electronic degrees of freedom within the energy proportional to A0 about the chemical potential have the largest weight in the gap function. Close to half filling, the gap function has four troughs extending along the diagonal directions, whereas, away from half filling, the fourfold symmetry of the energy gap is more crucial in determining the spatial relaxation of the gap function. That the quasiparticle states are concentrated along the horizontal and vertical directions at energies for which the density of states is the highest O ~ A 0 ) explains the latter finding. In the strong-coupling limit, some modifications are expected.29 A strong magnetic moment creates virtual-bound states whose spatial dependence has distinctive features due to (i-wave pairing symmetry. For example, in the neighborhood of the defect, both the particle and hole components of the spectral density remain nonzero along the diagonals, although the maximum value at a given distance is located off diagonally; see Fig. 8. Close to half filling, the Fermi-surface geometry has a strong focusing effect on the spatial variation of the virtual-bound states, and their lobes along the vertical directions spread only slowly. At half filling (/A = 0 ) , the lattice Green's functions G(,0)(r,&>) and Gf\r,a>) vanish identically at the sites for which (x+y)la is an even integer. Because, for |a>|) does not vanish at the same lattice sites for J- and

(r ± / f x ) 2 ] , (18)

r

r

where ±** \\ (by definition). At half filling, kF±r± = ir(x + y)l2a. Away from half filling (/t=-W72), the spectral density tends to vary radially demonstrating more clearly the effect of the k-space structure of the superconducting energy gap. The example shown in Fig. 8(b) is computed in the spin-polarized ground state which is obtained when the magnetic moment is large enough and the superconductor is not half filled. Neither does the spectral density obey particle-hole symmetry in this case. In general, quasiparticle excitations are qualitatively described by Eq. (17) which provides a straightforward method for determining the spectral density and the resonance energies. For example, the resonance energies are obtained from the equation G±(il) = l/v±. The twofold degeneracy of the virtual bound state is lifted both by the local Coulomb interaction and by the band structure which is asymmetric relative to the chemical potential. Because the lattice Green's functions are fast to compute numerically, we have been able study in detail the evolution of the spectral density as a function of the chemical potential and the impurity potential (not shown). In spite of the possibility of the quantum transition, A(r=0) does not change sign for any value of magnetic moment w. While, for c t s c K < l , the magnetic moment is strongly screened by the electron spin density at the impurity site (sz(r=Q))~~5, there is a compensating spin-density cloud of opposite sign in the neighborhood of the magnetic moment that leads to the vanishing spin polarization. Figure 9 shows the spatial variation of the electron spin density with a given magnetic moment both at and away from half filling, illustrating the two cases c w >c^(/i) (/u=0) and cw

122 SPECTRAL PROPERTIES OF QUASIPARTICLE .

55

*/a

«

*/ft

»?

12 657

1.2 "I

FIG. 8. The spectral density &4 ff (r,±fl 0 ) (in units of NF) for (a) /J. = 0 and (b) fi=-W/2 in a two-dimensional d-wave superconductor as a function of position r=(x,y) around a classical magnetic moment (irNFw = W and t/=0) located at r = 0 ; a is the lattice spacing. These results are computed self-consistently with f± = 10a. At half filling, the spectral density obeys particle-hole symmetry: &4T(r,fl0) = 5A i (r,-ft 0 )- The energies of the shown virtual-bound states are (a) ft0~0.05A0 and (b) (l0—0.5A0. < c*(/U') (jx, =£0). The magnetic moment perturbs electronic degrees of freedom on the energy scale of w which must be large compared to the superconducting energy gap A 0 to have pronounced virtual-bound states. At half filling, the behavior of the spin density is dominated by the anisotropy in the Fermi velocity, disguising the effect of pairing symmetry. Away from half filling, the virtual-bound states contrib-

ute at distances r> f0, while at shorter distances the nonzero spin density is mostly coming from quasiparticle states residing outside the energy-gap region. These considerations offer intriguing prospects regarding high-temperature superconductors. In addition to possible d-wave features, their low carrier densities exacerbate the role of the highly anisotropic Fermi-surface geometry that is

FIG. 9. The electron spin density {sz(r)) in a two-dimensional d-wave superconductor as a function of position r=(x,;y) around a classical magnetic moment (iriVFw= 10 and C/=0) located at r = 0 ; a is the lattice spacing. The chemical potential is (a) jt = 0 and (b) /*= — W/2. The peak (j z (r=0)) j in both cases is cut off in order to illustrate the finer details. These results are computed selfconsistently with £± = 10a. Note that (a) {sz) = 0 and (b) ( J 2 ) = - | .

123 12 658

M. I. SALKOLA, A. V. BALATSKY, AND J. R. SCHRIEFFER

inevitably mixed in the spatial dependence of the gap function and the quasiparticle states in the vicinity of impurities. These properties should be accessible to many experimental probes, such as scanning-tunneling microscopy, etc. In particular, it might now be possible to determine experimentally the coherence length in short-coherence-length superconductors by directly measuring the spectral density in the vicinity of defects. In low-temperature superconductors, this is not possible because the power-law prefactors kill the signal before the exponential cutoff set by the coherence length can be observed. Even though the coherence length in these materials is typically short (£ 0 ~20 A), it still is long enough that our mean-field results should be relevant in describing their properties qualitatively.30 V. MANY MAGNETIC IMPURITIES As an example, consider strongly scattering magnetic impurities in an s -wave superconductor. In the dilute limit of impurities, only nearest-neighbor impurity states interact due to their exponentially decreasing overlaps. Appendix B describes a mapping of a system of dilute impurities onto an effective model. Using known results for such a model, we conclude that all quasiparticle states in the impurity band in two dimensions are localized. In three dimensions, there is a critical density of impurities, below which all states are localized and above which a mobility edge appears. A similar problem of the impurity-induced quasiparticle states in a two-dimensional d-wave superconductor has also been considered.31 Although the assumption that the impurities do not interact leads to an equal density of states at every impurity site, the fact that the impurity-induced quasiparticle states may have large overlaps modifies this conclusion. Disorder with strongly interacting impurities yields large impurity- and energy-dependent variations in the quasiparticle wave functions. These variations may, in the worst case, obscure the detection of bound states — for example, by scanning tunneling microscopy — or facilitate the detection if the probe is close to impurity sites that have very large relative weights in the wave function. VI. FINAL REMARKS There is a qualitative difference between s- and li-wave superconductors regarding the quasiparticle states induced by magnetic moments. This difference arises from the local properties of the superconducting order parameter F(R,r) which is nonzero for j-wave pairing but vanishes for rf-wave pairing at r = 0 . Thus, in the s-wave superconductor, any local probe sees particle and hole degrees of freedom that are coupled by the pairing field. This feature suggests that the two intragap peaks in the spectral density should be interpreted as reflections of the same quasiparticle state. In contrast, because the pairing field in the d-wave superconductor vanishes at short distances, a local perturbation leads to two separate quasiparticle states which are either purely particle- or purely holelike at the impurity site. One can distinguish the two cases by their response to a local Coulomb interaction Un(0). The properties of the quasiparticle excitations are affected by U because their charges are not the

55

same. Using the ground-state charge as a reference point, define the charge operator Q as 2 = 2 [«(r)-<*ol«(r)|*o>]-

(19)

r

Clearly, < * _ u | g | * _ 1 J ) = «J-i;f. If the particle and hole features belong to two distinct states, the Coulomb interaction U will shift their energies differently because their charges differ. In contrast, if they are manifestations of the same state, the Coulomb interaction could not affect them independently, and the spectral density will have two intragap peaks that are symmetrically located below and above the chemical potential. Indeed, we always find that, in an s -wave superconductor, the spectral density contains two peaks at energies ± f i 0 regardless of the local Coulomb interaction whereas, in a d-wave superconductor, the Coulomb interaction lifts the degeneracy of the impurity-induced quasiparticle states and splits the corresponding peaks in the spectral density so that there are four peaks symmetrically located relative to the chemical potential. We conclude by noting that the special nature of the induced quasiparticle states has interesting implications for optical absorption in an s-wave superconductor containing a local moment. Because the bound state inside the energy gap exists only for a given spin direction (its time-reversed conjugate overlaps with the quasiparticle continuum), no sharp absorption feature is found at 2ft0 but the optical absorption begins at the energy A0 + ft0. ACKNOWLEDGMENTS We would like to thank J. Byers and M. Flatte for interesting discussions and, in particular, for helpful comments on the Fridel oscillations in the normal state, A. Yazdani for discussions on experimental issues, C. Kallin for introducing M.I.S. to the Bogoliubov-de Gennes technique, and A.J. Berlinsky for useful comments on the manuscript. This work was supported in part by Natural Sciences and Engineering Research Council of Canada, the Ontario Center for Materials Research (M.I.S.), by the U.S. Department of Energy (A.V.B.), by the NSF Grant No. DMR-9629987 (J.R.S.), and by the Many Body Theory Program at Los Alamos. APPENDIX A: IMPURITY SCATTERING IN BOGOLIUBOV-VALATIN FORMULATION The reasons for different behavior of various types of impurities in superconductors, as well as the effect of pairing symmetry, can be illuminated by rewriting the problem in terms of the elementary quasiparticle excitations of the clean superconductor, described by the Hamiltonian, Eq. (6). This Hamiltonian is diagonalized by the Bogoliubov-Valatin transformation ^k—>rk=5k^'k, where r k = ( 7 k T y t k | ) r and /cos0 k

-sin0 k \

In this parametrization, « k =cos6 k and i/k=sinfl,. Choosing sin2^=A k /£ k gives

124 55

SPECTRAL PROPERTIES OF QUASIPARTICLE .

H{,-2J

EkTkr3rk,

(A2)

where £ k = \Jek+Ak. In terms of these excitations, the Hamiltonian for the ^-function impurity is Himf=1aH(a), with (A3) iv

kk'

The three types of scattering interactions (a = w,u,A) magnetic,

are (i)

v £ ! = w[cos(0k-0k.)fo-sin(0k-0k-)if2];

(A4)

12 659

der the transformation in which ¥ k — » ¥ £ = T ^ ^ , where R denotes a rotation through an angle ir/2 about the impurity site. When the system is half filled and U=0, the symmetry group of H includes the charge-conjugation transformation. Because such a symmetry group is non-Abelian, it also has higher than one-dimensional irreducible representations. In particular, the impurity-induced virtual bound states belong to a two-dimensional irreducible representation; i.e., they are twofold degenerate. The degeneracy is lifted by a nonzero U, by an asymmetric band structure, or by a gap function which has a nonzero s-wave component. For w = 0 , the quasiparticle states are at least twofold degenerate because of time-reversal symmetry. APPENDIX B: T MATRIX FORMALISM APPLIED TO FORMATION OF IMPURITY BAND

(ii) potential, Vkk', = l/[cos( 6k+ dk,)r3 + sin( 0k+ 0k,) T]]; (A5) and (iii) local order-parameter suppression, V k £ i = - M [ s i n ( 0 k + 0 k O T 3 - c o s ( 0 k + < V ) f i ] , (A6) where <5A=A0 —A(r=0)5=0. These expressions form the basis for the following discussion of when to expect a bound state in the superconducting energy gap.

In this appendix, a mapping to an effective theory is presented, from which the consequences of a finite density of impurities can be explored. To formulate the problem of many magnetic impurities, we utilize the four-dimensional Gor'kov-Nambu representation, because the magnetic moments are allowed to be randomly oriented. In this representation, the effective Hamiltonian is H=H0+Himp, where

#0 = 2 *I(ek'3+ A k T2^2)*k.

1. s -wave superconductor First note that, for k~kF, 0 k ~(ir/4)sgn(A k ). For an j-wave superconductor at energies close to the chemical potential, the magnetic interaction term is diagonal: VM~WT0. Thus, the excitation energy is locally decreased for spin-down excitations and increased for spin-up excitations (recall that the effective w changes sign for opposite spins of quasiparticles). As a consequence, a bound state appears only for the spin-down quantum number. 32 For large enough w, the bound-state energy crosses the chemical potential. The potential scattering term is purely off diagonal: Vkk,~ UT\\ it causes level repulsion regardless of the sign of U. Therefore, potential scattering does not introduce intragap states. Local suppression of the order parameter leads to an attractive diagonal potential for both types of quasiparticles: V k k !~-<5Ar 3 . This produces two degenerate bound states in the energy gap, as expected from time-reversal symmetry.

and Vk= (0 k j
¥+(r)V(r)y(r),

(B2)

where

t/(r) = 2 vnSn

(B3)

Here, i>„ = w„-e+£/„T 3 is the interaction strength of the nth impurity moment located at r„ . Following the usual convention, we have denned e=<7 2 e 2 +T 3 (cr 1 e 1 + cr 3 e 3 ). In the absence of the impurities, the single-particle Green's function is

2. rf-wave superconductor In contrast to the j-wave case, the alternating sign of the d-wave order parameter yields quasiparticle scattering terms that are predominately neither diagonal nor off diagonal. While a large enough potential will lead to pronounced resonance states, they do not cross the chemical potential when die normal-state quasiparticle spectrum has particle-hole symmetry and the gap function average over the Fermi surface vanishes. Finally, the appearance of degeneracies in the spectrum of virtual bound states can be understood in terms of the symmetry properties of the effective Hamiltonian, H=H0 +Himp In general, a d-wave superconductor is invariant un-

(Bl)

(0)

e'kT

1

6 (r.«)=i2

.

.,

-.

(B4)

+

N k t o - e k T - 3 - A k T 2 o ' 2 + (0 Generally, the total Green's function in die presence of impurities is given by G(x,x') = G(0Xx-x')+ [ dydy'G(0Hx-y)t(y,y') X&0)(y'-x'),

(B5)

where t is a T matrix. We use the notation in which all the matrices are given in four-dimensional Gor'kov-Nambu

125 12 660

M. I. SALKOLA, A. V. BALATSKY, AND J. R. SCHRIEFFER

space, jr = (r,f), and }dx = ~ZTSdt. The T matrix is a solution of the Lippmann-Schwinger equation,

t(y,y') = V(y)S(y-y')+ j

dzV(y)G(0)(y-z)t(z,y').

55

impurities modeled by Eq. (B3) do not change the total number of states, JdioSAcr(r,(o) = 0. Equation (Bll) can be written in a compact form, if we consider the spectral density only at a given impurity site r„ :

(B6)

Mtr(r„,o>) = --Im<no-|G (0 »(a J )T(a.)G (0) ((u)|no-).

For impurities with ^-function interactions, it is straightforward to see that a T matrix of the form f(r,r';o)) = 2) tmn(w)S

6,.

(B7)

mn

(B12) The importance of Eq. (B12) becomes clear by a following example. Assume that for a given (o = a>a there exists a state | <pa) such that

solves the Lippmann-Schwinger equation. Indeed, denning

[ v - 1 - G ( o ) ( ^ J ] k a ) = 0.

<m|T(o))|n) = f m „(w),

Then, averaging over the random orientation of magnetic moments, the spin-unpolarized spectral density becomes

{m|G(0,(a>)|n) = G ( 0 '(r m -r„;o;), (m\y1\n)

(B8)

= v~JSmn;

the solution is T((o) = [v- 1 -G ( 0 ) ((u)]- 1 .

(B9)

This formula allows us to compute various physical properties of the system in the case of a finite number of random impurities for s- or d-wave superconductors. One such quantity of interest is the local spectral density. Equation (B9) also implies a useful correspondence between the current problem and that of noninteracting quasiparticles moving on an infinite lattice with random on-site energies and hopping amplitudes (see below). Consider the local spectral density, 1 i40.(r,w) = --ImG„
"

<
mn

XG (0) (r„-r,a>)|(7),

(Bll)

incorporates the effect of impurities. We have adopted the notation, according to which (o-|G|cr) = Gffo., etc. (Note that G is the particle component of G: G=GU.) Because the

>L. YU, Acta Phys. Sin. 21, 75 (1965). H. Shiba, Prog. Theor. Phys. 40, 435 (1968). 3 A.I. Rusinov, Sov. Phys. JETP Lett. 9, 85 (1969). "P.C.E. Stamp, J. Magn. Magn. Mater. 63-64, 429 (1987). 5 A.V. Balatsky, M.I. Salkola, and A. Rosengren, Phys. Rev. B 51, 15 547 (1995). 6 M.I. Salkola, A.V. Balatsky, and D.J. Scalapino, Phys. Rev. Lett. 77, 1841 (1996). 7 Y. Onishi el at., J. Phys. Soc. Jpn. 65, 675 (1996). 2

<5A(r„,ft>) = C „ 2 (n), (B14) nowhere C„ = (H<;;+t/2)/(iv2-) = (2TT)~'lfa<x>"J [(o)-&)^)2+a>'^2], oya=<x>'a+ib>"a, and / „ is the residue of the ath pole. This form is useful when fa depends only weakly on a. In particular, the matrix element |(no-|ipa)|2 incorporates the spatial information of the impurity-induced states; for example, whether the quasiparticle state at the energy (o=a>a is localized or extended. The above result suggests a fruitful formulation for exploring impurity-induced quasiparticle states in the superconducting energy gap. The central idea is to define the Hamiltonian,

(BIO)

and define Alr{r,) + 8AJj,ai), where A0(w) = -7r~ 1 ImG^(r=0,|>A0, and zero otherwise. The second term, 5ACT(r,o,) = - i l m E

(B13)

^Edii.'c,'-^!'./.. m

(B15)

mn

where cm is a four-component spinor and im„ = G ( 0 ) (r m -r„ ;o)). Noting the similarity with Eq. (B13), we can immediately conclude that the zero-energy eigenstate of H also determines the properties of a quasiparticle state in the impurity band at a given energy to. For example, in a three-dimensional j-wave superconductor where magnetic impurities yield localized states in the energy gap, the formation of the impurity band is mapped to the Hamiltonian H with short-range hopping amplitudes. Indeed, imn °cr~1e~r'»»/x, where rmn=\rm-rn\. In rf-wave superconductors, Eq. (B15) has been applied to study localization properties of quasiparticle states that are induced by unitary scatterers.31

8

L. Dumoulin, E. Guyon, and P. Nedellec, Phys. Rev. B 16, 1086 (1977), and references therein. 9 H.F. Hess, R.B. Robinson, R.C. Pynes, J.M. Valles, Jr., and J.V. Waszczak, Phys. Rev. Lett. 62, 214 (1989). 10 I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, and 0. Fischer, Phys. Rev. Lett. 75, 2754 (1995). "A. Yazdani, B.A. Jones, C.P. Lutz, M.F. Crommie, and D.M. Eigler (unpublished). 12 A. Sakurai, Progr. Theor. Phys. 44, 1472 (1970).

126 55

SPECTRAL PROPERTIES OF QUASIPARTICLE . . .

During the course of preparing this manuscript, we became aware of work by Flatte and Byers (Ref. 14) also considering magnetic impurities in .5-wave superconductors. Our results are partially in accordance with theirs in that the structure of the s-wave gap function is distorted and it changes sign at the impurity site at the transition point. 14 M.E. Flatte and J.M. Byers (unpublished). 15 0. Sakai et a/., J. Phys. Soc. Jpn. 62, 3181 (1993), and references therein; see also, J. Zittarts and E. Muller-Hartmann, Z. Phys. 232, 11 (1970). 16 P.W. Anderson, J. Phys. Chem. Solids 11, 26 (1959). 17 J.R. Schrieffer, Theory of Superconductivity (BenjaminCummings, Reading, 1983). 18 J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957). 19 V.J. Emery, Phys. Rev. Lett. 58, 2794 (1987). 20 In this expression, a term of order 0 ( a / f ± ) 2 relative to the maximum value of F is omitted as a small correction; it is obtained by interchanging rj and r± . 21 This relation follows from particle-hole symmetry: f,6 ( 0 ) (r=0,w)f 1 = (5 <0) (r=0,a.). ^Because the spin projection along the impurity moment }drsz(r) describes a conserved quantity, the quasiparticle states can be labeled according to the spin quantum number sz. 23 See, for example, P.G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, New York, 1989). 24 R. Kummel, Phys. Rev. B 6, 2617 (1972). 25 A.R. Bishop, P.S. Lomdahl, J.R. Schrieffer, and S.A. Trugman, Phys. Rev. Lett. 61, 2709 (1988). 26 A nonmagnetic impurity has an analogous effect on the j-wave

12 661

gap function; see T. Xiang and J.M. Wheatley, Phys. Rev. B 51, 11721 (1995). !7 In our numerical mean-field calculations, a superconducting ground state of rf-wave pairing symmetry is realized for an effective two-particle interaction which is repulsive for electrons at the same site and attractive between the nearest-neighbor sites; V(T) = VOS^ + V{2.VS„ , where u 0 = — U!>0 and v labels the four nearest-neighbor lattice vectors on a square lattice. !8 The d-wave gap function A(r) is defined as the amplitude of the partial-wave component that transforms as dxi-yi under the symmetry operations of the square lattice: A(r) = 4S,,(-l)"A(r,r 1 ,), where v sums over the four nearestneighbor lattice vectors and r_„=—r„. Because the magnetic moment breaks up locally the crystal symmetry, also other partial waves acquire nonzero amplitudes in the vicinity of the moment. 19 M. Franz, C. Kallin, and A.J. Berlinsky, Phys. Rev. B 54, R6897 (1996). 10 In contrast to Ref. 29, which focuses on nonmagnetic defects in strong-coupling d-wave superconductors with very short coherence lengths, our interest is mainly in understanding the consequences of magnetic defects when there is a clear separation between scales set up by the Fermi length kp1 and the coherence length f„. 1 A.V. Balatsky and M.I. Salkola, Phys. Rev. Lett. 76, 2386 (1996). 2 The BCS Hamiltonian transforms as HQ—* — H0 under the transformation in which ¥ k —•¥£= r^9^. However, because the exchange interaction is invariant under the same transformation, it is no longer necessary to have an even number of quasiparticle states inside the energy gap at nonzero energies.

127 JOURNAL OF APPLIED PHYSICS

VOLUME 84, NUMBER 10

15 NOVEMBER 15

Intrinsic limits on the Q and intermodulation of low power high temperature superconducting microstrip resonators R. B. Hammond, E. R. Soares, and Balam A. Willemsen Superconductor Technologies Inc., Santa Barbara, California 93111 T. Dahm Max Planck Institute for Physics of Complex Systems, 01187 Dresden, Germany D. J. Scalapino University of California, Santa Barbara, California 93106 J. R. Schrieffer Florida State University, Tallahassee, Florida 32306 (Received 11 May 1998; accepted for publication 18 August 1998) Cuprate superconducting thin films are being used to make compact low power microwave devices. In recent years, with improved materials and designs, there has been a steady improvement in device performance, notably an increase in resonator Q and a decrease in the nonlinear intermodulation. It is important to understand how much improvement can be expected. Here we discuss the intrinsic limiting behavior that one might achieve with a perfect film, review the present status, and discuss what the limiting behavior implies for high temperature superconducting filters. Our analysis indicates that increases in unloaded Q to ~ 1 0 6 and decreases in intermodulation by a factor of ~10 4 , compared with today's values, might be achieved. © 1998 American Institute of Physics. [S0021-8979(98)06322-l]

I. INTRODUCTION

II. LINEAR RESPONSE OF A MICROSTRIP RESONATOR: Q

Thin film high temperature superconductors are being used to fabricate passive rf filters for wireless communications.1 The low surface resistance of these materials at liquid nitrogen temperatures allows the design of compact high Q circuits, increasing the performance and reducing the size, compared to conventional devices. However, because of their small cross-sectional areas for current flow and high Qs, large current densities can be present, even at low power levels. These high current densities, especially near the edges of microstrip structures, lead to intermodulation. Filters are typically constructed from coupled arrays of microstrip resonators. The compact size and intrinsically high Q of the individual structures, which can be lumped or quasi-lumped element circuits, allows for complex filter designs. The filter response that can then be achieved is basically set by the unloaded Qs of the individual structures, which determine the insertion loss for a given filter function. At the power levels of interest, the intermodulation, which arises from the nonlinear properties of the superconducting film, is the sum of the intermodulation components produced by the individual elements transmitted through the filter. Thus it is important to determine the limiting characteristics of the elementary microstrip structures. Here we examine the limiting Q and intermodulation characteristics of a microstrip resonator which are set by the intrinsic properties of the superconducting film. We then review the status of present, state of the art, films and examine how much room remains for improvement. We conclude by discussing what this improvement can mean for rf filters. 0021 -8979/98/84(10)/5662/6/$15.00

5662

The cross section of a typical microstrip resonator is shown in the inset of Fig. 1. Such a resonator would be half a wavelength long, neglecting end effects. The current /, flowing in it, would be related to the power transmitted through it P, by w 1

0)

where the power dissipated within the resonator is assumed to be negligible. Here, QL is the loaded quality factor of the resonator and Z 0 the impedance of the microstrip. QL is usually defined as the inverse of the fractional 3 dB transmission bandwidth g ^ ' ^ A w / w . Suppose 6^=1000 and Z 0 = 50 ft, then if P,= 1 mW, the current flowing in the resonator 7 = 0.16 A. When used as an element in a multiresonator filter, this resonator would be nonlrivially coupled to other resonators and the problem becomes somewhat more complex. Throughout this article we will consider a microstrip on a ft = 508 fim thick substrate, of linewidth w = 4 0 0 ^ m and film thickness f = 0.6/u.m [of order several superconducting penetration depths X~0.3/u.m, for a Tl 2 Ba 2 CaCu 2 0 8 + I (TBCCO) film at 77 K], For such a strip, the current density would be j = Ilwt = 6.7X 104 A/cm 2 if the current was carried uniformly. However, the current density on a lossless microstrip line varies as the surface charge density, and thus we conclude from Ref. 2 that the current density on a thin microstrip line of width less than or of order of the substrate height varies approximately as © 1998 American Institute of Physics

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128 J. Appl. Phys., Vol. 84, No. 10, 15 November 1998

Dahm et a/.

5663

and j

30

i !

,

1/

i

R=

S; 20

^y

' ^

10

198

200

199 x(nm)

FIG. 1. Variation of current density j(x)lj(0) near the edge of a w — 400 fxm wide by (— 0.6 /xm thick superconducting line on a h — 508 /jm high substrate. A penetration depth of X = 0.3 fim was assumed for this calculation. The solid line is the approximate form of Eq. (2). The numerical results (crosses and dashed line) deviate from this approximate form at a distance of order X (indicated by the arrow) from the strip edge. The inset shows the cross section of this microstrip resonator.

>(*) =

7(0)

(2)

V1-(2JC/W) 2 2

J\RJ2dS (IjdS)2 •

(6)

Here the integrations run over the cross section of the microstrip structure and H and j are the magnetic field and current density respectively, with X the superconducting penetration depth and Rs the surface resistance. The X2/'2 term in L is associated with the kinetic inductance of the superfluid. It plays a negligible role in determining the overall size of L, and hence Q0, for microstrip structures with h>X. However, for the high Q structures of interest it is the inductive response of the high temperature superconducting (HTS) film rather than the resistive response that determines the size of the intermodulation. Thus when the nonlinear dependence of X on j is taken into account, it is the second term in Eq. (5) that is responsible for the intermodulation. The unloaded Q of the microstrip associated with the resistive losses in the superconducting film depends on the shape of the nonuniform current distribution. For a uniform current distribution, the resistance per unit length of a strip of width w is R = Rslw, while for the current density given by Eq. (2), the integral in Eq. (6) gives 2 w\ Rs Rs R= — In - — - 1 . 5 —

(7)

until x is within a distance of order 2X /r from the edge 7T" \ X/ W W (note that since f~2X this cutoff distance is of order \ for for the microstrip of dimensions described earlier. More dethe present case). The dashed curve in Fig. 1 shows the retailed numerical calculations show that this effect can give sults of a numerical calculation of the current density j(x) rise to as much as a factor of 2 in the reduction of Q0 due to following the method described in detail in Ref. 3 as comthe nonuniform j(x). pared to Eq. (2) shown as the solid curve. At a distance of Using an approximate form for the inductance per unit order X from the edge, the square root divergence of Eq. (2) length, suitable when the width w of the microstrip is less is cut off and j(x) linearly approaches the limiting edge than or of order its height h (Ref. 4) value. In this case, the total current/is equal to (Tr/2)wtj(0) and the current density at xc = (wl2)-(2\2lt)~(wl2) — X is (8) 2TT \ w enhanced by a factor of order Vw/4X so that 2

we have

/ (3)

y(-*c)~-

Linearly extrapolating j(x) to the edge with the slope given at x=xc gives w\ \2

J

i 2TT

I w wt V \

A/cm2

(4)

ioL

TT ln(8/!/u>) w

-0.8

(9) R, for the microstrip with dimensions defined earlier and with / in GHz and Rs in ft per square. Note that since R,<*-f2, Go a / - 1 . TBCCO films with /?s = 53 (JLD. at 3.7 GHz and 77 K have been reported.5 After frequency scaling this would correspond to Rs = 3.8fiQ, at 1 GHz and 77 K and thus Q0 -2X10 5 . Bonn etal have reported Ni (1.4%) doped YBa2Cu307_,s (YBCO) crystals with /J,= 1.13/ifl at 1 GHz.6 If a thin film with Rs~ 1 /j,Cl could be achieved, one would have Q0~ 106. These high Q values require substrates with extremely low loss tangents, t a n ^ l O - 6 , such as single crystal MgO (Ref. 7) or sapphire.

or a further enhancement of the edge current by a factor of 3/2 above that at x=xc. This gives rise to a peak edge magnetic field of order 45 G. In the following, we will be interested in power levels less than this so that, in the absence of defects, we will be in the Meissner regime. For example, at a power level of 10~5 W, the edge current density and magnetic field are reduced from the previous estimate by a factor of 10. In the following we will discuss both the unloaded Q and the intermodulation in this low power regime. The unloaded Q due to the losses in the superconducting III. NONLINEAR RESPONSE OF A MICROSTRIP film is given by Q0 =

129 5664

J . Appl. Phys., Vol. 84, No. 10, 15 November 1998

1 _ k2~

tionse-

Dahm et al.

(10)

m'

with ns the superfluid density, and m* is the effective carrier mass. For small values of j , at the temperatures of interest,8 one can expand ns(j)^ns[l — (JIJIMB)2] SO that X0')~x

i4M2 \7IMD/

/;2z

'di'

Here / is the length of the resonator (54.5 mm for a TBCCO film on MgO operating in its lowest mode at 1 GHz). At a frequency of 2G>I — io2, this becomes T(2ft)1-fc)2)/AZ.-r-.

JttiD Go

3.8 /jfl 8.3X106 A/cm2 2X10 5 2X10 5 W

- 1 fiCl -2X10 8 A/cm2 ~10 6 ~10 8 W

well separated stimuli is treated elsewhere. 13 Taking / , = 12, the ratio of the intermodulation power relative to the input power at o>2 is 14

QiPl (16) / „ ,

with '"ll^o'lMD

hi AT'

(17)

AL'

This characteristic power level, P I M D , is then proportional to the energy stored in the resonator when it is carrying a current / I M D , C/IMD= 'WiMr/2, and a relative measure of the strength of the nonlinearity in L, L 0 / AL. As such, it contains information about the resonator geometry through / and LQ, as well as the material properties through / J M D and AL.

(14)

We have measured the intermodulation products at 77 K in half wave resonators on TBCCO films from which we extract a current density scale of J\MD~ 8 X106 A/cm2. For / = 1 GHz, this corresponds to P I M D ~ 2 X 105 W for the microstrip with dimensions described earlier. For ideal films in which the intrinsic pair breaking processes determine the nonlinear response, one could have ,/IMD~ 2 x 10s A/cm 2 and Pmo~ 108 W. Current scales of the order of the depairing current have recently been shown experimentally to determine the nonlinear response of infrared transmission through thin BSCCO films.15 This provides evidence for nonlinear electrodynamics in cuprate superconducting thin films which are dominated by depairing rather than weak links. The extension of these results to microwave frequencies and YBCO or TBCCO remains a challenge to materials growers. Intermodulation products are often quoted in dBc, i.e., on a log scale referenced to the carrier signal. Then, taking e t ~ o ) / A o ) = 1 0 3 and P I M D = 2 x l 0 5 W , the intermodulation dBc value for an input signal power Pu = 1 0 - 5 W is

'IMD

10 log

For 2o)] — o>2 on resonance, this produces a current (15)

h

Ideal

IV. MATERIAL PROPERTIES: CURRENT STATUS AND IDEAL VALUES

^IMD~JiMD^t' Note that while one might have expected that A L / L 0 ~ \ / w , the edge peak in j(x), which enters to the fourth power, enhances this ratio by a factor of order w / 8 \ . If the current density could be flattened, one would not have this large enhancement. This could reduce AL by a factor of order 8 \ / w ~ 10~ 2 providing a significant reduction of the intermodulation as described below. When signal currents are present at (oi and a>2, the nonlinear term in the inductance generates an intermodulation voltage dl V(t) = lAL(13)

4

Current

(12)

with AL/L 0 ~0.15 and

V2a,-*^i

Parameter

(11)

Here j m D sets the scale of the nonlinear response. The nonlinear response of present day films appears to be dominated by weak links9 so that j i M D varies with position and may be characterized by a distribution.1011 This can lead to an intermodulation which does not scale with the third power of the input power. Weak links may also result in a magnetic flux penetration at very low fields which may lead to intermodulation that does not scale with the third power of the input power. However, at low powers, on good films, we find that the intermodulation power does scale with an approximate 3:1 slope, allowing us to characterize these films with a j I M D which is temperature dependent.12 When weak links do not dominate, the intrinsic pair breaking current density enters and for the device operating temperature of interest j m D ~neA0lpF} Here A 0 is the maximum value of the superconducting gap, PF is the Fermi momentum, and n is the carrier density. Substituting Eq. (11) into Eq. (5), one finds L = L n + AL

TABLE I. Relevant material parameters: Current status, ideal parameters. The current parameters are typical for TBCCO (see Ref. 5) and YBCO films at 77 K and —1 GHz, while the ideal parameters are extrapolated from theory and data on doped YBCO single crystals (see Ref. 6).

^10 log

Ql-

86 dBc, IMD.

(18)

if we estimate the intermodulation using VIMD—2X10 8 (2a),-a> )/L in the resonator. Here, we 2will0 consider only the limit o>\ A/cm2, this becomes —140 dBc. —>2 which produces the worst case intermodulation reWe have listed in Table I values of the surface resistance sponse for a single resonator; the more realistic case of two and intermodulation current density that have been achieved 2

UL

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130 J. Appl. Phys., Vol. 84, No. 10,15 November 1

Dahm et al.

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0 -10 -20 rt

-30 -40 -50 -60 -70 0 -10 -20

CO

3

-30 -40 -50 -

-60 -70

0.844

0.845

0.846 f(GHz)

0.847

FIG. 2. Calculated effect of varying Q on the shape of the transmission characteristics of a quasi-elliptic band-reject filter designed for the American cellular telephone system. Q is varied from 10 000 to 1 000 000. Results are shown for a 6-resonator filter (top) and a 10-resonator filter (bottom). The insets present enlargements of the data in the highlighted regions. The solid vertical lines represent the edges of the frequency band this filter is intended to reject.

along with the corresponding Q0 and PmD values. We have also listed the estimates of the limiting values which have been discussed. This shows that there is head room for further improvement. To what extent unloaded Qs of 106 and intermodulation products of -140 dBc (at input powers of 10"5 W; and for the geometry, frequency, and temperatures considered) can be achieved, depends upon the quality of the superconducting films. Clearly weak links will be particularly harmful to the intermodulation response. Nevertheless, the intrinsic limits are encouraging, particularly in light of

possible further improvements in circuit design. For example, if the edge singularity could be reduced, while one would see only a modest increase in Q0, there would be a significant improvement in the intermodulation. From the estimate above P ' IMD could be enhanced by 102 leading to a further improvement of 40 dBc of the intermodulation. V. INTERMODULATION IN FILTERS: INTRINSIC LIMITS

In order to understand what improvements in filter performance and design could occur if the intrinsic limits of

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131 5666

J. Appl. Phys., Vol. 84, No. 10, 15 November 1998

performance discussed above were reached, we have calculated the response of 6-resonator and 10-resonator realizations of a quasi-elliptic band reject filter.16 The results are presented in Fig. 2. This filter is designed to reject frequencies between 0.845 and 0.8465 GHz (the reject band) while introducing minimal loss outside of that range (the pass band), and is specifically designed for deployment in American cellular telephone systems. It is one of the most difficult filters to realize using conventional technology. In the 6resonator realization of this filter it becomes clear that one must sacrifice some rejection in the reject band to minimize the loss in the pass band, thus approximately one third of the reject band still has relatively poor rejection. In the inset it is also readily apparent that the gains that are possible by increasing Go above 30 000 are small. HTS resonators with such Qs at 77 K can be produced today and filters constructed from them now meet or exceed the performance of existing commercially available filters. However, in order to provide more complete rejection in the reject band without introducing additional loss in the pass band one can go to a more complex filter design such as the 10-resonator design in Fig. 2 (bottom). Here, it becomes clear that higher Qs are now needed before thefiltershape is independent of Q0, and a go between 100 000 and 300 000 would be desirable. Thus, with these high Qs a nearly ideal filter shape can be achieved for the real application. As previously discussed,13 the intermodulation produced by a filter operating at the low power levels of interest here can be computed by introducing the nonlinearity in one resonator at a time. The total filter intermodulation is then just the sum of these responses. Now, the optimum linear filter response is in conflict with the requirement of low intermodulation. In particular, the steep change in the transmission at the edges of the frequency band can give rise to high current densities in some resonators17 and thus a significant intermodulation response. These high current densities may be present in only a small number of the resonators which then dominate the intermodulation response.13 Assuming that one resonator is in fact dominant, Eq. (16) shows that the strength of the edge intermodulation varies as QpPiMo- A steep band edge will require a higher QL (i.e., weaker coupling) for that resonator. With this in mind, the remaining dependence of the intermodulation is on PjUD [see Eq. (16)] which varies as j*MD. Thus a factor of 10 increase in y'IMD implies a reduction in the intermodulation product by -40 dBc. Alternatively if one increases QL by a factor of 10, then one will need to increase j m r > by a factor of 10 to maintain the same intermodulation performance. In Fig. 3 we present a comparison of the calculated intermodulation for the two filters discussed above. One can clearly see that an increase in the steepness of the band edge is accompanied by an —26 dB increase in the worst case intermodulation product and a general increase in intermodulation of — 10 dB relatively far into the pass band. These increases in intermodulation can be mitigated by increasing ^"IMD • ^IMD c a n be increased by changing the geometry of the resonator; or by increases in 7IMD arising from material improvements. In this case, to preserve the peak intermodulation value when going from the 6 resonator design to the Downloaded 01 Aug 2002 to 146.201.234.113 Redistribution subjf

Dahm et al. -30 |

1

1

1

1

1

a f

-90

3

- 1 5 0

' — ' — ' — • — • — '

0.844

0.845

' — • — ' — " — ' — ^ — • — ' — • — ^

0.846 f(GHz)

0.847

FIG. 3. Calculated intermodulation for the two filters presented in Fig. 2. The solid line is for the 6-resonator filter, while the dashed line is for the 10-resonator filter. Here we have taken P., = P„. = 10"5 W, o>,—>w7, and />ftfD = 2X 105 W. Again, the solid vertical lines represent the edges of the frequency band the filter is intended to reject.

10 resonator design suggests the need for a —13 dB increase in PIMD or a factor of —5 increase in y'jMDIt is important to note that the intermodulation products generated by an HTS filter are extremely sensitive functions of frequency, in contrast to those introduced by conventional nonlinear elements such as amplifiers. Thus, the impact of integrating HTS filters into practical wireless communications systems must be considered differently than other broadband sources of nonlinearity and is quite sensitive to the particular needs of the application. Specifically, in order to estimate the two tone intermodulation products that might be generated in a real environment, one must consider the case of well separated signals (&>] =to>2), the precise frequency allocation and strength of the signals to which the filter will be subjected, as well as the contributions from all other sources of nonlinearity in the receiver. These issues fall beyond the scope of this paper.

VI. SUMMARY

In conclusion, the limiting surface resistance of both TBCCO and YBCO films at 77 K should allow a further increase in Q of microstrip structures. The present Q0~ 105 at 77 K and 1 GHz should approach 106 with improvements in films and circuit designs. With this improvement in Q we have seen that an increase in y'IMD by a factor of 10 would be sufficient to keep the peak strength of the intermodulation from increasing while allowing for a significant increase in the complexity of the filter. It appears that there is headroom for materials improvement through the elimination of weak links. In addition, flattening the current distribution could lead to an additional decrease in the intermodulation. From our discussion of the band reject filter it is clear that these improvements can be used to design significantly improved low power rf filters. to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

132 J . Appl. Phys., Vol. 84, No. 10, 15 November 1998

ACKNOWLEDGMENTS

The authors would like to thank R. Alvarez, A. Cardona, M. M. Eddy. G. L. Matthaei, G. L. Hey-Shipton, H. Casiella, J. R. Clem and D. E. Oates for valuable discussions. 1

M. J. Lancaster, Passive Microwave Device Applications of High Temperature Superconductors (Cambridge University Press, Cambridge, 1997), Chap. 5, and references therein. 2 P. Silvester, Modern Electrodynamic Fields (Prentice-Hall, Englewood Cliffs, NJ 1968), pp. 101-103. 3 T. Dahm and D. J. Scalapino, J. Appl. Phys. 81, 2002 (1997). "R. K. Hoffman, Handbook of Microwave Integrated Circuits (Artech House, Norwood, 1987). 5 B. A. Willemsen, T. Dahm, and D. J. Scalapino, Appl. Phys. Lett. 71, 3898 (1997). 6 D. A. Bonn, S. Kamal, K. Zhang, R. Liang, and W. N. Hardy, J. Phys. Chem. Solids 56, 1941 (1995). 7 D. A. Bonn (unpublished). *S. K. Yip and J. A. Sauls, Phys. Rev. Lett. 69, 2264 (1992); D. Xu, S. K.

Dahm et al.

5667

Yip, and J. A. Sauls, Phys. Rev. B 51, 16233 (1995). 'G. Hampel, B. Batlogg, K. Krishana, N. P. Ong, W. Prusseit, H. Kinder, and A. C. Anderson, Appl. Phys. Lett. 71, 3904 (1997). 10 J. S. Herd, D. E. Oates, and J. Halbritter, IEEE Trans. Appl. Supercond. 7, 1299 (1997). 11 J. McDonald, J. R. Clem, and D. E. Oates, Phys. Rev. B 55, 11823 (1997). 1J B. A. Willemsen, K. E. Kihlstrom, T. Dahm, D. J. Scalapino, B. Gowe, D. A. Bonn, and W. N. Hardy, Phys. Rev. B 58, 6605 (1998). 13 T. Dahm and D. J. Scalapino, IEEE Trans. Appl. Supercond. (in press). 14 Since we are considering only the limiting behavior we have taken Q0 %>QL and we ignore any higher order effects. These have been discussed in Ref. 3. i5 J. Orenstein, J. Bokor, E. Budiarto, J. Corson, R. MaJlozzi, I. Bozovic, and J. N. Eckstein, Physica C 282/287, 252 (1997). 16 G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching networks and Coupling Structures (Artech House, Dedham, MA, 1980). "G.-C. Liang, D. Zhang, C.-F. Shih, M. E. Johansson, R. S. Withers, D. E. Oates, A. C. Anderson, P. Polakos, P. Mankiewich, E. de Obladia, and R. E. Miller, IEEE Trans. Microwave Theory Tech. 43, 3020 (1995).

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II Solitons and Fractional Quantum Numbers

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135

SOME B A S I C A S P E C T S OF F R A C T I O N A L Q U A N T U M N U M B E R S

Frank Wilczek Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307,

USA

Quantization of charge is a very basic feature of our picture of the physical world. The explanation of how matter can be built up from a few types of indivisible building-blocks, each occurring in vast numbers of identical copies, is a major triumph of local quantum field theory. In many ways, it forms the centerpiece of twentieth century physics. Therefore the discovery of physical circumstances in which the unit of charge can be fractionated, its quanta dequantized, came as a shock to most physicists. It is remarkable that this fundamental discovery emerged neither from recondite theoretical speculation, nor from experiments at the high-energy frontier, but rather from analysis of very concrete, superficially mundane (even messy) polymers [l; 2]. In the process of coming to terms with charge fractionalization, we've been led to a deeper understanding of the logic of charge quantization itself. We have also been led to discover a whole world of related, previously unexpected phenomena. Exploration of this concept-world is far from complete, but already it has proved richly rewarding, and fed back into the description of additional real world phenomena. Bob Schrieffer's contributions in this field, partially represented in the papers that follow, started early and have run deep and wide. In this introduction I've attempted to distill the core theoretical concepts to their simplest, most general meaningful form, and put them in a broader perspective. Due to limitations of time, space and (my) competence, serious analysis of particular materials and their experimental phenomenology, which figures very prominently in Schrieffer's papers, will not be featured here.

2.1

The Secret of Fractional Charge

To begin, let us take a rough definition of charge to be any discrete, additive, effectively conserved quantity, and let us accept the conventional story of charge quantization as background. A more discriminating discussion of different varieties of charge, and of the origin of quantization, follows shortly below. The conventional story of charge quantization consists of three essential points: some deep theory gives us a universal unit for the charges to be associated with fields; observed particles are created by these fields, acting locally; the charges of particles as observed are related by universal renormalization factors to the charges of the fields that create them. The last two points are closely linked. Indeed, conservation of charge implies that the state produced by a local field excitation carries the charge of the field. Thus renormalization of charge reflects modification of the means to measure it, rather than of properties of the carriers. This is the physical content of Ward's identity, leading to the relation e ren . =

136 (Z3)2ebare between renormalized and bare charge in electrodynamics, wherein only wavefunction renormalization of the photon appears. This reasoning, however, does not apply to states that cannot be produced by local action of quantum fields, which often occur. Such states may, for example, be associated with topologically non-trivial rearrangements of the conditions at infinity. Simple, important examples are domain walls between two degenerate phases in 1 spatial dimension systems and flux tubes in 2 spatial dimension systems. These states are often associated additive quantum numbers, also called topological charges. For example, the flux itself is an additive quantum number classifying flux tubes, given in terms of gauge potentials at spatial infinity by
(2)

,

ft\

Given this form, quantization of q\Jre and q^re in integers does not imply that renormalized charges are rationally related. In particular, suppose that the first charge is associated with local fields, while the second is topological. Then the ratio of renormalized charges for a general state, and the state of minimal charge produced by local operations is a conventionally normalized charge. In the absence of topology, it would be simply q^\ an integer. Here it becomes ^normalized

=

Q

H

Q

•

\£)

ei

The ratio j - is a dynamical quantity which, roughly speaking, measures the induced charge associated with a unit of topological structure. In general, it need not be integral or even rational. This is the general mechanism whereby fractional charges arise. It is the secret of fractional charge. An important special case arises when the topological charge is discrete, associated with a finite additive group Zn. Then the renormalized charge spectrum for q^> — n must be the same as that for qW = 0, since a topological charge n configuration, being topologically trivial, can be produced by local operations. So we have the restriction P ^ = ~ (3) e\ n with an integer p. Then the fractional parts of the normalized charges are always multiples

The primeval case of 1 dimensional domain walls, which we are about to discuss in depth, requires a special comment. A domain wall of the type A —> B, going from the A to the B ground state, can only be followed (always reading left to right) by an anti-domain wall of the type B -¥ A, and vice versa; one cannot have two adjacent walls of the same type. So one does not have, for domain wall number, quite the usual physics of an additive quantum number, with free superposition of states. However, the underlying, spontaneously broken Z2 symmetry that relates A to B also relates domain walls to anti-domain walls.

137

Assuming that this symmetry commutes with the charge of interest, the charge spectra for domain wall and the anti-domain wall must agree. (This assumption is valid in the case at hand; indeed, the charges whose values are of most interest generally are those associated with unbroken symmetries.) At the same time, the spectrum of total charge for domain wall plus anti-domain wall, a configuration that can be produced by local operations, must reduce to that for vacuum. So we have 2 ^ = integer, and we find (at worst) half-integer normalized charges, just as if the domain wall charge were itself a proper Zi charge.

2.2

Polyacetylene and the Schrieffer Counting Argument

For our purposes, polyacetylene in its ground state can be idealized as an infinite chain molecule with alternating single and double bonds. This valence structure is reflected, physically, in the spacing of neighboring carbon nuclei: those linked by double bonds are held closer than those linked by single bonds. Choosing some particular nucleus to be the origin, and moving from there to the right, there are two alternative ground states of equal energy, schematically •••121212121212-••(A) •••212121212121 •••(B)

(4)

Now consider the defect obtained by removing a bond at the fourth link, in the form •••121112121212 •••

(5)

By shifting bonds down between the tenth and fifth links we arrive at ••• 121121212112-•• ,

(6)

which displays the original defect as two more elementary ones. Indeed, the elementary defect •••12112121-•• ,

(7)

if continued without further disruption of the order, is a minimal domain wall interpolating between ground state A on the left and ground state B on the right. The fact that by removing one bond we produce two domain walls strongly suggests that each domain wall is half a bond short. If bonds were electrons, then each wall would fractional charge | , and spin ± j . In reality bonds represent pairs of electrons with opposite spin, and so we don't get fractional charge. But we still do find something quite unusual: a domain wall acquires charge e, with spin 0. Charge and spin, which normally occur together, have been separated! This brilliant argument, both lucid and suggestive of generalizations, is known as the Schrieffer counting argument. In it, the secret of fractional charge is reduced to barest bones.

138 A simple generalization, which of course did not escape Schrieffer [3], is to consider more elaborate bonding possibilities, for example •••112112112112-••

(8)

•••111112112112-••

(9)

•••111211121112 ••• ,

(10)

Here removing a bond leads to

which is re-arranged to

containing three elementary defects. Clearly true fractions, involving one-third integer normalized electric charges, are now unavoidable. 2.3

Field Theory Models of Fractional Charge

While the Schrieffer counting argument is correct and utterly convincing, it's important and fruitful to see how its results are realized formally, in quantum field theory. First we must set up the field theory description of polyacetylene. Here I will be very terse, since the accompanying paper of Jackiw and Schrieffer sets out this problem in detail [4]. We consider a half-filled band in one dimension. With uniform lattice spacing a, the Fermi surface consists of the two points k± = ±7r/2a. We can parametrize the modes near the surface using a linear approximation to the energy-momentum dispersion relation; then near these two points we have respectively right- and left-movers with velocities ± | f f |Measuring velocity in this unit, and restricting ourselves to these modes, we can write the free theory in pseudo-relativistic form. (But note that in these considerations, physical spin is regarded only as an inert, internal degree of freedom.) It is convenient here to use the Dirac matrices 7°

0 1 10

71

0-1 1 0

where 7 X = 7°7 1 is used to construct the chirality projectors —j—. In the kinetic energy ^kinetic = tp(i-y-d)lp

(12)

the right- and left-movers —^—tp do not communicate with one another. However scattering on the optical phonon mode 0, with momentum ir/a, allows electrons to switch from one side of the Fermi surface to the other. This is represented by the local Yukawa interaction AL(x, t) = g
(13)

139

One also has kinetic terms for 0 and a potential V(4>) that begins at quadratic order. The wave velocity for of course need not match the Fermi velocity, so there is a violation of our pseudo-relativistic symmetry, but it plays no role in the following. This field theory description does not yet quite correspond to the picture of polyacetylene sketched in the previous section, because the breaking of translational symmetry (from x-$x + atox —> x + 2a) has not appeared. We need not change the equations, however, we need only draw out their implications. Since our optical phonon field 4> moves neighboring nuclei in opposite directions, a condensation () is subject to quantum corrections, which alter it qualitatively. Upon calculation of a simple one-loop vacuum polarization graph, one finds a correction

AV(<£) = ^Vln(<£ 2 /A

(14)

7T

where /x is an ultraviolet cutoff. (This cutoff appears because the assumed Yukawa interaction g this correction always dominates the classical to condense, no matter how small is g. Indeed, one finds the classic "BCS type" dependence (c/>)2 = fj,2e

m 7r

~P-

(15)

at weak coupling. This elegant example of dynamical symmetry breaking was first discussed by Peierls [5], who used a rather different language. It was introduced into relativistic quantum field theory in the seminal paper of Coleman and E. Weinberg [6]. In four space-time dimensions the correction term goes as AV(4>) oc g4 (f>A In cf)2, and it dominates at small <j> only if the classical mass term (oc (f>2) is anomalously small. 2.3.1

Zero

Modes

The symmetry breaking ($) = ±v induces, through the Yukawa coupling, an effective mass term for the fermion ip, which of course can be interpreted in the language of condensed matter physics as the opening of a gap. The choice of sign, of course, distinguishes between two degenerate ground states which have identical physical properties, since they can be related by the symmetry

(16)

140

With this interpretation, we see that a domain wall interpolating between (0(±oo)) = ±u, necessarily has a region where the mass vanishes, and we might expect it to be favorable for fermions to bind there. What is remarkable, is that there is always a solution of zero energy - a mid-gap state - localized on the wall. Indeed, in the background (x) = f(x) the Dirac equation for zero energy is simply idxi>i = gfip2 idxtp2 = gMi

(17)

i>\ (x) = exp(-g / * dyf(y)) ^a(x) = -»^i(x).

(18)

with the normalizable solution

Note that the domain wall asymptotics for (4>(x)) allows the exponential to die in both directions. It is not difficult to show, using charge conjugation symmetry (which is not violated by the background field!), that half the spectral weight of this mode arise from modes that are above the gap, and half from modes that are below the gap, with respect to the homogeneous ground state. When we quantize the fermion field, we must decide whether or not to occupy the zeroenergy mode. If we occupy it, then we will have occupied half a mode that was unoccupied in the homogeneous ground state, and we will have a state of fermion number \. If we do not occupy it, we will have the charge conjugate state, with fermion number — ^. It is wonderful how this delicate mechanism, discovered by Jackiw and Rebbi [l], harmonizes with the Schrieffer counting argument. 2.3.1.1

Zero Modes on Domain Walls

An abstract generalization of this set-up, with relativistic kinematics, is very simple, yet it has proved quite important. Consider massless, relativistic fermions in an odd number 2n + 1 of Euclidean dimensions, interacting with a scalar field according to Lmt, = gcpx^tjj as before. Let (cj>(z)) = h(z) implement a domain wall, with /i(±oo) = ±u. Then off the wall the fermion acquires mass 2 m2 = g2v2. But, guided by previous experience, we might expect low-energy modes localized on the wall. Here we must look for solutions of the 2n + 1 dimensional Dirac equation that also satisfy the 2n dimensional Dirac equation. With the factorized form ip(x,z) = f(z)s(x), where / is a c-number and s a spinor satisfying the 2n-dimensional Dirac equation, we must require 72n+1§~zf(z)s

= -gh{z)s.

(19)

For s± an eigenspinor of 7 2 n + 1 with eigenvalue ± 1 , this leads to f±(z)

<x e-/o<M±9Mv)).

(20)

Only the upper sign produces a normalizable solution. Thus only a particular chirality of 2n-dimensional spinor appears. This mechanism has been used to produce chiral quark

141

fields for numerical work in QCD [7], avoiding the notorious doubling problem, and it has appeared in many speculations about the origin of chirality in Nature, as it appears in the Standard Model of particle physics. A very much more intricate example of chiral zero modes on domain walls, in the context of superfluid 3He in the A phase, is analyzed in the accompanying paper of Ho, Fulco, Schrieffer and Wilczek [8]. (Note the date!) A very beautiful spontaneous flow effect is predicted in that paper, deeply analogous (I believe) to the persistent flow of edge currents in the quantum Hall effect. I'm not aware that this particular experiment, which is surely not easy, was ever carried through. But, especially in view of the advent of exquisitely controlled condensates of cold atoms, I'm confident that we haven't yet heard the last word on this subject, neither theoretically nor experimentally. 2.3.2

Vacuum Polarization

and Induced

Currents

To round out the discussion, let us briefly consider a natural generalization of the previous model, to include two scalar fields fa, fa and an interaction of the form Lint. = gifatptp + g2fa4>lxip-

(21)

Gradients in the fields fa, fa will induce non-trivial expectation values of the number current jP = tpj^tp in the local ground state. In the neighborhood of space-time points x where the local value of the effective mass 2 , that is g\4>\ + g\<$2, does not vanish, one can expand the current in powers of the field gradients over the effective mass. To first order, one finds /• \ _

W

-

l g\9i(&\dii.to-\)

2,r

£#+!&<%

h^

(22)

where 6 = arctan^^.

(23)

9191

We can imagine building up a topologically non-trivial field configuration adiabatically, by slow variation of the fa. As long as the effective mass does not vanish, by stretching out this evolution we can justify neglect of the higher-order terms. Flow of current at infinity is not forbidden. Indeed it is forced, for at the end of the process we find the accumulated charge Q = / j ° = ^(0(oo)-0(-oo))

(24)

on the soliton. This, of course, can be fractional, or even irrational. In appropriate models, it justifies Schrieffer's generalized counting argument [9]. Our previous model, leading to charge ^, can be reached as a singular limit. One considers configurations where fa changes sign with fa fixed, and then takes 52 —>• 0. This gives A9 = ±7r, and hence Q = ± | , depending on which side the limit is approached from.

142

2.4

Varieties of Charge

In physics, useful charges come in several varieties - and it seems that all of them figure prominently in the story of fractional charge. Having analyzed specific models of charge fractionalization, let us pause for a quick survey of the varieties of charge. This will both provide an opportunity to review foundational understanding of charge quantization, and set the stage for more intricate examples or fractionalization to come. Deep understanding of the issues around charge quantization can only be achieved in the context of quantum field theory. Even the prior fact that there are many entities with rigorously identical properties, for example many identical electrons, can only be understood in a satisfactory way at this level. 2.4.1

Bookkeeping

Charges

The simplest charges, conceptually, are based on counting. They encode strict, or approximate, conservation laws if the numbers thus calculated before and after all possible, or an appropriate class of, reactions are equal. Examples of useful charges based on counting are electric charge, baryon number, lepton number, and in chemistry 90+ laws expressing the separate conservation of number of atoms of each element. Using operators 4>j to destroy, and their conjugates 0+ to create, particles of type j with charge qj, a strict conservation law is encoded in the statement that interaction terms

^.^n^rn^r m

n

(25)

which fail to satisfy

Yl k0m Qjm = Yl K qjn m

(26)

n

do not occur. (In the first expression, already awkward enough, all derivatives and spin indices have been suppressed.) Alternatively, the Lagrangian is invariant under the abelian symmetry transformation (j)j->eiXqij.

(27)

An approximate conservation law arises if such terms occur only with small coefficients. One can also have discrete conservation laws, where the equality is replaced by congruence modulo some integer. In all practical cases effective Lagrangians are polynomials of small degree in a finite number of fields. In that context, conservation laws of the above type, that forbid some subclass of terms, can always be formulated, without loss of generality, using integer values of the qj. It will be usually appear simple and natural to do so. In a sense, then, quantization of charge is automatic. More precisely, it is a consequence of the applicability of local quantum field theory at weak coupling, which is what brought us to this class of effective Lagrangians.

143 Of course, the fact that we can always get away with integers does not mean that we must do so. For example, suppose we have a situation where there are two applicable conservation laws, with integer charges q]j ,q, for particles of type j . If I define the master-charge

Qj

= qf

+ wqf\

(28)

with w irrational, then conservation of Q encodes both of the prior conservation laws simultaneously. This semi-trivial trick touches close to the heart of the fractional charge phenomenon, as exposed above. 2.4.1.1

Gauge Charges; Nonabelian Symmetry

Substantial physical issues, that are definitely not matters of convention, arise for conserved quantities that have independent dynamical significance. The prime example is electric charge, to which the electromagnetic field is sensitive. Empirically, the electric charges of electrons and protons are known to be equal and opposite to within a part in 10~ 21 . Their cancellation occurs with fantastic accuracy despite the fact that the protons and electrons are very different types of particles, and in particular despite the fact that the proton is composite and is subject to the strong interaction. More generally, the accurate neutrality of all unionized atoms, not only hydrogen, can be tested with sensitive atomic beam experiments, and has never been found to fail. Neither pure quantum electrodynamics nor its embedding into the Standard Model of matter explains why electrons and protons carry commensurate charges, though of course both theories can accommodate this fact. Specifically, either theory would retain its intellectual integrity if the photon coupled to a modified charge Q = Q + e(B-L),

(29)

where B is baryon number, L is lepton number, and e a numerical coefficient, instead of to the conventional charge Q. If e is taken small enough, the modified theories will even continue to agree with experiment. To produce a mandatory unit of charge, that cannot be varied by small amounts from particle to particle (or field to field), we must embed the abelian counting symmetry into a simple, nonabelian group. Unified gauge theories based on the gauge groups SU(5) or 5*0(10) accomplish this; moreover, they account nicely for the full spectrum of SU(3) x SU(2) x U{\) quantum numbers for the particles observed in Nature [10]. This represents, at present, our best understanding of the origin of charge quantization. It indirectly incorporates Dirac's idea [ll] that the existence of magnetic monopoles would force the quantization of charge, since these theories contain magnetic monopoles as regular solutions of the field equations [12; 13].

144 2.4.2

Topological

Charges

Bookkeeping charges, as described above, reside directly in quantum fields, and from there come to characterize the small-amplitude excitations of these fields, that is the corresponding particles. These particles are, at the level of the effective field theory, point-like. In addition to these objects, the theory may may contain collective excitations with a useful degree of stability, which then become significant, identifiable objects in their own right. These are usually associated with topological properties of the fields, and are generically called solitons. Of course, at the next level of description, solitons themselves can be regarded as primary ingredients in an effective theory. Solitons fall into two broad classes, boundary solitons and texture solitons. Boundary solitons are associated with non-trivial structure at spatial infinity. A simple example is domain walls in polyacetylene, as discussed above. Texture solitons are associated with non-trivial mapping of space as a whole into the target field configuration space, with trivial structure at infinity. A simple example is a phason in 1 space dimension, as covered implicitly above (take <j>\ + (fy = constant). There the target space for the field 6 is a circle, and a field configuration that starts with 9 = 0 on the far left and winds continuously to 9 = 2TT on the far right has non-trivial topology as a mapping over all space, though none at the boundaries. Skyrmions [14; 15] provide a higher-dimensional generalization of this type. Texture solitons can be produced by local operations, but generally not by means of a finite number of field operations (creation and destruction operators) so their topological quantum numbers can also appear in fractionalization formulae.

2.4.3

Space-Time

and Identity

Charges

Each of the charges we have discussed so far can be considered as a label for representations of some symmetry group. This is obvious for bookkeeping charges, which label representations of phase groups; it is also true for topological charges, which label representations of homotopy groups. There are also symmetry groups associated with space-time transformations, specifically rotations, and with interchange of identical particles. And there are corresponding quantum numbers. For rotations this is, of course, spin. For identity it is Fermi versus Bose character - an additive, Z2 quantum number. These quantum numbers are quite familiar to all physicists. Less familiar, and perhaps unsettling on first hearing, is the idea that they can be dequantized. Let's focus on that now.

2.4.3.1

Space-Time Charges

In three space dimensions, rotations generate the nonabelian group SO(3). The quantization of spin, in integer units, follows from this. Actually, not quite - that would prove too much, since we know there are particles with half-integer spin. The point is that quantum mechanics only requires that symmetry groups are implemented "up to a phase," or, in the jargon, projectively. If the unitary transformation associated with a symmetry generator g

145 is U(g), then we need only have U(gi)U(g2) = v(gi,92)U(gig2),

(30)

where 77(31,(72) is a phase factor, since observables, based on inner products, will not depend on 77. It turns out that projective representations of 50(3) correspond to ordinary representations of SU(2), so one still has quantization, but in half-integer units. In two space dimensions the group is SO(2). We can parametrize its elements, of course, in terms of an angle 9, and its irreducible representations by the assignments U(9) = ets9, for 0 < 9 < 2TT. These are ordinary representations only if s is an integer; but they are perfectly good projective representations for any value of s. Thus in two space dimensions angular momentum is dequantized. 2.4.3.2

Identity Charges

Among all quantum-mechanical groups, perhaps the most profound is the symmetry group associated with interchange of identical particles. For the existence of this symmetry group, manifested in the existence of quantum statistics and associated exchange phenomena, permits us to reduce drastically the number of independent entities we need to describe matter. We teach undergraduates that quantum statistics supplies symmetry conditions on the wave function for several identical particles: the wave function for bosons must not change if we interchange the coordinates of two of the bosons, while the wave function for fermions must be multiplied by -1 if we exchange the coordinates of two of the fermions. If the interchange of two particles is to be accompanied by a fixed phase factor eie, it would seem that this factor had better be ± 1 , since iterating the exchange must give back the original wave function. Nevertheless we can make sense of the notion of fractional statistics; but to do so we must go back to basics [16]. In quantum mechanics we are required to compute the amplitude for one configuration to evolve into another over the course of time. Following Feynman, this is done by adding together the amplitudes for all possible trajectories (path integral). Of course the essential dynamical question is: how are we to weight the different paths? Usually, we take guidance from classical mechanics. To quantize a classical system with Lagrangian L we integrate over all trajectories weighted by their classical action elJLdt. However, essentially new possibilities arise when the space of trajectories falls into disconnected pieces. Classical physics gives us no guidance as to how to assign relative weights to the different disconnected pieces of trajectory space. For the classical equations of motion are the result of comparing infinitesimally different paths, and in principle supply no means to compare paths that cannot be bridged by a succession of infinitesimal variations. The space of trajectories of identical particles, relevant to the question of quantum statistics, does fall into disconnected pieces. Suppose, for example, that we wish to construct the amplitude to have particles at positions x\,X2, ••• at time to and again at time t\. The total amplitude gets contributions not only from trajectories such that the particle originally

146 at x\ winds up at xi, but also from trajectories where this particle winds up at some other Xk, and its place is taken up by a particle that started from some other position. All permutations of identity between the particles in the initial and final configurations, are possible. Clearly, trajectories that result in different permutations cannot be continuously deformed into one another. Thus we have the situation mentioned above, that the space of trajectories falls into disconnected pieces. Although the classical limit cannot guide us in the choice of weights, there is an important consistency condition from quantum mechanics itself that severely limits the possibilities. We must respect the rule, that if we follow a trajectory aoi from to to £1 by a trajectory ai2 from t\ to £2, then the amplitude assigned to the combined trajectory a.02 should be the product of the amplitudes for aoi and au- This rule is closely tied up with the unitarity and linearity of quantum mechanics - i.e., with the probability interpretation and the principle of superposition - and it would certainly be very difficult to get along without it. The rule is automatically obeyed by the usual expression for the amplitude as the exponential of i times the classical action. So let us determine the disconnected pieces, into which the space of identical particle trajectories falls. We need consider only closed trajectories, that is trajectories with identical initial and final configurations, since these are what appear in inner products. To begin with, let us focus on just two particles. In two spatial dimensions, but not in any higher number, we can unambiguously define the angle through which one particle moves with respect to the other, as they go through the trajectory. It will be a multiple of TT; an odd multiple if the particles are interchanged, an even multiple if they are not. Clearly the angle adds, if we follow one trajectory by another. Thus a weighting of the trajectories, consistent with the basic rule stated in the preceding paragraph, is p(a) = eie*/*,

(31)

where (f> is the winding angle, and 8 is a new parameter. As defined, 8 is periodic modulo 2TT. In three or more dimensions, the change in the angle <j) cannot be defined unambiguously. In these higher dimensions it is only defined modulo 27r. In three or more dimensions, then, we must have e1^^ = e%e<^ I* if (j> and (/>' differ by a multiple of 2-K. SO in three or more dimensions we are essentially reduced to the two cases 6 = 0 and 8 = TT, which give a factor of unity or a minus sign respectively for trajectories with interchange. Thus in three dimensions the preceding arguments just reproduce the familiar cases - bosons and fermions - of quantum statistics, and demonstrate that they exhaust the possibilities. In two space dimensions, however, we see that there are additional possibilities for the weighting of identical particle paths. Particles carrying the new forms of quantum statistics, are called generically anyons. Passing to N particles, we find that in three or more dimensions the disconnected pieces of trajectory space are still classified by permutations. With the obvious natural rule for composing paths (as used in our statement of the consistency requirement for quantum mechanics, above), we find that the disconnected pieces of trajectory space correspond to

147 elements of the permutation group Pn. Thus the consistency rule, for three or more dimensions, requires that the weights assigned amplitudes from different disconnected classes must be selected from some representation of the group Pn. In two dimensions there is a much richer classification, involving the so-called braid group Bn. The braid group is a very important mathematical object. The elements of the braid group are the disconnected pieces of trajectory space. The multiplication law, which makes it a group, is simply to follow one trajectory from the first piece, by another from the second piece - their composition lands in a uniquely determined piece of trajectory space, which defines the group product. The "braid" in braid group evidently refers to the interpretation of the disconnected pieces of trajectory space as topologically distinct methods of styling coils of hair. It may be shown that the braid group for n particles is generated by n — 1 generators ah satisfying the relations ajOk =

crjfco-j-, \j - k\ > 2

(Tj(Tj + l(Tj = (Tj + l(Tj(Tj + l, 1 < J' < fl — 2.

(32)

The as generate counterclockwise permutations of adjacent particles (with respect to some fixed ordering). Thus in formulating the quantum mechanics of identical particles, we are led to consider representations of Pn - or, in two spatial dimensions, Bn. The simplest representations are the one-dimensional ones. These are anyons with parameter 6, as previously defined. Higher-dimensional representations correspond to particles with some sort of internal degree of freedom, intimately associated with their quantum statistics. This discussion of fractional statistics has been at the level of quantum particle kinematics. Their implementation in quantum field theory uses the so-called Chern-Simons construction. This was spelled out for the first time in the accompanying paper of Arovas, Schrieffer, Wilczek and Zee [17]. 2.5

Fractional Quantum Numbers with Abstract Vortices

For reasons mentioned before, two-dimensional systems provide an especially fertile source of fractionalization phenomenon. In this section I'll discuss an idealized model that exhibits the salient phenomena in stripped-down form. Consider a U(l) gauge theory spontaneously broken to a discrete Zn subgroup. In other words, we imagine that some charge ne field <j> condenses, and that there are additional unit charge particles, produced by a field ip, in the theory. The case n = 2 is realized in ordinary BCS superconductors, where the doubly charged Cooper pair field condenses, and there are additional singly charged fields to describe the normal electron (pair-breaking) excitations. Such a theory supports vortex solutions [18; 19], where the <> / fieldbehaves asymptotically as a function of the angle 6 as (f>(r,9) ->• vew,

r ->• oo

(33)

148

where v is the value of cj> in the homogeneous ground state. To go with this asymptotics for (f> we must have for the gauge potential Ae(r,9)^—

(34)

ne in order that the covariant derivative DQCJ) = (de — ineAo), which appears (squared) in the energy density will vanish at infinity. Otherwise the energy would diverge. In this set-up the magnetic field strength B = V x y l vanishes asymptotically. Indeed, since the fields transform as <j>'{x) = exp(iQA(x))0(x) = ex.p(ineA(x))4>(x) A'fl(x)=Afi(x) + d,A(x) (35) under a gauge transformation we can, by choosing A = —9/ne, remove the space dependence of 4> and make AQ vanish altogether. We have, it appears, transformed back to the homogeneous ground state. However A is not quite a kosher gauge transformation, because the angle 9 is not a legitimate, single-valued function. The correct formulation is that the vortex asymptotics is trivial, and can be gauged away, locally but not globally. Since we can pick a well defined branch of 9 in any patch that does not surround the origin, all local gauge invariant quantities must reduce to their ground state values (this explains why D(f> and F vanish). But the line integral § A • dl of A around a closed loop surrounding the origin, which by Stokes' theorem measures the flux inside, cannot be changed by any legitimate gauge transformation. And it is definitely not zero for the vortex; indeed we have the enclosed flux $ = § A • dl = ^ . Another aspect of the global non-triviality of the vortex, is that our putative gauge transformation A = —9/ne transforms a unit charge field ip into something that is not single-valued. Since tp'(x) = exp(iA(a;))^(x)

(36)

we deduce ^{9 + 2TT) = exp(

)%l/{9).

(37)

Now let us discuss angular momentum. Superficially, vortex asymptotics of the scalar order parameter seems to trash rotational invariance. For a scalar field should be unchanged by a rotation, but ve%e acquires a phase. However we must remember that the phase of (p is gauge dependent, so we can't infer from this that any physical property of the vortex violates rotation symmetry. Indeed, it is easy to verify that if we supplement the naive rotation generator Jz with an appropriate gauge transformation KZ = JZ-Q-

(38) ne then Kz leaves both the action, and the asymptotic scalar field configuration of the vortex invariant. Thus, assuming that the core is invariant, Kz generates a true rotation symmetry of the vortex. If the core is not invariant, the solution will have a finite moment of inertia,

149

and upon proper quantization we will get a spectrum of rotational excitations of the vortex, similar to the band spectrum of an asymmetric molecule. This step, of course, does not introduce any fractional angular momentum. For present purposes, the central point is that passing from J to K modifies the quantization condition for angular momentum of quanta orbiting the vortex. In general, their orbital angular momentum becomes fractional. The angular momentum of quanta with the fundamental charge e, for example, is quantized in units of — ^ 4- integer. In two space dimensions the object consisting of a vortex together with its orbiting electron will appear as a particle, since its energy-momentum distribution is well localized. But it carries a topological charge, of boundary type. That is the secret of its fractional angular momentum. The general connection between spin and statistics suggests that objects with fractional angular momentum should likewise carry fractional statistics. Indeed there is a very general argument, the ribbon argument of Finkelstein and Rubenstein [20], which connects particle interchange and particle rotation. The space-time process of creating two particleantiparticle pairs, interchanging the two particles, and finally annihilating the re-arranged pairs, can be continuously deformed into the process of creating a pair, rotating the particle by 2n, and finally annihilating. Therefore, in a path integral, these two processes must be accompanied by the same non-classical phase. This leads to P = e2*iS,

(39)

where S is the spin and P is the phase accompanying (properly oriented) interchange. This gives the ordinary spin-statistics connection in 3+1 space-time dimensions, in a form that generalizes to anyons. For our vortex-i/> composites, it is easy to see how the funny phase arises. It is a manifestation of the Aharonov-Bohm effect [21]. Transporting charge e around flux 1/roe - or, for interchange, half-transporting two such charges around one anothers' fluxes - accumulates non-classical phase 2ir/n. 2.6

Fractional Quantum Numbers in the Quantum Hall Effect

Microscopic understanding of the fractional quantum Hall effect has been built up from Laughlin's variational wave function, analogously to how microscopic understanding of superconductivity was built up from the BCS variational wave function [22; 23]. To be concrete, let us consider the | state. The ground state wave function for N electrons in a droplet takes the form ^(Zl,...,zN)

= Ylizi-ZjfJlexpi-lzil2/!2) i<j

(40)

i

where I2 = -^ defines the magnetic length, and we work in symmetric gauge Ax — -\By,Ay = +\Bx. The most characteristic feature of this wave function is its first factor, which encodes electron correlations. Through it, each electron repels other electrons, in a very specific

150

(holomorphic) way that allows the wave function to stay entirely within the lowest Landau level. Specifically, if electron 1 is near the origin, so z\ = 0, then the first factor contributes z I I K J f- This represents, for each electron, a boost of three units in its angular momentum around the origin. (Note that in the lowest Landau level the angular momentum around the origin is always positive.) Such a universal kick in angular momentum has a simple physical interpretation, as follows. Consider a particle of charge q orbiting around a thin solenoid located along the z axis. Its angular momentum along the z axis evolves according to dL

Tt = qrE(f>

(41)

= &f

where E^ is the value of the azimuthal electric field and $ is the value of the flux through the solenoid; the second equation is simply Faraday's law. Integrating this simple equation we deduce the simple but profound conclusion that AL = ^ A ( g $ ) .

(42)

The change in angular momentum is equal to the change in the flux times charge. All details about how the flux got built up cancel out. ^From this point of view, we see that in the ^ state each electron implements correlations as if it were a flux tube with flux 3 ^ . This is three times the minimal flux. Now let us follow Schrieffer's idea, as previously discussed for polyacetylene, and remove the electron. This produces a hole-like defect, but one that evidently, as in polyacetylene, begs to be broken into more elementary pieces. Either from the flux point of view, or directly from the wave function, it makes sense to break consider an elementary quasi-hole of the type tl>(z2,...,zN) = l[z1ll(zi~zj)3llexp(-\zi\2/l2). i<3

(43)

»

(Note that electron 1 has been removed.) The first factor represents the defect. By adding three defects and an electron, we get back to the ground state. Thus the elementary quasihole will carry charge —e/3. Here again the Schrieffer counting argument is correct and utterly convincing, but a microscopic derivation adds additional insight. It is given in the paper by Arovas, Schrieffer, and Wilczek [24], through an orchestration of Berry's phase and the Cauchy integral theorem (see Chapter 3). At another level of abstraction, we can use the Chern-Simons construction to model the electrons as being vortices, quite literally, of a fictitious gauge field. This leads to a profound insight into the nature of the quantum Hall effect, which ties together most of what we've discussed, and provides an appropriate climax. A constant magnetic field frustrates condensation of electrically charged particles, because the gradient energy J \dv ~ iqei.Aen\2 ~ (qe)2\(v)\2 J A2

(44)

151

grows faster than the volume, due to the growth of A, and therefore cannot be sustained. This is the theoretical root of the Meissner effect. However if each particle acts as a source of fictitious charge and flux, then the long-range part of the total potential qe\.eA + qfict.a will vanish, and the frustration will be removed, if gei.e.B-r-tffict.ra^fict.

= 0,

(45)

where nv is the number-density of r] quanta and $fict. is the fictitious flux each carries. Given 2^ that is to say, a definite filling fraction - a definite value of 9fict.^fict. is implied. But it is just this parameter which specifies how the effective quantum statistics of the r) quanta have been altered by their fictitious gauge charge and flux. Condensation will be possible if - and only if - the altered statistics is bosonic. Identifying the 77 quanta as electrons, we require 7fict.$fict. = (2m + l)7r with m integral, to cancel the Fermi statistics. We also have ^elne fraction v. Thus we derive -

= 2m + 1,

(46) = ^

= ^, for filling (47)

accounting for the primary Laughlin states. These connections among superconductivity, statistical transmutation, and the quantum Hall effect can be extended conceptually, to bring in anyon superconductivity [25; 26] and composite fermions [27]; tightened into what I believe is a physically rigorous derivation of the quantum Hall complex, using adiabatic flux trading [29]; and generalized to multicomponent systems (to describe multilayers, or states where both directions of spin play a role) [30], and more complicated orderings, with condensation of pairs [31; 32]. In this field, as in many others, the fertility of Bob Schrieffer's ideas has been invigorated, rather than exhausted, with the harvesting. Acknowledgments This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. I would like to thank Reinhold Bertlmann and the University of Vienna, where this work was completed, for their hospitality. References [1] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976). [2] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979); Phys. Rev. B 22, 2099 (1980).* [3] W. P. Su and J.R. Schrieffer, Phys. Rev. Lett. 46, 738 (1981).* 4 R. Jackiw and J. R. Schrieffer, Nucl. Phys. B 190, 253 (1981).*

152 R. F. Peierls, Quantum Theory of Solids (Clarendon Press, Oxford, 1955). S. Coleman and E. J. Weinberg, Phys. Rev. D 7, 1888 (1973). D. B. Kaplan, Phys. Lett. B 301, 219 (1993). T. L. Ho, J. R. Fulco, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 52, 1524 (1984).* J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981). H. Georgi and S. Glashow, Phys. Rev. Lett. 32, 438 (1974); see also J. Pati and A. Salam, Phys. Rev. D 8, 1240 (1973). P. A. M. Dirac, Proc. R. S. London, A133, 60 (1931). G. 'tHooft, Nucl. Phys. B 79, 276 (1974). A. Polyakov, JETP Lett. 20, 194 (1974). T. H. R. Skyrme, Proc. R. S. London, 260, 127 (1961). Reviewed in R. Rajaraman, Solitons and Instantons, (North Holland, 1982). J. M. Leinaas and J. Myrheim, II Nuovo Cimento 50, 1 (1977); F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982); G. A. Goldin, R. Menikoff, and D. H. Sharp, Phys. Rev. Lett. 51, 2246 (1983). A. Arovas, J. R. Schrieffer, F. Wilczek, and A. Zee, Nucl. Phys. B 251, 117 (1985).* A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957). H. B. Nielsen and P. Olesen, Nucl. Phys. B 61 45 (1973). D. Finkelstein and J. Rubinstein, J. Math. Phys. 9, 1762 (1968). Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). Reviewed in R. Prange and S. Girvin, The Quantum Hall Effect, (Springer Verlag, 1990). D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984).* R. B. Laughlin, Science 242, 525 (1988); A. L. Fetter, C. B. Hanna, and R. B. Laughlin, Phys. Rev. B 39, 9679 (1989). Y.-H. Chen, F. Wilczek, E. Witten, and B. I. Halperin, Int. J. Mod. Phys. B 3, 903 (1989). J. K. Jain, Phys. Rev. Lett. 63, 199 (1989). B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993). F. Wilczek, Int. J. Mod. Phys. B 5, 1273 (1991); M. Greiter and F. Wilczek, Mod. Phys. Lett. B 4, 1063 (1990). X. G. Wen and A. Zee, Phys. Rev. B 46, 2290 (1992). G. Moore and N. Read, Nuc. Phys. B 360, 362 (1991). M. Greiter, X. G. Wen, and F. Wilczek, Phys. Rev. Lett. 66, 3205 (1991). (The symbol * indicates a paper reprinted in this volume.)

PHYSICAL REVIEW B

VOLUME 1 1 , NUMBER 9

1 MAY 1 9 7 5

Dynamics and statistical mechanics of a one-dimensional model Hamiltonian for structural phase transitions* J. A. Krumhansl t and J. R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19174 (Received 4 November 1974) We have studied thermodynamic and some dynamic properties of a one-dimensional-model system whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used to study displacive phase transitions. By studying the classical equations of motion, we find important solutions (domain walls) which cannot be represented effectively by the usual phonon perturbation expansions. The thermodynamic properties of this system can be calculated exactly by functional integral methods. No Hartree or decoupling approximations are made nor is a temperature dependence of the Hamiltonian introduced artificially. At low temperature, the thermodynamic behavior agrees with that found from a phenomenological model in which both phonons and domain walls are included as elementary excitations. We then show that equal-time correlation functions calculated by both functional-integral and phenomenological methods agree, and that the dynamic correlation functions (calculated only phenomenologically) exhibit a spectrum with both phonon peaks and a central peak due to domain-wall motion.

I. INTRODUCTION

In recent years, there has been considerable interest in systems in which structural phase transitions apparently take place due to the instability of some lattice displacement pattern, which takes the system from some stable high-temperature phase to a different low-temperature lattice configuration. The dynamics of such systems is frequently characterized by a vibrational mode whose frequency decreases rapidly near the critical temperature, as though the restoring force for that displacement pattern softens, thus "soft modes." The history of this viewpoint is generally well known, 1_3 particularly in the study of ferroelectrics, though many other systems show the behavior in some form or other. Peierls instabilities would share some of the features, although significant changes in the electronic properties occur simultaneously with the lattice distortion, and the coupled problem is more complex. While it is likely that such displacive transitions are at least accompanied by soft modes, the theoretical interpretation is not altogether satisfactory since formal analyses to date are all based on anharmonic phonon perturbation theory, using some set of self-consistent high-temperature lattice phonons as a basis. But at the transition temperature, the displacements relative to that lattice become large and no perturbation scheme is expected to be satisfactory. Computer simulations 4-6 have been carried out to shed light on these matters; and, indeed, are very informative. In addition to showing features which are expected as some order parameter develops a nonzero value, there are two other interesting features: the appearance of clusters of 11

locally ordered regions, and the development of a "central peak" near ci> = 0 in the dynamic response function S(q0, w). The central peak which accompanies the soft mode experimentally 2 has received a variety of interpretations, 2,7,e which also remain somewhat open to question. We thought that it might serve a useful purpose to see whether one could approach these problems from other than a perturbation or mode-mode coupling point of view, and the work here is a first step in that direction. To date, the development has been restricted to a one-dimensional model, for which there cannot really be a phase transition for finite range interactions. On the other hand, we have been able to treat strong nonlinearity in some detail, making contact with an exact (in principle) calculation of the equilibrium statistical mechanics using functional integral methods. Several interesting features appear in the results, principally of an interpretive nature. The most important result potentially is that the Fourier (phonon) representation commonly used in perturbation calculations is inadequate to discuss one important type of excitation that can occur in highly nonlinear systems, and which we refer to as "domain walls." These were postulated by Takahashi9 some time ago on phenomenological grounds, and now appear to us to be a natural consequence of strong anharmonicity in the statistical mechanics of this model system. The plan of the paper is as follows: In Sec. II we present the model Hamiltonian, and discuss the solutions of the resulting equation of motion for the displacement field; from the small-amplitude phonon modes to some limiting large displacement patterns, including time-dependent solutions. In 3535

3536

J.

A.

KRUMHANSL AND J .

WWW <°)

(b)

(c)

FIG. 1. Eigenvalue spectrum (schematic) of the transfer operator, from Eq. (30), for temperature-dependent effective mass m*-m{,c\ll'lK\T2): (a) low temperature, 177* large; (b) intermediate temperature; (c) high temperature, m* small. Sec. Ill we use the functional-integral method to calculate the partition function for this Hamiltonian, adapting and extending the work of Scalapino, S e a r s , and F e r r e l l , 1 0 including the calculation of c o r r e l a tion functions. In Sec. IV we do the statistical mechanics of a random a r r a y of domain walls on a background of small-amplitude phonons, and we can make a complete identification with the functional-integral r e s u l t in the l o w - t e m p e r a t u r e r e g i m e . In Sec. V we show that static correlation functions can be calculated either way, and that one is led to a model for dynamic correlations which can yield a central peak in an appropriate scattering function—because of the motion of dom a i n s , not because of coupling to entropy fluctuations or hydrodynamic modes. Conclusions and discussion a r e contained in Sec. VI. II. MODEL AND EXCITATIONS OF THE SYSTEM A standard model Hamiltonian for a system which might undergo a displacive phase t r a n s i t i o n ' - 6 a s sumes that the Hamiltonian is of the form

+

Lm
•

(1)

Here i, j indicate lattice s i t e s ; A, B, Ctj a r e potential coefficients; and « , , u{ a r e displacements and velocity of the displacing ion with respect to some heavy ion or reference lattice. Typically, A might be determined by attractive interactions of the mobile ion with the reference lattice, B by short-range repulsion of those near ions, and C i} by interactions between the displacing atoms. In the situations where this is presumed to r e p r e sent a lattice which is unstable against a displacive transition, A i s negative, B is positive, and Clt a r e positive. This Hamiltonian is a tremendous oversimplification of any r e a l three-dimensional s y s tem, particularly of symmetry r e s t r i c t i o n s and long-range forces, which a r e important in real f e r r o e l e c t r i c s . None the l e s s , we find that even

R.

SCHRIEFFER

11

in one dimension t h e r e a r e results which a r e interesting and nontrivial. Before proceeding with the a n a l y s i s , we note two approximations which a r e often m a d e in d i s cussing the finite t e m p e r a t u r e behavior of the model s y s t e m : (i) H a r t r e e approximation: (uz? = u](u\)T usually {u\)T*> \A\(T/Ta) in the high-temperature region. This yields a psuedoharmonic Hamiltonian 1 ' 2 ' 1 0 (also derivable by lowo r d e r anharmonic phonon perturbation theory) with A* = \A 1{T/T0~ 1). This d e s c r i b e s a s t a b l e lattice for T > T„, and vice v e r s a below T 0 . Many studies have been made with this effective Hamiltonian as a point of d e p a r t u r e , but the approximate nature of its basis should not be forgotten, (ii) Mean-field approximation: Onedera " h a s studied the statistical mechanics of this system with the - M approximation us = {u)T in iZctMt p 2 - This s u p p r e s s e s all dynamic information which depends on the details of interion d i s p l a c e m e n t s , and amounts to a collection of anharmonic o s c i l l a t o r s coupled only by their mean thermal d i s p l a c e m e n t s . Thus, no phonons a r e considered, and i n t e r p a r t i c l e fluctuation effects a r e certainly omitted. We have tried to avoid either of t h e s e a p p r o x i mations; and p a r t i c u l a r l y in contrast to the analys i s in Ref. 10, we take A s - ! AI to b e independent of t e m p e r a t u r e , as in the original Hamiltonian, thus not putting in the c r i t i c a l behavior artificially. We do not attempt to employ r e n o r m a l i z a t i o n methods to obtain an effective Hamiltonian. The one approximation we will make i s to a s s u m e that the Hamiltonian (1) can be replaced by a continuum representation „

rdxTpix)2

A

, ,,

B

, ,4

mc\

/du\zl

(2) where I i s the lattice spacing and xs =jl = x locates an element (ion) in the continuum r e p r e s e n t a t i o n . This approximation limits us to displacement fields which do not change radically over a l a t t i c e s p a c ing. In the above, c 0 i s the velocity of l o w - a m p l i tude sound waves (phonons) which would occur if A and B were negligible (i. e . , only interaction b e tween displacing ions a r e important). We now p r o ceed with the analysis. Taking A = ~\A\, B>0, the " o n - s i t e " potential is a double well potential with minima at (see Fig. 1), and depth v(±u0), u = ±u0 = ±(\A\/B)l/* z

v(±u0) = -i\A\ /B z

v{u±Uv) = -k\A\ /B

,

(3a)

,

(3b) z

+ 2\A\(u±u0) /Z+---

. (3c)

Two different physical r e g i m e s of the p a r a m e t e r s occur under the names of " o r d e r - d i s o r d e r " or

11

DYNAMICS AND S T A T I S T I C A L M E C H A N I C S O F . . .

"displaclve" systems. If the depth of the wells is so great that an intersite energy mc\ (2u0/l)z, which is the interaction energy between nearest neighbors displaced to opposite wells, is not great enough to lift the particle over the barrier, then only large thermal fluctuations at individual sites can do it. Effectively, one has a collection of weakly coupled anharmonic oscillators, randomly displaced to «<* ±w0, as one expects in a disordered system. This order-disorder regime occurs for i\A\s/B>4mcUo/l!

•

(4)

In the opposite limit to this inequality, there is strong intersite interaction and extended lattice modes determine the physics. Here the system is said to undergo displacive transitions and \\A\z/B<4mc%u\/lz

.

(5)

We will be concerned entirely with the displacive case, that being more relevant to the soft-mode situation. Applications are made at high temperature, and no essentially quantum effects are involved. Therefore, we first consider the classical equations of motion and their solutions, then in Sec. Ill, the classical statistical mechanics of the system. The equation of motion for the displacement field u(x) which follows from (2) is mu + Au + Buz-mc%u"

=0.

(6)

We note that quite generally, if u=f{x- vt), then / must obey Mv*-cl)f'

+ Af+Bf3 = 0 .

(7)

Introduce the dimenslonless variables m(c\-vz)/\A\

= S 2 (length squared) ,

(8a)

f/u0 = ri ,

(9b)

(*-trf)/l = s .

(8c)

The dimensionless form of the equation is

-0+>?-»? s = O .

(9)

Both static and time-dependent solutions may be constructed from the solutions of (9). We discuss first the limiting forms of solutions. A. Small amplitude, ^ « i | « l Solutions of v" + V= 0 are of the form V=asin(s+e),

(10)

where a is the amplitude and S is the phase. Sub-

3537

stituting physical variables, u=auosin[{x-vt)/Z

+ 0] ,

(11)

which is simply a phonon with wave number q = £ " l , frequency vlt, and phase velocity v (which is q dependent) that satisfies the equation vzqi = clqz-\A\/m

= wl ,

(12)

which is also a dispersion relation. Of course, since A is negative, the frequency uiq will only be real for finite q»(lA\/mc\). These phonons are small amplitude oscillations about u = u0. Another set of small amplitude oscillations can occur if all particles are displaced and lowered in energy to the bottom of one of the wells. Then V = 1 + y, where y is a small dimensionless displacement. To terms linear in y, the equation of motion is y " - 2 y + O(r2) = 0 . With a slight revision in the definition (8a) the solutions are small oscillations about ±u0 of the form u = ±u0+auasin(qx—


(13)

with the dispersion relation c 2 o9 2 + 2 | A | / m = o ) |

(14)

instead of (12). The frequencies are real for (all) q & 0. It should be noted that for N particles, this state is very much lower in energy (NA 2 /4B) than the configuration vibrating about « » 0. In the cases above, the nonlinear term ij 3 has been omitted or linearized about 77= 1. Thus, the modes found can be superimposed in lowest-order calculation of the partition function. But, of course, as soon as the nonlinear terms are considered, the phonon modes are coupled to each other; and the thermodynamics is quite nontrivial. For the most part, Green's-function decoupling approximations, or perturbation methods, have been the only methods applied to the interacting phonon system. However, we proceed somewhat beyond those formulations in the present case. To do so, we now look at the solutions of (9) in another regime, which we call the large amplitude strong anharmonic regime. B. Large amplitude regime, ij 3 =t;=+ 1

Equation (9) is formally identical to that governing the order parameter in Ginzburg-Landau theory for a one-dimensional (1 - D) superconductor. One type of solution, for which V is not small, is *? = +1 or - 1 for all s. This is mostly an uninteresting solution, the order parameter constant throughout the system; but it is the lowest-energy state since all particles are at rest at the bottom of a potential well. The small oscillations in the second case

156 3538

J.

A.

KRUMHANSL

AND J .

above are one kind of low energy excitation above this lowest energy s t a t e . But there a r e other, intrinsically nonlinear, field patterns which a r e also important in the low-lying excitation s p e c t r u m . More important, they cannot be represented by any reasonable o r d e r of p e r t u r bation theory based on phonons. Such field patterns a r e well known in type-II superconductors. F o r example, one simple p a r t i c u l a r solution of (9) i s J7 = t a n h ( s / / 2 l .

(15)

R.

SCHRIEFFER

11

the notation being s t a n d a r d . n This is an elliptic integral of the f i r s t kind, and g e n e r a t e s the family of solutions T]=asn(bcr)

(20)

where sn(6c) is the elliptic sine function. F o r ba = 0, i . e . , s = (x- vt)/t = 0, ri=0, and sn i s an odd function of its argument. With a m o d e s t amount of algebra, it is easy to find the s m a l l - a m p l i t u d e and large-amplitude l i m i t s . Equation (20) may be written in further detail

This corresponds to a family of solutions in physical variables (16)

tt = M 0 t a n h [ ( x - t f r t / V 2 £ ]

where 4 is defined in (8a). In this pattern, the displacement is constant at - u 0 over nearly all the semi-infinite region (x- vt) < 0: it is + u 0 for (x- vt) >0. The transition takes place through a domain wall of approximate thickness 2-/2 £., and the wall moves with a velocity v. F r o m the definition (8a), ! = ( m / ! A I ) 1 / 2 {c\-vz)in, it is seen that c 0 is the upper limit on the drift velocity (perhaps more precisely that £ must not be l e s s than a lattice constant). The excitation energy required to produce this pattern is localized in the domain wall; it will be calculated in detail in Sec. IV. But it i s apparent that this kind of excitation is quite the converse of phonon excitations where the energy is distributed throughout the lattice. In one sense, phonons a r e independent in q space while these domain walls a r e independent in r space, their interaction falling off exponentially when separated by m o r e than a wall thickness. T h e r e i s , however, the all i m portant difference that the small amplitude solutions above w e r e approximate solutions of (9), while (15) i s an exact solution. This last observation prompted us to see whether we could connect the two types of solutions. This has been partly possible, a s follows, in t e r m s of elliptic functions. Equation (9) may be converted by quadrature into an implicit integral relation between V and s. LetCT= S / / 2 ~ and

-Kg)!.]' -[HSU

(17)

then (18)

( g ) 2 = (a 2 -n 2 )(6 2 -^ 2 ) and (7 is found from the elliptic integral [(a'-y^bz-y*)]1'*

>

(19)

»-(h[-«(2ur*)-«» The elliptic sine i s periodic in 4if, w h e r e K is the complete elliptic integral of modulus k,

=

a 6

/ l - 1l - 2 \1+1

(dv/ds)*^1'2

')

(22)

The small-amplitude solutions a r e found in the limit (rf7]/ds) o=0 « 1, where k^0, and 4if = 2n. We r e c o v e r (10), 7)^ or sins . But for large amplitude, i f - 1 then if— °° and the period of the solution becomes very long (and not related to any r e n o r m a l i z e d fundamental period). In fact, if (dv/ds)^a= (2) l,z then one finds that (21) approaches t) = tanh(syT) and we recover the domain wall solution above. In all of the above, the phase 6, a choice of origin, has not been explicitly written in; but because the Hamiltonian is translationally invariant, it is clear that this freedom e x i s t s . Also, since s = (x- vt)/k, a whole family of stationary and moving fields is included in (21). It would be nice, now given various dynamic solutions of the equation of motion, to proceed to e x p r e s s the Hamiltonian using them as a b a s i s , then do the statistical mechanics. This i s what i s usually done with phonons, because a harmonic Hamiltonian s e p a r a t e s in mode s p a c e . Unfortunately, as is obvious from the nature of nonlinear s y s t e m s , it is absolutely impossible to do the same superposition by simply adding two solutions for the displacement field. Indeed, the whole subject of nonlinear oscillations in extended s y s t e m s is extremely complex; and we must admit to having been unable to c a r r y forward a rigorous program to connect these exact solutions of the equation of

11

DYNAMICS AND S T A T I S T I C A L MECHANICS O F . . .

motion to an evaluation of such thermodynamic quantities as the partition function. On the other hand, we believe that the analysis above suggests certain features which should be included in a proper phenomenological discussion of the statistical mechanics—at least in the lowtemperature region where low-energy excitations dominate. Specifically, we believe that one will see two different patterns in the displacement field; one, small amplitude oscillation motions about the potential minima u<*±ua; and then the other, occasional domain walls where the displacement flips from one minimum to the opposite over a region of length » 2-/SI. There will be a thermal mixture of these two kinds of excitation. It is apparent that phonons (extended) and walls (localized) interact weakly; and if the density of walls is low, they interact weakly with each other. Can one go from one language to the other in a systematic way? Again, the complex physics of nonlinear oscillations in extended systems (e. g. , laser oscillations) often seems to allow the pumping of energy from a number of weakly excited extended modes, via the nonlinearity, into a local pulse or shock; and our conclusion is that this happens here. To test these ideas, one must resort either to experiment or to some related theoretical calculation which can be carried out exactly. It is not possible to find a real system which is accurately represented in nature by our model Hamiltonian. Computer simulations4"* are of considerable use in a quantitative way, and we will comment on them in the concluding discussion. However, a formally exact solution for the thermodynamic quantities would be much more useful for calibration purposes. Fortunately, using functional integral methods 10 ' 13 it is possible to evaluate the partition function for this one-dimensional model exactly—conditional on a knowledge of solutions of a one-dimensional anharmonic oscillator Schrodinger-like equation—which can be solved numerically if necessary. We carry out that exact calculation in Sec. Ill, and then compare it in Sec. IV with the phenomenological thermodynamic behavior deduced from a mixture of phonons and domain walls.

The classical partition function follows from the Hamiltonian (2) as a functional integral in the field variables u(x), p(x) , Z = \{Hu)f>{.p)]e'an'"')

•

evaluated in terms of the eigenfunctions of transfer integral operators. We have followed SSF generally, except in two respects: (i) we do not put an explicit temperature dependence into A, e. g., a{ T- T0), which in a sense puts in the phase transition "by hand," but take A = - \A\ constant; (ii) instead of solving the anharmonic oscillator Schrodinger equation numerically to determine the eigenvalues of the transfer operator, we studied the lowest states In a formal (WKB approximation) way; this preserves subtle interpretive features that can be easily lost in numerical studies. In the classical approximation, for the given Hamiltonian, the momentum and field displacement integrations for Z factor completely. Z - ZtZu, with Zp = (2nmKBTf n as usual for N particles. We are left with the potential energy term Zu [ V{{u})} to compute. Dividing x into stations x{ separated by I, one writes the partition function, for nearest neighbor interactions, as Zu=jJJ{duie-ef'Mi'ui-^)

,

<23>

Scalapino, Sears, and Ferrell 1 0 (SSF), and Kac and Helfand13 have shown how this expression may be

(24)

where/(«(,«(_!) is that part of the potential depending on a, and UJ.J. The integral may be evaluated exactly in the limit of a large system using the eigenfunctions and eigenvalues of the transfer integral operator J d « i . 1 e - a " « i . " ' - i , * „ ( « j . 1 ) = e-s«»*„(«,) . (25) The *„ are distribution functions for the field amplitude u, which are not only useful to compute Z, but also to compute expectation values of various quantities, assuming that their properties are such that the *„ form a complete set, i. e., 6 («-!;)=£*„<«) * * ( » ) .

(26)

Thus, supposing at first that at xu ui = v, then the integral Zu can be rewritten ^» = S*n(«)(rid« i e- 8 / ( "f"i=i'*„(«!>

(27a)

H

= E * ; M *.(««,) (He -6,»).

111. EQUILIBRIUM THERMODYNAMICS OF THE ONE-DIMENSIONAL MODEL: "EXACT"

3539

(27b)

Next, integrating over all possible initial and final displacements v, uNtl, and replacing N by L/l yields Z^e''*"*'*

.

(28)

n

Obviously, as £-«>, Z„ is dominated by the lowest eigenvalue c 0 ,

3540

J.

A. KRUMHANSL

«a,c-(x/«*0t

z

AND J . R .

(29)

This p r o c e d u r e and calculations of moments or correlation functions a r e discussed further in the references cited. The burden is now t r a n s f e r r e d to finding solutions of the t r a n s f e r - o p e r a t o r equation. These a r e found 10 from the solutions of an equation which i s Schrodinger-Uke (of course, quantum mechanics i s not really Involved). Applying the method of SSF, we find the effective oscillator equation, in which K= 1 (

A

2

B

4

~Wmc\

d* duz

*„(«)=£„*»,

(30) where s 0 is a z e r o of energy from normalizing certain integrals and plays little role in the t h e r modynamics. Note one important difference between (30) and SSF equation (2. 23); we maintain the strong temperature dependence of an effective m a s s m* on t e m p e r a t u r e , a s defined by Z2 ZP'mcl

d* du*

1 2m*

d2 dtr

m*^m{c\/lzK\Tz) A variety of r e s u l t s w e r e obtained by SSF from numerical evaluation of the solutions of their version of (30). More insight can be gained, at least for low-temperature region, by an interpretive examination of the low-energy solutions of (30), a s a function of t e m p e r a t u r e . The potential is shown in F i g s . 1(a)-1(c), with an indication of the way in which the energy levels might be d i s tributed for low, intermediate, and high t e m p e r a t u r e . F o r low temperature, the effective m a s s m* is large and the eigenvalues begin near the bottoms of the wells, split into p a i r s by "tunneling— in the sense of this effective Schrodinger equation. At high t e m p e r a t u r e , m * may become so small that the lowest eigenstate e 0 lies well above the potential hump. This does not provide an exact criterion for obtaining a t r u e phase t r a n sition, but it does suggest that below some intermediate t e m p e r a t u r e the t h e r m a l distribution i s such a s to find the displacement pretty much near ± ua, while well above this temperature the d i s placements range over the whole region in the lowest eigerifunction * 0 ( M ) . An examination of computer solutions would provide further detail, if done with high precision. However, in the l o w e r - t e m p e r a t u r e regime, approximate solutions can be constructed, as for standard quantum-mechanical double-well probl e m s . To a first approximation we have harmonicoscillator states in each well, for which the

SCHRIEFFER

11

(doubly degenerate) spectrum E„~(«+i)(2|/l|/>M*)1/2

(31a)

12

~(n+iKlKsT/cB)(2\A\/m) '

.

(31b)

The potential near the minima i s Vug^i(2\Al) x {u -u0f. This double degeneracy i s split by tunneling En.a^En

+ tn,

B„,^En-tn,

(32)

where t„ i s the matrix element connecting the nth s t a t e s in opposite wells. Taking the lowest s t a t e s , n= 0, a s lying on either side of a potential hump of height A 2 / 4 B and average width M0, a WKB approximation yields ,

E.

f

u„ (Az » » c g V / 8 1

*%«*[—(•£ m*Yl-

(33)

Thus, the two lowest levels have eigenvalues [from (31a) with n = 0 ]

•--:&n'-M-(£-n}(34)

with * 0 . . = ( 1 / V 2 ) [ * 0 ( K - « O ) ± * O ( K + «O)J ,

(35)

where * 0 (u) i s the n=0 h a r m o n i c - o s c i l l a t o r s t a t e . F o r low t e m p e r a t u r e , but finite, the "tunneling" splitting of the lowest oscillator level a s given by (33) may be very s m a l l , but upon taking the t h e r modynamic limit L/l = N~°° only the lower of t h e pair of states s u r v i v e s . We have the s e r i e s of equations using (29), (31b), (32), Zu = exp[-{L/l)p{s0+E0-t0)]

,

F = - KB T/n(Z„Zu) = F„ + FU ,

(34)

Fp = -\NKBTln(2TiMKT)

(35)

F„ = NKBT

rj_/2Mjy

, /2

s^-i

[2c0 V m )

KT

\-

(36)

Upon substituting and collecting t e r m s after some algebra (37)

^.^(^p,^,)], .1/2

(38a)

11

DYNAMICS AND S T A T I S T I C A L MECHANICS O F . . .

«-[-&(^n- <-» The terms in (38a) are shown in Appendix A to be the free energy of the phonons specified by (13) and (14). W h a t i s £ t u M ? We show in Sec. IV that it is just equal to the free energy of a thermodynamic distribution of domain walls, together with the phonons. Thus, this detailed functional integral calculation shows that two qualitatively distinct low-energy excitations are present, particularly at low temperature. IV. STATISTICAL MECHANICS OF DOMAIN WALLS

Let us assume the viewpoint that the domain walls can be considered as weakly interacting elementary excitations if, say at low temperature, they are distributed at random in low concentration along the one-dimensional model system. We then compute the thermodynamic properties, and compare the result with the exact calculation in Sec. m , to see whether this view is plausible. The point is that if separated by much more than a wall thickness, the strain and kinetic energy fields do not interfere between domain walls. Actually, there are some interesting kinematic r e strictions, in that domain walls of the same sign cannot be adjacent to each other nor pass through each other while walls of opposite sign can pass through each other and annihilate. For the present, we assume that the density of walls is so low that these finer details will contribute only an "excluded volume" type of correction to an otherwise dilute gas. To proceed, we need the excitation energy associated with the wall, which from (1) comprises kinetic and potential energy terms. These are to be evaluated for the field given by (15) and (16). The potential energy, relative to the lowest energy where u = ± u 0 is given by

^=/f[f ( " 2 -" ? ) + f ( " 4 - M j ) +

mclfduYl -Z^W) J

might (a) replace tanh y by y if I y I < 1, and by ± 1 otherwise, (b) considering only low-energy excitations, assume that the slow moving walls, v2«cl, dominate; then | =* £„ independent of v. With these approximations15 £ DP - 2V2a0/l)(Az/2B)

(l-k)

,

(39)

HO)

with u = u0 tanh[(x - vt) /yjTl 1. These integrals appear to be intractable, but in fact can be carried out exactly. However, for interpretive purposes one

(41)

The factor 7/60 from integrations will be neglected. Each of these expressions is easily interpreted, defining A = 2V2 |„ as the thickness of a domain wall. The number of particles in a wall is {2J2 l o / ' ) , with mean potential energy approximately (i42/2B)(relative to the ground state). Similarly, the kinetic energy may be associated with a kinetic effective mass of the domain wall roD given by m* = m(Z^Z^/l)(u%ml)

.

(43)

The statistical problem is then that of a "gas" of "quasiparticles" having the above potential and kinetic energy, distributed in a one-dimensional volume. The partition function is that found for placing these particles on a line. So that they may be considered distinguishable, we divide the line into n, segments having thickness of a domain wall A = 2-/2" | , then n, = L/A. The partition function is then

where B is an appropriate phase space normalization. In the approximation that Z is dominated by the most probable » „ which if e"fl£DP « 1 is also the average n„ =na e"S£DP, then the expression can be evaluated to yield

ZD - ZDKZDP - ( j g * £ J"" *•••-.

and the kinetic energy by

*«=Jf(fl')'

3541

(45)

The same result can be obtained more elegantly using a grand canonical distribution. From (45) the free energy is FD = -KTlnZ D = - KBTnj^

1+ i m(-|!*a|^ , (46)

^=-M^k)( 1 + i l n ^f) e " £ D p / " (47)

3542

J.

A.

KRUMHANSL

AND J .

We now compare this with the "tunneling contribution" to free energy found from the exact functional integral calculation as given by (38b). Inserting the definitions for l 0 , EDP from Eq. (41) (but neglecting ^ compared to unity), and a 0 , we find that (38b) may be rewritten Ftmn = - N K E m / 2 ^ 2 ^ e - E D P

/K T

B

.

(48)

Remarkably, except for a prefactor of the order of unity, FD=Ftmn. Thus, a qualitatively important part of the exact free energy is associated with the excitation of domain walls. We take this agreement between the phenomenological statistical mechanical model and the exact calculation a s confirmation of the proposal in Sec. I that when nonlinearity plays an important r o l e , both phonon and localized domain wall excitations a r e to be found in the thermodynamic behavior. Of course, the phonon free energy v a r i e s slowly (linearly) with T i n this classical approximation, while the free energy and concentration of domain walls drops rapidly (exponentially) with decreasing t e m p e r a t u r e . However, as we will see in Sec. V, a number of experimental quantities can depend strongly on the presence of domain walls.

R.

SCHRIEFFER

11

but, of c o u r s e , does not ever become infinite for any finite t e m p e r a t u r e , for a one-dimensional s y s t e m . So much for the functional integral r e s u l t for the equal time displacement-displacement c o r r e l a tion function. Can the phenomenologieal domain-wall model be used to calculate this correlation function? Consider the following model: A t # = 0, M-Mn, except for small phonon oscillations, but between x-0 and a finite value of x t h e r e may be nw(x) domain w a l l s . At each wall u = ±u<s flips to =f «„. Thus, the c o r r e lation function «(*)«(0)=<M0(-l)"«.t*>

(52)

At low density of domain walls, a P o i s s o n d i s t r i bution should apply, and

-P(nBM) = [ ( « j V n J ]«""«>,

(53)

where n„(x) = (x/A) e~iEDP from (44). The average value of the correlation function is <«M«(0)> = «02<(-1)"»<*'>

(54)

and <(- 1)"«<*>> = ( - 1)° e-**> + ( - \)nwe-** +(-D2%^e-R»+...,

(55)

V. APPLICATIONS OF THE PHENOMENOLOGICAL MODEL <(-l)"» ( "> = e - 2ii «.,

A. Equal-time correlation functions—low temperature

whence, where A = 2V2~£„, As discussed by Scalapino, Sears, and F e r r e l l , 1 0 the two-point equal-time correlation function may be written eU

tan n>

z

(u(Q)u(x))=J2e- "- ' \(0\u\n)\ ,

(49)

n

where again the €„ and states I n) are those of the eigenfunctions of the transfer integral operator defined for the functional integral in Sec. III. If we include higher oscillator states in the sum, we can find the correlation function characteristic of phonons about ±Mo. But a much larger displacement is associated with a jump from - UQ to + a 0 ; at low t e m p e r a t u r e , sum (49) is then dominated by the lowest pair of tunneling s t a t e s . It is straightforward using (33) and (35) to find then that W O ) K W ) « «„2 e - * W '

- «? e-"x*,

(50)

where the correlation length i s found from (33) to be \c=n^eEop"CT

(51)

in the low t e m p e r a t u r e region where the tunneling approximation holds. At higher temperature, when the lowest eigenstates of (30) a r e above the saddle point between wells, one expects an algebraic dependence of \c on t e m p e r a t u r e . Thus, below some intermediate temperature, the correlation length begins to i n c r e a s e dramatically (exponentially);

(u(x)u(0)) = 4 e x p [ - (2x/A) e -* E op], x

{u(x)u(Q)) = t%e->" c,

(56) (57)

with Xc being identically the same c o r r e l a t i o n length given by (51) from the functional integral calculation. (The exact r e s u l t differs by | . ) This r e s u l t is one more indication that the domain-wall model i s both formally and practically valid for obtaining information about t h e r m o d y namic a v e r a g e s in the l o w - t e m p e r a t u r e r e g i m e . B. Dynamic correlation functions

For s c a t t e r i n g experiments in which s o m e probe excites a displacement H(0, 0) which in t u r n induces emission proportional to u(x, t), a relevant quantity is <«(0,0)u(x, /)>

(58)

and its F o u r i e r t r a n s f o r m S(q, u), % , u>) = ^ p jjdxdtei{"-"){u(0,

0)u(x, /)>.

(59)

This correlation function description is an idealized model of real photon or neutron s c a t t e r i n g . We have not yet been able to make an exact c a l culation of this dynamic quantity in the s p i r i t of the functional integral for equilibrium quantities.

11

DYNAMICS AND S T A T I S T I C A L MECHANICS O F . . .

Therefore, we now rely completely on the domainwall model (again for low temperature). Considering a particular point x, then the phenomenological picture we have is that for a while u(x, t) is approximately + UQ + a cosu>gt, then along comes a domain wall flipping the displacement to - «o + «' cosojo/, and so on. Here, a cosw0t is a small amplitude oscillation with o>0- (2 \A\/m)112. These domain walls have random spacings and random velocities, according to the distribution discussed in Sec. IV. Some important features of the frequency spectrum can be found as follows: Assume that over a correlation length X,., we can ap-

3543

proximate (M(0, 0)U(X, t)) by (K(0, 0)U(Q, t)), then

S(g,u>)cc^3l jyte-""(u(0,0)u(0,

/)>,

(60)

with ox(q) an approximate spatial transform over a correlation length. But by the convolution theorem, the a) transform of the correlation function is just the power spectrum (w(0, OJ)«*(0, u)), where M ( 0 , W ) = ^ - f dte-iu'u(0,t). "11

(61)

•'-BO

This Fourier transform in our model becomes

where the tt are the random arrival times of the domain walls. As usual, the average of the sum of random phased terms in K(0, OJ) is negligible, but not so for ), which is found to be M

W ^ ( T

V

\

W

]

1

W

+

/ « j [Bln(a.- Wt Hfc-/,.,)] «\

/ D .W. V

(a>-
The average is to be carried out over the distribution of arrival intervals [tlH - t{) domain walls. This distribution is calculated as follows: (i) At x =0, walls move in from right and left, (ii) The number reaching x = 0 from one side between 0 and t and having velocity v is the number lying in the interval l=vt. Thus, JV>W=f vtn„(v)dv, Jo

H9{v)=^e-">'e*mi',t".

(65)

The number arriving from both sides between 0 and / is f° ve-^o'2'2

where rD1 = (Z/0^mi)e-eB'>p,

dv = r~ ,

(66)

(67)

(iii) The probability P(t) that no domain wall has yet reached x - 0 up to time t obeys dp__dn_ J_ dt~ dt ~~tD ' whence P(t)=(l/tD)e-"t'>

\17 2

1(0+Wo) 8

/ „ .

and this is the probability distribution to be used for the intervals in (63). We then must calculate quantities like f'dt

Jo tDe

-tit* suVW

2tD

~17~~ 1+W*5 *

From (63), in the spirit of (60), we find

(64)

where n„(v) is the average number of walls per unit length having velocity v. From Sec. IV,

N(t) = ^retE"p

/a![stotM+Wj)Ur^)r\

/„,_

(68)

+?

Vi+4(u,-w 0 m + i+4(JVw,)tf)J (69)

Here az is a mean-square thermal amplitude of the phonons with frequency u>0 (as specified by q). We see that the spectral function contains not only the expected peaks at the phonon frequency o>0, but also a "central peak" whose height increases exponentially with inverse temperature as tD increases. This central peak is a manifestation of the strongly nonlinear domain-wall-type of displacement field, not of coupling to entropy or hydrodynamic modes. It is tempting to say that this "central peak" is that seen in computer simulation experiments, *'5 or in actual neutron scattering experiments. However, we can only say that it is provocative, for several reasons. On the one hand, one might argue that a phase transition takes place in this one-dimensional model at Tc = 0, since there correlation lengths become infinite; if that is so, the central

162 3544

KRUMHANSL

AND J .

peak i s appearing above t h e " t r a n s i t i o n . " On the other hand, if one could solve the analogous t h r e e dimensional model, and obtain finite Tc, it may well turn out that the tunneling approximation given here for one dimension may be applicable only to the low-temperature regime T« Tj in that case, the relevance of domains to the experimental c e n t r a l peak, T ? Tc would be uncertain. Of course, it i s t h e region T>TC where the "central peak" question i s most interesting. 1 B VI. SUMMARY AND DISCUSSION We have studied thermodynamic and some d y namic p r o p e r t i e s of a one-dimensional model s y s tem whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used t o study displacive phase t r a n s i t i o n s . - By studying the classical equations of motion, we find important solutions (domain walls) which cannot be r e p r e s e n t e d effectively by the usual phonon p e r t u r bation expansions. The thermodynamic p r o p e r t i e s of this system can be calculated exactly by functional integral methods. M No H a r t r e e o r decoupling approximations a r e made nor i s a t e m p e r a t u r e d e pendence of the Hamiltonian introduced artifically. At low t e m p e r a t u r e , the thermodynamic behavior a g r e e s with that found from a phenomenological model in which both phonons and domain walls a r e included a s elementary excitations. We then show that equal-time correlation functions calculated by both functional integral and phenomenological methods agree, and that the dynamic correlation functions (calculated only phenomenologically) exhibit a spectrum with both phonon peaks and a central peak due to domain-wall motion. Much r e m a i n s to be done to examine the extent to which the ideas discussed h e r e apply to r e a l systems, and how they r e l a t e to or a r e in c o n t r a diction with conventional t h e o r i e s . Nonetheless, it s e e m s that such features a s clusters (i. e . , r e gions bounded by domain walls) which appear in computer simulations of model systems, the " c e n t r a l p e a k , " and the consistency of exact and p h e nomenological thermodynamic calculations i s encouraging. However, we also must note that the lack of any general methodology for discussing the finite-temperature behavior of extended nonlinear s y s t e m s p r e s e n t s a formidable obstacle to the p o s sible extension to higher-dimensional s y s t e m s , o r to do exact dynamics at finite t e m p e r a t u r e . Finally, we r e c o r d a few speculative ideas, which may be worth further development. F i r s t , if these domain walls a r e p r e s e n t in the l o w - t e m p e r ature phase of psuedo-one-dimensional c r y s t a l s which have undergone P e i e r l s transitions, the P e i e r l s energy gap in those walls could go to z e r o , the m a t e r i a l becoming locally metallic. One could then have a distribution of conducting sheets (walls)

R.

SCHRIEFFER

11

in an insulating matrix; the low-frequency e l e c t r i cal p r o p e r t i e s and optical p r o p e r t i e s would not be simply related, a s in a homogeneous m e d i u m . Second, t h e r e is the question of whether a soft mode going to z e r o frequency is the exact condition for a s t r u c t u r a l p h a s e transition. T h i s question cannot be answered properly until adequate d y n a m ic extensions of the analysis h e r e can be m a d e . On the other hand, the functional integral a n a l y s i s is suggestive that t h e r e i s a t e m p e r a t u r e range in which the collective dynamic behavior will change from that of oscillations in either of two wells to that of a single nonlinear oscillator, whose period becomes very long in the transition region. In our model, this would occur in t h e region of F i g . 1(b), when the effective m a s s m* = m(c%/lzK%Tz) i s such that the lowest eigenvalue of (30) lies n e a r the s a d dle point of the potential. Thus, while it i s not c e r tain that a phase transition will occur exactly at the t e m p e r a t u r e where the soft mode frequency goes to zero, it i s very likely to be n e a r b y . We appreciate the advice and c o m m e n t s of c o l leagues, p a r t i c u l a r l y T. Lubensky and J . W. W i l kins. One of u s (J. A. K . ) gratefully acknowledges the hospitality of the University of Pennsylvania during a sabbatic visit and support by the Mary Amanda Wood Grant for this work during that period. APPENDIX: PHONON FREE ENERGY FOR VIBRATIONS IN WELL The dispersion relation (14) is (Al) The density of s t a t e s in q space i s L/ir, whence t h e density of states in to i s dn

dw

(A2)

nc0(^-2\A\/m)in'

It may be rewritten in t e r m s of N= L/l, normalization

->Umim=(UAI/m>l'*

Will/

N

and a Debye

(A3)

leads to d

"- = N l dui irc0 ( u E - 2 U l / m ) 1 / 2

(A4)

u>la = 2\A\/m

(A5)

and + ir*cl/?

The free energy p e r oscillator is F(u>) = KBT ln(l - e " u ' * ' r ) in the classical limit KT» ergy i s then

KBThi(Hw/KBT)

Ku>. The total free en-

11

DYNAMICS AND S T A T I S T I C A L

Substituting from above, after some tedious algebra, we find

MECHANICS O F . . .

The first term identifies with the lowest oscillator level in the functional integral, while the logarithmic term is given by (35) and the s 0 term of (34), when H = l, yields -KBTiXaZp -N/Ss„)=NKBTln{Co/2iiKBTl).

In the limit I- 0, which i s in the spirit of the functional integral method used, (TTC 0 //) 2 » 2 \A \/m and sec" 1 - |ir; so

*Work supported in part by the V. S. Atomic Energy Commission under Contract No. ATG.l-D-3161, Technical Report No. C00-3161-27. ^Permanent address: Dept. of Physics, Cornell University, Ithaca, N. Y. 14853. *W. Cochran, Adv. Phys. 9, 387 (1960). 2 G. J. Coombs and R. A. Cowley, J. Phys. C 6, 121 (1973); R. A. Cowley and G. J. Coombs, ibid. £, 143 (1973); R. A. Cowley and A. D. Bruce, ibid. 6, 2422 (1973). 3 J. F. Scott, Rev. Mod. Phys. 4JL, 83 (1974). 4 T. Schneider and E. Stoll, Phys. Rev. Lett. 31., 1254 (1973). 5 S. Aubry and R. Pick, presented at "International Conference on Ferroelectricity," Edinburgh, 1973 (unpublished). S Y. Ishibashi and Y. Takagi, J. Phys. Soc. Jpn. 33, 1 (1972). 7 F. Schwabl, in Proceedings of NATO Advanced Study Institute on Anharmonic Lattices, Structural Transitions, and Melting, Geilo, Norway, 1973 (to be published). 8 J. Feder, Solid State Comm. 9, 2021 (1971).

3545

(A9)

Note added in proof. We can now relate this problem to several of the mathematically similar situations in both magnetism and q>* field theory, as we shall discuss in a separate Comment. Comparison with 1-D computer simulation has been possible, and following formal methods developed in these related areas we have carried through exact 2-D, and mean-field 2-D and 3-D solutions.

B

H. Takahasi, J. Phys. Soc. Jpn. JjS, 1685 (1961). D. J. Scalapino, M. Sears, and R. S. Ferrell, Phys. Rev. B £ , 3409 (1972). "Y. Onedera, Prog. Theor. Phys. 44, 1477 (1970). IZ M. Abromowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). 13 M. Kac and E. Helfand, J. Math. Phys. J , 1078 (1963). "During the course of this work, we recognized some similarity between our model problem and various nonlinear-field theory studies [cf. S. Weinberg, Phys. Rev. D.2, 3357 (1974); L. Dolan and R. Jackiw, ibid. 9, 3320 (1974)). We were also aware of various nonlinear wave studies [for a review, see Nonlinear Waves, edited by S. Leibovich and A. R. Seebass (Cornell U. P . , Ithaca, N. Y., (1974)] in which both spatially localized (e. g., "solitons") and extended types of solutions are found. We have also received a communication from C. M, Varma who studies the connection between soliton waves and the "central peak" problem. ,s The exact integration yields f instead of 1 - ^ ; also, m$ is multiplied by f. 16 See Note Added in Proof. 10

164 PHYSICAL

REVIEW

B

VOLUME

18, NUMBER

8

15

OCTOBER

1978

Brownian motion of a domain wall and the diffusion constants Y. Wada'and J. R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received 6 June 1978) We have studied interactions between a domain wall and phonons in a one-dimensional-model system of a structurally unstable lattice with a double-well local potential and nearest-neighbor coupling. We find that a nonlinear effect in the interacting phonon amplitudes gives rise to a Brownian-like motion of isolated domain walls at low temperatures as well as higher harmonic generation of transmitted and reflected phonons. When there is a domain wall at rest, it was known that the linearized equation of motion has three types of independent solutions: "translation mode," "amplitude oscillation" of the domain wall, and propagating "phonons." In the second-order approximation, these modes interact. An incoming phonon produces a translation of the wall, giving rise to its Brownian motion. The magnitude of the translation is computed together with the amplitude and phase shift of the higher harmonics. We estimate the diffusion constant of walls, using the fluctuation-dissipation theorem and the thermal average over the phonons, to be D — 0.516 l)7, where / is the lattice spacing and ai0 is the frequency of small oscillation of the ion (with mass m) around u,„ a minimum point of an isolated double-well potential.

I. INTRODUCTION

In recent years, considerable progress has been made in the study of nonlinear equations in various fields of physics. Of particular importance are nonlinear equations which admit large-amplitude solitary-wave solutions. The usual perturbation calculations, which postulate small-amplitude deviations, are inadequate to obtain such solutions. One example of solitary-wave solution is the domain wall which was discussed by Krumhansl and one of the present authors (JRS)1 (hereafter KS). They studied thermodynamics and some dynamic properties of a one-dimensional-model system whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used to study displacive phase transitions. It is the model system of a structurally unstable lattice, having a double-well local potential and nearest-neighbor coupling. A domain-wall excitation is characterized by the following distribution of ions at each lattice point. Over nearly all the semi-infinite region of the left-hand side of the domain wall, the ions are uniformly at the potential minimum of the same side of the lattice point. Nearly all ions on the right-hand side of the wall occupy the minimum of the other side. The transition takes place through a domain wall of a finite thickness. The wall can move with a constant velocity which is less than the velocity c 0 of low-amplitude sound waves (phonons). At low temperature, the thermodynamic function was evaluated in two ways: exactly by the functional integral methods and phenomenologically by regarding both phonons and domain walls as elementary excitations. The agreement of the results of the two evaluations confirmed the idea that phonons and domain-wall excitations play an 18

important role. In addition to this static property, a number of dynamic properties could depend strongly on the presence of domain walls. The dynamic correlation function is one of them. A possible relation between the development of a "central peak" in the dynamic correlation function and the distribution of domain wall was pointed out. It was later shown by Bishop, Domany, and Krumhansl2 that the appearance of phonons and domain walls as elementary excitations survives the passage from classical to quantum mechanics. Molecular dynamics computer simulations have been carried out for this problem. Koehler, Bishop, Krumhansl, and Schrieffer3 showed the motion of the linear chain as a series of snapshots in time for various temperatures. The results showed pronounced domain structure at low temperature— a feature emphasized in the analysis of KS. They further found that there is a phonon dressing of domain walls. Domain-wall potential energy EDP, determined with the help of correlation length, turned out to be smaller than the bare value, estimated by KS, at low temperature. It was also observed that the domain walls do not keep moving freely between collisions with other walls. Rather, isolated walls appear to undergo Brownian-like motion. Aubry4 also performed molecular dynamics calculations to obtain the dynamic correlation function of the one-dimensional system. He pointed out the possiblity that the domain walls participate in developing the central peak. Simulations of a twodimensional system were done by Schneider and Stoll.5 They found clustering phenomenon to be very important in this case. This clustering phenomenon is equivalent to the appearance of twodimensional domain structure. 3897

i 1978 The American Physical Society

165 3898

Y . WADA AND J . It. S C H R I E F F E R

The s a m e model has been investigated by e l e mentary-particle p h y s i c i s t s . It i s called a (p1 model in one-plus-one dimension. The domain-wall solution describes the Fourier t r a n s f o r m of the form factor of an elementary p a r t i c l e . It was shown that, 6 when t h e r e is a domain wall at r e s t , the equation of motion linearized with respect to the deviation from the domain-wall solution has three types of independent solutions: a " t r a n s l a tion m o d e , " and "amplitude oscillation" of the domain wall, and propagating "phonons." The t r a n s lation mode is a Goldstone mode 7 which a r i s e s due to the breaking of translation s y m m e t r y by the p r e s e n c e of the domain wall. The a m p l i t u d e - o s cillation mode gives r i s e to a modification of the form of the domain wall. It costs some energy and thus corresponds to an excited state in the language of elementary-particle physics. The phonon modes a r e the continuum solutions which except in the vicinity of the domain wall r e s e m b l e the l i n e a r ized solutions of the original equation in the a b sence of domain walls. By examining the a s y m p totic form of the solutions for x—±°°, one finds that the phonons suffer only a phase shift when p a s sing through the domain wall. In this s e n s e , the phonons do not interact with the domain wall. This is the reason why Koehler et al.3 suggested that the observed diffusive wall motion might be a r e s u l t of effects nonlinear in the phonon amplitudes, a s well as possible d i s c r e t e - l a t t i c e effects. The purpose of this study is to show that the nonlinear effect, in fact, can give r i s e to the Brownian-like motion of the domain walls. When a wave packet of phonons, with typical frequency ui, i s incident on a domain wall, the nonlinear effect gene r a t e s two harmonics: one with almost vanishing frequency and the other with 2u>. The former e x cites the translation mode, shifting the domain wall a finite distance. The wall moves as if the effective interaction with the incoming phonon is attractive. Since phonons make collisions with the wall randomly, it behaves as a Brownian particle. The other component with the frequency 2u> i s p a r tially transmitted and partially reflected with a phase shift. The diffusive wall motion is characterized by a diffusion constant. It can be estimated using the fluctuation-dissipation theorem. If we denote the position of the domain wall at time t by 6(t), the diffusion constant D i s given by 6 D = <[BMP>/2t,

(1.1)

for the one-dimensional system. Here the angular brackets mean an average over the distribution of phonons. The time t should be long in comparison with the " m e a n - free t i m e " of the collisions. Since we a r e interested only in the lowest o r d e r of non-

18

linear contribution to 6(/), we may ignore nonlinear t e r m s in the phonon-distribution function which contribute to h i g h e r - o r d e r c o r r e c t i o n s to the diffusion constant. This is a good approximation when the number of excited phonons is s m a l l at low t e m p e r a t u r e . The diffusion constant, thus obtained, t u r n s out to be proportional to T 2 , indicating that the diffusive wall motion is a s e c o n d - o r d e r effect. In Sec. II, we review the one-dimensional model of KS, the linearization of the equation of motion when t h e r e is a domain wall, and the t h r e e types of the solutions. In Sec. HI, nonlinear collisions of a phonon with a domain wall a r e investigated. The shift of the domain wall, the coefficients of t r a n s m i s s i o n and reflection of t h e higher h a r m o n ics and their phase shift a r e calculated as a function of the wave number of the incoming phonon. In Sec. IV, the statistical mechanics of phonons in the presence of a domain wall i s developed. Using the fluctuation-dissipation t h e o r e m , it is applied to obtain the diffusion constant. Finally,in Sec. V, we d i s c u s s the obtained r e s u l t s and some r e maining problems. II. MODEL AND LINEARIZED SOLUTIONS The Hamiltonian proposed by KS has a continuum representation „

fdx\p(xf

A

. ,2

B

. ,4

mcl/du\2~) (2.1)

in the displacive c a s e . Here I i s the lattice s p a c ing and x locates an ion with m a s s m. The fields u(x) and p(x) a r e the displacement and momentum of the displacing ion at x with r e s p e c t to s o m e heavy ion or reference lattice. A and B a r e p a r a m e t e r s which c h a r a c t e r i z e the local potential. KS discussed the case of a structurally unstable l a t tice, taking A = - \A\ and B > 0. The local potential i s a double-well potential with minima at u = ±u0 = ±(\A\/B)l/*.

(2.2)

The classical equation of motion for t h e d i s p l a c e ment field u(x) which follows from (2.1) i s mB2u/at2+Au

+ Bu3 -mc&2u/ax2

= 0.

(2.3)

This has a domain-wall solution u = u„ t a n h [ ( * - i r f ) / / 2 | ] ,

(2.4)

with

? = m(cl-tn/\A\.

(2.5)

Suppose a domain wall i s located at x = 0 and not moving. Deviation from the domain-wall solution would be small at low t e m p e r a t u r e s . If we put

166 B R O W M A N MOTION OF A DOMAIN WALL A N D T H E . . .

18

u(*,r) = u 0 tanh(x/V2U +
(2.6)

the deviation ip satisfies the equation

where 42 = mc%/ \A | and u2a = 2 \A | /m. It has been shown* that the eigenvalue problem

3899

where Q = / 2 g | 0 . Because the solutions
y_\2(*)d*=!/2«0

(2.11a)

f~
(2.11b)

- c^dVrf* 2 + [w» - (3u>g/2)cosh2(x//2«0)lcp = w V (2.8)

f_m
= 2ir(l+Q*)(4 +

Q*)6(q-q'),

has eigenvalues and eigenfunctions: 2

u> = 0, «•-*«•.,

2

r

^ 0 W = cosh- (x/v 2g,

(2.11c) (2.9a)

while the completeness relation has the form 3 3 47^
< Pl W = s i n h ( 7 | - ) C o s h - » ^ | i - y

, f"

(2.9b) 2

2

w ^ + c f o ^ , ( Wf >0),

(2.9c)

<*>,(*) = ^ » [ 3 t a n h ^ ) -3v^940tanh^-~-\-l -292|^. These three solutions have simple physical i n terpretations. When 2 = 0 solution yields the "translation mode" of the domain wall. It i s a Goldstone mode7 which arises due to the breaking of symmetry by the presence of the domain wall. When = (2 - Q 2 - 3 i Q ) / ( 2 -Q2+ 3iQ), (2.10) A( 9 ) = 2 t a n - 1 [ 3 Q / ( Q 2 - 2 ) ] )

dq(p,(x)
_.,..

•V_.2,r(l + V)(4 + (? 2 )- 6 (

,. }

-

. , . (

'

}

The relations (2.11c) and (2.12) are proved in Appendix A. Let us suppose that, at time t = to«0, a wave packet of phonon i s at x « 0 moving in the direction of the domain wall. It gives the initial condition for 0: *<*»') =£• 2 > X S » ( < 7 * - < » „ ' + e „ > >

( 2 - 13 )

when t**t0. Here u>4>0 and a , i s nonvanishing only at q > 0. L i s the length of the system. The quantity a , gives the amplitude of the incoming phonon which we assume to be small in comparison with unity. Since a , would be a localized function around a typical wave number q, the initial form (2.13) can be rewritten i/i(x, t) = Im exp[t(?x - 5J„<+e„)]/(x - vjt - * 0 ) , (2.14a) where

/(*)=f £ a « ex Pl'te -*)*]>

(2.14b)

and a>a and 0, are the values of u>c and 6a at q = q. The quantity v i s the group velocity v, =

(do>,/dq),*=>c%i/ut.

(2.15)

The quantity x0 i s defined by x0 =

-(dejdq),^.

(2.16)

It specifies the initial position of the wave packet at * 0 + V o The first-order approximation of the equation of motion for ip (2.7) i s

3900

Y. WADA AND J. R. S C H R I E F F E R i

p , -! *«J 1 1cosh=(V/2« 0 ) fc = 0. * £ - £ g # i +T [«§

e?"

" "'

(2.17) We can immediately write down the solution which satisfies the initial condition (2.13)

4.0

:>•

3.0

-

1.0

-

"0

<2>»=
(2.19)

After transmitting through the domain wall, the function tj)1 takes the form at x» 0 and t» 0 <MM)~f- £

i

>

i

i

\Ax,/f0

-

Ax 2 /f 0

"~~-^-J!^—.

i

i

0.5

(2.18)

where the function $ 0 is defined by

i

2.0

*,(*, O - 3 - I - £ { « . * . < * > e x p [ i ( - u . / + «„)] 2ii -«>*<*) e x p ^ - e , ) ] } ,

i

18

i

i

i

i

i

i

—r

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Q

FIG. 1. Shift of the phonon wave packet due to the interaction with a domain wall as a function of the wave number <j. Solid curve is the shift of the incident packet A*, and the dashed curve that of the higher harmonics A#2. The abscissa is Q=Vir?| 0 .

We expand I/I2 in terms of the
a , s i n [ ? * - w , i + e,+ A( ? )] Q

M*. 0 = 8 a[t)
0k(t)
= Im exp{t[g* - ay + 5, + A(g) ]} x

/(*-f,'-*o-A*i)t

Substitution into (3.2) and use of the orthonormality of the
where ttAfa)\

•-ft").

A*

_ 6 v f 2 U 2 + g !l ) 4 + 5<J-2 + S 4

(3.3)

(2.20)

(2.21)


Here we have used a notation (2.22)

9Bus_f ~4/2mtB

dxtanh I

(3.4a)

9Bu„ tanh g^*-S*- 27^r (7fe)^' 0 /.^

The shift Ax, is illustrated in Fig. 1.

(3.4b)

m. NONLINEAR PROCESSES When the amplitude of the incident phonon a , is small in comparison with unity, the deviation from the domain-wall solution can be written in a form of power series in a Mx,t) = l(x,t) +

M*,t)+-

(3.1)

where the function tp2 i s bilinear in a a ' s . Substituting into the equation of motion (2.7), we equate the bilinear terms on both sides to obtain

afy c i»+*2 -$lj? N

-iu>20coSh*(*/v-2yfe

2

= - (ZBuJm) tanh(x//2£<,) # 2 .

ipl(x,t) = j

(3.2)

*/]d*tanh (j^-j

cpfft,

(3.4c)

where K= /2k£0. These equations should be integrated so that the (3's and dB/dt's are zero at t = t0, since the initial condition is completely satisfied by #,. Instead of the above condition, we introduce adiabatic hypothesis, taking 20— -«° and assuming that the nonlinear interaction on the right-hand sides of Eqs. (3.4a)-(3.4c) have gradually increased from zero as if there i s an additional factor exp et, € > 0. To simplify the integration, we introduce the Fourier transform of
du>e'-"""\(x,w),

(3.5a)

u'

x(*,w) = - ^ r £«,«,,'{5(^-^-a>„.)(?>„^ 0 . exp^e.+ e,.)] + 6(u) + w, + u. ( ,.)0J<|)*.exp[-»(e, + e,,)] -28(u)-a)o + u)„.)0o0J.exp[t(e„-e,.)l}.

(3.5b)

168 BROWNIAN MOTION OF A DOMAIN WALL AND T H E . . .

18

$„(t) = ZBuJm(\ + K2)(4 + if 2)

Then, integration of (3.4) gives

x exp[-tu>i/(u> + i«)2],

B t)=

dxdwiwl

^ ^rJ'.L

^k)

3901

(3.6a) x exp{-i(i)//[(ui + ie) i ! - w2.]}.

x

*>

(3.6c)

x exp[- iu) + U.Y - Wo\}, (3.6b)

Substituting the expression (3.5b) of x into (3.6a), we can carry out the u> integration with the help of the S functions to obtain

»•'''°-I«^ ? .S.°•°••£ J '^^(7fe)>•'4^ | -'*^ W ^(^X.'t'a^ l '^^') .•frw-w ^(""•- ( :;; , :;,^> t - > )

The integrand of the g and «' integrals is nonsingular except for the last term in the parentheses at «,= ui,.. The contribution of these nonsingular terms corresponds to a forced oscillation of the translation mode by the incident phonon. The frequency OJ^ + O) , is not the eigenfrequency of the translation mode. The forced oscillation would r e lax quickly as soon as the incident wave packet passes. On the other hand, the singularity of the integral does give rise to the generation of the translation mode to which we shall confine our discussions hereafter. After some calculations, which are given in Appendix B, we can show

//*

tanh

(7ife)

<w

/-l(lD,-li)..)ttt(8,-t,.)

\

' eXp V(4+5Q 2 + J 4 + 5Qz +"Q2 9ug y> a y . l i t Q ' ) "V"2V- ^u)?(4 + 5Q2 + Q4) Q

9«g v 01(2 + 9") " 4 7 2 ^ 1 *-* 1 + Q2 '

(3.10)

°

2/2sinh[7KQ+Q')/2] ' ' where Q' = -f2q' £0. At the singular point q' ~q, the x integral in (3.7) takes the form

_/2t(Q-Q')(? 8 U2+Q') 4 + 5Q* + Q4

(3.9)

It is important to note that the singularity is not of second order but of first order, since the numerator (3.9) vanishes linearly there. Contribution of the region in the vicinity of the singularity can be evaluated as follows (at t»0):

The quantity 0O is related to the shift of the domain wall 6, which is given by

i=£H&

9u2 ' 4X

a2.(2 + Q2) i+e2 •

(3.11)

The domain wall moves in the negative direction by a finite distance. The discussion so far can be reiterated for the case of a phonon coming in from the positive x side of the domain wall. The shift of the wall is again given by (3.11) with the opposite sign. It moves in the positive direction by the same distance. The interaction between the domain wall and the incoming phonon is effectively attractive. The motion of the domain wall should look like a random walk in very low temperature where the phonons collide with the wall in a random way. If

3902

Y. WADA I

0.7

I

1

I

1

I

AND

I I

1

-

0.6 g a J =°

s

J. R.

0.5 0.4

-

°- 3

-

0.2 0.1 i

i

i

i

1

1

1

1

1

"0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Q FIG. 2. Magnitude of the translation of the domain wall 115 | as a function of the wave number of the incident phonong. The abscissa is
the phonon i s almost coherent, we may put al = aMq-Q).

(3.12)

Equation (3.11) then becomes e=-(9nSaV8ir)(2 + 9 V f l + 9»),

(3.13)

18

SCHRIEFFER

which is shown in Fig. 2. Substitution of the expression (3.5b) for x into (3.6b) gives Q^t) which i s composed of t h r e e t e r m s . One of them involves (pQ,>, another <$>*$*,, and the last <pa„+u>0, ±ieY -fui 2 , which does not vanish since OJ^ » u>0. The denominator of the last t e r m (u^ - ai a , + ie) 2 - {ui2, does not e i t h e r , if the incident phonon wave packet has enough coherence so that the distribution of the function cta i s well localized. The wave n u m b e r s q and q' should be close to each other in t h e localized region and \<jiq-iji0. T h e r e f o r e , the m o tion of the amplitude oscillation i s the forced o s cillation by the incoming phonon, since the i n t e grand of (3.6b) is nonsingular. It would r e l a x very rapidly. Substituting (3.5b) into (3.6c), we again obtain three t e r m s for 0k(t) which involve 0 „ 0 , . , $*<(>*•, and <£>„<£*., respectively. The above argument can be applied to the last t e r m . Contribution by t h e singularity of the integrand i s due to the first two t e r m s which can be written

xexpH^^+f'^'*).

(3.14)

We may denote the last t e r m of 2, given by (3.3), by ^J", since it will be shown to be related to higher harmonics. With the help of (3.14), it takes the form

xexp{-i(wa

+ wa.)t

+

He^eil.)

ji[A(q)

+

+

^(q')]}/[(UK2H4

+

K!)}i/:!

x[(u>„ + u v + i € ) 2 - u ) J l , where F(ktq,q') F(k,q,q')--

(3-15)

is defined by -i X . dx tanh(y//2^ 0 )(p t y„
(3.16)

The integrationn is c a r r i e d out in the Appendix C to show shoi that F(k,q,q') J'dx

tanh (-£A

is real and

P
2

-|P4

+

7 < 8 2

3

4

2

3P 5 P - %Pt P 2 - I P 2 _ f P , P 2 P

-\Pip3+£r$P*

+ %P*P4--&Pe))

, (3.17a)

18

BROW M A N MOTION OF A DOMAIN WALL AND T H E . . .

where P i s the principal part and the various quantities are defined by P n =Q" + Q",+ if",

P = Q+Q' + K,

Singularities in the k integral of (3.15) are at

_

to

-0.3

0 2

i

i

i

i

i

-

0

^ ,

i

6.5

1.6

i 1.5

i 2.6

i 2.5

i 3.0

i 3.5

i 4.0

FIG. 3. Amplitude of the transmitted higher harmonics t(q). The abscissa is (?=/2?towhere K=J2K$0.

(3.24)

3* -u0sinh[-n(Q-iK)]

t(Q)

(Q 2 + 4 ) 3 / 2 ^ ( ^ + l ) ( 4 ^ 2 + 13) 1 ' 2 ' 2

xexp{iifx-j(ai,+ io1I.)< + j(e„+ 0„.) + !t[A(«)+Ato)+Ata')]}. (3.19) The phase function is expanded at the point q = q' = q. The wave number K can be approximated by -q + q'-q),

(3.20a)

when can be derived with the help of (3.18). Here 7? and vK are defined by ic = (lc 0 ) ^ - o . ? ) " 2 , » . » ( * * . / < * « ) , . , .

0.20b)

The transmitted wave (3.19) finally takes the form *?(*» ' ) * - • - - <(3)

R e ex

P <*{** - 2 ^

+ 2

(3.25) The function t(q) i s illustrated in Fig. 3. The transmitted wave is shifted forward by the amount A*2 = 3/2S 0 [(2 + $ 2 )/(4 + 5Q2 + (?) + 2Q( 2 +/T 2 )//?(4 + 5if a + X 4 ) ] , (3.26) which is shown in Fig. 1. The reflected wave at the region with a large negative x can be obtained by the k integral in (3.15) with an additional contour of a large s e m i circle on the lower half of the complex * plane. The residue at the other p o l e - K - i e g i v e s the contribution. Iterating the discussion above, we find il>l"(x, t), . . . ~ r { q ) R e exp(i{- KX - 2 ( 5 / + 26,

^«

+ [±A(K) + A(J)1}).

+ [!*(*)+ A » ) ] } ) x [/(- (v Jv,)x x

7 4.5

The amplitude of the transmitted wave t(q) i s found to be

IJncJZ*

K^K+(VJVK)(q

r

-

-0.4 -0.5

(3.18)

x- •.

i

-0.1 ior

«=(l/c 0 )[(o. ( ,+ a ) 0 . ) 2 - ^ ] ' / 2 .

We shall first discuss the transmitted wave at the region with a large positive x. The k integral in (3.15) can be closed by a contour of a large semicircle on the upper half of complex k plane. The residue at the pole K+it gives the only contribution to the integral which takes the form

i

0.0

(« = 2 , 3 , 4 , 5 ) . (3.17b)

* = ±(« + t e ) ,

1

3903

t/t(f ,/«,)*• " l>,< - *o - A * 2 ) F,

- v„t -xB-

Ax 2 )] 2 , (3.27)

(3.21) where t(q)= ~(Z/iu0ilt)F(-K,q,q),

(3.22a)

A*2 = (iA*,) -(^/2t>«)tdA( K )/dK].. l ! .

(3.22b)

The constant B was replaced by mcl/ulU, using (2.2) and a relation below (2.7). After a lengthy calculation, making use of (3.17a) and a relation ff2 = 4 $ 2 + 1 2 , we obtain • /"....

,./

//TI, » *

a

2 3 4V27rU4 4 V 2 T T | 0 ( 4 ++ g( ? 2)) 3

(3.23)

"0 0.1 0 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Q FIG. 4. Amplitude of the reflected higher harmonics r(q). The abscissa is $ = / 2 ' ^ 0 .

Y. WADA AND J. R.

3904

with the amplitude of the reflected wave 2

r(

3/2

,_, 3ff (g + 4) «)=«0Biiiiitir(9+iX)]Jf(5,+ i)(45*+i3)1'» •

we can show that the transformation of variables in a functional integral (/>(*i)

(3.28)

£(*„),"(* x )

IV. STATISTICAL MECHANICS AND DIFFUSION CONSTANT We have studied in Sec. Ill the elementary proc e s s of collision between a domain wall and a wave packet of phonon, which i s nonlinear in the amplitude of the phonon. It gives rise to a motion of the domain wall. It should be one of the motives of the Brownian-like motion. However, the result of the simulation 3 shows that the wave trains of phonons are so long and overlapping that the concept of a wave packet hardly applies even at the low temperature T = kBTB/A2- 0.117. It is necessary to generalize the discussion of the elementary proc e s s e s to the case that the phonon has an arbitrary distribution. We are interested in the lowest-order effect of the nonlinear processes, that i s , the effect to second order in the phonon amplitude. This means that it is unnecessary to take into account nonlinear effect in the phonon distribution. The latter contributes a term, to the diffusion constant, which i s of the same order a s that due to higherorder elementary processes. In this sense, our discussion is valid only at low temperature. The displacement and momentum fields u(x) and p(x) are expanded in terms of the
u tahh

- (A»/>i> • --PaypK,-

J J dp{Xi)du{Xi)

y, 1/ /22

/ 3

/

33 \\ '* "

/

jtsM.

3

[ 2 L ( l + / f 2 ) ( 4 + A' 2 )] 1/ + ^T

(4.1a) \

1 / 2

(4.1b)

(4.2) Pi'Plt

/>?=/>-»•

Introducing the real and imaginary parts of qt and p, by qt=q„. + iqa,

Pt = P&+iPu>

lidp^dp^dq^dq^. 7»

(4.3)

The proof of (4.3) i s given in Appendix D. Substitution of (4.1a) and (4.1b) into Hamiltonian (2.1) gives

•S&^E-ikl').

(4-4)

where higher-order terms in the qt are neglected, since the nonlinear effect in the phonon distribution is irrelevant. The constant EDf i s a domain wall potential energy introduced by KS. Relation (4.4) is derived in Appendix E. Corresponding to (2.13), initial condition for the fields, # r ) a n d p ( x ) , a U = / 0 , can be written

/

3 Y"

[2L(l+/f 2 )(4+if 2•2\ll/2 )) ; > p(x,l^ =

(4.5a)

mi(x,ta)

3 V"

(*) ZP'ip*[2Hl+K'){4

i (4 + K )Y'>-

5b)

Distribution of the p, and qt i s determined by Hamiltonian (4.4). In comparison with (4.1a) and (4.1b), we have chosen that q0 = po=0 since the domain wall i s at the origin and not moving initially. We can write the first-order solution of (2.17) as

Since these fields are real, the coefficients satisfy the relations

PS = P„,

• • 9M>
(b.xydpdipldqjlqi

x

+

2

.-loQi-

_ _

y/2

qk
=

(•i

I

• » hh) 1

«(*„))

is canonical and

It is illustrated in Fig. 4.

u(x)

18

SCHRIEFFER

+ a* exp ('

Z

a t 2

'° jJ <M*)

fo,, exp(-tu),<) + gjk exp(iuy) ]
since the function i/\ i s real. The initial conditions (4.5a) and (4.5) are satisfied if we require

172 18

BROWNIAN MOTION O F A DOMAIN WALL A N D T H E . . .

(4.7) Here were some trivial factors expfciu),/,,) and exp(±iz'V~3a>(/0) which were removed by a proper redefinition of the a,. The second-order equation (3.2) is solved making use of the expansion (3.3). The unknown ^(t) satisfies (3.4a) where the function ^ on the right-hand side is now given by (4.6). Introducing the notations

F0(k,k')=l-iJ"dxUah x

3905

-fi~ (p0
[(l + /(7(4 +Jr)(l+if' )(4 + if' )]

, (4.8a)

p / M _ f'dxtanhl?* ]

' ,. .. ^217-1/4 ^ ' we can rewrite (3.4a),

(4.8b)

§ * = - 87|*p ( f SFo^*')(a,e-'u»'+aV,u»')(«^-i"»''+«V"k") 00 »»• +

( S J" 2:Foi(*)^"'^0'/2 + a ^' V S , , , 0 " 2 fV ( "" + a-V'"*,]) •

(4.9)

Use is made of the fact that /

dxtBnh

( 4 -l°)

7f£~ W i = °-

We will again use the adiabatic hypothesis in order to integrate (4.9). If we put

^,t)-.fjtlfy2e-^^-^i^,

(4.11)

the integration of (4.9) gives

+ B f 0 .»(i|2« + «„*)*]

j.

(4.12)

Thermal average of various quantities are easily evaluated with the help of Hamiltonian (4.4). Since ==0. = 0, (a*ak,)=(lkBT/mut)6„k.,

(oj>=0,
(4.13)

Thermal average of the translation of the domain wall vanishes; <6(0>=-vr240/«0 = 0 , since it is the sum of the term F0(-k,k) which vanishes as (3.8) shows. Fluctuation of the translation (6*(t)) turns out to be

<»«>,>-(i^),E(^~),[i»K*«...or*i*K-«r.fli']

(4.14)

3906

18

Y. WADA AND J. R. SCHRIEFFER

As discussed in Sec. I, the diffusive wall motion is characterized by a diffusion constant D. According to the fluctuation-dissipation theorem, the knowledge of the fluctuation (6(/)2) gives us the magnitude of the diffusion constant by relation (1.1):

bility to give such a contribution. It can be transformed as

,

dkdk>{{F9{k,k')f+ [F„(ft,-*')'} J a

2

Z>=<6(0 >/2<,

when t is long in comparison with the mean free time. We are therefore interested in a component of (6(02) which increases linearly with time. In other words, a component of (6(f)2) which diverges linearly with e - 0 is important. With the help of (4.11), we have |4.(OJ,0I 2 = (W 2 +£ 2 )- 2 .

o

(4.17) 2

Since F„(k,ft')vanishes as (ft - ft') atft~ft',as (3.8) shows, this term in the numerator of the integrand can be neglected. On the other hand, we obtain with the help of (3.8) and (4.8a),

(4.18) Substitution into (4.17) gives

f

dk (1 +Jf*)a(4 + tf2)2u>2(du),/rfft):

_ / J ^ l V l / - ^ t ( 2 ^2- ^ ) 2 u>«(4u> _3u>*)

81 , , 4ir

The definite integral is evaluated to give a constant W z + Z S ) - ^ *0.160. The diffusion constant turns out to be }

(6(0 >. D=- Zt -=(6(tr)/ (2/° dte")

2

/tJ\M

/: \mM2io2/

= 0.516cV 2 ('-¥sV.

dui.

( " > • - " . ) '

f(u.,.-o. ll ) 2 + e 2 P

f'dx&x'-l)*

(4.19)

gives rise to a translation of the domain wall, the effective interaction being attractive. The distance of the translation is rather insensitive to the wave number of the phonon. It decreases as the wave number increases. The value at the limit of short wavelength is one half of that of the limit of long wavelength. The order of magnitude of the translation is 161 ~«„a2 as given in (3.13). With the help of (2.13) and (3.12), we can find

•UW)2) = 0.160^,

J

(4.16)

It is now evident that among the four terms in (4.15) the term with |*(tt) t - uk.,t) I2 has a possi-

2

/9uUkBT\

|) 2 dx <* •

in

(5.1)

which gives

(4.20)

It is worth remarking that we have obtained the diffusion constant making use of the same mechanism as the finite translation of the domain wall was derived in the discussion of Sec. III. The singularity of energy denominator is somewhat suppressed by the vanishing numerator. It led to the finite domain-wall translation in Sec. III. It gives in this section the fluctuation of the translation which is proportional to t. V. DISCUSSION We have studied interaction between a phonon wave packet and domain wall. The second-order process in the amplitude of the incoming phonon

- f ifidx

(5.2)

In order to estimate the translation, we may calculate the produce of mean-square deviation of ions and the width of the packet. Its ratio with «2 gives |6 The higher harmonic generation is another interesting phenomenon. This does not happen in a situation with no domain walls. It is essential that the wave number of phonons may not be conserved during the generation process because of the presence of the domain wall. Suppose we try to observe the transmitted waves at x>0. We will observe two signals. The stronger one is that of the incident phonon. It reaches x at a time"(x - x0 - Ax,)/ua which is earlier by the amount &xt/va than in a system without the domain wall. The

18

BROWNIAN

MOTION

OF A D O M A I N

weaker signal is due to the higher harmonic. Its "total intensity" /_" (ipp'fdx is of the order of Jliftdx/ul as shown with the help of (3.21), (3.25), and (2.20). It arrives at #when t = x/vK- (x0 + Ax2)/ va. Since &xl>&x2, the stronger signal is observed first if the distance x is so close that xK(b.xl - Ax2)/(/f _2^), since its group velocity v, is faster than v„. We can observe also the reflected wave at x<0. In this case, there is only one signal due to the higher harmonic. It arrives at - 1 * | when t = \x\/v, - {x0+£^x1)/v03m'2. In order to understand why the diffusion constant does not depend on the wall thickness $0, we consider the dependence on «0. It is important to note that the eigenvalue problem (2.8) does not involve « 0 as a parameter. Neither does the Hamiltonian (4.4). Therefore, the first-order solution (4.6) does not depend on u0. Equation (3.2) shows that the second-order solution is proportional to Bu0 ^u'0l, leading to the relation BB{t) " a j 1 . The translation of the wall 6 is thus proportional to tt02. Finally, the diffusion constant turns out to be inversely proportional to

WALL

AND

THE...

3907

u*. The independence on the wall thickness is characteristic to the second-order nonlinear effect as the temperature dependence which is T2. Measurement of the diffusion constant should be more feasible than the elementary process since we do not need an ideal configuration of one domain wall and the constant is related to a dissipation process. The difficulty we have to meet is to obtain a good one-dimensional sample. The temperature and wall thickness dependence of the diffusion constant should be the most interesting properties to be measured. It is also important to obtain quantitative information on the magnitude of the diffusion constant with the help of molecular-dynamics computer simulation. It will clearly show if the nonlinear effect, discussed here, is really the most dominant mechanism which gives rise to the Brownian-like motion of domain walls. Much remains to be done even when the nonlinear effect would turn out to be the dominant mechanism. Correction due to quantum effects in dynamics and statistics would have to be discussed at low temperature. It would be important to derive a Langevin equation for the motion of domain walls. It would give us information about the random force acting on the domain wall. We should be able to derive the fluctuation-dissipation theorem of the second kind which is a relation between the diffusion constant and fluctuation of the random force. The phonon dressing of domain walls, discovered by the simulation work,3 should also be an interesting problem. It should have a close relationship with the nonlinear process discussed in this paper. We would finally like to point out a new feature of our discussions. Unlike the usual Brownian particles, our domain walls and phonons are made of the common constituents—the ions. The difference is only in the manner of motion of the constituents. This situation is more universal in solid-state physics than that corresponding to the usual Brownian particles. Magnetic domain walls and spin waves are another example. One can find other examples in so-called soliton phenomena of various problems. Our work shows a possibility of investigating the interactions between the soliton and its surroundings when both are composed of the same constituents. Namely, we can use "normal modes" when there is a soliton, in which the Goldstone mode plays an important role. It is very likely that the method developed in this work has a wide range of application. ACKNOWLEDGMENTS

We appreciate the advice and comments of colleagues, particularly T. Lubensky for his critical reading of the manuscript. One of us (Y.W.) grate-

175 Y. WADA AND J . R . S C H R I E F F E R

3908

fully acknowledges the hospitality of the University of Pennsylvania during a sabbatical visit, and he would like to thank the Japan Society for the Promotion of Science for the travel grant. This work was supported in part by the NSF under Contract No. DMR77-23420.

18

APPENDIX A: ORTHONORMALITY RELATION (2.11c) AND COMPLETENESS RELATION (2.12)

With the help of the definition of
/" ( 4 ) + 9P/,(3) -(3Q 2 + 3
where P = Q' - Q. We define the quantities Ip(n) by 7,(2n)=2 f

rfycosPy(tanh2"y_

(XI)

Substitution of (A5) into (A4) gives 7,(2) = - P i r / s i n h ( i f f P ) ,

1), (A2)

-'n

[p(2n +1) = 2 f rfy sinPvttanh 8 "- 1 )) _ 1). Integrating by p a r t s , we can obtain a recursion formula /,(2n) = 7 A ( 2 n - 2 ) - 2 / ( 2 « - l ) -[P/(2n-l)]lp(2n-l),

+ 9QQ' + 4) cosPy + 3P(QQ'+ 2) s i n P y ] ,

(A3)

7,(3) = - 2 / P + (1 - i P ' W s i n h ^ T r P ) , /,(4) = ( - | P + iP 3 )7r/sinh(i7TP) , 7,(5) = - 2 / P + ( l - | P 2 + i P 4 ) i r / s i n h ( i i r P ) , 7,(6) = ( - | | P + i P 3 -

(A6)

^PWAohiliP), 2

7,(7) = - 2 / P + ( l - . i § P - , l P 4

If(2n + 1) = I„(2n - 1) + (P/2n)7,(2«).

-iflj/sinhfitf).

This is solved to give 7,(2) = _ 2 _ P 7 , ( 1 ) ,

The integrals of the trigonometric functions a r e

7,(3) = ^ + ( 1 - ^ ) 7 , ( 1 ) , I dycosPy=lim

/,(5) — • J i ' + ^ + ( l - i i J , + i/' , )/,(l). +

(A7)

,A4\

(_»P+.iP»_-i-i>X(l)>

( l _ i £ 7 *+ ^ P * - ^ P % ( l ) .

The integral 7,(1) i s given by 9 /,(1)=—(-1+J

J d y s i n P y = lim f rive"" sinPy = P / P , -'o <—o» ->o where P is the principal part. Substituting (A5)-(A7) into (Al), we find the right-hand side cancels each other except the integral of cosPy which gives

7,(7) . - - P + i j - . - J y J * +

I dy e~,y cosPy = n6(P).

J"dx
dy cosPy sech y J

(A8)

2 7T ""P+Sinh(ijp)-



~Qi3 ta»hiik)+ + 3

3

With the help of (2.9c), the third term on the left-hand side of (2.12) can be rewritten

^ ( v f e ) -9 tanh(ik)

tanh

(7frJ +3 ]

^H(7fe)[3ta^(7fe)-1]-tanh(7fe)[3tanh2(7fe)-1]

176 18

BROWNIAN MOTION OF A DOMAIN WALL AND THE...

3909

If x>x', (*<*'), the q integral along a large semicircle on the upper (lower) half of the complex q plane can be added to form a closed contour. The poles, atq = i/J2t0,J2i/i0,(-i/f2$0,-J5i/U make contribution to give

- tanh(vfe) tanh (^r 0 ) ±tanh(7fe) t a n h fe)[ U n h (7fr3 - ^(vfe)]}

3 3 = 6(* - x') - jj^jr
(A9)

APPENDIX B: DERIVATION OF EQ. (3.8) With the help of the definition of cp,(x) (2.9c), we can rewrite the integral (3.8) f" dx tanh(x/V7|>0
APPENDIX C: DERIVATION OF EQ. (3.17a) Calculation is lengthy but straightforward. We shall write here an intermediate expression for the integral (3.17a) y""rfxtanh(x//I«> t

+ |Ps)/<,(4) + (9 + f Pt + \P\ + i ^ + fP 2 - fPaP2+ 1^)7,(3) + 3(-P-J' 2 P + P 5 - i P P | + P^3 + iP 4 P-P 5 )/ > (2) -[l+P2 + iP22-iPi

+ bP3 +

+ 2VT«0t(8 + 5P2 + \Pt-\P\-%P\

$P2P-iP3):i]lp{l)} - 9P»

-fP'P, + |P 4 + iPP3P2 - ^P3 + 6/2"«oi( P,+P2P3 - P5) f dy cos Py .

- iPlP* + \P,P* ~hP*)f~ "y •»"» Py

(Bl)

3910

Y. WADA AND J .. RR. .S C H R I E F F E R APPENDIX D: PROOF OF EQ. (4.3)

space, the completeness relation (2.12) r e a d s

We introduce a lattice space in o r d e r to c a r r y out the functional integral in a thermal average over the phonons. The length of the system L is divided into AT p a r t s by the points (xl,x2,... ,xs) which are distributed uniformly with the distance Ax; L = JVAx. The 'distance Ax would physically be identical with /, but mathematically we may a s sume that Ax is independent of I. In this lattice

p(xt)

fjrf/>(x,)rfu (x{)=J

•••

\P0pi

'"

18

Tranformation of integration v a r i a b l e s generally gives a relation

P(xK)\

PkiPkz'"

I

u(xK)

MxJ

T

V?o<7i • • •

x dpadp1dq0dq1 JJ^dp^dp^^q^dq^

9*i?*2

,

(D2)

Both Jacobians have the same value since the transformation (4.1a) i s identical with (4.1b)except for a constant term u 0 tanh(x)/-/2£ 0 ). It is easy to find that M(x„n

/

3

\1/2

I

3

\1/2

/ J

3

V /a

, J

3

Y/2

M

SfTTQ * \?mj

/ ^

i

*

\

l( 2>

/wp,(t|)

Rey,^)

Re
s

-Imb{x,)

• "U(i + K*foZx*/2W U(i

^fizVh/z)^

With the help of the completeness relation (Dl), we obtain (Ax:)"1 j 2 _ jji

raaspoa ed _

0

0

(Ax)"

0

0

0 1

0

= (Ax)"".

(D3)

(Ax)"1 • • •

APPENDIX E: DERIVATION OF EQ. (4.4) With the help of the expansion of p(x) (4.1b), it is easily seen

where the orthogonality relations (2.11a)-(2.11c) a r e used with 27r6(fe-fe') = I 6 M , . Neglecting h i g h e r - o r d e r t e r m s in the qu similarly obtain

(E2) we can

18

BROWNIAN

MOTION

/_" dxu(xf = / [ dx{«*tanh'{^y

OF A DOMAIN

2«0tanh^

WALL

AND THE...

3911

[ ( ^ j \ * i +

? [2i(i + ^K4 + ^)FJ} + ^ + ^ + , 5 k*'2'

(E3)

/ " dx u(xY <*f ' dxL% tanh4 (gj^j + 4„3tantf

(^Jj^Jjl,lqi
£ [ a i ( 1 + 5J* +/ ^ )]1/2 ]

+6

"°2tanhZ (TyfevkJ ^ ^ fefej"'1*1

Substitution of (E3)-(E5) into the Hamiltonian (2.1) shows that the linear terms with respect to qx and qk cancel each other, since ($A)2uBtzah(g^-J

? [a^i+^W'Jl' ( E 4 )

Similarly, the coefficient of qtqk term is proportional to (iB)Gulf'

+ (iB)4«30tanh3^-)

+

dx t a n h 2 ( ^ - ) . p ^

rwri f" J^d(pl dcpt %J..dx dx where the last term is obtained by a partial integration. The cross terms in the quadratic form also vanish. For example, the q0qk term has a coefficient proportional to

^xtanh

2

^)^

wcg +

=

^[3/>tanh2(7?I>^

r"^d
T^JJX dx dx

(E10)

which vanishes with the help of the relation

\A1 •\3f

dxt&nh2(^-J
d*
2

(Ell)

(E7)

Thus we also have the coefficient of q\ term which vanishes with the help of the relation dV„/d^ = -*>0/«? + (3 1 p 0 /^)tanh a (x/2| 0 ).

(lB)6ulf~

dxUmtf(JL-)
(E8)

This relation gives the coefficient of q% term

(^K/^-anh^^^/X^)2 ,^fjx<JJI^l.

3/f

(E9)

(E12)

The coefficient of ?,?,. term is proportional to

179 Y. WADA AND J. R.

3912

SCHRIEFFER

IB

,/W£W8/>£)£i) = /2? 0 {45/ p (6) + 4 5 i ' / p ( 5 ) - [ 5 4 + 45KK"+18(if2 + if' 2 )]f / ,(4)-[39P+18i 3 /i:if'+3Uf 3 + A:'3)]/J,(3) + [21 + 12{K* + Kn) + 24KK'+ 3(K* +Ka)KK'+ + [12P+3lK3+K'3) + 2S2~Z0[12-21KK'-6(J(t

+ 6PKK' +

1

SK2K'1]IP(2)

lPK1K'2]Ip(l)}

+ Kn) + 3<J(*+K'2)KK'+1lK2Ka] 2

1 + 2f2Z0(16P-12PKK'+lPK*K" ) 2

J

f" dxsii>Px-2S2irl0[:WK' o

f"dxcosPx -'o

+ 2KK'(J{*+K'2) + 2l-K3K'!>]6(P),

(E13)

where P=K+K'. Substituting (A5)-(A7) into (E13) and using (E2), we obtain

3 / ~ d ^ a n h 2 ( — W * + £ 2 y*"d*(^

(E14)

e constant term in the Hamiltonian is

= £„

rf*/l2

(E15)

where EDP is the domain-wall potential energy introduced by KS. Substituting the various relations obtained above into (2.1), we finally find

(E16)

•Permanent address: Department of Physics, University of Tokyo, Tokyo, Japan, ' j . A. Krumhansl and J. R. Schrieffer, Phys. Rev. B 1^, 3535 (1975). 2 A. R. Bishop, E. Domany, and J. A. Krumhansl, Phys. Rev. B 14, 2966 (1976). 3 T. R. Koehler, A. R. Bishop, J. A. Krumhansl, and J. R. Schrieffer, Solid State Commun. V7, 1515 (1975). *S. Aubry, J. Chem. Phys. 64, 3392 (1976). 'T. Schneider and E. Stoll, Phys. Rev. B 13, 1216 (1976).

6

J. Goldstone and R. Jacklw, Phys. Rev. D U , 1486 (1975); R. F. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D 10; 4130 (1974). 7 J. Goldstone, Nuovo Clmento 19_, 154 (1961). B See, for instance, Foundations of Modem Physics, Vol. 6, and R. Kubo, In Statistical Physics edited by M. Toda and R. Kubo (Iwanaml, Tokyo, 1972), p. 185 (In Japanese). 'Tables of Integral Transforms, edited by A. Erdelyi (McGraw-Hill, New York, 1954), Vol. 1, p. 30.

180 PHYSICAL REVIEW B

VOLUME 22, NUMBER 4

15 AUGUST 1980

Soliton excitations in polyacetylene W, P. Su,* J, R. Schrieffer,* and A. J. Heeger l)('l>tinineni oj Physics. Utthvrsiiy <>}'Pennsylvania. Philmlrlphia. Pftuisyluinia 11/7*1 (Received 3 December 1979) A theoretical analysis of the excitation spectrum of long-chain polyenes is presented. Because of the twofold degeneracy of the ground state of the dimerized chain, elementary excitations corresponding to topological solitons are obtained. The solitons can have three charge states Q = 0. + c. The neutral soliton has spin one-half while the charged solitons have spin zero. One electronic state is localized at the gap center for each soliton or antisoliton present. The soliton's energy of formation, length, mass, activation energy for motion, and electronic properties are calculated. These results are compared with experiment.

1. INTRODUCTION Because of the degenerate ground state of the bond-alternated polyene chain, one expects excitations to exist in the form of a topological soliton, or moving domain wall. In an earlier paper, 1 we suggested the possibility of such soliton formation in the conjugated organic polymer (CH)*, polyacetylene, and we outlined some of the implied experimental consequences. Related theoretical studies have been carried out in a Ginzburg-Landau 2 scheme as well as in a continuum approximation. 3-5 Magnetic-resonance studies of undoped irans(CH)„ have shown the existence of highly mobile neutral magnetic defects in the polymer chain.*"10 Since a charge neutral soliton in a long-chain polyene would have an unpaired spin localized in the wall, it was suggested that the motionally narrowed spin resonance might arise from bond-alternation domain walls induced upon isomerization. Moreover, analysis of the transport" and magnetic properties 8 in lightly doped samples led to the suggestion that doping may proceed through formation of charged domain walls. Thus, the concept of soliton formation and the detailed evaluation of properties are of direct interest to the continuing development of this novel class of conducting polymers. Our theory assumes the existence of bond alternation along the polymer chain (i.e., alternating "single" and "double" bonds) and relatively weak interchain coupling (i.e., quasi-one-dimensional behavior). There is experimental evidence that both these assumptions are valid in (CH)„. Raman studies' 2 1 3 have detected the splitting of the carbon-carbon bond stretch vibrations resulting from bond alternation consistent with the normal mode analysis.14 Weak interchain coupling is suggested by the observation of considerable anisotropy in physical properties after polymer orientation.' 5 Moreover, recent nuclearmagnetic-resonance studies have' 6 demonstrated 22

one-dimensional electronic spin diffusion in the polymer, both undoped and heavily doped. In this paper we present a detailed theory of soliton formation in long-chain polyenes in the one-electron approximation. The model Hamiltonian is described in Sec. II and solved for the perfect dimerized chain in Sec, Ml. Soliton excitations and their properties are derived in Sec. IV, and doping effects are considered briefly in Sec. V. Section VI includes a brief comparison with experimental results.

II. MODEL HAMILTONIAN To simplify our description of (CH) X , we assume, as described above, that to lowest order one can neglect interchain electron hybridization. Also, we assume that the o- electrons can be treated in the adiabatic approximation since the gap between the 0, leading to a short ("double") bond between groups n — I and n, and a long ("single") bond between n and n + 1. These displacements have the reverse signs if double and single bonds are interchanged, as in Fig. K b ) . Let a be the equilibrium spacing between the v coordinates of successive CH groups in the undimerized structure (i.e., all bond orders equal to 1.5). a is approximately 1.40x-/3/2 A = 1.22 A. Because of symmetry, the remaining five coordinates for each CH group are not coupled into the dimerization structure to first order in phonon coordinates and will be neglected. Since the change of bond length due to dimeriza2099

©1980 The American Physical Society

181 22

W. P. SU, J. R. SCHRIEFFER, AND A. J. HEEGER

2100

H

(a)

H

H

A

shown in Fig. 2. We note that Eq. (2.2) is the standard form of the electron-phonon coupling in metals. Finally, the kinetic energy of nuclear motion is given by

H i

\

£* = 7 %Miil

.

(2.3)

where M is the total mass of the CH group. The model Hamiltonian is the sum of these energies

H

(b)

www I

9

H

H u fl

M

ns

I

+ \lK(u„^-u„)2

If u,., 9 un+,7

H

H

H

FIG. I. Perfectly dimerized /VflHS-polyacetylene showing the dimerization coordinate u„ for the two degenerate ground states: A phase [Fig. 1(a)] and B phase (Fig. Kb)). tion is small, of order 0.08 A, we assume that the obonding energy can be expanded to second order about the undimerized state,

E,~\%K(um

+ \2M«l

-o2

(2.1)

where K is the effective a spring constant. We assume the n electrons (pz orbitals) can be treated in the tight-binding or Hiickel-type approximation with a hopping integral iH+\,„ which can be expanded to first order about the undimerized state (2.2)

=/0-«("n+|-"n)

i0 is the hopping integral for the undimerized chain and a is the electron-lattice displacement (phonon) coupling constant. Model calculations' 7 " indicate that this linear approximation is valid since the bond-length changes are small. A sketch of l,+L„ is

(2.5)

lp*.u,]=6XH

We note that as-(CH) x can be treated in a similar manner using the configuration coordinate u„ shown in Fig. 3. For the pattern shown in Fig. 3(a), ;/„ and u„+2 are negative and «„_| and u„+l are positive, while the signs are reversed for the configuration in Fig. 3(b). Missing from / / are the explicit Coulomb interactions between n electrons. They are partially included by using screened values of fo and a. We also treat Coulomb interactions between the charged soliton and impurities. However, if the »r-7r Coulomb interactions are very strong, our approach is invalid and one should start from the "large- U " limit for the ir electrons. Finally, we note that H should be supplemented with the constraint of fixed total length of the chain, since we assume that a is the equilibrium lattice spacing of the undimerized, including ir bonding.

H

\

//

'o-a d w u j

=c

C^,

H

H

H

c=c \

\ C-

//

vc—C

\un"1 /

/ H

H

/

// «vA

\ C

\

H

\

V,C—CNfn'i

-C

H

H

/ C—C

FIG. 2. Nearest-neighbor hopping integral as a function of the dimerization coordinate difTerence. The undimerized chain has » n + | — u„ = 0, while this difTerence is equal to 2u0 and— 2t<0 for a single and a double bond, respectively.

(2.4)

where <„, and c„s create and destroy IT electrons of spin s ( ± -i-) on the n th CH group. The c' and c satisfy Fermi anticommutation relations while u„ and p„ " Mil „ satisfy canonical commutation relations

H

n+1

•

n

n

H

y

\

H Un-1

\ /

H H

H

c=c

/

H

c

/ u^,\ / u V 2 \ / H H H H ' H H FIG. 3. Two dimerization structures for c/.v-polyacclylene, the dimerization coordinates are perpendicular to the C-H bonds, as shown. The A and B structures are not degenerate in energy as they are for the nam phase.

182 SOLITON EXCITATIONS IN POLY ACETYLENE

22

III. PERFECTLY DIMERIZED CHAIN

2101

The conduction- and valence-band states are given by

We begin by investigating the perfectly dimerized chain in the Born-Oppenheimer approximation, where the configuration coordinates n„ are constrained to be (3.1)

H„-(-!)"«

The kinetic energy EM will be treated later. By evaluating the ground-state energy as a function of (/, one can determine the values of the displacement amplitude u which minimize the total energy E$. By symmetry, if (i0 minimized £ 0 , so does — u0- Hence we expect a twofold degenerate ground state corresponding to the two-bondings configurations shown in Figs. 1(a) and Kb). For fixed «, the first two terms of Eq. (2.4) are //'(«)«-2l'o+(-l)"2au] + w„+l.,)+2NKu2

(3.4b) where „ is the ir orbital on the // th group. We use periodic boundary conditions on a chain of N CH groups. The chain length is L - Na. The operators (»" and ck for the zero-order bands are related to the operators cns by I

(3.5b)

.

(3.2) ks

Since the hopping potential is periodic with a period 2a, we use a reduced-zone scheme with zone boundaries at ±7r/2o. In Fig. 4 the zero-order (« = 0 ) bands are shown, where. v

Ek = -2lacoska 0c

--e

£* = + 2t0coska = tk

(3.5a)

< k.'

By inverting these transformations //" can be expressed in the k representation

ns

xU-l+uc„

(3.4a)

t

(3.3) .

+ 4ai/ sinArt ( t ^ <•*• + <-t*M,) ] + 2NKu 7 (3.6) Finally, Hd can be brought to diagonal form by defining operators «£ = «*<•<£ +£*<& •

(3.7a)

a'to-aWto-Pici,

(3.7b)

•

where | a j 2 + |0»|2«l

.

(3.8)

By inverting Eq. (3.7) and requiring that / / ' ' be diagonal in the a operators, one finds H'-%Ek(nl,-n&)

+ 2NKu2

•

(3-9)

ks

where Ek=(it

+ Ak2)in

(3.10)

and the gap parameter A* is defined by bk = 4aii sinAn .

(3.11)

+

with n = o ( j being the occupation number operator, as usual. The transformation coefficients, for ak = real positive, are given by

FIG. 4. v bands for undimerized iram-(CH), with a rone scheme for a unit cell having two CU groups. When dimerization is included, gaps occur at A = ±TT/2U, with C and V becoming the conduction and valence bands of a semiconductor.

(3.12a)

«* =

t-k

Pk =

1L Ek

sgnA

(3.12b)

Thus, the single-particle energy eigenstates of the

.

183 W. P. SU, J. R. SCHRIEFFER, AND A. J. HEEGER

2102

The ground state of the chain for exactly one ir electron per atom on average is given as a function of u by Eq. (3.9) where nks -= I and «& = 0. One has

perfectly dimerized lattice are

«=

i 2

1

**•=

2

1/2

,+

t| ,+ t]

X* -fsgnA

,,/2

1-f

I 2

xf .

f,(») = - 2 2 f t + 2 i V A V

(3.13a) 1/2

.

(3.I5)

1/2

-i

X* - sgnA 2

22

where A is summed over the first Brillouin zone of the dimerized lattice — ir/2a S t S ir/2o with the two terms in Eq. (3.15) corresponding to the ir and oenergies. By replacing the sum by an integral, one obtains

(3.13b)

with eigenvalues £ » • - - £ * . Ei- + Ek . (3.14) Phases are chosen so that tylc—'X£'c as Ak — 0. i

Eg(u)

2

= -1L

f* °[(2t0coska)

= _ i^° IT

+

C'n [ 1 _ ( 1 _ ,2)

sin2ka

{4ausmka):iY'2dk+2NKu2 Jl/2 (,k

a +

(3.16a)

2NKu2

(3.16b)

J

»

4/V/0 _2^ _ NKliz1 '-E(l-z2) +-

(3.16c)

where £ ( 1 - z 2 ) is the elliptic integral and (3.17) For small z, £(l-z2)sl+j(ln4/UI-{)z2+

•• •

(3.18)

Therefore, the n energy is always more negative than the a energy and £ 0 ( « ) has a local maximum at // = 0 , corresponding to the Peierls theorem. Taking the value" K = 2 1 eV/A 2 , the ir bandwidths 20 as H/lt = 4fo= 10 eV, and choosing a so that £o has a minimum when the dimerization gap is £&- = 4f|

Po<£) =

2nl(/Ek/ifk\

W7r)|£-|/l(4/02 0, otherwise,

= 1.40 eV, we find E0(u) shown in Fig. 5, with minima at ±M0 where « 0 —0.04 A. With this choice of parameters, we find a = 4.1 eV/A, a value comparable to that given by quantum chemical calculations." The bond-length change due to dimerization is + VJ u 0 = ±0.073 A, close to the value used by Baughman el al.2' in their calculations. The condensation energy per site in eV is ^

= -^[£„(«0)-£-„(0)] = -0.015

(3.19)

The density of states per spin of the perfect dimerized bands is

-£2)(£2-A2)],/2.

fE0(u)/N

A=

;2'o.

(3.20)

PoM

•E c /N FIG. 5. Born-Oppenheimer energy per CH group plotted as a function of the staggered dimerization coordinate II = (— 1 )"II„. The two stable minima correspond to •4(-t-i/0) and B( — t<0) phases.

-2tr

-2t, 0 2t,

2tr

FIG. 6. One-electron density of states for A or B phase /ra;M-
184 SOLITON EXCITATIONS IN POLY ACETYLENE

22

has

where the gap parameter A is defined as

-2-

(3.21)

A = A„ / 2 < ,-4aM 0 = 2/, •

p0(E) is plotted in Fig. 6. Since our treatment of the soliton requires the Green's function C'(a) for the perfect dimerized lattice, we next derive this quantity. Using the eigenfunction expansion one has

**(»)*^*("') C.U)-? k,\ w-Ef + il

H-H'odd.

.

.

[Ur^-ioW-A.2)]"2' <#"£(«>-

-co

[(4tt-2)V/2

0 < Ul < A |oi| > 2 / 0 .

This expression is consistent with the density-of-states Eq. (3.20) expression since (3.23b)

(/ 0 + ' i ) + ' 'l

A S |b*| « 2 / „

[U2-4/02)U2-A2)]l/2'

(3.23a)

„ ' ( £ • ) , — -§£"£ 2lmG„"„(£) .

1+fl 1-fl

(3.26)

We note for later use that

Thus, combining Eqs. (3.8), (3.22), and (3.23), one

2
(3.25)

For the diagonal element, one finds

1 )"-/8j<'*°7V/V

= - / ( - l )>;(«)

:. n - n even (3.24) ih)2-Ek2' 2Eklat+iBk^-\)"]2t^'k',"-"', (u) + / ' S ) 2 - t V

(3.22)

= [a*+//8 t (-l)"](<*'"*A/A 7

*l>'k(n ) = liak(-

N f (w + _ J v

C,U).

where 8 — 0 + . 0" for A = c. v, respectively, and the wave functions
2103

1/2

-'o

1-fl 1+fl

1/2

(3.27a)

2/,SS|O)|SS2»D

C'.+i (<")' 1/2

('o + ' i ) + ' i

B+\ 8-1

B- I + /. fl + 1

1/2

I =S2/,. or \w\ > 2»0

(3.27b)

•n orbitals do not overlap, one has,

where

(>ti\l>*\kv) =>2&Mxsin2ka/Ek /i odd

f

2£*-
(3.30)

where Mx is the dipole matrix element

and (| = y A , as defined above. Equations (3.27) and (3.28) apply for n odd while the same expressions hold with t\ replaced by - / i for n even. As an application of these expressions we calculate the frequency dependence of the optical-absorption rate. In the long-wavelength limit, the dielectric tensor is

m/cu2

.

(3.28)

Mx--

/

J

((x.v.z)-$-4>(x+a.v.z)iPr o.v

.

(3.31)

Inserting Eq. (3.30) into Eq. (3.29), one finds

l m e „ ( c u ) oc

2 [(4/„) -
(3.32)

(3.29)

where n is the n electron density and me is the electron mass. For polarization of the electric field along the chain, j = x, and assuming next-nearest-neighbor

This expression is plotted in Fig. 7 in comparison with experiment. The experimental results are obtained from analysis of the inelastic electron scattering data of Ritsko ei al.IS on nonoriented (C\\)x films.

185 W. P. SU, J. R. SCHRIEFFER, AND A. J. HEEGER

2104 1

1

25

1

1

-

~

J

4

1

1 1 1 1

13

7

1

i

19

10

1

-

22

16

1

1 1

,— Theory

-

r -

i

-

\ \

-

\

-

Observed

r Y/^i

0

j

i

2

p ^ —

4

6

i

i

i

-

8 10

w (eV)

FIG. 7. Infrared absorption plotted as a function of photon energy for undoped iiom-ICH),. The experimental results are obtained from analysis of the inelastic electronscattering data of Ritsko ci til. (Ref. 15) on nonoriented (CH)„ films.

IV. SOLITON EXCITATION As we discussed above, the classical ground state of the dimerized chain is twofold degenerate with "on — ± ( - 1 )"»(>• Associated with this degeneracy, we expect there to exist an elementary excitation corresponding to a soliton. 22 It is useful to define an order parameter iji„ associated with any displacement u„ •K = ( - ! ) " « „ .

«„ =

(-!)>„

(4.1)

The two ground states are then defined by — uQ. A phase.
ua.

B phase.

(4.2)

Suppose that
22

and a soliton is created so that the left-hand portion of the chain is in the B phase. The system increases its energy not only because of the increase of local energy in the vicinity of the soliton but also because the left-hand boundary of the chain terminates Bphase material rather than .4-phase material. This end-effect energy shift is inconvenient since we want to treat the solitons as localizable elementary excitations. One scheme to eliminate the difficulty is to create an antisoliton ( S ) followed by a soliton ( 5 ) so that the material goes from A to B to A as one moves from left to right along the chain. If 5 and S are widely separated they do not interact and the energy change is twice the soliton creation energy Es. To simplify the numerical work, we found it convenient to calculate the energy of a single soliton in a finite chain for various shapes of the wall. Since the end effect is independent of wall shape, a calculation of the absolute energy of 5 + 5 for any single wall width serves to fix the zero of soliton energy when calculating the energy of a single soliton in a finite chain. This is the procedure we use. Alternatively, one can calculate the end-effect energy analytically by determining the energy difference between an A- and fl-phase end of a chain. To determine the minimum energy configuration which approaches the perfect A and B phases as n goes to + oo and - oo, respectively, we isolate a segment of the chain surrounding the soliton whose center is located at n = 0. Let the segment extend from n = - v to + v. We decompose the full BornOppenheimer Hamiltonian H into // = Ha + V

(4.3)

where the zero-order Hamiltonian H0 corresponds to perfect fi-phase displacements for n S — v and perfect /(-phase displacements for n ^ v. The hopping is defined to be zero between groups in the segment —v € n ^ v. Therefore, the hopping integrals defining H0 are /0-(-l)"~Vi. 0. -v^n
iK-v IIZSV.

(4.4)

Here for convenience, we have taken v to be odd. Physically, H0 describes 1v — 1 isolated atoms between the semi-infinite B and A chains. The perturbation V is the missing hopping in the isolated segment for any set of displacement order parameters •v*.

-U.-'«+(-i)"«(i»,ti + i )

(4.5)

-vS n < v The i>„'s are to be varied so as to minimize the total system energy.

186 SOLITON EXCITATIONS IN POLY ACETYLENE

22

A convenient method for determining the groundstate energy shift A£ of an extended system due to a localized perturbation V is the relation 23 A£

IT */-oo

Imlndet[l-G°U)]K
.

(4.6)

G° is the Green function in the absence of K, and n is the chemical potential, which is zero for our system. This method has the advantage that one need not diagonalize // to determine A£. Also, the determinant has a size limited by the spacial extent of V. In our case the determinant has dimension 2v + 1. We find good convergence for the soliton in (CH), with 2 v + 1 of order 41—61. To evaluate Eq. (4.6) we must determine G°. Since the H0 hopping breaks the chain into three noninteracting segments A

»S»

S —v < B n H-v

The determinant in Eq. site representation. Since and B, only the boundary ments enter, i.e., G"„ and and (4.11) one finds G?viu>) =

.

G° is block diagonal in these segments. In S, one has ,. - v < n.n'<

v .

(4.8)

n.n

m

GtJw)

(4.13)

where G^, and G^„+i are given by Eqs. (3.25) and (3.27). The integral in Eq. (4.6) was carried out numerically with the trial function /lo.

n S —v

— Hotanh(«//). "

^

v < n
•'•

(4.7)

< v .

C 0 , ( o ) ) = (l/a))8

(4.6) is evaluated in the V is zero in segments A limits of G" in these segG°„ _„. From Eqs. (4.10)

G°v.-,(">)

= GiA>

— »o-

II

2105

To determine G° in A, we consider a perfect /(-phase chain and place an infinite-site diagonal potential U on group v - 1 so that the chain to the right of this site is uncoupled from site v — 1 and thereby uncoupled from the rest of the chain. In segment A, G° satisfies

G°, = G", +Gi.^UG°

, ,

At" is plotted in Fig. 8 for three values of the energy gap £",, = 1.0, 1.4, and 2.0 eV. As discussed above, the zero of energy was shifted to give the correct answer for a widely separated soliton-antisoliton pair with / = 1. For the value 1.4 eV, the soliton formation energy is £"s = 0 . 4 2 eV and the width parameters / = 7. We have tried other trial functions with little change of energy and shape of the soliton. Thus, for (CH)„ the wall is quite diffuse and one expects small lattice

I I I I n.n'^v

.

(4.9)

I I I I I I I I

0.9

where G ' i s given by Eq. (3.24). Taking the limit V —* oo, one finds for the A segment G".-G*

(4.10)

,--

In the B segment, G° is determined in the same manner from a B-phase chain by placing an infinite potential on site — v + 1, G\=G"

7** Ct- +|G"

(4.11) where G is the Green's function for a B-phase chain, corresponding to l\ — — i\ in the expression for Gd. We note that as a consequence of symmetry, one has the relation. G°,

(4.12)

2 4 6 8 10 12 14 16 18 20 22 24 26

FIG. 8. Soliton energy HI) plotted as a function of an assumed half-width ' for several values of the energy gap /:.

187 22

W. P. SU, J. R. SCHRIEFFER, AND A. J. HEEGER

2106

periodicity effects in pinning the soliton. Preliminary calculations show that £s varies by roughly 0.002 eV as the center of the soliton moves between lattice sites. This indicates that relatively free translation of the soliton would occur in an otherwise perfect lattice, down to temperatures on the order of 20—40 °K. In passing we note that while the wall is diffuse, this does not imply that a Ginzburg-Landau-like theory involving a gradient expansion of the freeenergy functional is appropriate. Rather, the wall thickness to is of the order of the nonlocality distance £o — Kv//ir& and a more accurate analysis, including nonlocality and umklapp processes, must be carried out, as we have done here. To investigate the electronic structure of the soliton we have calculated the change in the density of states Ap(£) due to the presence of the soliton. The results are plotted in Fig. 9 and show a single sharp state <M'i) at f o = 0, the gap center, for each spin orientation. Below, we derive the explicit form of fodi). Since the total number of electronic states in the ir band is conserved (i.e., independent of l<Ji„ I ) . it follows that:

f~^bp(E)ctE = 0

(4.15)

From this relation and the fact that ^p(E) is a symmetric function of £, it follows that the valence band has a deficit of one-half a state for each spin, as does the conduction band. Since the valence band is fully occupied both with and without the soliton, the valence band is missing a total of one electron in the presence of the soliton. It follows that in a neutral soliton the missing electron occupies the state i>0. Note that the valence band remains spin paired while the electron in d>0 is spin unpaired. Thus, a neutral

-2t,

-2t,

2t,

soliton has spin one-half. As we discuss below, the low-energy-charged soliton states correspond to removing the unpaired electron from o or adding a second electron of opposite spin to <£0. Therefore, the spin of a charged soliton is zero. In summary, one has Qo = 0,

Qfte,

s0=T.

s±-0

(4.16)

It would appear that in forming a soliton we have violated Kramer's theorem, which requires that the spin of a system with an even number of electrons be an integer, and with an odd number be half an odd integer. Since the total number of electrons is conserved in creating a neutral soliton yet the soliton has spin ~, a compensating spin must occur somewhere else in the system. The situation is clear for a ring of N CH groups, where N is very large compared to /. Because of the single-valuedness of the order parameter, a soliton 5 which separates B and A phases must be followed by an antisoliton 5 which separates A and B phases, as illustrated in Fig. 10. In this figure the order parameter is plotted radially, positive outside and negative inside the ring. If 5 and S are widely separated, their interaction is exponentially small and they are independent excitations. However if S and 5 are neutral, they each have spin ~ and Kramers theorem is satisfied. This model also clarifies the counting of electronic states, since there must be an integer number of states in the valence band with and without S and S. Since for each spin direction S and S each remove one-half of a state from the valence band, one complete state is removed by S plus 5, satisfying the integer-state counting rule. This result also holds for the conduction band. For a finite chain discussed above, a similar argument holds if S or S are created far from the chain ends and are themselves widely separated. If only 5 or 5 is created, it can be shown that an extra spin of one-

2t,

FIG. 9. Changes of density of suites Ap(£) due to the presence of a soliton. The gap center state gives a 5 function of strength unity. This is compensated for by densities missing from the valence and conduction bands each integrating to one-half a state. In essence, the gap-center state is a nonbonding state between the bonding and antibonding bands.

FIG. 10. Soliton Sand antisoliton S occurring in a ring of (CH);,. The order parameter Ji is plotted radially.

188 SOLITON EXCITATIONS IN POLYACETYLENE

22

half is created or destroyed at a chain end, ensuring that Kramers theorem is satisfied.24 At each site n, the electron density missing from the valence band is exactly compensated by the den' sity \o(ii) I2 of the unpaired electron. This can be proved as follows. For any configuration of the chain, the local density of states p „ , ( £ ) satisfies the sum rule

£

p„„(£)rf£=]

Ap„„(£)rf£ + |<M»>l 2 = 0 •

n= 1 n= 5 n= f 1

0.125

(4.17)

which follows from the completeness of the eigenstates of H (( uV J ). [Note Eq. (4.15) follows from Eq. (4.17) by summing on n.] Therefore the change of p„„(£) due to the presence of the soliton integrates to zero. Furthermore, Ap„„(£) = Ap„„(- £ ) , so that we find the local compensation sum rule l\

2107

-2tn

2t,

FIG. 12. Same as Fig. 11 except results for « — I, 5, 11 are shown.

4>0in) of the gap center state satisfies (4.18) '»+I.„<M")+V-H.»+2<M" + 2 ) = 0

which proves that the missing electron density in the valence band is exactly compensated at each site by the 4>o electron density. Thus, a neutral soliton is both globally and locally charge neutral. An analogous relation of local charge compensation has been derived by Brazovskii4 in the continuum model. Since the energy of the system is the same if the occupancy of
by ±H<M»)I 2 .

Ap„„(£) for the valence band is shown in Fig. 11 for it = 0, 6, and 12, and for ;; = 1, 5, and 11 in Fig. 12. The smallness of Ap„„(£) for odd n is consistent with the fact that | # 0 ( ' ' ) l 2 vanishes for odd », as we now demonstrate. Since £n = 0, the wave function

(4.19)

.

where 'n + l.n

—

'n.«

•l0 +

(4.20)

(-\)"al*n+i+*„)

and i/<„ is given by Eq. (4.14). Since $<)(») for even and odd n are uncoupled, there are two linearly independent solutions, one of which decreases exponentially as n — ± oo, while the other diverges and is not normalizable. If the soliton is centered on II = 0 . ± 2 , . . . the normalizable state only involves even », while it only involves odd n for a soliton centered at ± 1, ± 3, . . . . The state is a linear combination of even and odd n if the soliton's center is between sites. In the present case, the soliton is centered on n = 0 and 0(") = <M — " )• For even n one finds 0(n + 2 ) = -

« + !,» 'n+2.n + l

0()("> "'n-l.n-2

-'i.o

*o(0)

'n.n-l

(4.21)

FIG. 11. Change in the local density of states Ap„„(fc) due to the presence of a soliton, plotted for the valenceband region —2i0< E <-2it. Symmetry ensures i\pB„(£") — Ap„„(— £'), so the conduction- and valence-band changes are mirror symmetric about the gap center. Not shown is the 8 function at the gap center of strength lo(«)l2. Results for a —0, 6, 12 are shown.

while 0{n) = 0 for odd it. < M O >s determined so that "in is normalized to unity. li>(ii) | 2 is plotted in Fig. 13, for / = 7 and / = 5, showing little variation with /. Roughly speaking, <£o(») behaves like a WKB state, tunneling through a barrier of height A and modulated by the zone-edge wave function for k = ± it 12a, 0(n) = — sech—cos-j-irn

.

In passing we note that Coulomb interactions

(4.22)

189 22

W. P. SU, J. R. SCHRIEFFER, AND A. J. HEEGER

2108

-t8

-16

-14

-12 -10

-B

14

-6

16

18

FIG. 13. Gap-center state probability density U 0 ( H ) |2 plotted for a soliton centered on n =0. Two soliton widths / = 5 and 7 iire considered. By symmetry, <£0(") =0 for odd /i if the soliton is centered on even n. between electrons split the <£o level for different occupancies in a nonintuitive fashion. 25 The mass Ms of the soliton can be determined by calculating the energy of a slowly moving domain wall, iK(/) =«otanh[(/ia —v,t)/la

]

(4.23)

From time-reversal symmetry, any change in wall shape, e.g., /, must be of order v,2 and does not contribute to M, for small vs. Continuing to work within the adiabatic approximation, we find

±M.vl-±M2,il 2/V

5) sech4

(4.24)

Therefore, using the parameters for EG = 1.4 eV one obtains, 2

"•-•h

M — 6m,

where me is the free electron mass. The small value of M, is a consequence of the smallness of the dimerization length ua compared to the lattice spacing a. One would expect that the soliton would have high mobility because of its small mass, and it must be treated as a quantum particle.

V. DOPING EFFECTS In the traditional semiconductor picture of doping, an impurity donates an electron (or hole) to the conduction (or valence) band of the solid and no structural change of the solid occurs. In (CH) X , one must consider whether the state of lower energy is a free electron (or hole) as in the semiconductor picture, or

a charged soliton (Q — ± e ). Choosing the center of the gap as the origin of energy, the minimum energy to inject an electron (or hole) is A, while the energy to make a charged soliton is Es, Thus, if £S
(5.1)

soliton doping occurs through the formation of charged solitons, while if E, > A

(5.2)

semiconductor band doping occurs. For a range of gap sizes we found £, ~ 0.6A. Therefore, soliton doping is favored in (CH)j-iike systems. This result implies that for each donor (K, Na, etc.) or acceptor (CI, AsF 5 , etc.) which transfers an electron or a hole to the chain, 26 one charged soliton is formed. Since charged solitons have zero spin, no spin resonance or Curie-law susceptibility would be associated with the charge carriers, as is experimentally observed. One might ask, how is it possible to transport a single charge + e without having spin transport. In essence, the charged soliton carries one missing electron, half of which is in the up-spin valence band and half in the down-spin valence band. This is accomplished by slightly deforming all of the states in the valence band so as to reduce locally the up- and down-spin electron density each by a total of half an electron in the vicinity of the soliton. Far from the soliton, the electron density returns precisely to its value without the soliton. For Q = - e, the soliton has two electrons spin paired in 4>a and the missing electron density from the valence band is doubly compensated for by the 4>a electrons. Next we calculate the interaction energy of A£, of a charged soliton Q = ±e interacting with an impurity of opposite sign of charge Q' — + e. For simplicity we assume the impurity to be a point charge located a distance il from the chain and centered at n = 0. When the soliton is centered on site ns, the Born-

22

SOLITON EXCITATIONS IN POLYACETYLENE

Oppenheimer energy has the additional term 2

2

p T A £ , — — s

|d>o(»-",)l . ' J " ' : , •••;:.» 2

£ r

,

l(r,aV+d V»

(5.3)

where we assume the interaction is screened by the macroscopic dielectric constant e. For £"0 - 1.4 eV and € = 1 0 , one finds for \x,\/d < 1, \E, (v, ) - - £ , + -i-A-»x,2 + O (.v/)

(5.4)

where Vj = nsa. One finds £* = 0.33 eV and kb = 0.0029 eV/A 2 for
plotted are ESU) for the free soliton and E,(l) - £(,(/). For the bound soliton, the Coulomb attraction shrinks wall thickness to approximately / = 5 and increases the binding energy to Eb =0.32 eV. The shifted vibrational quantum isfru>s = 0.O7 eV. The dipole oscillator strength of this excitation is large because it corresponds to a full electronic charge being excited. Another excitation mode of the soliton is the / or shape oscillation. If one writes 1//„(f)

(5.5)

Treating the soliton as a quantum particle, the zeropoint motion reduces the equilibrium binding energy to Eb - jHa>s. In a more complete treatment of the solitonimpurity interaction, / must be allowed to vary to minimize the system energy since the Coulomb energy is reduced as / is decreased. In Fig. 14, — Eb(l) for xs = 0 is plotted as a function of / as calculated from Eq. (5.3) for Ec•- 1.4 eV and rf = 2.4 A. Also •

200 160 -1 120

-1

80

E UJ

E

40

\

A-*

s"E b

Js*/r

O

/

-40 N

-80

.

O

/-Eb

' '

4

i

8

t

i

i

i

1 1 «

12 16 20 1

FIfJ. 14. Frce-soliton energy £",, the Coulomb energy of charged soliton-charged impurity interaction - A t , and the sum t", — Lb are plotted as a function of the soliton width /. When bound to an impurity the soliton width decreases somewhat, although the binding energy changes relatively little due to this effect.

= » u tanh|«fl/[/« + 8(r)] 1

(5.6)

then an effective Hamiltonian can be written //, = {A,8 2 + yM|fi 2 .

(5.7)

fw,=/r(A,//w,)l/2 .

(5.8)

and The coefficients A| and Mt are given by ti=-

0.06 eV, f/ = 2.0 A 0.05 eV, , / - 2 . 4 A.

2109

I 0 2 £,

M, = -

a/2 a2l'

X"2sech'

(5.9) (5.10)

For E0- 1.4 eV, and / - 7, Ka>] =0.09 eV. Infrared absorption experiments on doped (CH) X have been carried out by Fincher el al.v and phonon modes in the presence of a soliton have been treated by Mele and Rice.28 VI. COMPARISON WITH EXPERIMENT; CONCLUSION The existence of paramagnetic defects in undoped iraii.v-(CH), is well established. 6_ '° Isornerization of c/.v-films results in a spin resonance signal whose intensity grows with increasing /ra».v-isomer content. 6 The results imply that the magnetic defects in irans(CH), are not due to impurities but are on the polymer chain and are induced by isornerization. Goldberg el al.7 suggested that the narrow roomtemperature electron-spin resonance (ESR) in trans(CH) X results from motional narrowing due to a mobile bond-alternation domain wall. Recent ESR studies 2 ' of ( C H ) , and (CD)* have demonstrated that the ESR linewidths are determined by the unresolved hyperfine splittings; the temperature dependences of the measured linewidths in w m - ( C H ) , and trans-{CD)x are consistent with the picture of a mobile defect. The mobility of the neutral defects has been confirmed through observation of the Overhauser effect in lrans-(CH)x by Nechtshein el al. ' 6 However, they observed no such enhancement in ra-fCH), (i.e., containing a small Irons content), but rather the "solid-state" effect due to coupling to immobile electron spins. Thus the neutral defects induced by isornerization are highly mobile

2110

W. P. SU, J. R. SCHRIEFFER, AND A. J. HEEOER

22

only in the fully isomerized trans-(CH)x in agreement with the ESR studies. The spin resonance of the trapped immobile defects in the partially isomerized polymer has been used to obtain information on the spatial extent of the magnetic defect.29 Analysis of the results implies a derealization of the spin over a region with half-width of about seven lattice constants. Thus the theoretical description of these neutral defects (in the undoped polymer) as mobile solitons, with the unpaired spin spread over the extended domain wall, is in good agreement with all aspects of the experimental results. Experimental studies of the doped polymer are also qualitatively consistent with the concept of doping through soliton formation. Light doping of (CH) X produces a dramatic increase in conductivity with no accompanying increase in Curie-law spin susceptibility.8 It has been suggested 30 that this may be attributable to nonuniform clustering of dopants into metallic regions, resulting in a small Pauli, rather than Curie, magnetic susceptiblity. While such clustering may play a role at dopant concentrations approaching the semiconductor-metal transition ( ~ 1 mole%), complete clustering is unlikely especially at the lightest doping levels. Moreover, compensation of the charge carriers by many orders of magnitude in undoped trans-(CH)x produces no decrease in Curie-law intensity. 7 Thus the spins associated with the Curie

law and the charges associated with the conductivity are apparently decoupled in the lightly doped regime. Furthermore the strength of the Curie-law contribution decreases 8,31 on doping consistent with the expected ionization of isomerization-induced neutral solitons upon doping. We conclude that the solitons, 32 or bondalternation domain walls, described theoretically in this paper appear to play a fundamental role in the properties of polyacetylene, especially at very light doping levels. However, detailed experimental studies are required, especially as a function of dopant concentration, to clarify the role of solitons in the doping mechanism and in the subsequent electrical transport.

'Address after January 1980: Dept. of Phys., Univ. of California. Santa Barbara, Calif. 93106. 'W. P. Su, J. R. Schricffer, and A. J. Heeger, Phys. Rev. Lett. 42. 1698 (1979). 2 M. J. Rice, Phys. Lett. A 7L 152 (1979). 3 Hajime Takayama, V. R. Lin-Liu, and Kazumi Maki, Phys. Rev. B2J, 2388 (1980). 4 S. A. Brazovskii, JETP Lett. 28, 656 (1978); and private communication. 5 B. Horovitz and J. A. Krumhansl. Solid State Commun. 26, 81 (1978). *II. Shirakawa. T. Ito, and S, Ikeda, Die Macromol. Chcm. 179, 1565 (1978). '1. B. Goldberg, H. R. Crowe, P. R. Newman. A. J. Heeger, and A. G. MacDiarmid, J. Chcm. Phys. 70, 1132 (1979). 8 B. R, Weinberger, J. Kaufer, A. Pron, A. J. Heeger. and A. G. MacDiarmid, Phys. Rev. B 20. 223 (1979). 'A. Snow, P. Brant, and D. Weber, Polym. Lett, j j , 263 (1979). •I(IJ. C. W. Chien, F. E. Karasz, G. Wnek, A. G. MacDiarmid, and A. J. Heeger, Polym. Lett, (in press). "Y. W. Park, A. Denestein, C. K. Chiang. A. J. Heeger, and A. G. MacDiarmid, Solid State Commun. 29. 747 (1979): Y. W. Park, A. J. Heeger. M. A. Druy, and A. Ci. MacDiarmid, Phys. Rev. B (in press). I2 S. Lcfrant. L. S. Lichtman. H. Temkin. D. B. Fitchcn. D. C. Miller, G. E. Whitehall, and J. M. Burlitch. Solid State Commun. 29. 191 (1979). I3 I. Ilarada, M. Tasumi, II. Shirakawa, and S. Ikeda, Chem.

Lett. (Jpn.) U, 1411 (1978). C. Trie. J. Chem. Phys. 5J_. 4778 (1969). C. R. Fincher, Jr., D. L. Peebles. A. J. Heeger, M. A. Druy, Y. Matsumura. A. G. MacDiarmid, II. Shirakawa, and S. Ikeda. Solid State Commun. 27, 489 (1979): Y. W. Park, M A . Druy, C. K. Chiang, A. J. Heeger, A G . MacDiarmid, II. Shirakawa, and S. Ikeda, Polym. Lett. JJ. 195 (1979). J. J. Ritsko. E. J. Mcle, A. J. Heeger. A. G. MacDiarmid, and M. Ozaki (unpublished). '^M Nechlschein. I". Dcvreux, R. L. Greene. T. C. Clarke, and G. B. Street, Phys. Rev. Lett. 44, 356 (1980). "L. Salem, The Molecular Orbital Theory of C'imiuxtitetl Systems (Benjamin, New York, 1966). I8 R. E. Pcierls, Quantum Theory of Solids (Clarendon, Oxford, 1955). p. 108. "Y. Oshika, J. Phys. Soc. Jpn. 12, 1238, 1246 (1957). 20 P. M. Grant and I. P. Batra. Solid State Commun. 29. 225 (1979). 2I S. Hsu. A. Signorelli, G. Pez, and R. Baughman, J. Chem. Phys. 68. 5405 (1978): R. Baughman and S. Hsu. Poly. Lett. 17. '85 (1979). 22 Solitons and Condetised Matter Physics, edited by A. R. Bishop and T. Schneider (Springer-Verlag, New York, 1978). "T. L. Finstein and J. R. Schrieffer. Phys. Rev. B 7, 3629 (1973). 24 W. P. Su (unpublished).

ACKNOWLEDGMENTS We would like to thank Dr. Sergi Brazovskii and Dr. Steven Kivelson for stimulating discussions, and Dr. J. Ritsko for permission to use the Inrudata prior to publication. We are also indebted to Dr. David Campbell for an informative discussion on solitonantisoliton collisions in d>4 field theory. This work was supported in part by NSF Grant No. DMR 7723420 and by NSF-MRL program under Grant No. DMR 76-80994.

I4

IS

25

J. R. Schrieffer (unpublished).

26

For references see A. J. Heeger and A. G. MacDiarmid, in

22

SOLITON EXCITATIONS IN POLYACETYLENE

Proceedings of I he Dubromik Conference on Quasi-OncDiinensional Conductors, Lecture Notes in Physics vo\ edited by S. Barisic (Springer-Verlag, Berlin-Heidelberg, 1979), p. 361; A. J. Heeger and A. G. MacDiarmid, in NATO Conference Series VI: Materials Science edited by W. Hatfield (Plenum, New York, 1979), p. 161. See also accompanying paper in this volume by A. G. MacDiarmid and A. J. Heeger for more details on chemical aspects of the problem. 27 C. R. Fincher, Jr., M. Ozaki, A. J. Heeger, and A. G. MacDiarmid, Phys. Rev. B ]j>, 4140 (1979). 28 E. J. Mele and M. J. Rice (unpublished). 29 B. R. Weingberger, J. Kaufer, A. J. Heeger, and A. G. MacDiarmid, (unpublished). 3 °Y. Tomkiewicz, T. D. Schultz, H. B. Brown, T. C. Clarke, and G. B. Street, Phys. Rev. Lett. 43, 1532 (1979). 3I P. Bernier, M. Rolland, M. Galtier, A. Montaner, M. Regis, M. Candille, and C. Benoit, J. Phys. Lett. 40, L297 (1979).

32

2111

While we have used the term soliton to refer to a shapepreserving nonlinear excitation in the absence of other such excitations, their integrity may be questioned when soliton-antisoliton collisions become important as their density increases. At present, little work has been done on soliton-antisoliton collisions in the coupled electron phonon model discussed above. However, judging from studies in the related problem of *fieldtheory [T. R. Koehler (unpublished); C. Wingate (unpublished); B. S. Getmanov, JETP Lett. 24, 291 (1976); and A. E. Kudryavtscv, JETP Lett. 22, 82 (1975)], collisions between soliton and antisoliton rarely lead to their mutual annihilation. Rather, the excitations reflect approximately elastically or, for small relative velocity, can form a bound state. Investigation of this problem for the above Hamiltonian is being carried on by our group at present.

193 Proc. Natl. Acad. Sci. USA VoL 77, No. 10, pp. 5626-5629, October 1980 Physics

Soliton dynamics in polyacetylene (electron-phonon interaction/broken symmetry ground state/nonlinear dynamics of quantum fields) W . P. SU* AND J. R. SCHRIEFFER Department of Physics, University of California, Santa Barbara, California 93106 Contributed by J. Robert Schrieffer, June 30,1980 ABSTRACT The equations of motion of the coupled electron-phonon system are integrated in real time for the model of polyacetylene recently proposed. To illustrate the physical behavior of this nonlinear system we consider the time evolution starting from three physically relevant configurations: (/) end generated soliton, (a) electron-hole pair generation of a charged soliton-antisoliton pair, and (Hi) the dressing of an injected electron. The calculations show that the system relaxes within a time of order 10 - 1 3 see, converting excited electron-hole pairs into soliton-antisoliton pairs.

H

H

,',

'

'

H

H

H

H

Polyacetylene (CH) r is a simple linear polymer formed as a chain of CH groups. The trans configuration, corresponding to a herringbone structure of CH groups, is the stable phase at room temperature and below. Because undoped (CH)X has exactly one IT electron per CH group, the traditional view is that the ground state exhibits a lattice distortion or dimerization in which bond lengths between CH groups are alternately longer and shorter than the average bond length a (Fig. 1). In the language of condensed matter physics, trans (CH) r has undergone a commensurate Peierls distortion of index 2. The doubling of the size of the unit cell introduces a periodic potential acting on the ir electrons which opens a gap 2A in the ir energy band structure at the Fermi surface, converting the one-dimensional metal into a semiconductor (1), as is observed. However, evidence is growing (2-4) that, instead of the familiar electron and hole excitations characteristic of a conventional semiconductor, the stable low-energy charge-carrying excitations in (CH), are charged solitons, S*. These excitations are in essence charged domain walls (2,3,5) separating regions with different ground state order as illustrated in Fig. 1. The A and B ground states are related by interchanging double and single (short and long) bonds. As Su et al. (2, 3) have shown, S + and S~ have zero spin in contrast with holes and electrons which have spin % Also, the neutral soliton S° has spin % The width of die soliton is approximately 14 lattice spacings for (CH), and its mass is remarkably small, roughly six electron masses. There is considerable experimental evidence supporting these results. In order to gain further insight into dynamical processes in (CH) r , such as electrical conductivity, spin diffusion, optical absorption, photoconductivity, etc., we have carried out a real-time integration of the equations of motion describing the coupled electron and phonon fields. Following Su et al. (2, 3, 6), we adopt the model Hamiltonian

# - - £ [ « o + (-D"«(*«+i + *»)]

x[c + B + 1 ,c„ + c+„c. +1 ,,] + f E(*,,+ 1 + >A»)2 M

^

•

n

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.

H

u

u

n-l

u

I

H

H

n-i

'

H

I

H

U

IWI

H

H

,1,

H

u

n*l

H

I

i

H „

H

H

H

|

|

i

H

H

H

FIG. 1. Structure of the (CH)X chain, illustrating the pattern of displacement coordinates |u„| in the A (Upper) and B (Lower) phases.

to describe a chain of N groups, where K is the effective spring constant for the undimerized system, which is equal to 21 eV/A 2 for (CH),, and M is the CH mass. The x band width is 4t 0 =* 10 eV and the electron-phonon coupling is a ^ 4.1 eV/A. The staggered displacement field \p„ is related to the physical displacement % of the nth CH group along the symmetry axis (i) of the polymer by \f>n = (-l)"Un

[21

as illustrated for the two degenerate classical ground states in Fig. 1 Upper (\pn = —ti 0 ) and Lower {\pn = u 0 ), where u 0 * 0.04 A. The parameter A is chosen such that ipn = 0 for all n corresponding to the ground state in the undimerized sector Adiabatic approximation To integrate the equations of motion, we assume that the adiabatic (Born-Oppenheimer) approximation holds. As Brazovskii and Dzyaloskinskii (7) have shown, this approximation is justified if die Peierls gap 2A [»1.4 eV for (CH),] is large compared to die optical phonon energy hwo » 0.14 eV. A restriction on their proof is that the magnitude of order parameter |i/<| must not be reduced to a small fraction of its equilibrium value UQ anywhere along the chain. Because we are interested in amplitude solitons in which \j/ passes dirough zero at the soliton center, a more general justification of the adiabatic approximation is required. In conventional metals, the adiabatic approximation works well in treating lattice dynamics because the phase space for creating electron-hole pairs is very small at the low energies corresponding to phonon destruction. We argue that the same justification holds for (CH)„ an assumption which can be checked self-consistently from the electron spectrum one obtains from the adiabatic approximation. Within the adiabatic approximation, the coordinates \\f/n} * Permanent address: Department of Physics, University of Pennsylvania, Philadelphia, PA 19104.

5626

194 Proc. Natl. Acad. Sci. USA 77 (1980)

Physics: Su and Schrieffer move in a potential V(\\f/„ \) given by the sum of the ground state energy EodV'n)) of the electronic system plus the harmonic lattice interaction in Eq 1,

v(kW) = Eo(hU) + ~ Z Wwi + W 2

5627

equations Fn(0) , M ' F„(l) u>„(2) = « , ( ! ) + - ^ r , « „ ( ! ) = io„(0) +

n(l)T, +n(2) = ^ n ( l ) +

wn(2)r,

£* n -A[^O + (-1WN-I1-

[12)

[31

For given \\pn), E0 is the sum of the energies e„ of the occupied one-electron energy eigenstate
tu„(»") = u)„(m •

1) +

Ffcl2 ^n(m) = ^ „ ( m - 1) + io B (m)T.

[4i

E<Mn\) = Z 'Mn))**-

where n ^ is the occupation number of orbital u with spin orientation s. The total number of electrons is

US

The c./s are the eigenvalues of an effective one-electron Hamiltonian ft whose matrix elements in the site representation are

;¥„„' = •

- ( 0 - ( - l ) n a ( ^ „ + i + iAn)

n' = n + l

- t o + ( - l ) n a ( ^ „ + "/'n-i) 0

n' = n - 1 otherwise

. fl .

so that the eigenvalue equation for pm is H o - ( - l ) n O # n + l + &i)]*Wl + [~to + ( - l ) " « ( f n + &i-l)l«>,«

= «„¥>«„•

[7]

For numerical convenience, we consider a finite length chain having N C H groups. W e find instructive results for N as small as 30. For N of this order, it is convenient to find the
D(<-)»det[JrVB-««B>]-0.

In Eq. 12, time is expressed in units of r and F„(m) denotes the force on group n as calculated from the order parameters evaluated at time m r . In order to facilitate the calculations, we have chosen the spring constant K so that the soliton width is reduced from £ = 7 for (CH)X to £ = 2.75. In this case, UQ corresponds to =0.1 A compared to 0.04 A for (CH)X. The time step size T is chosen to be 1.25 X 1 0 - 1 5 sec compared to the period of the k = 0 optical phonon 2ir/u)o = 36 X 10~ 15 sec, so that the configuration changes smoothly with respect to time. We measure 4*n in A. units. Results To illustrate the richness of the nonlinear dynamics of this system, we consider three distinct initial configurations. In the first, the normalized order parameter is initially l/'n (0) =_1 for all sites in the chain with an odd number of sites, where\p n (t) 3 '/'nM/uo- The second example traces the dynamics of the system, following the initial creation of an electron-hole pair by a photon. Finally, the third example shows the time evolution of the lattice distortion following the injection of an electron at the conduction band edge with \j/n initially equal tol. Example 1: End-Kink Generation. In Fig. 2 the normalized order parameter is plotted as a function of n j o r several different times starting from the initial condition \p„(0) = 1 for all n. As was shown by one of us (6), the ground state of a chain with an odd number of sites has a soliton located near the center

[8]

For an even number of electrons, one fills the N/2 lowest energy states with two electrons each and for an odd number of electrons adds the extra electron to the next higher state to obtain the electronic ground state energy £0([
[9]

[ioi

M The derivative of V with respect to ipn i s calculated as F

n

= -^-

[ill

«*„

for suitably small 8\pn. To integrate the equations of motion, the initial coordinates iA„(0) and the staggered velocities to n (0) must be specified. Then, using time steps of length r, one has the set of iterative

FIG. 2. Time evolution of ^„, illustrating an end-generated soliton for time t = 0,10, 40,100, 226, and 400 in units of T = 1.25 X 10"15 sec.

195 5628

Physics: Su and Schrieffer

Proc. Natl. Acad. Sri. USA 77 (1980)

of the chain. In chemical language, the initial state \^„(0) = 1 corresponds to die chain ending in a double bond on the left and a single bond on the right. This is a high-energy state, and the system distorts to form a double bond at each end of the chain plus a soliton. As one sees from the plots, part of the energy lowering goes-into phonons. For example, in curve D the rapid oscillations of the staggered order parameter \f/n corresponds to large-amplitude long-wavelength acoustic phonons. Also, in curves B, C, and D, the order parameter at the left-hand end is increased above t*o corresponding to an enhanced double bond strength at a free end. Curve E illustrates the soliton bouncing against the left-hand end of the chain and finally in curve F the soliton has moved back toward the right-hand end. As in Fig. 3, if one plots the position of the center of the kink (i.e., the position where the interpolated value of \f/n vanishes) as a function of time, one finds that the soliton's speed is essentially constant except when it is in contact with a chain end. The speed is found to be of order 1.3 X 10 6 cm/sec, which is close to the speed of sound in_this model. Example 2: e + h —• 5 + 5. Suppose that an electron-hole pair is suddenly created at time zero, with the electron at the bottom of the conduction band and the hole at the top of the valence band. We assume that at this instant the order parameter \p is that for the ground state of the chain before the pair was created. The system is unstable in this electronically excited state and will evolve in time so as to create a soliton-antisoliton pair. The time evolution of \j/ for this process is shown in Fig. 4. As one proceeds from curve A to B to C to D, the electronhole pair continuously distorts the lattice, self-consistently localizing the electron-hole pair near the center of the chain in states split off from the conduction and valence band edges. As time progresses, phonons are generated as seen in curve E, and the soliton and antisoliton begin to separate. It is interesting to note that, according to this calculation, the time required to dress a bare electron-hole pair by forming a soliton-antisoliton pair is of order 10 - 1 3 sec rather than a long incubation time that one might guess based on the large change of the order parameter as well as the multiphonon generation which is observed. 400

300

200

FIG. 4. ^„ vs. a for £ - 0,6,12, 20,30,50,60, and 130 in units of r = 1.25 X 10 -15 sec, describing the generation of a soliton-antisoliton pair from an alectron-hole pair created at t = 0 at the band edges. A matter of principle arises in the above discussion—namely, is it proper to consider that an electron-hole pair is created by photon absorption at an instant of time, even though the light intensity is weak so that the mean time to absorb a photon may be long compared to the relaxation time of 1 0 - 1 3 sec found above? The reason that the sudden creation picture is correct is familiar from the theory of measurement: precisely when the photon is absorbed is highly uncertain; however, when it is absorbed, the coupled electron-phonon system evolves in time as discussed above, even though we take the classical limit of

251-

FlG. 3. Position of center of soliton in units of lattice spacing vs. time in units of T = 1.25 X 10 -15 sec. The graph is folded in the time axis and illustrates the uniformity of the velocity.

FtG. 5. n vs. n for t = 0, 26, 50, 74, 96, and 116 for T = 1.25 X 10"15 sec for an electron injected at the conduction band edge.

196 Physics: Su and Schrieffer the lattice displacement equations of motion. Alternatively, treating both the electronic and lattice displacement coordinates quantum mechanically, the system's state vector ,fr, when acted upon by the electromagnetic potential A is given to first-order in A by

!*(*)>

i +

iJ^V)-A(f')A']|*(0)) [13]

where j is the total electric current and c is the speed of light. Thus, to first-order in A, the time evolution of the component of | Sf'(r)) due to A{t')dt' is unaffected by A at other times. Example 3: Electron Injection. To investigate the dynamics of the system after the injection of an electron (or a hole)—e.g., in a tunneling experiment—consider placing an electron at the bottom of the conduction band at time zero, withi/' n (0) = 1 for all n. Fig. 5 illustrates how the electron self-consistently distorts the lattice and is localized in a split-off state, forming a "strong polaron." As time progresses, shake-off phonons_appear, as evidenced by the large amplitude oscillations of \p in curves C-F. In Fig. 6, the kinetic energy of lattice motion has been removed and the system is allowed to relax adiabatically to its ground state configuration in the presence of the added electron. This figure is analogous to the conventional picture of a strong polaron in a system without symmetry breaking. The binding energy of the polaron is —0.3 eV.

FIG. 6. \j/„ vs. n for the ground state of one electron added to the conduction band.

Proc. Natl. Acad. Sci. USA 77 (1980)

5629

Discussion The conventional picture of a rigid band semiconductor should be applied with care to polyacetylene, in which the gap parameter is -determined self-consistently from the coupled electron-phonon system. For electron or hole injection as well as electron-hole pair creation, the order parameter distorts so rapidly that the electronic spectrum is significantly broadened. This distortion drastically alters the transport properties of electrons and holes because the excitations become charged solitons through this dressing process. One important result is that the charged solitons have a limiting velocity of approximately the speed of sound, which is very small compared to limiting velocities of electrons and holes in conventional semiconductors. Another is that the nonlinearity of the dynamics tends to suppress the recombination of soliton-antisoliton pairs. We conclude that real time integration of the field equations is a powerful tool in studying various physical properties of the coupled electron-phonon system. The calculations are easily extended to include impurity effects. The authors are grateful to Profs. A. J. Heeger and D. J. Scalapiro for stimulating conversations. This work was supported in part by National Science Foundation Grant DMR 80-07432 and by the National Science Foundation MRL program at the University of Pennsylvania by Grant DMR 76-80994. 1. Peierls, R. E. (1955) Quantum Theory of Solids (Clarendon, Oxford). 2. Su, W. P., Schrieffer, J. R. 4 Heeger, A. J. (1979) Phys. Rev. Lett. 42,1698-1701. 3. Su, W. P., Schrieffer, J. R. 4 Heeger, A. J. (1980) Phys. Rev. B22, 2099-2111. 4. Ozaki, M., Peebles, D. L., Weinberger, B R., Chiang, C. K., Gau, S. C, Heeger, A. J. 4 MacDiarmid, A. G. (1979) Appl. Phys. Lett. 35(1), 83-85. 5. Rice, M. J. (1979) Phys. Lett. 71,152-154. 6. Su, W. P. (1980) Solid State Commun, in press. 7. Brazovskii, S. A. 4 Dzyaloshinskii, I. E. (1976) Z/i. Exp. Tech. Fiz. 71,2338.

VOLUME 46, NUMBER 11

PHYSICAL

REVIEW

LETTERS

16MARCH1981

Fractionally Charged Excitations in Charge-Density-Wave Systems with Commensurability 3 W. P . S u t a ) and J . R . S c h r i e f f e r Department of Physics and Institute for Theoretical Physics, University of Santa Barbara, California 93106 (Received 29 December 1980)

California,

A theoretical study of topological excitations (kinks) in a one-dimensional one-thirdfilled Peierls system is presented. The charges associated with the kinks a r e found to be fractional Q=±ie, ± § e . Calculations of the spatial widths and electronic structure of different types of kinks a r e carried out numerically. Possible applications to tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) are mentioned. PACS numbers: 71.45.-d

Recently, some novel physics have emerged from studying the nonlinear topological excitations in a simple one-dimensional coupled electron-phonon system. 1 It is known that in this model for the half-filled-band case (one electron per site), the lattice will undergo a commensurate Peierls distortion, i.e., dimerize, thereby opening a gap in the electronic spectrum at the F e r m i surface, kf =±ir/2a, where a is the mean lattice spacing. The dimerization is due to the energy lowering of the occupied electronic states being initially larger than the lattice strain energy, with the total energy reaching a minimum for two displacements ±uB, These two degenerate 738

dimerization patterns a r e termed the A (+ u0) and B (-u0) phases. B is simply a translation of A by one lattice spacing. A topological soliton excitation or kink is formed by a domain wall separating regions of A and B phase material. Associated with each kink is a localized electronic state
© 1 9 8 1 The American Physical Society

198

PHYSICAL REVIEW LETTERS

VOLUME 46, NUMBER 11

mitian Hamiltonian. If the electrons were spinless and the gap-center state were empty, then because the deficit in the valence band i s one-half of a state, the net charge of the kink would be + i | e | . This i s related to the fermion-number-2 object discovered by Jackiw and Rebbi 2 in a r e l a tivistic field theory. The fractional charge is masked in polyacetylene by the spin degeneracy of the electron but leads to unusual charge-spin relations, 1 namely, charged kinks a r e spinless and neutral kinks a r e spin 1 in contrast to the charge ± e and spin i of electron and hole excitations in solids. In the one-third-filled-band case, however, the local charge deficit is no longer l i e I per spin but rather \\e\ or \\e\ and one is left with a fractional charge for kinks Q =± j j e \, ± | | e | . For commensurability n, the kinks have charge which is a multiple of ± e/n? To perform quantative calculations, we consider the model Hamiltonian originally used for polyacetylene, 1 H = - £ [ f 0 - aJ r c n > , +H.C.) + 5-E ( « „ + i - M „ ) 2 + S i M " „

(1)

where u„ is the displacement of the wth unit from its equilibrium position, c* is the electron c r e a tion operator, andAf is the m a s s of one unit of the chain. For model calculations, it is convenient to take f 0 =2.5 eV, a =4.81 eV/A, * =17.4 eV/A 2 , M = m a s s of CH. These values lead to a gap of 4.0 eV and a coherence length | 2 = 2.75c in the dimerized case 4 (compared with a gap of 1.4

16 MARCH

1981

eV and £ 2 s ? a f ° r actual polyacetylene). Following Ref. 1, we plot in Fig. 1 the total energy per site as a function of the t r i m e r i z a t i o n amplitude u and phase 9 in a perfectly t r i m e r i z e d chain: u„ = u cos(fir«-0).

(2)

The three ground states A, B, and C a r e obtained by taking u =u0 =0.07 A and 9 - $v = 0 , f-ir, and •fir, respectively, as illustrated in Fig. 2, where dashed bonds mean shorter bond length than undashed bonds. Notice the B -phase pattern is just the A -phase pattern displaced to the right by one unit, while the C-phase pattern is the A -phase pattern d i s placed to the right by two units o r to the left by one unit. Also notice that in Fig. 1 t h e r e a r e two different condensation energy scales for going fromj4 t o B phase; the energy i n c r e a s e is about 0.002 eV per site for purely unwinding of the phase, while 0.011 eV for reducing the amplitude to zero a s well. These two energy s c a l e s a r e important in determining the structure and energy of the kinks in the trimerized system. For a perfectly trimerized chain, let the hopping integral associated with a short bond be - if0 +6) and for a long bond, - (t0 - 26), where 6 =yfiiau0/2, from Eqs. (1) and (2). The total charge on each site is given by the integral over occupied states of the imaginary part of the diagonal Green's function. One finds that the charge oscillates around the average value of - f-lel with p e r iod of 3a. For example, for the A phase the charges on sites 2 and 3 a r e equal and a r e smaller than that on site 1, by a fraction of order 36/f 0 . Knowing the degenerate ground s t a t e s , we can study the structure of kinks connecting them. We distinguish two classes of kinks: type I, which

K

2

.

3

4

W-

5

A

5

150"" 270°

390°

B FIG. 1. Total energy per site plotted as a function of the phase angle 6 for three different values of the amplitude of trimerization u.

K

M

H

K

K

n=-l

0

1

2

3

- x

4

5

6

*-•

6

7

C

FIG. 2. The three degenerate ground states of a perfectly trimerized chain.

739

VOIUME 46, NUMBER 11

PHYSICAL

REVIEW

leads from .A to B, B to C, or C to A a s one moves from left to right; and type II, which leads from A to C, C to B, or S to A with increasing n. To determine the charge of a type-I kink, consider an infinite chain of pure A phase. Suppose that one deforms the lattice displacement pattern by maintaining the left-hand portion of the chain n <*> in the A phase (0 =^n) while 8 - \n increases to fir, \TI, and |ir a s one moves to the right of three widely spaced points nlt n2, and n3, respectively. Since the phase shift is 2ir for n » » 3 , the system returns to the A phase for large n, with type-I kinks centered at nlt n 2 , and n 3 . The shape of the kinks i s irrelevant to this argument so long a s the kinks have finite width. Because of the 27r phase shift in the bonding pattern, relative to the initial perfect .A phase, during the deformation processes a total charge - 2| e | flows past a Gaussian surface located far from the kinks. 3 From charge conservation and the fact that symmetry ensures that the charge of the three type-I kinks a r e identical, it follows that the primitive charge of a type-I kink is +1 \e\ . This simple result agrees with that of Green'sfunction calculations a s well as numerical calculations on.chains of finite length. Thus, stable elementary excitations of fractional charge can occur a s a consequence of ground-state degeneracy. A similar argument shows that the primitive charge of a type-II kink is - f | e | since the phase shift for n - + °° is now - 2ir and a charge - 21 e \ is accumulated in the region containing the three kinks. It is natural to identify type-II kinks as the antiparticles of type-I kinks, i.e., KBA =KAB> etc. Let the ordered pairs AB, BC, and CA be denoted by 1, 2, and 3, respectively. Then, without disturbing the ground state at large distance, one can create K{Kt pairs or the triplets KxKJi3 and KjKJ(3. One cannot help but notice the analog with the quark structure of mesons and hadrons. In this regard, related relativistic field-theory studies are being pursued by Goldstone and Wilczek. 5 We have investigated the electronic structure of sharp kinks using Green's-function methods for an infinite chain and numerical calculations for finite chains with a number of sites of order 80. As illustrated in Fig. 3(a) for type-I kinks, there is a localized state cp, in the upper half of the lower gap and symmetrically a state
LETTERS

16MARCH1981

FIG. 3. Gap states associated with (a) a sharp type-I kink and (b) a sharp type-II kink.

die band. The symmetrical situation occurs for the state in the upper gap. If , of the kink, in contrast with the dimerization case. Kinks with n, =0 or 2 tend to relax with pure phase distortion and fixed amplitude, with the energy of the localized state moving up or down in the gap until it approaches the band edge. On the other hand, for n, = 1 , the energy of
200 VOLUME 46, NUMBER 11

O

20

PHYSICAL

40 n

60

RE

80

FIG. 4. Minimum energy displacement patterns u„ for finite chains with Ns sites and Ne electrons: N3 = 81, 79, 80, 81, and 81, andiv~ e =54, 53, 53, 56, and 55 in (a)-(e), respectively.

for the minimum energy kink configuration and relaxation takes place primarily by modulation of the amplitude of the charge-density wave. For example, Fig. 4(a) shows the relaxed configuration of a chain of 81 sites and 54 electrons, i.e., exactly one-third-filled band, a is the line connecting the u3„ displacements for the first group in each unit cell with n =0, 1, . . . , 26. 0 and y are the corresponding curves for the displacements of the second and third groups, u 3 n + 1 and M

3 n +2»

Figure 4(b) is the relaxed configuration of a chain of 79 sites and 53 electrons. These two numbers a r e chosen to produce a half-occupied type-I kink. Compared with Fig. 4(a), we see the end configurations a r e about the same; therefore, we conclude that the transition region intrinsic to the kink occurs within about 15 sites, 2{ 3 =* 15a, and £3=* 7a. It is interesting to note that

IEW

LETTERS

16MARCH 1981

this number agrees with that calculated from the usual relation | =Sv F /irA, where 2A = 4 6 - 1.2 eV is either of the gaps in Fig. 3 . Figure 4(c) shows the corresponding result of a half-occupied type-II kink, while Fig. 4(d) is the result of adding two extra electrons to the chain in Fig. 4(a). Instead of creating two kinks a s doping in polyacetylene does, the period of the displacement pattern d e c r e a s e s by the right amount so that t h e r e i s a new state close t o the valence band to accommodate the extra electrons. We might conjecture that if we have a chain long enough, we would see one of type I plus another of type II, both fully occupied. The fact that we do not see this in Fig. 4(d) is because the coherence length in this case i s comparable or longer than the chain itself. Finally, Fig. 4(e) is the r e laxed configuration for a half-occupied polaron. Again, the localized state is located at the center of the gap. We conclude that stable excitations of fractional charge should exist in quasi-one-dimensional systems, with commensurability 3 being particularly favorable. Tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) under 19 kbar pressure 6 is such a case since it i s observed to have a commensurable charge-density wave of period 3a. This work was supported in part by the National Science Foundation under Grant No. DMR-8007432 and by the National Science FoundationMaterials Research Laboratory program at the University of Pennsylvania under Grant No. DMR77-23420.

Permanent address: Department of Physics, University of Pennsylvania, Philadelphia, P a . 19104. 'W. P . Su, J . R. Schrieffer, and A. J . Heeger, Phys. Rev. Lett. 42, 1698 (1979), and Phys. Rev. B 22, 2099 (1980). 2 R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976). 3 M. J . Rice, A. R. Bishop, J . A. Krumhansl, and S. E Trullinger, Phys. Rev. Lett. 36, 432 (1976). 4 W. P . Su and J . R. Schrieffer, P r o c . Natl. Acad. Sci. U.S.A.77, 5626 (1980). 5 J. Goldstone and F . Wilczek, private communication. 6 A. Andrieux, H. J . Schulz, and D. J e r o m e , J . Phys. (Paris) Lett. 40, L385 (1979), and Phys. Rev. Lett. 43, 227 (1979).

741

Nuclear Physics B190[FS3 ] (1981) 253 - 265 © North-Holland Publishing Company

SOUTONS WITH FERMION NUMBER \ IN CONDENSED MATTER AND RELATIVISTIC FIELD THEORIES R. JACKlW and J.R. SCHRIEFFER2 Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA Received 29 January 1981

States with fractional fermion charge have been discovered in relativistic field theory and condensed matter physics. In the latter context they lead to unexpected but experimentally verified predictions for one-dimensional electron-phonon systems like polyacetylene. We examine the common basis for this fortunate convergence between condensed matter and relativistic field theories.

1. Introduction In a study of the spectrum for a one-dimensional, spinless Fermi field coupled to a broken symmetry Bose field, Jackiw and Rebbi (JR) [1] noted the occurrence of a localized zero-energy, c-number solution \pQ to the Dirac equation when a soliton is present. Furthermore, they proposed the interpretation that in the soliton-fermion system there is a twofold energy degeneracy (beyond the broken symmetry degeneracy), and that the two states carry charge ± 4 . In other words, they found that introducing the soliton changes the number of fermions present by a fractional amount, namely ± 3 , depending whether x(/0 is occupied or not. Independently, Su, Schrieffer and Heeger (SSH) [2], studying a coupled electronphonon model for the quasi one-dimensional conductor polyacetylene (CH)V, found a dynamical symmetry breaking of the system, which leads to degenerate vacua and soliton formation. In the presence of a soliton. there is a c-number solution i^0 of the electron field, localized near the soliton, with energy at the center of the gap. As a consequence, one-half a state of each spin orientation is removed from the sea in the vicinity of the soliton*. Thus, if one neglects the electron spin, the existence of the zero-energy state and fermion number ± 4 are common to the two situations. In this paper, we outline the two theories and point out the fundamental reasons for similar behavior in different models. We hope that this will stimulate interaction between condensed matter and particle physicists. 1

Also at Center for Theoretical Physics, MIT. Cambridge. MA 02139. Also at Department of Physics, UCSB. Santa Barbara. CA 93106. * See also ref. [3]. 2

253

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Schrieffer / Solilons withfermion number {

2. Solitons in a one-dimensional electron-phonon system 2.1. PRELIMINARIES

There are materials, such as polyacetylene (CH) t , in which electrons move primarily in one dimension. Typically, these materials consist of parallel chains of atoms, or groups of atoms, in which electrons hop preferentially along the chains, while hopping between the chains is strongly suppressed. This anisotropy renders such materials quasi one-dimensional. The structure, called trans-(CH)x, is shown in fig. 1In solids, the matrix element /„.„, giving the probability amplitude of an electron hopping from site n to n\ depends on the distance between these sites, whose average spacing is a. The atoms can be displaced from their perfect crystal lattice position for a variety of reasons, e.g. zero-point motions, thermal excitations, broken symmetry effects, etc. These displacements alter the matrix elements /„•„, leading to the so-called electron-phonon (or electron-lattice displacement) interaction. Using the mean-field approximation, in which the phonon field un is treated as an unquantized c-number, Peierls [4] has shown that the one-dimensional electronphonon system is unstable with respect to spontaneous breaking of the reflection symmetry M„ <-» — w„, for any non-zero electron-phonon coupling strength. The distortion gives rise to a charge density wave (CDW). in which the electron density and nuclear displacements oscillate periodically in space, with wave vector K = 2AF. where hk? is the Fermi momentum of the valence electrons in the undistorted system. Bragg scattering of the electrons from the CDW potential opens a gap 21. at the Fermi surface ±A' F . in the electronic energy spectrum £: see fig. 2. For (CH)V. there is exactly one electron per site, so k? = x/2a. In the ground state, all one-electron states with |A| < AF are doubly occupied (spin up and down) leading to an insulator. In this case K = 2AF = it/a. and the CDW. with wavelength 2TT/K = 2a. is commensurate with the lattice period. From the invariance of the system under discrete translations by ±a.±2a it follows that the ground state is twofold degenerate. This is so since a translation by ±a is equivalent to a reflection, but reflection symmetry is spontaneously broken. Consequently, states translated by a

H

H

H

I

I

I un I

H

H

I

c

vViv v\A i

i

!

i

i

i

Fig. I. Trans configuration of (CH)V: a= 1.2 A. The coordinate displacement of the ;ith group, the phonon field, is denoted by »„.

R. Jackiw, J. R. Schrieffer / Soliions with fermion number j

255

Fig. 2. Band structure due to the Peierls instability in (CH)V.

are distinct from untranslated states. Furthermore, a soliton is formed by the domain boundary between the two ground states. A one-dimensional, /V-site. lattice hamiltonian for the above physical situation has the form

# = 2 ( jff+Hi',,-1',,^))-

2

^ , „ ( ^ i . / - „ . , +
(2.0

Here u„ is a real, scalar. Bose field describing the coordinate displacement along the symmetry axis of the nth group (see fig. 1): pn is the conjugate momentum. M being the group's mass (CH mass for polyacetylene). The first sum gives the phonon energy, kinetic and potential: the second, describes electron hopping from site /; to site n+ 1. with amplitude /„ + I „. The fermion operators cl v and cn y create and destroy electrons of spin s(= ± T ) at site/;. In the SSH analysis, the potential energyis taken in a quadratic approximation. K«„.«„+i)=rA'(«„+,-»„)2.

(2.2)

and the hopping amplitude is expanded to first order: '»+i n = ro - « ( » , , - 1 - » „ ) •

(2.3)

For (CH),. a = 4.1eV/A,

K=21eV/A:.

r0 = 2.5eV.

(2.4)

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Thus the SSH hamiltonian is

n= 2 I 2M T I J '+2T K + I - " * ) 2 n=\ l

0

2l

\Cn+\,sCn,s

~*~Cn.sCn+l,s)

+ « 2 K+i - " J ( ^ + , , / n , i + 4 - J c n + l J .

(2.5)

The first sum in (2.5) is- the harmonic vibrational energy of the free phonons; the second describes band electrons moving on a lattice in a tight binding approximation; the last sum provides a linear coupling of the phonon field to the electrons. In the absence of interactions, a = 0, the ground state is non-degenerate. It consists of the phonon vacuum and a Fermi electron sea, with all states doubly occupied up to k = \kF\. In this limit the gap parameter vanishes, A = 0. 2.2. BROKEN SYMMETRY GROUND STATE

To understand the symmetry breaking, it is convenient to introduce a staggered displacement field „=(-l)\.

(2.6)

The ground state is determined by making an adiabatic (Born-Oppenheimer) approximation: the phonon's kinetic energy is ignored, and $n is set to a constant value u. The ground-state energy is determined as a function of u, and one finds E0(u) = E0( —«); finally, E0(u) is minimized with respect to u. A non-vanishing minimization value for u signals spontaneous breaking of the reflection symmetry; thus, if u0 is one such value, so is — u0. The adiabatic approximation is justified a posteriori by noting that in the broken symmetry ground state, the typical phonon energy yJK/M is much smaller than the gap 2 A in the electron spectrum, for interesting values of a. A straightforward calculation gives [2] (2.7) where / is the elliptic integral. E0(u) is plotting in fig. 3. Two minima are seen at u = ±u0, corresponding to doubly degenerate ground states, A and B; while u = 0 is a local maximum, thus verifying Peierls' theorem. With numerical values for the

R.Jackiw, JR. Schrieffer / Soli tons with fermion number \

257

"E0(u)/N

Fig. 3. Born-Oppenheimer energy per CH group, plotted as a function of the staggered displacement field u = 4>„ = (— I)" u„. The two stable minima correspond to A( + u0) and B( — u0) phases.

parameters appropriate to (CH) t , one finds « 0 = 0.042 A. Because of the periodic spatial variation of the electron hopping matrix element when uj^Q, the electronic plane wave k of the undistorted chain mixes with the state k — K (or k + K if k < 0), leading to a gap 2 A in the electron spectrum, as shown in fig. 2, where K — ir/a is the first reciprocal lattice vector of the broken symmetry lattice. One finds A = 4a|«i 0 |.

(2.8)

2.3. SOLITON EXCITATIONS

As a consequence of the ground state's twofold degeneracy (<£„ = ±u0), there exist topological solitons, which act as boundaries between domains having different ground states. It is convenient to work with a chain formed as a large ring, having an even number of atoms; see fig. 4. Let the staggered field be plotted radially, with 0 outside the ring and „<0 inside. For <£„ to be single valued, it must change sign an even number of times, in going around the ring. Thus when „ =« 0 tanh[(n - « , ) / / ] - « 0 t a n h [ ( n - / i 2 ) / / ] - « 0 ,

(2.9)

where the soliton width / is approximately 7 for (CH)X. Near the solitons, the above reduces to the familiar topological kink, with hyperbolic tangent profile, which is appropriate for a double-well potential of the if,4fieldtheory [see (3.3)]. The energy of this solution, interpreted as the soliton creation energy, is found to be Es = 0.42 eV~0.6A.

258

R- Juckiw, J.R. Schrieffer / Solitons with fermion number 5

Fig. 4. Soliton S and antisoliton S occurring in a ring with an even number of atoms. The staggered field (j>„ is plotted radially.

In a continuum version of the SSH model. Takayama. Lin-Liu and Maki (TLM) [5] have shown that the hyperbolic tangent exactly satisfies the coupled mean field equations (M-»cc). They find Es = 2&/ir ~ 0.63A. in good agreement with the results of the discrete calculation. If the coupling constant a is increased, the soliton width / decreases and discrete lattice effects become increasingly important. The electronic spectrum in the presence of S and S can be readily determined, both in the lattice and in the continuum models. One finds that the change of the electronic state density p(E) in the presence of solitons relative to that in their absence exhibits two discrete states $0± , whose energies are symmetrically located about the center of the gap. £ = 0. As |/i, — « : | -» oc. the energy splitting between the two states vanishes as e - | " | - " ; l / / , so that one can form zero-energy eigenstates ^ os and i/^. localized about S and S. as linear combinations of the two states \f/0± . ^ os and ^0s are given by [2] ^o(") = S:i=l,

-^sech[(n-ni)/l]cos[{-rr(n-ni)]. S:/ = 2.

(2.10)

To see how the continuum states are altered by S and S. we note that the completeness of electronic eigenstates \j/„. at each site, implies that the energy integral of the local density p„ „(£), at any site n, is unity: T d E f t , „(£) = !. Pnn(E) = P(E)=2pnn(E).

2\Un)\2S(E-Er), (2.11)

R.Jackiw.J.R.

Schrieffer / Solitons with fermion number \

259

Also the electronic hamiltonian is odd under charge conjugation, c* ^± c„, so that P„„(£) = P „ „ ( - £ ) .

(2.12)

Therefore, breaking up the spectral integral in eq. (2.11) into contributions from the negative and positive energy continua plus the discrete zero-energy state, and using the symmetry (2.12). gives 2/J 0 ~d£p,;„(£) + | ^ ( , 0 | 2 = 2 / °J d£p„„(£). -x

(2.13)

~x

Here the primed quantity is the local density in the presence of the soliton; the unprimed, in the vacuum where the soliton is absent. Thus, the local deficit in negative energy states satisfies /0"d£[p;„,(£)-p„„(£)] = - i | ^ ( / 0 | 2 .

(2.14)

Summing over all sites, and using the normalization condition on ^ 0 . one finds that the total deficit from the negative energy valence sea is precisely 3 a state per spin. Including spin, a total of one electron is missing from the Fermi sea. due to the soliton. If ^ 0 is unoccupied, all spins are paired and the soliton has charge Q— +e and spin .s = 0. Correspondingly, when ^0 is singly occupied, the soliton is neutral 0 = 0. but the spin is ± 4 . The two states are degenerate in energy, since ^0 is a zero-energy eigenstate. for infinite separation between S and S. (However, one does not usually compare these two situations since they involve a different number of electrons.) These local charge-spin relations would appear to violate Kramer's theorem, since one cannot go from integer spin to integer spin and remove or add one electron with spin T- Nevertheless, these peculiar charge-spin relations have been observed experimentally in (CH)V *. Conventional fermion excitations in solids (electrons and holes) have charge ±e and spin ± 4 . The resolution of the apparent paradox is that the antisoliton also has these spin-charge relations, so the difficulty is removed by the topological requirement that S and S be created in pairs, even though they act as independent excitations when widely separated. In other words, global constraints relating charge and spin are valid, but they do not fix local charge-spin relations. 3. Fermionic solitons in a relarivistic field theory 3.1. PRELIMINARIES

In a study of the spectrum for a Dirac field coupled to a broken symmetry Bose field, JR discovered in various models the existence of zero-energy fermion eigen* For evidence of charged, spinless solitons. see ref. [6a]. Evidence for neutral, spin-i solitons is contained in ref. [6bJ.

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Schrieffer / Sotilons with fermion number j

states, localized in the vicinity of a soliton of the Bose field. Their model, involving a scalar field $ coupled to a Fermi field ^ in one continuous spatial dimension, is closely related to the above discussed solutions from condensed matter theory. In the one-dimensional model, JR consider a hamiltonian [1]

P = -^~(3-D v i ax ' Here II is the momentum conjugate to $, and F($) is a potential energy density for $. The Dirac spinors have two components, hence the fermions carry no spin — a simplifying option available in one dimension. Correspondingly, the two Dirac matrices a and /? are two-dimensional:

The model is analyzed in an adiabatic approximation*. The ground state is determined by minimizing /djc[4((d/d:0$) 2 + K($)] with constant , while the soliton is found by minimizing with an .t-dependent $. subject to the condition that the total energy be finite. The Dirac hamiltonian is then quantized, with $ taken as an external, prescribed, c-number, background field. For present purposes V(<&) is a symmetric, double-well potential, as in fig. 3, but of simpler analytic form: J W = 7 7 ( M 2 - * 2 ) 2 = >'(-*)•

(3-3)

3.2. BROKEN SYMMETRY GROUND STATE

The minimum of (3.3) is at <E> = ±p; thus the reflection symmetry $ <-* — $ is spontaneously broken. The Dirac hamiltonian describes a free particle with mass gap A = g/i = m. The fermion modes satisfy the free Dirac equation (ap + fim)ui-) =

±\e\u(-\

\e\ = )/k2+m2

.

(3.4)

Charge conjugation is implemented by the a 3 matrix. It takes positive energy solutions of (3.4) into negative energy solutions, and vice versa. Quantization is * It can be shown that this approximation is the starting point for a systematic, weak coupling expansion; see ref. [7].

R. Jackiw, J.R. Schrieffer / Solitons with fermion number j

261

achieved by the usual expansion in modes:

f = S(e- i!, Vr^) + e"lW^)} •

(3.5)

k

Here u[+) is a positive energy solution, and u(i") is the charge conjugate of the negative energy solution:

The operators b\ (bk) create (annihilate) particles, while d\ (dk) do the same for antiparticles. The charge operator

= i/d* 2 {*?(*)%(*)-*,(xW(x)),

(3.7)

becomes

Q = 2{b\bk-dtdk).

(3.8)

k

The spectrum is elementary and can be built on either the A vacuum ( $ =/x) or the B vacuum ( $ = —/i). Of course the vacua are charge neutral, 2|0> = 0.

(3.9)

3.3. SOLITON EXCITATIONS

The existence of two ground states leads to topological solitons, which interpolate between them. To find the soliton shape of $ , we solve the equation d2 r * ( j c ) + r ( * ) = 0. ax1

(3.10)

For (3.3), finite energy solutions are given by* $(x)=±/UanhA.x.

(3.11)

* The origin has been arbitrarily set at x = 0, the location of the soliton. Of course, the soliton may be located at any x = -t0, and the proper quantum mechanical treatment of this degree of freedom is given in ref. [7]. For a review, see ref. [8].

26?

R. Jackiw, J. R. Schrieffer / Solitons with fermion number j

Note that as x passes from negative infinity to positive infinity, $(*) interpolates between the two vacua. With the positive sign, (3.11) describes the soliton S; with the negative, the antisoliton S. The energy of the solution (3.11), interpreted as the soliton formation energy, is

= djc

=} 2

/ (£*F * *-

3 i2

<- >

The quantization of the Dirac equations, proceeds as in the vacuum sector, except that modes in the presence of the soliton satisfy (ap + Pmtanh\x)Ui-)

= ±\S\U{-).

(3.13)

Again charge conjugation insures that the positive eigenvalue is paired with a negative one, and the modes may be explicitly found. Additionally. (3.13) admits a zero-eigenvalue solution, >/' 0 (.v)a:n)exp[-wy A d.v'tanhA.v').

(3.14)

which is charge conjugation self-conjugate. +S(x) = oM(x)=h(x).

(3.15)

Thus the spectrum exhibits at the center of the gap an additional state. Quantization again proceeds by an expansion in modes, * = a%(x) + 2 [ e-'^'5,f4 ( + )(-v) + e^'DlV^x)].

(3.16)

k

U£+) being a positive energy solution, and KA(-) the charge conjugate of the negative energy solution,

vr=Avry.

(3.1?)

The operators B\ (Bk) and D\(Dk) create (annihilate) conventional fermions and antifermions in the soliton sector. However, the further operator a when operating on the soliton state produces another state of the same energy; hence the two states

R. Jackiw, J. R. Schrieffer / Solitons with fermion number 5

263

are degenerate in energy. To distinguish them, we may label them as | ± , 5 ) and fl| + ,5> = | - , S > ,

J\-,S)

= \ + ,S),

a|-,5>=0, at| + , 5 > = 0 .

(3.18)

In analogy with the physical situation encountered in the polyacetylene system, we may call the "plus" state occupied and the "minus" state unoccupied. The charge quantum number of | ±,S) is evaluated by substituting the expansion (3.16) into (3.7). One finds Q = a*a-$ + Z(BiBk-DlDk).

(3.19)

k

Consequently it follows that

e|±,S> = ±4|±.S>.

(3.20)

i.e.. each of the two soliton states carry 7 unit of charge. 4. Discussion The two models under discussion—one drawn from a realistic situation in condensed matter physics, the other from a formal, mathematical investigation in relativistic quantum field theory—obviously differ in detail. The SSH hamiltonian is on a lattice, and also the fermions carry a spin degree of freedom. The JR hamiltonian is in the continuum; the fermions are spinless. A closer comparison can be made with the TLM hamiltonian—a continuum approximation to SSH. if the fermion spin is ignored for simplicity. The structures of the JR and TLM fermion hamiltonians coincide, but differences remain in the boson parts. Nevertheless, in crucial respects the two models are similar: both give rise to spontaneous breaking of the field reflection symmetry, and as a consequence, have doubly degenerate ground states. They possess soliton excitations which interpolate between the degenerate vacua. Moreover, the fermion equation in both cases admits a localized zero-energy solution, which then implies charge fractionalization: 3 unit of charge is gained or lost depending whether \p0 is filled or empty. In the condensed matter example, the fractionalization is obscured by a doubling of degrees of freedom due to spin, but an experimentally observed signal remains in unusual charge-spin relations: charged solitons are spinless and neutral solitons carry spin 4.

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R. Jackiw, J. R. Schrieffer / Solitons withfermion number j

In both models the presence of charge conjugation symmetry allows the pairing of positive energy modes with negative energy modes, leaving the zero-energy state unpaired. The common phenomena are in fact universal. It has been shown under very general mathematical hypotheses, that the Dirac equation in the background field of a topologically interesting configuration, like a soliton, always possesses zeroeigenvalue modes whose number is related to an integer which characterizes the non-trivial topology*. Since the result is valid in any number of dimensions, the possibility exists that effects similar to those observed in polyacetylene may be found in higher-dimensional systems, e.g. associated with vortices in superfluid helium three. The SSH-TLM model is also interesting for particle physicists in that it realizes its symmetry breaking due to the Peierls instability through quantum-mechanical dynamics. In the classical, tree approximation where fermions are ignored, the only dynamics for the bosons are the harmonic lattice vibrations; in contrast to the JR hamiltonian, for which even in the absence of the fermions and without quantal effects for the bosons, the classical solutions break the symmetry. The effective potential of fig. 3 for the Bose field is generated dynamically in the condensed-matter application, while it is arbitrarily posited in the mathematical field theory example. The possibility of realizing spontaneous symmetry breaking by dynamics, rather than by assumption, is widely discussed but rarely achieved by particle physicists. It is truly remarkable that a phenomenon as esoteric and peculiar as charge fractionalization should have been discovered in two different contexts: mathematical investigations of model field theories by particle physicists; description of experimental phenomena by condensed matter physicists. That this should happen is strong reaffirmation of the unity of all branches of physics and another example of the power of mathematics to uncover unexpected physical behavior. The ideas on charge fractionalization can be carried further. It has been suggested that arbitrary charge fractions can be obtained in fermion-soliton systems, provided charge conjugation is abandoned and the vacuum structure is sufficiently complex. An example with \ units of charge, which is not obscured by the two spin states, has been discussed by Su and Schrieffer [10] in the condensed matter context of TTF-TCNQ; while related ideas for particle physics have been investigated by Goldstone and Wilczek [11]. We thank our colleagues for many discussions which clarified for us each other's results. RJ especially acknowledges instruction on Peierls' instability by Y.R. Lin-Liu and I. Ventura, while JRS is grateful to W.P. Su for help with the manuscript. This research was supported by the National Science Foundation under grant nos. PHY77-27084 and DMR80-07432. * A review of these "index theorems" is found in ref. [9].

R. Jackiw, J.R. Schrieffer / Solitons withfermion number \

265

References [1] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398 [2] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; Phys. Rev. B22 (1980) 2099 [3] M.J. Rice, Phys. Lett. 71A (1979) 152 [4] R.F. Peierls, Quantum theory of solids, (Clarendon Press, Oxford. 1955) [5] H. Takayama, Y.R. Lin-Liu and K. Maki, Phys. Rev. B21 (1980) 2388 [6] (a) S. Ikehata, J. Kaufer, T. Woerner, A. Pron, M.A. Druy, A. Sivak, A.J. Heeger and A.G. MacDiarmid, Phys. Rev. Lett. 45 (1980) 1123: (b) LB. Goldberg, H.R. Crowe, P.R. Newman, A.J. Heeger and A.G. MacDiarmid, J. Chem. Phvs. 70 (1979) 1132 [7] J. Goldstone and R. Jackiw, Phys. Rev. Dl 1 (1975) 1486 [8] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681 [9] T. Eguchi, P. Gilkey and A. Hanson, Phys. Reports 66 (1980) 213 [10] W.P. Su and J.R. Schrieffer, Phys. Rev. Lett. 46 (1981) 738 [11] J. Goldstone and F. Wilczek, in preparation

214 PHYSICAL REVIEW B

VOLUME 25, NUMBER 10

15 MAY 1982

Fractional charge, a sharp quantum observable S. Kivelson and J. R. Schrieffer Institute for Theoretical Physics and Department of Physics, University of California, Santa Barbara, California 93106 (Received 28 December 1981) The magnitude of quantum fluctuations of the charge of a fractionally charged soliton is calculated. The soliton charge operator is defined as Q, — Q/ — (01 Qj | 0), where §f is the integral of the charge-density operator sampled by a function / peaked at the position of the soliton, and falling smoothly to zero on a scale L. 10) is the ground state of the system in the absence of solitons. It is shown that the mean-square fluctuation of Q, taken about its fractional average value Q, vanishes as O(go/L) for L » £ & where £0 is the width of the soliton. Thus, as L —• oo, the soliton is an eigenfunction of the charge operator with fractional eigenvalue. We also show that the portion of the charge fluctuations that are due to the soliton falls as exp( —L /| 0 ) as L —• oo. Nonetheless, the charge of the entire system, including all solitons, is integral.

&

1

Recently, Su and Schrieffer (SS) deduced that in quasi-one-dimensional charge-density-wave (CDW) systems of commensurability n = 3, there exist soliton excitations of charge Q, = ± e / 3 , ±2e/3, and ± 4 e / 3 and spin y , 0, and 0, respectively. These tively. These results are consistent with the fermion number y solitons discovered by Jackiw and Rebbi2 for a one-dimensional Dirac field coupled to a q>* bose field and with solitons having peculiar charge-spin relations discovered by Su, Heeger, and one of the present authors 3-5 for the linear polymer (CH)X, an n=2 CDW system. For general commensurability n, the simple counting arguments o{ SS associate with a soliton a fractional charge eN/n where N is the number of allowed spin polarizations. While the expectation value of the soliton charge was confirmed to be fractional using charge conjugation arguments' and by direct calculation for the n = 2 and n = 3 cases, questions have been raised6 as to whether the charge of a soliton is in fact a sharp quantum observable. That is, are the quantum fluctuations of the soliton charge about its fractional average value vanishingly small or is the fractional value Qs simply a quantum average of several integer values? In the latter case, each individual measurement of the charge would yield an integer value, and only the mean of these observed values would be fractional. More precisely, one may ask if an operator Qs exists such that (1) the state \s) containing a soliton of fractional charge Q, is an eigenfunction of Q„ 25

!

*

>

=

&

!

*

>

.

» • »

and (2) the force F on the soliton due to a slowly varying (screened) electric field, E, is

A central complication in answering this question arises from the quantum fluctuations of the band electrons measured over any finite length of chain, whether or not the soliton is present. These fluctuations are precisely analogous to the vacuum fluctuations which complicate the definition of the charge on an electron. As in that case, the charge only has a well-defined value in the infinitewavelength limit. We are thus led to define the charge Q/ in a region sampled by a smoothly varying sampling function / ( x ) of range L, for instance, f(x)=e-'1/Ll

,

(1.3)

so that Qf= f"j(x)f(x)dx

,

(1.4)

where p(x) is the charge-density operator. In order to show that the charge is a well-defined observable, it is necessary to show that: (1) the expectation value of the charge, (s \Q,\s), approaches a unique limit for large L for any f{x) and (2) that the quantum fluctuations of the charge, [5C]2=<s|[e,]2|*>-[<s|&|s>]2,

(1.5)

vanish for large L. The first property follows 6447

©1982 The American Physical Society

215 S. KIVELSON AND J. R. SCHRIEFFER

6448

directly from results in Refs. 1—3 where it is shown that the charge associated with a soliton is localized in a region of width 2£o, so that for

{s\Q,\s)=Q,+0(e~Uk),

(1.6)

where £0 is the correlation length defined in the next section. In Sec. II, two results concerning the nature of the quantum fluctuations are obtained. Firstly, it is shown that for any smoothly-varying sampling function of range L, such as the one in Eq. (1.3), the charge fluctuations vanish as

[8fi]2«(fo/D.

(1.7)

as long as £0«L «d, the soliton-antisoliton spacing. It is only in this limit that a sharp quan turn number Q„ can be defined. It is, however, only in this limit that the macroscopically observable charge is defined. Secondly, it is shown that the charge fluctuations which are due to the presence of the soliton (as distinct from the vacuum fluctuations that are always present) fall off exponentially with L /|o- Specifically, one finds [SG,] 2 -[oGo] 2 ~e

(1.8) 2

where for a given sampling function, [&QS ] and [6Q 0 ] 2 a r e the mean-square charge fluctuation in the presence and absence of a soliton, respectively. Thus, we see that the fluctuations of the soliton charge vanish exceedingly rapidly in the limit of large L. II. QUANTUM FLUCTUATIONS OF THE SOLITON CHARGE In this section we consider a specific model which allows us to calculate explicit upper bounds on the fluctuations of the soliton charge about its expected value, Q,. We thus consider the continuum model of Takayama, Lin-Liu, and Maki7 (TLM) which represents a system with commensurability « = 2 . For simplicity we consider the case of spinless electrons {N= 1) so Qs = ± je. The electronic part of the Hamiltonian for the continuum model is

25

sponding to right-going and left-going waves near the Fermi momentum, ±kF. The perfectly dimerized state has A(x) = A0 while in the presence of a soliton AU)=A 0 tanhU/£ 0 ) where £ 0 =fe! F /A 0 . In the following calculation we will adopt units ftvF = l. The model in (2.1) is unbounded below unless a cutoff is introduced into the fermion spectrum. However, the charge fluctuations remain finite in the limit that the cutoff goes to infinity, so we will neglect the cutoff and compute the charge fluctuations in this limit.8 In the same spinor representation the charge density operator is

P(x)=^(x)Wx) .

(2.2)

The charge operator, Qf, is then defined in terms of p as in Eq. (1.4). The mean-squared fluctuation of the charge about its expectation value may be computed in the ground state \G) according to

[SC] 2 =2 l«?IC/|a>l :

(2.3)

where | G) is the electronic ground state for a given lattice configuration, AU), and { j a ) J is a complete set of excited states. The advantage of using the continuum model arises from the fact that all the one-electron wave functions are known (see Refs. 7 and 9), both in the case of a perfectly dimerized chain and of a soliton-bearing chain. Thus, an explicit expression for [oQ]1 can be written in each case. In the perfectly dimerized case, A(x)=AQ, Eq. (2.S) can be written in terms of a double sum over the occupied valence-band wave functions, \kv), and the unoccupied conductionband wave functions \k'c),

[seo] 2 =22K*»IG/l*'<>r

(2.4)

where

eik* V2ek(l

<xlfct>> = -

Vek—k -Vek+k (2.5)

e'k*

<x|*c> =

Y2ek£l

Vtk+k Vek-k (2.6)

8TLM=fdxti,Hx)

-ifivFaz — ox

ek is the one-particle excitation energy, ek={k2+£$)l/2

MX)(TX ix),

(2.1)

where t/> is a two-component spinor field corre-

,

(2.7)

and O is the total length of the chain. For large 0 , the sums in Eq. (2.4) can be converted into integrals,

216 FRACTIONAL CHARGE, A SHARP QUANTUM OBSERVABLE

25

2

raw -/££!*«•>

and

[B(k,g)-ej+q2] B(k,q)

B(k,q)^ei-q2 (2.8)

where B(k,q)=ekjtqek~q transform of / ,

and F(k) is the Fourier

.

It is then straightforward to show that in the absence of a soliton, [SCo] 2 <

F(k)= JdxfMe ifcx

6449

In the presence of a soliton, AU)=A 0 tanh( AQX), an analogous expression for the charge fluctuations, [bQ, ] 2 , can be obtained from the known one-electron states. There are two contributions to

(2.16)

»o .

(2.9)

where A0 is a number of order unity, given by the expression dx 8

Ao=f^-[g'M]2

(2.17)

Similarly, in the presence of a soliton, [5G,]2=[6Gt]2+[oe,]2.

(2.10)

2

[&Qt,] involves only band-to-band excitations, with the midgap state associated with the soliton having the same occupation in the excited state, | a), as in the ground state, \G), and [5g,] 2 only involves excitations to and/or from the midgap state. [SQ*]2 can be written as

[»M/f/*

[B{k,q)-elW) B(k,q)

F(2q)

A 0 r(2g) B(k,q)

(2.11)

where T(k)= / dx e,fa7'(x)tanh(Aox) ,

(2.12)

f'(x) is the derivative of/, while [bQg]2 is given by

[5e,]2=/f-^|m)|2,

(2.13)

where S(k)=

f dx e'^f'WuecW&ox)

(2.14)

Note that we have not specified the occupancy of the midgap state. This is because for the case we are considering (n=2), the charge fluctuations are the same whether the state is occupied or not. We will now use Eqs. (2.8)—(2.14) to establish the desired results. First we establish a rigorous upper bound to the charge fluctuations. To do this, we adopt a sampling function with a single characteristic length, L, so that f(x)=g(x/L), whereg(0)=l andg(x)-*0 as x-*±x>. To establish an upper bound, we simplify the expression for bQ by using the inequalities

Bik,q)< \f?k-q2\ +*t>}q2/B{k,q) , B(k,q)^ei+q2

,

(2.15)

12

[SQ,}2
(2.18)

+

where Ai=

dx

/ -^-[g'(x)] 2 [l+(7/2)sech 2
is equal to A0 plus higher-order terms in §0/L, and

^ = / y dx_ l4 « ' W W | 2 .

(2.20)

Equations (2.16) and (2.18) show that the charge fluctuations vanish in the limit of large L. Note that this result depends critically on the existence of a gap, since £ 0 diverges as the gap goes to zero. Indeed it is easy to show that for A 0 = 0 (a onedimensional metal) the charge fluctuations, [&Qm ] 2 , approach a constant value in the limit of large L, [5e„,] 2

/2(0)_ 1 as L / | o - + oo 2ir 2ir

(2.21)

The second result involves distinguishing the part of the charge fluctuations that are explicitly due to the presence of the soliton from the fluctuations that are present, even in the absence of a soliton. Thus, as in Eq. (1.8), we are led to define

l*Q,)2=[BQ,V-W>o]2

(2.22)

To simplify the computation of [Ag,] 2 it is convenient to consider a sampling function, fix), such that fix)«1 over a region of width L and then falls to zero in a distance /. Specifically, we will consider a function fix) whose derivative is of the form

f'(x)=h(x +L/2)~h(x

-L/2),

(2.23)

64J0

S. KIVELSON AND J. R. SCHRIEFFER

where h (x) is smooth and nonzero in a region of width I about the origin and J" dx h (x) = 1. We would like to show that, for large L » | 0 , /, Aft falls exponentially to zero with increasing L. There are two sorts of terms that appear in the integrand of the integral expression for Aft

[Ag,]2= / -£•[/,(«>+/,(«)]

25

localized gap state defined in Eq. (2.13). For h (x) given by Eq. (2.29), we can find [8ft] 2 explicitly:

[8ft ] 2 =*

-L/io

iVTrr

(2.24)

2ir'

l+O

(2.32)

i] contains terms that are exponentially small for all q since it contains terms proportional to \S(2q) | 2 defined in Eq. (2.13) and | f (Iq) | 2 , III. CONCLUSION

f(*)= / d x e ' ^ / ' M I t a n M A o x ) - ^ * ) ] , (2.25) where ij(x) = l for x^O and —1 for x <0. Because /'(x) is localized in the vicinity of x= ±L /2, 7" is proportional to e ° and S is proportional to e~Ln*°. Thus, the dominant contribution to Ix comes from the term proportional to | S(2q) | 2 . /2(g) contains terms that are not particularly small, but are rapidly oscillating functions of q, t^oHHlq) cos(4gL) Ao+92

h(q) =

The definition of the charge of a soliton requires care, since as in quantum field theory, two complications arise. First, the ground state of the system in the absence of solitons exhibits quantum vacuum fluctuations of the local charge which must be subtracted when considering the fluctuations due to the presence of a soliton. Secondly, the chargeform factor of a soliton is spacially extended over the width £0 of the soliton. As is conventional in quantum field theory, it is useful to define the soliton charge operator ft as ^ l

J*<.

ft=<2/-<0|ft-|0>, -T(q)sm(4qL)

(2.26)

where

jft [*(M)-d+*'1

(2 2g)

J

IT * BHk,q) Because of the rapid variation of I2(q)> the integral of 72 is negligibly small. For instance, consider the case in which A(jc)=e-x2/'Vv^/.

(2.29)

It is easy to see that JdqJ2(q)~e-1Li/'2,

(2.30)

which is much smaller than the terms from Ix. We conclude that the dominant term in [Ag 0 ] 2 is the term proportional to \S | 2 ; that is to say [S&] 2 -[oeo] 2 =[8&J 2 +0(e +0{e-2Ll/'2), 2

,

(3.2)

(2.27)

and ;

where |0) is the ground state without solitons and Qf= fflx)pix)dx

H(k)= f dxelk*h(x) ,

(3.1)

-2L/J,,

(2.31)

where [8ft] is the charge fluctuations due to the

where p(x) is the linear charge density operator and fix) is a smoothly varying sampling function centered on the soliton and falling off on a scale L. ft is defined as the limit that L approaches infinity, with L «d, the distance between solitons. Thus, ft samples the change in charge when a soliton is created in a given region of the system, with vanishing contribution from other solitons which may be created in other regions. We have shown that not only is the expectation value of ft fractional in the presence of a soliton, but also that the change in the mean-square fluctuation of ft about its fractional mean vanishes as e as L —* oo. Furthermore, the mean-square vacuum fluctuations of ft in the absence of the soliton vanish as £o/L as L —* oo. It follows that, even if one does not remove the vacuum fluctuation contribution, the mean-square fluctuations of ft about its mean vanishes as \/kFL as L —»• oo. The soliton approaches an eigenstate of the charge operator ft as L —* oo as long as L is small com-

218 FRACTIONAL CHARGE, A SHARP QUANTUM OBSERVABLE

25

pared to the spacing between solitons. The remaining question is whether the operator ft couples to accessible external fields. Since the coupling of the chain to a (screened) electric field E(x) is

H'=-

f EMpMdx

(3.3)

then if E(x), likej'ix), is slowly varying in space it will couple to Q„ which is insensitive to the explicit form of fix) as long as L » £0. Thus, Q, is the operator which enters in determining the electric force of F, on a soliton due to a slowly vary-

>W. P. Su and J. R. Schrieffer, Phys. Rev. Lett. 46, 741 (1981). 2 R. Jackiw and C. Rebbi, Phys. Rev. D 13_, 3398 (1976). 3 W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979); Phys. Rev. B 22, 2099 (1980). 4 R. Jackiw and J. R. Schrieffer, Nucl. Phys. B 12Q, 253 (1981). 5 M. J. Rice, Phys. Lett. Z1A, 152 (1979). 6 A. K. Kerman and V. Weisskopf (private communication).

6451

ing (screened) electric field E,

£(*)=&£(*) .

(3.4)

ACKNOWLEDGMENTS We would like to thank Professor A. K. Kerman and Professor Weisskopf for raising the question of the quantum sharpness of fractional charge. Also, we would like to acknowledge many illuminating discussions with Dr. W. P. Su. This work was supported in part by National Science Foundation Grants Nos. DMR80-07432 and PHY77-27084.

7

H. Takayama, Y. R. Lin-Liu, and K. Maki, Phys. Rev. B 21, 2388 (1980). S. Kivelson, T. K. Lee, Y. R. Lin-Liu, I. Peschel, and L. Yu, Phys. Rev. B 21, 4173 (1982). The charge fluctuations have also been computed in the discrete model of SSH (Ref. 3) where no artificial cutoff is required. All the continuum model results obtained in this paper can be obtained as the first terms in an asymptotic expansion for [&Q]2 within the SSH model. [S. Kivelson (unpublished).] 8

219 P H Y S I C A L REVIEW L E T T E R S

VOLUME 52, NUMBER 17

23 APRIL 1984

Solitons in Superfluid 3He-A: Bound States on Domain Walls

T. L. Ho, (a) J. R. Fulco, J. R. Schrieffer, and F. Wilczek Institute for Theoretical Physics and Department of Physics, University of California, Santa Barbara, California 9 (Received 16 January 1984) The effects of solitons on the spectrum of fermion excitations in superfluid 3He-/4 are investigated. It is found that there is a two-dimensional manifold of bound states with energies within the gap of the bulk superfluid. The bound-state spectrum lacks inversion symmetry parallel to the wall. PACS numbers: 67.S0.Fi

There is considerable interest in brokensymmetry fermion systems exhibiting topological soliton excitations. It has been shown that the phenomenon of topologically generated fermion bound states is quite general. For example, in relativistic quantum field theories1 and in quasi-onedimensional conductors2 there are zero-energy fermion states bound to the soliton, which thereby acquires fractional or even irrational fermion number depending on the ground-state degeneracy of the system.3,4 Fermion bound states also exist in vortices in type-II superconductors,5 where the vortex singularity provides the trapping potential. On the other hand, in the case of superfluid 3He, where there are nonsingular topological solitons, it is not obvious whether all (or any) of them bind quasiparticles. In this Letter, we study the quasiparticle bound states associated with planar solitons in 3He-/4. We find that the spectrum of bound states has many branches which are confined to a two-dimensional manifold in momentum space. The spectra have some features similar to those of charge-densitywave (CDW) systems,2 in that for certain values of the momentum there is a zero-energy bound state. We also find that there is a lack of inversion symmetry of the bound-state spectrum in momentum space.

fd3r'[wb(TH(T.T')-

T')-H(.T,

7')]

G(r,r',w) F{r,r',(»)

3

He-/l is a p-wave BCS superfluid consisting of Cooper pairs with unit orbital angular momentum along an axis /. These are spin triplets (parallel spin pairing) which can be regardedas a state having spin perpendicular to an axis d.6 The excitation spectrum has an anisotropic gap A ("p) = [ 1 -(/•p) 2 3 1 / 2 A at the Fermi level |"p*|=pF. Unlike the CDW and relativistic theories mentioned above, the condensate field of 3 He-^ is a tensor (in spin and orbit space) rather than a scalar.6 As a result, there are many types of solitons in 3He-/4. In the following, we consider a simplified version of the "composite" soliton.7 It is a planar structure where d is uniform and / rotates through an angle 7T — 290 as one moves from — oo to +oo. The width of the soliton is fixed by the dipolar force and is of order 1 f*m. To understand qualitatively why bound states exist at all in composite solitons, consider the case in which 0O=O and /rotates continuously from — .pto x to y in the x -y plane as z varies from — oo to 0 to + oo, with A_—* A0—'A + . Near 2 = 0, where / = Jc, the "local" excitations traveling nearly along the x direction with If? I — p F w ' " have energies E(p) < |A 0 (/0| < |A ± |. Since these states lie in the forbidden region of the spectra at ±oo, they form bound states around z = 0. Consider the time Fourier transform of the Gor'kov-Nambu equations for p-wave superfluids:

•8(r-T)

(1)

h0 A(r,r*) A (/•,/•') -h0 '

(2)

+

where G, F, and A are spin matrices; GhV(a>) and r F^yito) are time Fourier transforms of the Green's tial. The gap function is given in terms of the twofunctions body potential as

GM„(T,r,/)«-i, Fllv(T,7',t)=-i(T*l(7,t)>l>l(T',0))\ hQ-= ( - V 2 /2WJ)-/X; and /J. is the chemical poten-

1524

A Mr (r,r)--K(r-r)<*„(r)iMD>. In the A phase, when the spin variable is a constant (say, d = y in the conventional notation) ,6 we have

AM„(r, r ) = A(r, r)8 M „

© 1984 The American Physical Society

[A(r, r ) — A ( r ,

220 VOLUME 52, NUMBER 17

PHYSICAL REVIEW LETTERS

T)]. The quantities G, F, and A in (1) and (2) then become scalars describing the parallel spin pairing. The Green's functions (7 and /"can be constructed from the eigenfunctions of H. For a spatially varying A, the integral equation HX = EX is difficult to solve. Considerable simplification can be made if we focus on the excitations near the Fermi level. The spatial transform of the gap function can therefore be approximated by8 Jrf3/-A(R+r/2,R-r/2)e-'7'r (3) =

where 4>(R) = wl(R)+iw2(R) (ivj • iv2 0, wf = n>2 — 1) is the local order parameter, and l=wxxw2 is the "angular momentum" texture. With (3), the integral equation HX = EX can be written as (£,(,-AO)«»-(A//»F)(T^-* + *-V)W».

(4)

( £ , + A 0 ) v „ = - ( 4 W ( T V ^ * + f ' V ) « n , (5) where X= (u„,vn). We search for solutions of the form (un,vH)=[u?(z),v°n(z)]e'T!r. For bound states, u° and vJ decay as exp( — « + |z |) as \z\ — ±oo, with K ± ' of order of the coherence length ^o^pf/irmA, which is large compared to pf1. Thus, for bound states or scattering states near the Fermi surface we neglect terms of order ^ ' P F 3 5 10~3 and Eqs. (4) and (5) become

(6)

[E„- i{pjm)bz + ep]v° = -(A//7F)(|A-^+/^-p)M„0,

(9)

(w 0 .v 0 )_,= (v°*,t,0*)-.

In addition, for <£ given by Eq. (8), there is the added formal symmetry E(~Px> -Py,P2)

E(px,py,pz),

It is essential to realize that the physical excitation spectrum is given by the positive eigenvalues E(JS) > 0, while the negative eigenvalues E(-p)<0 correspond to the energy lowering when an excitation of momentum + ~p is destroyed. Because of Eqs. (9) and (10) we consider only the case p • z 3= 0. Defining f = z/£ 0 , it follows that (pl/m)d2 = p-zA&(, and Eqs. (6) and (7) become ( £ + l ) f l - [ 8 { + /'({)]*. (.E-l)b-[-dt

(7)

(11)

+ FU)]a.

where £ ( ? ) = £ - z A £ , F(0 = (p-wl)/(p-z), with a = u°+v°, b=* - i(u°-v°). one obtains (£2-l)

m)B2-tp]u°

= (A//> F )(|V-0 + /0-p)v°,

Equations (6) and (7) have solutions with the formal symmetry

(10)

= /(A//> F )p-0(I),

[£„ + /' {pj

23 APRIL 1984

(12)

From Eqs. (11)

•8f+

(13)

where Va(b)~F2+(-)diF. For a sharp wall one finds the remarkable result that a physical excitation exists only for px > 0, with P2 = PP, and has energy E(V)-]p-z\L.

(14)

2

where ep = p /2m-n. We first consider order parameters in which w2 is fixed along z and wt rotates in the x-y plane (case a), 4>(z) = wx(z) + iw2{z) = x cosfl (z) + y sine (z) + &,

(8)

with the boundary conditions 0( — oo)=ir — 0O and 0(oo)-0 o , where Os=0 o s=|7r. We note that, at z— ±oo, the bound state solutions of Eqs. (6) and (7) are of the form exp( — K + |z|)(w,V) ± , where K+ and K_ > 0. The fact that E„ = £Cp) is real implies that ep = 0, i.e., p2 = p^.

This result is a consequence of Vb being an attractive (repulsive) delta function for px positive (negative). When the width of the wall increases, this "zeroth branch" solution persists but a finite number of higher-energy branches appear for both positive and negative px (see Fig. 1). The components of the wave function for the zeroth branch are a=0,

b(O = b(.0)exp[-£dsF(s)].

(15)

For this solution to exist, the following conditions must be satisfied: p-Wii + oo) > 0 ; £ - i v , ( - o o ) < 0 .

(16)

Note that as long as £(£) £ 0 for £ — ±oo the 1525

221 PHYSICAL REVIEW

VOLUME 52, NUMBER 17

LETTERS

23 A P R I L 1984

E

COS

p. 2

FIG. 1. The spectrum of bound states £(]?) for case a: i(z) = xcose(z) + ysin0(z) + /rand n = 0,l,2.

FIG. 2. The spectrum of bound states E(\S) for case b: (j>(z)- - z cosS(z) +j> sine(z) + ixand /i =0,1,2.

zeroth branch is independent of the shape of the soliton. Next we consider order parameters in which iv2 is fixed along x (case b),

and

4>(z)= - z cos9 (z) + y sm0(z) + ix, (17)

V<£;*0.

Since Eq. (17) is obtained from Eq. (8) by rotating $ ( z ) by w/2 about the y axis, it is easy to see that Eqs. (11) and (12) still apply and Eqs. (13) are replaced by (E+y)a

= (b( +

F+i&d(cos9)b,

( £ - y ) i > = ( - d{ +/=•+/'8d ( cos0)a,

(18)

where 8 ~ ( A / 4 £ F ) ( p - z ) ~ \ y=(p-x)/(p-z), and / " ( { ) - ( / > • M>i)/(p"z). All other quantities are defined as before. Again |"P*|2 = P F and we take p • z 5* 0. The zeroth-branch solution is 6 = 0, a ( O = e" C O 8 'exp[-J* 0
(19)

and the excitation spectrum is (see Fig. 2) E„-0(V)=-P-X&.

foTpx<0.

(20)

We next calculate the change Ap of the bare fermion density integrated over pz for fixed px > 0, due to the existence of the soliton centered at z — 0. To this end we introduce rigid-wall boundary conditions at z = ±a and periodic boundary conditions on x and y. For example, for case a and a sharp wall

where |v0j*)P=-Hi-€(?)/£(?)] 1526

(22)

ftB=(2n

+ l)7r/2a,

pjn = {In + l)n/2a

(23)

+a(?')/a,

where a(?')=-tan-1[(p-w1+)/(p-z)]. In the limit pFa » 1, the change in the bare fermion density arising from the scattering states for px > 0 is

A

^-iX° dp.

d(a\v(p)\2) dpz

1 2'

(24)

while the corresponding quantity for px < 0 is — y . Therefore the integrated bare fermion density arising from the scattering states is unchanged. However, since | f p | 2 = y for the bound states there would be an accumulation of half a bare fermion state, in the ground state, for each pL. In conclusion, we find that a planar soliton in 3 He-^ exhibits a two-dimensional manifold of bound states in the quasiparticle spectrum. This spectrum has two types of branches: the n - 0 branch and n > 1 branches. The former has the surprising property that its spectrum lacks inversion symmetry for px —• - px, where x is parallel to the domain wall. As a consequence, at T > 0 these quasiparticle bound states will be occupied and quasiparticle current will flow parallel to the wall in the direction /_ x z, where /_ is the angular momentum in the region z < 0 far from the wall. In the typical experimental situation in which many solitons are produced, there are broad antisolitons separating the solitons. Our solutions show that, at the antisolitons, an anomalous quasiparticle current will flow in the reverse direction. These currents are connected at the container wall, so that total

222 VOLUME52, NUMBER 17

PHYSICAL REVIEW LETTERS

current is conserved. This work was supported by National Science Foundation Grants No. PHY80-18938, No. DMR82-16285, and No. PHY77-27084 (supplemented by funds from the National Aeronautics and Space Administration), and the Physics Department of the Ohio State University. One of us (T.L.H.) would like to thank the Physics Department and the Institute for Theoretical Physics at the University of California at Santa Barbara for their hospitality during the summer of 1983. One of us (J.R.F.) would like to acknowledge the hospitality of the Centre de Physique Theorique-Ecole Polytechnique and the University de Paris-Sud, Orsay, France where part of this work was done.

'"^Permanent address: Department of Physics, Ohio State University, Columbus, Ohio 43210. 1R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).

23 APRIL 1984

2\V. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979), and Phys. Rev. B 22, 2099 (1980); R. Jackiw and J. R. Schrieffer, Nucl. Phys. B190 [FS3], 253 (1981). 3\V. P. Su and J. R. Schrieffer, Phys. Rev. Lett. 46, 738 (1981). 4 J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981). 5 P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966), p. 153. 6 See, for example, A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975). 7 K. Maki and P. Kumar, Phys. Rev. B 14, 118 (1976), and Phys. Rev. Lett. 38, 557 (1977); C. M. Gould and D. M. Lee, Phys. Rev. Lett. 37, 1223 (1976). 8 The Fourier transform of (3) should be /(S., \?t\)ibp - ^ ( R ) , where /== 1 near pF and vanishes beyond a cutoff a>c about the Fermi level. Our approximation reproduces the bulk spectrum near \f\=pF. It amounts to suppressing the spatial variation of the magnitude of the gap and requiring a>c/A » 1. The approximation is reasonable since, as we shall see, the bound states only involve states with |p"| = p F .

1527

Synthetic Metals, 9 (1984) 451 - 465

451

COLLECTIVE COORDINATE DESCRIPTION OF SOLITON DYNAMICS IN /7L4JVS-POLYACETYLENE-LIKE SYSTEMS S. JEYADEV and J. R. SCHRIEFFER Department of Physics, University of California, Santa Barbara, CA 93106

(U.S.A.)

(Received February 21, 1984)

Abstract The two-soliton field configuration for the 0 4 field in the absence of phonons is approximated with the soliton positions and the amplitudes of the shape oscillation mode as collective coordinates. The potential energy of the field is calculated for this configuration as a function of soliton separation and the amplitude of the shape oscillation mode. The equations of motion for these coordinates are derived, and the collision of two solitons is studied by numerically integrating the coupled equations.

1. Introduction Recently there has been considerable interest in the subject of solitons in condensed matter physics. Soliton models have been proposed for the study of excitations in commensurate quasi-one-dimensional conductors such as fnms-polyacetylene [1 - 3] as well as structural phase transitions [4], magnetic phenomena [5] and textures in superfluid 3 He [ 6 ] . Our primary interest is in studying the dynamics of soliton-antisoliton (SS) collisions in commensurate quasi-one-dimensional conductors. Such processes are of importance in the recombination kinetics of photogenerated SS pairs in pristine trans-poly acetylene [ 7 ] , as well as transport effects in the doped material [8]. Since the dynamics of the coupled electron-phonon system in frans-polyacetylene are quite complicated, we consider the simpler 0 4 field theory [9], This theory has many features in common with the adiabatic approximation to the coupled electron-phonon system in frans-polyacetylene and provides a framework for understanding SS collisions in transpolyacetylene. We use the collective coordinates introduced by Gervais and Sakita [10] and extended by Tomboulis [11] as the dynamical variables and derive equations of motion for these variables. The formalism is of more general applicability than the * SS collision. Initial computer simulations by Wingate [12] revealed that, if the initial velocities of a separated soliton-antisoliton pair were above a certain critical velocity, the pair always separated after one bounce. Below this critical 0379-6779/84/$3.00

© Elsevier Sequoia/Printed in The Netherlands

452

velocity there exists a spectrum of values of initial velocities for which the pair could bounce more than once before separating. Campbell et al. [13] have carried out extensive computer simulations of the two-bounce case and have constructed a phenomenological theory for it. More recently, Guinea [14] has studied collisions of solitons and breather-like modes in irans-polyacetylene by performing computer simulations directly on the Su, Schrieffer and Heeger model [1].

2. Equations of motion Following ref. 4, the Lagrangian of the 0 4 field is expressed, in the continuum limit, as

L=

A , B . 1 ,/du r dx 1 . , — mu-(x, t) — —u(x, t)~ — — u(x, ty — — mc,, — , 2 2 4 2 \dx!

IT

(1)

with A < 0 and B > 0. m is the mass of the medium per length /, and c 0 the speed of harmonic sound for ^ 4 = 5 = 0. A < 0 leads to a two-fold degenerate broken symmetry pound state, analagous to the two bonding configurations in trans-(CK)x [1]. B > 0 stabilizes the Peierls distortion. The local potential (.4u2/2) + (Su 4 /4) is a double well with minima at u0 =

±(\A\lB)v\

(2)

The Euler-Lagrange equation of motion for the field is mu +Au + Bu3 — mc02u" = 0

(3)

In addition to conventional small-amplitude harmonic phonon modes about the ground state, this non-linear equation has travelling soliton solutions of the form x — vt u - uQtanh (\y/2^0y/l-v2lc02

(4)

where £02 = mc02/\A\. Introducing the dimensionless variables jc = x/So. t = Cof/Soi 0 = vl°o and 17 = u/u0

(5)

eqns. (1), (3) and (4) can be rewritten as I•-

\mc0-

r- J d * ^

fj-17" -T? +TJ3 = 0,

and

+

-^ - 7 * - r -

.'2

(6)

(7)

225 453

Tj = tanh

•pi

(8)

y/Zs/V^

respeetviely, where the dot represents differentiation with respect to i and the prime differentiation with respect to x. This notation will be used henceforth. We are interested in investigating the influence of internal vibrational modes when solitons collide. Thus we seek to determine the form of smallamplitude oscillations about the soliton solution [15]. Consider a stationary soliton located at the origin and express the deviation from the static solution by \jj(x, i) so that ri(x, r) = tanh —= + i//(x, t)

(9)

Substituting (9) in (7) yields 3 , x 4, -ip" + 2 1 — — seclr —— x = 2 y/2

3 t a n h | - T = U 2 + i/r

Iv™

(10)

For \\j/(x, t)\ < 1, we can neglect the non-linear terms on the right-hand side of eqn. (10). Setting \jj{x, t) =
(11)

we find the eigenvalue problem ^ —- 4> = F2<j> - 0 " + 2 1i — — sech2 v^. 3

x

(12)

whose eigenvalues and eigenvectors are given by the well-known expressions (i)F2 = 0

^ 0 (x) = sech 2 Je/v^

3 (ii) F- » -

^I(JC)

(iii) F2 = 2 + —=Fl

(13a)

x x = tanh -= sech —

(13b)

0 Q (x) - exp(iQie/ v /I) 3 tanh 2 V^

— 3i£ tanh —2 V

Fo> 0

—1

Q:

(13c)

These solutions correspond to (i) translation of the soliton, (ii) vibration of the soliton width and (iii) continuum phonons scattered by the soliton. For a soliton moving with velocity 0, the eqns. (13) should be Lorentz transformed, i.e.,

x-pi

x = —,-

-,

(14a)

454

£' =

i—fic

(14b)

and (14c)

M*', f ) =
In order to study the collision process, we assume that initially we have two pure solitons travelling towards each other with equal speeds, 0O; i.e., the initial amplitudes of the <j>i(x) and
Ftf + Px)]

^

(+)COS

+

7T^F]

SSQ(0

Q

Fy(i-0x),

0Q(-)COS

i

FQ(i + 0x) PQ(1-)COJ

—

QK

,

V

H

1 —

'

P" (15)

JV"_O2

where x y± =

±a(t)

v^Vl-^'

<* ~ 0,

0i(*)

=

tanh y ± sech y±

(16)

and, similarly, i(x) and (j>Q{x) modes and obey the initial conditions

Pit

°°) = j30 A ( £ - " - ° ° ) = 0

B<,(F->-°o) = o

(17)

The use of the more general a(t) rather than j30t takes into account the 0o(*_) modes. We now take as our new dynamical variables a(t), A(t) and BQ(i) and seek equations of motion for these variables. From eqn. (6) we have for the Lagrangian density n* — JC[T7,7?,T7']=£O-1?

|2 "

+

2

1?>2

1

1

2*

IT ^

-n2-

4

(18)

where £0 = mc02u02/£0l. Using eqn. (15) in eqn. (18) gives £ = Z[x, i; a(t), a(t), &(t), A(i), A(t), B(t), B(t)]

(19)

455

We use eqn. (19) in Hamilton's principle 5/J"d*dF£ = 0

(20)

to derive our equations of motion. The equations of motion are

fdx fdx

oa

dt (

(77 — T7" — T7 +1? 3 ) — = 0

6a

= 0

(21) (22)

and

fdx

(17-17

- 1 7 +T) )

dB Q

=0

(23)

We use the total derivative in eqn. (21) to indicate that the operator acts on both implicit and explicit time dependence. As can be seen, the method can be generalized to the study of other problems and systems. Given a Lagrangian density £=£[17,17,17'], we choose a trial solution 17 = T7(*, i; {a,(F)}, (a,(F)}) j = 1, 2, 3, ... where the { a j are the new dynamical variables. Then we have

da,

di T"

da( \\

(24)

where (25)

3. Potential energy calculation In order to gain a qualitative understanding of the collision process, we first calculate the potential energy of the field (15) with BQ = 0. We thus evaluate

P.E. = fdx

1_ 1 1 -T? 2 + - 7 ? 4 + - V 2 2"

(26)

for 17 = 1 + tanh y_ — tanh y+ + A\$x(—) — 0i(+)]

(27)

456

The calculation is tedious but straightforward, and the result is given in Appendix A. We note from eqn. (A.l) that the 'potential energy' of two solitons separated by a distance 2a depends explicitly on the speeds of the solitons. Hence eqn. (A.l) resembles a dynamically generated 'potential' more than a simple potential energy between two particles. However, in the limit 0 = 0, eqn. (A.l) can be interpreted as the potential energy between the solitons in the usual sense, plus twice the potential energy of a single soliton. In the limit d -»•<», we get from eqn. (A.l) P.E.-V„c /mCoW\ / r - ^ \

2(2 -p) 3(1 - 0 2 )

1

7T02 15-802 , A 2 2 2 4(1 - 0 ) A + 1 5 ( 1 - | 3 — ) A +

%o I

+ — A3 + — A 4 8 35 which is just twice the potential energy for the configurations

Th(x, t) = tanh

JCj1_g2

(28) '

K

+ A

*iW

(29)

Treating A as a variational parameter, the linear term in A in the expression for the potential energy of the one-soliton case (i.e., the field configuration given in eqn. (29)) shows that there is a non-zero value of A which minimizes the potential energy of the field. Since the Lorentz contraction is taken into account in the definitions of y ± , this implies that eqn. (8) is not the solution which minimizes the potential energy for a given value of the speed j3, as is consistent with relativistic dynamics in which the centre of mass and interval variables of an extended object are coupled. Continuing to treat A as a variational parameter, we calculate the kinetic energy of the field configuration (29): K.E. 2

2

mc0 u0 \

nr

1

n

r ) ^

55

=

/ dx — T?i = 2

LL

2

2

-

1l --0i L 3

7 * . , + - A +• — A 2

8

30

(30)

Hence the total energy, EAaeie, of a single soliton with respect to the vacuum is given by

imcoW)^

+ A3+ A4 r6 h

/

^

m-P)

4(1 -02)

30(1-02)

(31)

457 10

1

I

i

0 =0 s

-

-

*3 N,

—6

(MO 3

<

i~ >

1 4-

_

|

UJ

a. A'-I.O -0.5

2

0 0

1

i

Fig. 1. Potential energy of field as a function of separation, d, for . 4 = 0 , 0.5, 1.0, 1 5 and 2.0 with j3 = 0. Fig. 2. Potential energy of field as a function of separation, d, for A = 0, —0.5 and - 1 . 0 with (3 = 0. !0

*

i

!

8--0

/3=0

a

-

_

11

1

•s.

a; a

I

o

u* -

mo

<

MO A£-3J3

ICVJ

ICVJ

-3.5

U

UJ

zZO

a.

|

2

2l-

1

OL

10

d

-

i

-2.5

A»-4.0

15

0

-5

I

0

i

I

5 d

10

15

Fig. 3. Potential energy of field as a function of separation, d, for A = —1.5, —2.0, —2 5 and—3.0 with 0 = 0. Fig. 4. Potential energy of field as a function of separation, d, for .4 =» —3.5 and —4.0 with 0 = 0.

Again, we note, that A = 0 does not minimize the total energy of the field; i.e., eqn. (8) is not the minimum total-energy field configuration. However, the coefficient of the linear term A in both the potential and total energies is proportional to 02(1 — 02)1/2 and hence vanishes for a static soliton and hence eqn. (8) w the minimum energy solution (3 = 0. The fact that there are field configurations with lower potential and total energies than eqn. (8) is due to the fact that the 'minimum energy principle' is not valid as we have time-dependent solutions, as pointed out by Jackiw and Rossi [16]. The potential energy is plotted as a function of d for various A in Figs. 1 - 4 for

458

Fig. 5. Potential energy of field as a function of A for d = 1, 2, 4, 6, 8 and 10 with (3 = 0. Fig. 6. Potential energy of field as a function of .4 for d ~ —2, —4, —6, —8 and —10 with 3 = 0. 10 —

-7

'

i

;

£ = 0.8 3

-

-

r ^

o

A=,o

,3 = 0.8 31-

a? «l-

MO

< as

< 4

2h

-5

|

1

/ /

•? 4 A=0

2 _

-0.5

10

oL. -5

Fig 7. Potential energy of field as a function of separation, d, for A = 0, 0 5 and 1.0 with 0 = 0.8. Fig. 8. Potential energy of field as a function of separation, d, for A » 0, —0.5 with 0 = 0.8.

(3 = 0 and for /J = 0.8 in Figs. 7, 8 and 9. The plots show that the speed of the solitons does not greatly affect the qualitative nature of the curves. Of special interest is the development of a local minimum for negative values of A for 3 ~ 5. This is when the solitons just begin to overlap (their width in our units is 2 >/2~) and hence this feature can play an important role in trapping the outgoing solitons and causing them to bounce as observed by Wingate [12]. From Fig. 8 we also see that as 3 -*•<», the potential energy for A = —0.5 is lower than that for A = 0 as noted above. In Figs. 5 and 6 we plot the poten-

231 459

Fig. 9. Potential energy of field as a function of separation, d, for .4. = - 1 . 0 , - 1 . 5 , - 2 . 0 and —2.5 with 0 = 0.8.

tial energy of the field as a function of A for various 3. We see that for 3 > 2 the potential energy is minimized for A = 0. The 3 = 2 case has a broad minimum, while for 3 = 1 (i.e., appreciable overlap) we see that the minimum occurs at a finite negative value of A. This shows that interacting solitons flatten themselves to reduce the potential energy of the field. 4. Equations of motion for the collective coordinates In order to study the collision of the two solitons we have to derive the equations of motion for the dynamical variables a, A and {BQ}. However, as a first approximation we set BQ = 0 for all Q, i.e., we eliminate all the phonon degrees of freedom from the problem, and retain only terms up to first order in A. Then eqns. (21) and (22) will yield the equations of motion for a and A, correct to order A. Since relativistic effects make the equations of motion lengthy, we only retain terms of leading order in j3. Thus our trial solution now takes the form r}i(x, i) = 1 - tanh y + + tanh y_ + A f F ) ^ - ) - *,(+)]

(32)

where the common time dependence of the 0 t (±) modes has been absorbed in the definition of A(t) (the spatial dependence of the cosine factors is due to Lorentz transformation and is hence neglected), and x ±a

'•--7T

(33)

232 460

Then, as d: no longer enters the trial solution rj1,we have / d*(f)i - n\-

Vi + 77i3)(sech2 y_ + sech2 y + ) = 0

(34)

for the equation of motion for a and / d*(Th - r t f - T h + T h 3 ) ^ - ) - 0 , ( + ) ) = 0

(35)

for the equation of motion for A under the approximations mentioned above We thus obtain fv(d) A +

3 2

• f2(d)A + f3(d)A + f4(d) = fs(d)ii

-A

(36)

for the equation of motion for a, and 3 2

i(d) A + - A

-£ 2 (
for the equation of motion for A. The coefficients f^d), f2(d), f3(d), given in Appendix B, and d = 2a/y/2

(37)

etc., are (38)

In eqn. (36) f5(d) plays the role of the mass of the coordinate a. Since a is the position of the soliton, fs(d) gives the dependence of the soliton mass on the separation. f4(d) is the interaction term which acts the force on the coordinate a. Similarly, gs(d) and g4(d) are the mass and force terms of A. If we set A = 0 in eqn. (32) we have only the equation of motion for a, and from eqn. (36) and Appendix B we get 2>/2

1+

3(d cth - 1 ) 1 a =6 2 13" snh

— cth + cth 2 +• 3

4 cth + 4(cth - 1) d cth (39)

snh 1 i.e.

M,(2a)(2a) = -

P.E.

3(2a) J

\ where

M.(2a) =

vttl 1 +

(40)

ImcpW

3(dcth-l) snh2

'so

A = 0

461

behaves as the reduced mass of the soliton-antisoliton system. Note that /xs(0) = 2JI S (°°) = 2 -«/2/3. The mass of the soliton in these units is 2 \ / 2 / 3 .

5. Numerical results We take as boundary conditions two solitons with speed /30 = 0.05 approaching each other with A(0) = A(0) initially being zero. As shown in Fig. 10(c, d), the soliton and antisoliton accelerate towards each other and develop a non-zero value for A. This value is negative, and its magnitude increases monotonically with decreasing spacing, i.e., the soliton and antisoliton tend to flatten themselves as they approach each other, as expected. The solutions clearly indicate that no matter how small /30 is, the problem becomes relativistic when there is strong overlap between the solitons since the attractive potential increases the kinetic energy to values greater than the soliton rest energy. Further, in this region \A\ £ 1, and thus the linear approximations made in deriving eqns. (34) and (35) break down. An

(o

a(0) = 4

(d)

a(0)=4 20

40

60

80

0

J_

10

20

30

40

7 T Fig. 10 (a, b) Speed of solitons, /3, and amplitude of shape mode, A, as a function of time for the initial conditions a ( 0 ) = 1.5, 0(0) = 0.5, A(0) = A(0) = 0. (c, d) /3 and A as a function of time for the collision process with initial conditions a(0) = 4, 0(0) =—0.05, A(0) = A(0) = 0.

462

analysis of the numerical values shows that the neglect of non-linear terms and relativistic effects causes a large build-up of A: This, in turn, causes a spurious acceleration of the outgoing solitons. Further, the mass term for A, gi(d), vanishes as d -*• 0, causing a large increase in A. While the approximations used make it impossible to study the collision process of the solitons in detail, the method is still applicable to cases where the approximations are valid. Such is the case when soliton— antisoliton pairs of the same charge in frans-poly acetylene collide. The Coulomb repulsion prevents them from overlapping completely and this makes an analysis of the collision possible. In Fig. 10(a, b) we show the results for the initial conditions a(0) = 1.5, j3(0) = 0.5 and A(Q) = i ( 0 ) = 0. The members of the pahmove away from each other and their interaction causes a development of the width oscillation mode. This couples to the velocity which also exhibits oscillations. It can be seen that the asymptotic speed is lower, j3(°°) ~ 0.35. which is a result of the kinetic energy going into the potential energy of the field. The oscillations in ]3 and A have a time period T ~ 5 which is the period of the shaped mode in our units (CJ = >/3/2). Further the maxima in 0 occur at the minima of A and vice versa as expected, since the minima of A correspond to a lowering of the potential energy of the field. When a £ 1, our equations are not valid and the computer simulations of Campbell et al. [13] and Guinea [14] are, at present, the only method to study collisions of neutral and oppositely charged solitons in trcms-polyacetylene. Lastly, we note that the present method can be used to study the dynamics of the scattering of low energy solitons by impurities.

Acknowledgement This work was supported by National Science Foundation grant DMR82-16285. References 1 2 3 4 5 6 7 8 9 10 11 12

W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. B, 22 (1981) 2209. M. J. Rice, Phys. Lett., 71A (1979) 152. S. Brazovskii, Zh. Eksp. Teor. Fiz., 78, 677 (Sov. Phys. JETP, 51 (1980) 342). J. A. Krumhansl and J. R. Schrieffer, Phys. Rev. B, 11 (1975) 3535. J. Tjon and J. Wright, Phys. Rev. B, 15 (1977) 3470. K. Maki and P. Lumar, Phys. Rev. B, 14 (1976) 118;Phys. Rev. Lett., 38 (1977) 577. Z. Vardeny, J. Strait, D. Moses, T.-C. Chung and A. J. Heeger, Phys. Rev. Lett., 49 (1982)1657. Proceedings of the International Conference on the Physics and Chemistry of Conducting Polymers, LesArcs, J. Phys. (Paris) Colloq. 3, 44 (June) (1983). C. Itzykson and J. B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1980. J.-L. Gervais and B. Sakita, Phys. Rev. D, 11 (1975) 2943. E. Tomboulis, Phys. Rev. D, 12 (1975) 1678. C. Wingate, SIAMJ. Appl. Math., 43 (1983) 120.

13 14 15 16

D. CampbeU, J. F. Schonfeld and C. Wingate, Physica D, 9D (1983) 1. F. Guinea, Institute for Theoretical Physics, Santa Barbara, preprint. J. Goldstone and R. Jackiw, Phys. Reu. D, 11 (1975) 1486, R. Jackiw and P. Rossi, Phys. Reu. D, 21 (1980) 426.

Appendix A 2(16 - 1 7 0 2 ) 3(1 ~ 0j 2 ~) - + 8
P.E. - Vvac 2 2 !mc0 u0 2(4 -5j3 2 ) 1-02

d dcth-1 32 - 35ff2 + 12 (cth I + nA lah2} snh2 4(1-/J2)

1 j snh

6 cth +

1"

5-402/ 1

*

srH:

~cth

1 -13 Isnh2 1

asfl 45 — - 38(3

+A . L 15(1 - | 3 2 ) +

3 ( 1 + CSh

snh

2(2-30a)i

\

2

3 - 24 cth + 12 cth 2 snh

6 + 3 cth + 24 cth 2 - 12 cth 3 +

+

—;—M—

,, 3 cth

2-3cth3

7

~ TT c t h

+

2

I)

1 -(3 ( 2 snh 1 l-
* +3(1 -acth)

cth

( )

cth 2 +6 3 c t h 4 - — c t h 2 + 1 + ) \ 3

c t h - - 5 ( 4 cth 2 - 2) | 24 j 1 - 2 d cth L7 + -— cth + 2 cth 2 - & cth 3 + > + snb/ ) snh( snh 2 12 cth (3 J cth - 2

1/ iL2 - 2 cth 2 + d cth 3 - — } + \ snh 2 3J Ucth(
+ snh 1 +

-rr 2

1-0 |

>+

TT~2

snh

- 6 Cth3 + 4 Cth +

| (l-j3 )snh| )| r i 3 f + d(6 cth 4 - 6 cth 2 + 1) f + 2TTA3 - — + 4 cth 4 - 4 cth 2 +

)J

+ +

1 2

I

16

snh |

2(2 c t h 2 - 1 ) cth | \ 3 3 , , 13 cth / —— + - - + - cth 2 - cth 4 + — cth snh | | 8 2 snh 3 \

3cth( . 5 cth 4 snh )

11 , ,2 7 ( 3 - 5 cth 2 ) ) 1 ( cth + - + -cthf + A 2 8 snh snh 2

464

2 51 - 3 - — + — cth 2 - 12 cth 4 + csh 2 (12 cth 2 - 11 + 2 tnh 2 ) c s h - 3 6 cth 2 , 2 cth / 11 + 7> cth 2 + 6 cth 2 - — + * snh \ 2 1 2 cth 6 + 2A4 — + + tnh 2 - - 7 cth 2 + 6 cth 4 35 snh 5 3 3(2 c t h 2 - 1 ) cth ) 3 (

-— ^ — I

+

i2 cth2 io cth4) cth

^ r-

*

26 / 4 2„ 10 cth + — cth - 1 [

~ (A.l)

where d=

2a

(A.2)

snh = sinh d

(A.3)

csh = cosh d

(A.4)

tnh = tanh 3.

(A. 5)

and cth = coth d.

(A.6)

Appendix B: coefficients of the equations of motion The coefficients in the equation of motion for a are given by fi(d)

cth 1 + 2 — - 2 cth 2 snh snh

2 - 6 cth 2 5 - 6 cth 2 — ; + cth 4 snir snh 3 3 f3(d) = 3 — — + — cth + 3 cth 3 — 2 cth 4 + 2 cth 5 4 4

20 1

2" + 11 cth - 22 cth 2 - 14 cth 3 + 22 cth 4 snh

(B.l) (B.2)

465

3 - 10 cth - 12 cth 2 + 20 cth 3

(B.3)

snh2 6 1

4

Ud) = -

,

7T

fs(d) =

V5

1+

3ir

, 3-4cth + 4(cth-l)dcth cth + cth 2 + snh 2 3(d cth - 1 )

(B.4) (B.5)

snh2

The coefficients in the equation of motion for A are given by 1

d + 2(1 —d cth) cth

Slid) =

(B.6)

snh

20

8-A.d) = y/2

ST±

g,(d) - 3 3 23

3 - 5 d cth + 6(d cth - 1) cth :

(B.7)

cth - — cth 2 + 3 cth 4 3 3 + 12 cth - 16 cth 2 | cth + d ( l + 8 cth — 13 cth 2 - 12 cth 3 + 16 cth 4 ) - 4 snh

4 - 2 cth + 3 cth 2 + 2(1 + cth - 3 cth 2 ) d cth

(B.8)

snh2

Ud) -

3TT

- 4 cth 4 - 4 cth 3 - 3 cth 2 + 3 cth +

(1 + 4 cth - 4 cth 2 ) cth - 1 snh (B.9)

8s(d) -

2>/2" snh

1 - 2 cth 2 +

2 cth snh

(B.10)

where we have used the notation d = 2a/ v ^

(B.ll)

0 = 0!

(B.12)

cth = coth (d)

(B.13)

and snh = sinh (d)

(B.14)

Nuclear Physics B251 [FS13] (1985)117-126 © North-Holland Publishing Company

STATISTICAL MECHANICS OF ANYONS Daniel P. AROVAS Department of Physics, University of California, Santa Barbara, CA 93106, USA Robert SCHRIEFFER and Frank WILCZEK Department of Physics, University of California and Institute for Theoretical Physics, Santa Barbara, CA 93106, USA A. ZEE* Institute for Advanced Study, Princeton, NJ 08540, USA Received 31 July 1984

We study the statistical mechanics of a two-dimensional gas of free anyons - particles which interpolate between Bose-Einstein and Fermi-Dirac character. Thermodynamic quantities are discussed in the low-density regime. In particular, the second virial coefficient is evaluated by two different methods and is found to exhibit a simple, periodic, but nonanalytic behavior as a function of the statistics determining parameter.

In two space dimensions, a continuous family of quantum statistics interpolating between bosons and fermions is possible [1,2]. Two examples of particles obeying exotic statistics have been discussed. The soliton of the (2 + l)-dimensional 0(3) nonlinear a-model has a spin which is neither integral nor half-odd integral, and obeys a statistics which is neither Bose-Einstein nor Fermi-Dirac [3]. In condensed matter, one finds that the Laughlin quasiparticles [4] in the anomalous quantum Hall effect system also possess fractional charge and fractional statistics [5], a result recently derived from the adiabatic theorem [6]. We first discuss a method [1] by which the statistics of a two-dimensional system of charged particles can be changed (continuously) via the introduction of a fictitious "statistical gauge field." It is well known that the wave function of a charged particle interacting with a magnetic flux tube will acquire a phase change due to the motion of the particle. If the charge is e* = ve and the flux is * On leave from the University of Washington 1983-1984. 117

118

D.P. Arooas et at. / Statistical mechanics of anycms

<> / = a0 = ahc/e, a complete revolution will induce a phase change of e / 4 r with Ay = lirav. The basic idea of this method is then to introduce a flux tube at the position of each particle whose magnitude <£ will determine the phase associated with relative particle motion. The gauge field associated with this flux is nondynamical, i.e., its evolution is completely determined by the motion of the particles. It is expected that physical quantities are periodic in the statistics determining parameter a. In the symmetric gauge, the vector potential due to a flux tube of magnitude = a0 takes the form A(r)-£*x±-¥i-&ve. 2-n r2 2ir r

(1) v '

2n

The many-particle generalization we seek is therefore

AM-l&SxZ'li-pZ'v*; *

'

r2

2ir J

r

U

2ir

'

,J

(2) w

j

where 0,y = tan - 1 ((^ - # ) / ( x } - xt)) is the relative angle between i and j , and the prime on the sum indicates that the term j — i is to be excluded. This leads to the following many-body hamiltonian: HW-Z^Pi-^Aoiri-^A^f+Vir,,...,^),

(3)

where A0 is the physical vector potential, if present. Suppose one knows an eigenfunction yp0 of the bare hamiltonian H0 = H(0) with energy E0. Then since •«
a

0 -^EV^-^r,),

(4)

we see that »//a = exp(/'a£pairs0,7)t/'o is an eigenfunction of H(a) also with energy £ 0 . The problem is that the function tpa will not in general be a single-valued function oi its arguments. For the two-particle problem with no external potential and no interparticle interactions (free particles), the situation becomes eminently tractable. Recall that the wave function can be decomposed into a product i//„(r, R) = x(J?)£(r) where R is the center-of-mass position and r is the relative coordinate vector. We find 2 2 2 2 (M = particle mass). X(R) = e"-* *(r) = e * % , ( * r ) , E = h K /4M + h k /M Imposition of Bose (Fermi) statistics then requires that m be even (odd). The introduction of a statistical gauge field then provides us with new eigenfunctions +a(r,R) = eia%(r,R) = eiKWm+a)9J.mXlcr). Here, Bose (Fermi) statistics re-

D.P. Arovas etai. / Statistical mechanics ofanyons

119

quires that m + a = / be even (odd) and we obtain wave functions t.(r,*)-e"-WM(tr).

(5)

If we now introduce a circular boundary at some radius R, we find that the allowed energies are *l,n = *2Xf,-aUn/MR2,

/,(*,.»)-°-

(6)

Hence, choosing a to be an odd integer merely shifts the energy spectrum from Bose-like to Fermi-like. Note also that the spectrum is periodic in a with period Act = 2. This is in fact true for the iV-particle system, although explicit (single-valued) wave functions are difficult to obtain due to the fact that there are \N(N - 1) relative angles and only (N - 1) non-CM angular degrees of freedom. For M = 2, these numbers are identical. This result, eq. (6), can be used to evaluate the second virial coefficient. In two dimensions, it is easily shown that [7]* B{T)=\A-A~lX\Zz,

(7)

where A is the area of the system \ T = (2irh2/MkT)l/2 is the thermal wavelength, and Z 2 = T r e - ^ 1 is the two-partiele partition function. The virial expansion is an expansion of the equation of state in the density n: P = nkT[l + Bn + Cn2 + • • •]. In performing the trace to obtain Z 2 , the center-of-mass freedom is trivially separated, yielding a factor Z 2 = 2A\j2Z2, where Z 2 is now the single particle partition function for the relative coordinate problem: Z 2 = Trrele~^Wrd. This will again be area divergent, and it is therefore convenient to calculate the virial coefficient B(a, T) in terms of a known quantity, i.e., B(2j,T) = - \\2r or B(2j + l,T)= + \\\, the familiar result for Bose and Fermi systems, respectively (je Z). Thus, we obtain 5 ( a ' , r ) - 5 ( a , r ) = 2A2T[Z2(a)-Z2(a')].

(8)

We now appeal to the result (6). Clearly B(a, T) must be periodic in a with period Act = 2. We will take our original particles to have Bose statistics and expand about even and odd values of a. For a = 2j + 8, \S\ < 1, corresponding to quasi-bosons, the allowed values of \l- a\ are |5|,2 ± 5,4 + 5, etc. For a = 2j'+ 1 + 5, |5| < 1, corresponding to quasi-fennions, die allowed values of \l— a\ are 1 + 5,3 + 5, etc. Since B must be independent of the cutoff R in the limit R-* oo, and since R * To be precise, we should write B(T) = A\\ - Z-JZ\\ with Zx = Tre - *", the single particle partition function. With no externalfields,we have Hx ~p2/2M and Z{ = .4Ay2, which then yields eq. (7)

120

"•"• Annas et at. / Statistical mechanics ofanyons

appears only in the combination MkTR2/hz, it is desirable to rescale R 2 -> ]jh /MkTR. Expanding about the Fermi point, we find that (8) and (6) give + 2X2T

B(2j+l+S,T)^^X\

x.lim £ R

x

~*

£ [ 2 e - ^ . " / « ) 2 _ e - ( - - . . / « ) 2 _ e - ( - ^ . , / * ) 2 j . (9)

1-1 n - l odd

The factor in brackets resembles a second derivative. By expanding in S, one can then perform the /-sum by means of the celebrated Euler-MacLaurin formula [8]. This leaves 5 ( 2 j + l + S , r ) = H 2 T - 2 X 2 T S 2 lim ~R~2 £ JC1 „

^

^

-<*i../")"

(10) As n -» oo, xx „~* 00 and dx,,n _ dv

1 3Jj^c d -l+l(x>,n) "

) St.

The value of n at which this approximation JC„ „ - ^PTT + «ir - \ir becomes vahd is n0, say, which is certainly independent of R. The sum will then be completely dominated by the terms n0 < n < 00, the begirining terms being suppressed by the R~2 factor. Making this replacement, and writing E„ -» jdxUn/ir, we obtain 5 ( 2 ; + 1 + 8,T)-

iX2T-X2T52.Iim R~2 T dxxe^x^1 R -»00

=\\\-

$82\\.

'n 0 w

(ID One can check that all other terms in the expansion in S and in other approximations employed are formally of order R'1 as R -* 00. Thus, we predict 5(2y + l + 5 , r ) = iA 2 T (l-2S 2 ) p e r ,

(12)

where the subscript indicates that we are to extend this function for \S\ > 1 in a periodic fashion. The complete result has a cusp at Bose values a = 2j due to the required periodic extension. This in fact follows from the general formula (8). The only difference between the Fermi and Bose expansions is the existence of the

D.P. ArovasetaL / Statistical mechanics of anyons

121

4B(g,T)

A

Fig. 1. The second virial coefficient B(a,T) as a function of the statistics determining parameter a (T fixed).

\l-a\

= |5| term: B(2j + S,T)=

-\\\-\S2\2T

+ 2^T lim £ [ e -<-W*> 2 _ e-(*,.,.,,/*>2]

= -i\ 2 T +|fi|\ 2 T -iS 2 A 2 T-

(13)

This is exactly the form predicted above. Thus, we obtain a picture of B(a,T) for fixed T as in fig. 1. This result is also derivable from a path integral approach. The general lagrangian for the many-body system is

L=ZhMr? + aY,k

(14)

pairs

For a system of bosons, the partition function takes the form Z.v=^7/d2r1...d2rJVE(r1...rv|e-^|Pr1.../»rA,>,

(15)

which may be cast into a path integral form, as done by Feynman [9]. Again, the case N = 2 is considered, and the CM contribution is directly integrated out. This leaves Z 2 = i / d 2 r [ + < r | e - ' " - | - r>], L rel = \Mrz + ad.

(16)

The d term in L introduces a winding number-dependent phase in the path integral. This problem is in fact equivalent to the Bohm-Aharonov effect [10], the path integral formulation of which was studied extensively by Gerry and Singh [11-14].

122

D.P. Arovas et al. / Statistical mechanics ofanyons

The matrix element K(r, r'; T ) = / Dr (r)e'*~'£ drun r(/-0)«r,

= (r\c-iH^h\r')

r(r + r ) - r ' ,

,

0-ir,

(17)

can be decomposed into a sum over contributions of different homotopy sectors, with $' - 8 = + 2wn: K(r,r';r)-

eM"KH(r,

£

r'; r ) ,

^ ( ' • . » " ; T ) = / D r ( / ) e r ' l d<'L £„(/•, r';T)

M

=

AirhiT

(18)

-9--2nn),

(19)

-A/ 2 2 exp 4A/T ( r + r ' )

x C dAe' A <* + 2 *"V a */,J—) •'-00

(20)

^'Ufc'T/'

where 7„(Z) is the modified Bessel function. In our case, we have \r'\ = \Pr\ = \r\ and = 0, ir: K(r,r';r)=

M \ / -Mr2\ inAr expj ••„,.. |e"**/, 4-nhir Ihir

£

(Mr 2hn

'

(21)

Therefore, we arrive at the result Z5-l

L

^°°dxe-Vin_a|(x).

(22)

n— —oo even

This is, as expected, formally divergent. A convergence factor e ~ " is inserted in the integrand, with e -» 0 at the end of the calculation. We use the result [15]

J

o V2e

]/a2-l (23)

As before, we appeal to eq. (8). Expanding about Fermi statistics, a = 2j; + 1 + S,

D.P. Arovas ei aJ. / Statistical mechanics ofanyons

\22

and \n - a\ = 1 ± S, 3 ± 5, etc. Thus,

B(2j + 1 + 5, T) = \\\ +1\\lim e —0

yz -pr I (1 + v^7)-"(2-(l + V^")S-(1 + V27)-S) v2e „_! odd

= |A 2 T -i5 2 A- T .

(24)

Expanding about Bose statistics introduces a term |5|X2T due to the \n - a\ = \Sl piece in the sum. Making the required periodic extension recovers the earlier result of eq.(12). In some sense, the path integral result is more satisfying, because, although one still is presented with the delicacy involved widi extracting the (finite) difference of two divergent expressions, there is no necessity to impose a finite volume constraint, which was originally effected in order to perform the mode counting. One might object to our original calculation on the grounds that the virial coefficient might possibly be sensitive to the manner in which we perform the mode counting, since the dominant terms in the sum of eq. (9) are those at the tail end. As we have seen, this fear is unfounded. A striking result is the nonanalyticity of eq. (12). It would be interesting to know whether cusps also arise in higher-order virial coefficients. Due to the proliferation of the number of relative angles, such higher-order virial coefficients are exceedingly difficult to evaluate. In the high density limit, one might consider averaging the statistical flux over the entire system, and then consider the effect of a net statistical uniform magnetic field of magnitude B = na0, where n is the particle density. It is possible to reproduce the correct form of the free energy to leading order in n in this manner, however, one loses periodicity in a, and only certain values of a actually yield the correct result. The most significant feature of the statistical interaction is that it is long ranged, hence perturbation expansions in a yield divergences and resummation is necessary, a situation reminiscent of the electron gas. Nevertheless, this statistics transformation process does yield a viable method for interpolating quantum statistics. The representation of a Fermi gas in terms of a Bose gas may be useful in other contexts, such as lattice field theory. Unless the statistical interaction is treated nonperturbatively, however, divergences may be difficult to handle. Finally, it is interesting to derive the lagrangian of eq. (14) for the solitons of the nonlinear a-model. Let us briefly recall that the model in question has a unit-vector order parameter n"(x), a = 1,2,3 and a conserved topological current J***~trh'ben"drnbdxne. off

(25)

124

D.r. Arovas et at. / Statistical mechanics of anyons

The conservation of J* licenses us to manufacture a U(l) "gauge potential" by the curl equation J^e^3vAx.

(26)

The crucial point is that we could include a topological term H=^fd'xA^

(27)

in the action, with & a real number (a = &/ir), which is analogous to the ©-parameter in quantum chromodynamics. H is, in fact, the Hopf invariant describing maps of S 3 to S2. In a suitable gauge, such as dA = 0, we can solve for A and so write H as a nonlocal interaction among the n" fields. The solitons are bosons for 0 = 0 (a = 0) and fermions for & = IT (a = 1). In a more general context, any conserved current J^ can be coupled to the vector field A^. If the only other appearance of A^ in the lagrangian is the Chem-Simons term [16] ell,pAlid„Af), then A^ represents a nondynamical field [17] which can be eliminated to give a nonlocal interaction, which will impart anomalous statistics to particles carrying charge associated with the current. In ref. [3] the statistics of the solitons in this model were determined by invoking the linking number theorem. Here we will determine the statistics directly by interchanging two widely separated solitons, and in the process elucidate the linking number theorem. For separations large compared to the sizes of the solitons we can approximate the solitons by point particles and die topological current by •/"(*)= L / d r S ^ x - ^ T ) ) ^ ,

(28)

with a = 1,2 and qa(r) describing die trajectories of the two "point solitons." We evaluate H by inserting eq. (28) into eqs. (26) and (27) and keeping only the cross-terms. The divergent self-interaction terms are evidently artifacts of die point approximation. To best understand the situation, we go to euclidean 3-space and think of eq. (26) as one of the time-independent Maxwell's equations V XB=J with the identification of A^ as the magnetic field B. Then H can clearly be interpreted as the work done on a magnetic monopole moving along die trajectory ?i( T ) by m e magnetic field generated by an electric current flowing along the curve <72(T). Widi suitable normalization, this is just the number of times curves " 1 " and "2" wind around each other. We have thus made contact with the explicit form for the linking number between two curves given in mathematical texts [18]. This discussion also defines die linking number between two curves which are not closed. To evaluate H explicitly, it is easiest to distort one of the curves, say "2," to a straight line <7£(T) = T5'I0> as we are allowed to do. We find by eq. (26) that

D.P. Arovai et al / Statistical mechanics of anyons

125

,4,. = euXj/r2, A0 = 0, a pure (but topologicals nontrivial) gauge. Once again, we could have appealed to (2 + l)-dimensional electrodynamics, this time interpreting j° as B. These remarks make clear that the effect here is essentially the BohmAharonov phenomenon. It is sometimes convenient to transform to a singular gauge wherein A — 0 except along string singularities attached to each particle, across which A has a jump discontinuity of constant magnitude. In summary, the action describing N of these point particles is just N

'dx„\2

a=l»

6 -

,.

-_

(29)

a
Here x a is a two-dimensional vector locating particle a and 0 ai is the angle of particle b relative to particle a, measured from the x-axis, say. The preceding discussion has boiled QH down to the second term in this equation. As we have seen, although this term is a total time derivative and appears as an interaction, it determines the statistics of the particles. In the original model, the solitons have topological charge Q = /d 3 *./ 0 taking on all integer values. The \Q\ > 1 solitons are unstable against breakup. In writing down eq. (29) we have included only Q = + 1 particles. It is easy enough, however, to include Q= - 1 particles as well by noting that the (+ - ) "interaction" has opposite sign from the (+ 4-) and ( ) "interactions." DPA would like to thank Stefan Theisen for making die work of Gerry and Singh known to us, and for many useful discussions. This work was supported in part by the National Science Foundation under grants DMR82-16285 and PHY77-27084, supplemented by funds from the National Aeronautics and Space Administration. One of us (DPA) is grateful for the support of an AT&T Bell Laboratories Scholarship.

Note added This work supersedes the preprint "Interpolating quantum statistics", NSF-ITP84-25, by two of the authors (F.W. and A.Z.).

References [1] [2] [3] [4] [5] [6] [7]

F. Wilczek, Phys. Rev. Lett. 49 (1982) 957 Y. Wu, US Dept of Energy preprint 40048-09P4 (1984) F. Wilczek and A. Zee, Phys. Rev. Lett 51 (1983) 2250 R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395 B.I. Halperin, Phys. Rev. Lett 52 (1984) 1583 D. Arovas, R. Schrieffer and F. Wilczek. preprint NSF-rTP-84-66. submitted to Phys. Rev. Lett. J.G. Dash, Films on solid surfaces (Academic Press, 1968)

126 [8] [9] [10] [11] [12] [13] [14] [15] [16]

D.P. Arovas et al. / Statistical mechanics ofanyons

M, Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, 1972) R.P. Feynman, Statistical mechanics (Benjamin, 1972) Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485 C.C. Gerry and V.A. Singh, Phys. Rev. D20 (1979) 2550 A. Inomata and V.A. Singh, J. Math. Phys. 19 (1978) 2318 C.C. Gerry and V.A Singh, Nuovo Cim. 73B (1983) 161 S.F. Edwards and Y.V. Gulyaev, Proc. Roy. Soc. London A279 (1964) 229 I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products (Academic Press, 1980) J. Schonfeld, Nucl. Phys. B185 (1981) 157; S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Y.Wu, Washington preprint (1983) [17] C.R. Hagen, Rochester preprint (1983) [18] H. Flanders, Differential forms (Academic Press, 1963)

248 Physica Scripta. Vol. 32, 372-376, 1985.

Lattice Relaxation Effects on the Midgap Absorption Edge in 7/ans-Polyacetylene S. Jeyadev* Department of Physics, University of California, Santa Barbara, CA 93106, USA.

J. R. Schrieffer Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Received March 18th, 1985; accepted March 28th. 1985

Abstract The effects of lattice relaxation on the midgap adsorption due to a soliton in a ring of frartr-polyacetylene is studied within the adiabatic approximation. The Franck-Condon factors are calculated for processes involving the emission of up to three phonons. The absorption edge is found to be broadened by processes involving an even number of phonons. On this background appear extremely sharp peaks, due to processes involving an odd number of phonons. In the latter case, the shape oscillation mode of the soliton is found to be the dominant mode. Our results also indicate a third localized phonon mode about the soliton, in agreement with the results of [to et al. (24].

1. Introduction

levels at mid-gap results in optical absorption occurring at energies close to half the gap as well as that due to band to banc electronic transitions. The problem of optical absorption in frans-polyacetylene has attracted much experimental [13—15] and theoretical interest. Early theoretical treatments [11, 16. 17] ignored both Coulomb correlation effects and lattice relaxation. Though, more recently, the effects due to Coulomb correlations have been studied by various workers [18—20], the effects due to lattice relaxation have not been considered. Since lattice relaxation following an electronic transition is extremely rapid [21] occurring in about 10" 13 s, multi-phonon radiation can be expected to broaden the absorption edges. In this paper we study the effect of lattice relaxation on the mid-gap absorption in frans-polyacetylene in the absence of Coulomb correlation. We consider the process where the electron makes a transition from the edge of the valence band to the soliton level. The hole in the valence band drives a self-consistent lattice distortion, forming a soliton-polaron complex. We employ the effective phonon Hamiltonian technique developed to study the interchain tunneling of polarons in fra/ts-polyacetylene [22]. Since the method is explained at length there, we merely sketch it here. We first obtain the ground state phonon wavefunction in the presence of a soliton and then calculate the overlap with the four lowest phonon states around the equilibrium configuration of the lattice after photoexcitation.

Polyacetylene [1] is one of a wide variety of conducting polymers which possess more than one possible bonding configuration, referred to as "phases" in condensed matter physics. Such polymers can support excitations that interpolate between regions which are in different phases. These excitations have come to be called solitons. For solitons to be stable excitations, the phases between which they interpolate must be degenerate in energy. Though polyacetylene shares a number of properties with other conducting polymers (e.g. upon doping, there is a large increase in electrical conducting [2] new optical absorption [3] and spin properties [4]) it is unique because its trans form is the only experimentally known polymer to possess degenerate phases. Thus, solitons are expected to play an important role in determining the solid state properties of trans 2. The effective phonon Hamiltonian and overlap integrals polyacetylene. The soliton model proposed by Su, Schrieffer and Heeger [5] (SSH), Rice [6] and Brazovskii [7] has met The SSH Hamilitonian for a ring of N CH groups is given by with much success in explaining a wide variety of experimental tl n a results such as spinless electrical conductivity [8], photoconductivity [9], thermopower [10] and optical absorption (2-D [11]. The continum version of the SSH model proposed by + C.C 1 M . a + T A:£(* 1 , M + ! M 1 + :E n IT i W Takayama, Lin-Liu and Maki [12] (TLM) has also been where extensively employed to derive analytical results. ^=(-l)"«n (2.2) In the absence of solitons, the electronic spectrum of trans is the staggered order parameter andu„ the actual displacement polyacetylene is that of a direct gap semiconductor with a gap of the n"1 CH group. 4f0 is the bandwidth, a the electron of about 1.4eV. SSH showed that in the presence of well phonon coupling constant, K the tr-electron spring constant separated solitons the spectrum is still that of a semiconductor, and Cna(C^a) annihilates (creates) a ir-electron at site n with but associated with each soliton is a localized state that lies at spin orientation s. P„ =fij'\. 3/3i//„ is the staggered momentum of mid-gap. In the case of a neutral soliton this state is singly the /i m CH-group. Within the adiabatic approximation [21], occupied; it is doubly occupied in the case of a negative soliton the coordinates {\pn}are taken to move in a potential K({i//„}) and empty in the case of a positive soliton. The presence of given by the sum of the ground state energy E(.{4in}) of the irelectron system plus the harmonic lattice interaction in (2.1),

H = - 1 to.+(- o" «(*„•, + *,,)] (.c: ,„c , +

PflO = f(l*i + Ul(w„„ + i ) Present address: Xerox Corporation, Webster. NY 14580 Physica Scripta 32

Webster Research Center,

!

(2-3)

n

For given {
249 Lattice Relaxation Effects on the Midgap Absorption Edge in Trans-Polyacetylene

37:

one-electron energy eigenstates, (2.4) where nva is the occupation number of orbital v with spin a. The e„ ar.e .the eigenvalues of an effective one-electron Hamiltonian H' whose elements in the site representation are given by -to-i-lTaW^

+ tJ

n = M + l,

a

(2.12b)

R,TUSR,

(2.13)

n' = n — 1, 2 < n < /V

#„„ =

S+^

where A, =

<*&„ + *„-!)

(2.12a)

or

1< n
+ —o r A,o

H,

- r o - ( - l ) J V « ( * . v + ( - I ) J V * I ) «' = 1,

is a diagonal matrix containing the eigenvalues of U,, namely A£. The wavefunction when all the normal mode oscillators are in the ground state is then given by

n - N and, Wi,<7;-

n = /V, «' = 1

(2.141 0

otherwise

(2.5)

where n = 1, 2, . . . N. Since we need a single soliton jVmust be odd to satisfy boundary conditions. Using the molecular dynamics method [22], the order parameter for a single soliton {yj|}was determined. Let (2.6)
(2.8) MnWm

y,}={t?s

-WM!!«)i'A,x

(2.15i

where

A. =

ur

fl.A.'"*.1

a n d the n

°™alization constant ,VJ is given by

N% =

«<£'•

(2.16,1

(2.17i

Now let us consider the photoexcited state. Let {i'°} be the equilibrium order parameter of the lattice and X'n =

—

.\n> XI

(Xi ,X2, .

(2.18)

*n-ft

tne

^e displacements about equilibrium. Then the effective phonon Hamiltonian, Hc, is given by

^

Vs{{'ii J ) is the adiabatic potential energy given by

i -M

I

(2.19)

J. nm

where (2.9) u

EXi^nh is the /T-eiectron contribution obtained by filling the (yV— l)/2 lowest level of the it -electron Hamiltonian doubly and the next higher (i.e. mid-gap) state singly. The last term in (2.9) avoids the numerical inconveience caused by the zero eigenvalues of the dynamical matrix U'„m arising due to the translational invariance of H, i.e. a uniform change of \bn. The phonon Hamiltonian (2.7) can be represented in matrix form as H, =

2M

—+-xTU,x 2

(2.10)

where P and x are now column vectors. If R, is a matrix containing the eigenvectors of U, in its columns, then H, is diagonal in the normal coordinates q given by the orthogonal transformation R.q

(2.11)

nm

(2.20)

~ , ~ .

W-{*J} Now Vt({-J/n)) is given by . ,. _ .( . L _ , , _ ,, ( "•(I*'"'-' ~ EMVnf) +2* L(.Vn-n + VB)_ + i A e 2.v„ (2.21) where the ir-electron contribution, E„({
|\[,«)

=

_ L e-(v/M/2fl).«'TAe.t' No

(2.22)

where Fhysica Saipta 32

250 374

S. Jeyadev andJ.R.

Schrieffer (2.23)

phonons can be written as

and

K

det

(2.35)

(2.24)

& • )

in the normal mode representation, and.

Introducing the deviation in the order parameter of the two equilibrium lattice configurations 8„ = K -

ft

a;,*;....!*;«,> =

(2.25)

x'n = xn-8„

(2.25)

Hence the ground state-ground state overlap is given by

e-(V?/!«)i

T

I*J>

ZRlRlx'ix}-2 ti T

A e j('

(2.36)

in the JC' basis. The normalization constant is given by (2.37)

A'Sa = V I L A ' S and the overlap with l*5> is given by

r

.Ve-(V«/!*)t'-i)' 'le('-S)

= Jd* ld * ; .^

4?2

e -
yields the tranformation between t h e * andx' bases, i.e.

<*Si*S> = jdxidxl...Wa\xl,x2....)<XiJCi

1 M„

—

[det(,4,)det(.4.)]'•"

iyfrMHTa

<*3I*S> -7 4 > A | <*%&& =

, , ,fi)

where

C = ^.(.4, + ^ J - 1 ^ ,

(2.27)

r

A,-

(lKf«(i7r-S()J

+

—

X

(2.38)

'J

where \% is the 0 t h eigenvalue of Ue as before and thefl,are the eigenvalues of the matrix A, + At. The wave function when there are 3 phonons in the a mode is given by

The excited states of (2.19) can be handled in an identical manner. The phonon wave function when the a * normal mode contains one phonon is given in the normal coordinates by

<4'..«i • • • • l*W - J r [ S T j ^ - l ^ f t ] e-<^*>«' TA i"«

&Z.,
i.e.

•^ 3a

(2.39)

(2.28)

/V| a

_1

where A, is a diagonal matrix containing the eigenvalues of Ua and

ra

OC,,JC2,.

A"3a

-

«s-

12yaI.R?ax]

-lyfSl2n)x'TAex'

(2.40)

(2.29)

where

Xa being the a * eigenvalue of U„. The normalization constant is given by N\a = s/2N%

(2.30)

Inserting this into (2.28) yields

Ala =vX2TX3!)A'eo integi is found to be and the overlap integral ,1/2

<*8I*S«> = -

q'a = Y,Rl,x\

(2.31)

(2.41)

v^53T)\ *

M)" 4 Vxax

i

Inserting this into (2.28) yields <xi,^j.

••**-;£

2TaI«ta*i

+ I: e-(,/M/«l)x'

T

Aai'

(2.32)

?/»te(Ui-*«)

(2.42)

3. Results The equilibrium lattice configurations before and after photoThe overlap with the state |Sf'o>can now be calculated in a excitation, {#{} and {${} respectively, were determined as straightforward manner and we get explained in Ref. 22 using the SSH parameters f0 = 2.5eV, a = 2.5 eV, a = 4.1 eV/A, K=2l eV/A 2 . The size of the ring W W ) = i"^27«lf«S,('?*-8|) (2-33) was chosen to be N= 59. {<\i%} was found t o be in excellent agreement with those obtained by Okuno and Onodera [23] where the column vector 17 is defined by when the soliton and polaron are not separated (fi = 0 in their notation). The dynamical matrices, U, and Ue, were evaluated V = {A.+A^A.S (2.34) numerically by treating the derivatives in (2.8) and (2.20) as Next we consider the case where one of the normal modes, finite difference equations. The matrices A„ At, R„ R, and C in the photoexcited state, is in the second excited state. The were obtained by using standard routines. Ka was chosen to be phonon wave function when the a" 1 mode contains two O.leV/A 2 so that KC
251 Lattice Relaxation

"ects on the Midgap A bsorption Edge in Trans-Polyacetylene

375

uniform. Parity considerations, again, produce a sharp peak in the thiee phonon case. Due to the relaxation of the lattice, the <*'ol*o> = 0 27 (3.1) midgap absorption edge occurs at an energy lower than half of The overlaps for the one phonon processes, obtained from the gap..Our calculations show that it is lowered by 0.1 eV. (2.33) exhibit a remarkable result: the only significant overlap However, the absorption at the edge is weak due to the small occurs for the mode a = 18. The overlap for this mode is —0.37 Franck—Condon factor for the zero phonon process. while the next largest overlap is —0.03. Thus the FranckCondon factor for the mode a = 18 is 150 times larger than the S. Concluding remarks next largest! We discuss the implications of this result below. The overlaps for the two phonon processes given by (2.38), on The discrete SSH model has been used to obtain the phonon the other hand, are more evenly distributed over the phonon spectrum about a soliton and the soliton-polaron complex spectrum, their absolute values lying around 0.3. The three formed due to photoexcitation from the band edge. The phonon overlaps, given by (2.42) again exhibit a sharp peak: for spectrum in the former case if found to be in good qualitative the mode a = 18 the overlap 0.6 while the next highest is 0.07. agreement with those obtained from the TLM model [17, 2 4 ] . The overlap between the zero phonon state about the soliton These results are also discussed below. and the ground as well as the excited states about the photoexcited lattice configuration are evaluated. These show that the shift in the equilibrium position of the normal mode oscillator 4. Discussion is significant only for the shape oscillation mode. Though lattice The phonon spectrum of franspolyacetylene in the presence of relaxation broadens the edge and lowers it in energy, the a soliton has been calculated for the TLM model by a variational absorption at the edge itself is weak. The sharp peaks at the approach [17]. These calculations show that in the presence of edge are reminiscent of those occurring at the edge for the soliton there exists a localized zero-frequency mode (i.e. the interband transitions [25]. Finally we note that Coulomb Goldstone mode corresponding to the translation of the soliton) correlation effects have been neglected in this analysis. Though and a localized mode with finite frequency (corresponding to these effects may move the edge away from midgap, the the shape oscillations of the soliton). Our calculations also vibronic effects discussed above are expected to survive and exhibit these two modes but the energy of the shape mode contribute significantly to the observed optical absorption edge CJJ = 0.09 eV is lower than the value of 0.14eV calculated in [13—15] corresponding to the valence band to midgap state. Ref. 17. We ascribe this difference to the fact that A/f0 =0.28 for rrans-polyacetylene while the continuum approximation Acknowledgements assumes A/f0 <S 1. Ito, et al. [24] have studied the phonon eigenvalue problem numerically based on the TLM model. In This work was supported in part by the Department of Energy grant addition to the two localized modes mentioned above, they find DE-AT03-83ER45008. One of us (SJ) would like to thank Dr. M. Grabowski for many useful suggestions and discussions. a third localized mode of even parity about the centre of the soliton with an energy of OJ3 ~ 0.17eV. Our results also show the presence of a third localized mode of even parity. While the References shape and extent of this normal mode is in very good agreement 1. Proceedings of the International Conference on the Physics and with that shown in Ref. 24, we find that its energy w 3 ~ 0.1 eV Chemistry of Conducting Polymers (Les Arcs), Journal de Physique. is lower than that of Ref. 24. Colloque 3, 44, June 1983. The Franck-Condon factor for the zero phonon-zero 2. Shacklette, L. W., Eckhardt, H., Chance, R. R., Miller, G. G., Ivory, phonon process is K*ol*o>l2 ~ 0 . 1 . This shows that phonon D. M. and Baughman, R. H., J. Chem. Phys. 73, 4098 (1980). emission is significant and can broaden the midgap absorption 3. Ciechins, G., Stamm, M., Fink, J. and Ritsko, J. J., Phys. Rev. Lett. SO, 1498 (1983). edge. As noted above, the only significant mode for the one phonon process in the a = 18 mode. An analysis of R, (which 4. Peo, M., Roth, S., Dransfeld, K., Tieke, B., Hocker, J., Gross, H., Grupp, A. and SixiL H., Solid State Comm. 35,119 (1980). contains the phonon eigenvectors) revealed that this is just the 5. Su, W. P., Schrieffer, J. R. and Heeger, A. J., Phys. Rev. B22, 2209 shape oscillation mode. The dominance of just this one mode (1980). can be understood as follows. The lattice configuration of the 6. Rice, M. J., Phys. Lett. 71A, 152 (1979). photoexcited state, {i?J}, is similar to that of the soliton but for 7. Brazovskii, S., Zh. Eksp. Teor. Fiz. 78, 677 (1980); Sov. Phys. - JETP, 5 1 , 342 (1980). a large flattening near the centre of the soliton [23]. Since such 8. Chung, T. -G. Moraes, F., Flood, !. D. and Heeger, A. J., Phys. Rev. a deformation can be mostly accounted for by the shape mode, B29, 2341 (1984). the dominant relaxation process occurs by the emission of this 9. Etemad, S., Mtani, T., Ozaki, M., Chung, T. •€., Heeger, A. J. and phonon. An alternative statement of our result is that the shifts MacDiarmid, A. G., Solid State Comm. 40, 75 (1981). in the equilibrium positions of the normal mode oscillators for 10. Kivelson, S., Phys. Rev. Lett 46, 1344 (1981). the two lattice configurations {inland {i?J}are neglibible but 11. Kivelson, S., Lee, T. K., Lin-Lin, Y. R., Peschel, I. and Yu, L., Phys. Rev. B25, 6447 (1982). for the shape mode. Due to the different parities of the zero 12. Takayama, H., JJn-Lin, Y. R. and Maki, K-, Phys. Rev. B21, phonon and one phonon wave functions, the only significant 2388 (1980). overlap occurs for the one shifted oscillator. The two and three 13. Blanchet, G. B., Fincher, C. R-, Chung, T. -C and Heeger, A. J., Phys. Rev. Lett. 50, 1938 (1983). phonon cases are now easily interpreted. In the two phonon case the wave functions are of the same parity but are not 14. Weinberger, B. R., Roxlo, C B., Etemad, S., Baker, G. L. and Orenstein, I., Phys. Rev. Lett. S3, 86 (1984). orthogonal to each other because the force constants are 15. Vardney, Z., Orenstein, J. and Baker, G. L., Phys. Rev. Lett. 50, different. The overlap integrals are more uniform since it is 2032 (1983). reasonable to expect that the change in the force constants in 16. GammeL J. T. and Krumhansl, J. A., Phys. Rev. B24, 1035 (1981). overlap is found to be

Pfryma Scripts 32

252 376 17. 18. 19. 20. 21.

S. Jeyadev and J. R. Schrieffer Maki, K. and Nakahara, M., Phys. Rev. B23, 5005 (1981). Grabowski, M., to be published. Kivelson, S. Heim, D. E., Phys. Rev. B26,4278 (1982). Soos, Z. G. and Ducarce, L. R., J. Chem. Phys. 78, 4092 (1983). Su, W. P. and Schrieffer, J. R., Proc. of the Nat. Acad. Sci. 77, 5626 (1980).

Physica Scripta 32

22. Jeyadev, S. and Schrieffer, J. R., Phys. Rev. B30, 3620 (1984). 23. Okuno, S. and Onodera, Y., J. Phys. Soc. Japan, 52, 3495 (1983). 24. Ito, H., Terai, A., Ono, Y. and Wada, Y., J. Phys. Soc. Japan, 53 (in press). 25. Mete, E. J., Synthetic Metals 9, 207 (1984).

253 Physica Scripta, Vol. 33, 282-283, 1986.

Shape of Solitons in Classically Forbidden States: "Lorentz Expansion" F. Guinea. R. E. Peierls,' and R. Schrieffer Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA. Received 'September SO. 1985; accepted October 7. 1985

Abstract The shape of extended objects in classically forbidden regions is shown to undergo expansion analogous to Lorentz contraction of a relativistic body of finite velocities. The problem of two interacting Diiac particles moving in one dimension is solved explicitly and the results are generalized to soliton solutions of field theories. An estimate of the effect on tunneling rates is also given, including solitons in (CH),.

can neglect vacuum fluctuations, assuming that X < m. Defining xt =x±x and the energy as e, the functions f„.b(x+> x-) satisfy (« (e-V+X5(x.))fn

V)fn

-i^+m(/i2+/2i) = j. 3- i/ M i±+ „,(/„+/„) dx

It is well established that nonlinear excitations, such as solitons, (3) play an important role in quasi-one-dimensional systems. These ( e - K + A 6 (*_))/„ = - i ^ + m ( / „ + / 2 2 ) ox_ excitations are frequently treated as point particles, although their internal degrees of freedom are coupled to their center of mass dynamics. For example, in the relativistically invariant 0 4 i ^ + m(/ I 2 +/ 2 1 ) (fi-V)f„ ox+ theory, the width of a soliton undergoes a Lorentz contraction as its velocity increases, with an accompanying change in its which can be solved by the ansatz effective mass. = Aab(x.)ei"x'e-a,xA (4) In this letter we investigate how these effects change the Ut(x.,xl) quantum behaviour of a section when it propagates through a where Aab(x.) is constant for x. > 0 and x. < 0, and has a disclassically forbidden region. It will be shown that this coupling continuity in its imaginary part in order to satisfy the matching between the motion of the center of mass and the internal conditions at x_ = 0. From eqs. (3) and (4) we obtain exmodes induces a "Lorentz expansion" of the excitation with a pressions for the energy e and the width a"1 of the bound state corresponding decrease in the effective mass and an increase in 4m2 the tunneling rate over the value calculated for a point particle ( e - K ) 2 = (5a) + yf el + k" 1 + X2/4 without internal structure. We first consider a simple system which exhibits this coup, X 2 / 4m 2 ling in one dimension, namely two distinguishable, spinless, a 2 = —I r— +t (5b) 4 ^1 + X2/4 massive Dirac particles interacting through an attractive 6potential. The second quantized Lagrangian is k is the total momentum and eb is the bound state energy in the rest frame. As expected, e and a are relativistically invariant. 0 J'ti-id^-m-Vy )*! Moreover, it is easy to see that when e—V<eb (classically forbidden region) it is possible to build a solution like (4) with an imaginary value of k. If k = i«, we have (1)

= 1f

xJ^y^X^'M

K2 = el-(€-Vf

where V is an external potential and the coupling is the same as in the Thirring model. Since in this one-dimensional mode! only 7° and y 1 occur, we can use a two-component representation, and choose for 7° and 7 1 the matricies normally representing az and a x . We look for a bound state of the form

|u» = I

\ixix,f^(x,x)i,\.a{x)^Ax')\0)

(2)

where a and b are spinor indexes and |0> is the vacuum. We can view 4i*a,{x) as creating a physical particle or, alternatively, we

Permanent address: Nuclear Physics Laboratory, Keble Road, Oxford University, Oxford 0X1 3RH England. Physica Scripta 33

(6)

which govern the penetration of the bound state in the forbidden region. The dependence of a on K2 is the opposite as that on k2, so the bound state expands in this region. Expression (6) can be derived from the requirement of Lorentz invariance only, and gives the tunneling rate in the WKB approximation. For shallow potential barriers (|e — V\ < eb) this result coincides with the corresponding value for a nonrelativistic point particle, and is enhanced for higher barriers, in agreement with the intuitive picture that the effective mass decreases when the momentum is "imaginary" so that the tunneling rate increases. We will now generalize these restuls to soliton solutions of field theories with an infinite number of degrees of freedom. Given a one-soliton solution of the classical equations of motion,
254 Shape of Solitons in Classically Forbidden States: "Lorentz Expansion " construct a canonical transformation to a new set of dynamical variables X, P, x(x), and rr(;r) such that X is the cm. position discussed earlier and P its conjugate momentum [1, 2], The field x(*) can be expanded in terms of small oscillations around the classical solution; the amplitudes of these modes are coupled to P and induce the velocity dependent changes in the width of the .soliton required, for instance, in relativistically invariant theories. On the other hand, P is a conserved quantum number, which can be used to classify the eigenstates of the Hamiltonian \P;n), with energies en(P) (the index n is used to specify the states of the internal modes of the soliton). We can build localized wavepackets in terms of the position eignenfunctions: l^> = 1 \

f„(X)\X;n)AX

\X;n) = j

iPX

(7) e \P;n)dP.

The momentum operator acting on |V) replaces fi(X) by (bli)f'j(X). Considering only the internal ground state, n = 0, and neglecting the index n, we can construct wavepackets with energy
283

action is imaginary, and the main contribution comes from the least action path calculated when the sign of the potential energy terms are reversed [4, 5 ] . This is the WKB solution, which can be parametrized in terms of an imaginary velocity v as €{V)

vTT7

(12)

a(v) = a 0 V l + v1 which reproduces the results discussed before [6]. So far, we have only discussed theories with Lorentz invariance but the scheme presented here is very general and can be applied to other cases. In particular, we can analyze solitons in polyacetylene, making use of the effective Lagrangian: _ mx a0 2ct

X/a 2 1 a0

(13)

similar to (11). which explains many features obtained in numerical simulations [7]. If e is the height of the barrier, we can write

f>

a = a0 Jl + l*> = \ e"

f x |;OdJc

(8)

(H)

2m

J * L/i'1 + Y - '

with e(itf) = e.

(9)

These states correspond to particles propagating in a classically forbidden region and can be normalized by matching them to standard solutions outside the potential barriers [3]. For relativistically invariant theories, eq. (9) reproduces the result (6) discussed earlier, with the corresponding increase in tunneling rates. The width of the soliton is related to the amplitudes of the internal oscillations, which are coupled to the dependence of the wave function on the cm. coordinate, given by eq. (8). This solution can be thought of as a transformation to a state with imaginary momentum so that, for systems with Lorentz invariance, the width should be related to K by an equation equivalent to (5b) and it will increase in the forbidden region. This point can be further clarified if we only keep the cm. position and the width of the soliton as the relevant dynamical variables, effectively freezing the amplitudes of the high frequency oscillations. Classically, it amounts to consider only solutions of the form
J(f *4/(£)'-/™

L =

dx

(10)

we obtain an effective Lagrangian 2

a

2a 0

(11)

where a 0 is the value of a for a classical static solution, m is the effective mass of the soliton, and we are neglecting a kinetic term associated with d, because only the lowest energy state for each total momentum configuration will be considered. We can use eq. (11) to solve for the properties of the soliton in the forbidden region using the path integral formalism. The

For 6 = 0.2 eV the tunneling rate through a barrier 30 A wide is increased four-fold over the corresponding value for a point particle, and the width is expanded by 409t. In conclusion, we have analyzed the tunneling properties of composite excitations whose internal degrees of freedom are coupled to the cm. dynamics, like solitons in nonlinear systems. The internal modes relax in the classically forbidden region, making the size of the excitations increase, enhancing the tunneline rate.

Acknowledgments One of us (FG) acknowledges the Spanish Ministry of Education Fullbright Commission joint committee for a postdoctoral fellowship. This material is based upon research supported by the National Science Foundation, Grant No. DMR82-16285, and in part by Grant No. PHY7727084. supplemented by funds from the National Aeronautics and Space Administration.

References Gervais, J.-L. and Sakita, B, Phys. Rev. Dll. 2943 (1975). Tomboulis, E., Phys. Rev.D12, 1678 (1975). Field theories have an intrinsic degree of nonlocality associated with the decay length of correlation functions. We assume the barriers to be smooth over this length scale. Langer. J. S., Ann. Phys. (NY) 41. 108 (1967). Coleman. S., in "The Whys of Subnuclear Physics." ed. by A. Zichichi (Plenum, New York, 1979). These results are exact because the ansatz made about the possible classical solution does include the correct shapes at any velocity. It is, however, only an approximation in the following application. Bishop, A. R., Campbell, D. K., Lomdahl, P. S., Horovitz, B. and Phillpot, S. R.,Phys. Rev. Lett. 52, 675 (1984).

Physics Scripia 33

255 VOLUME 82, NUMBER 8

PHYSICAL

REVIEW

LETTERS

22 FEBRUARY 1999

Topological Excitations of One-Dimensional Correlated Electron Systems M.I. Salkola12 and J.R. Schrieffer1 'NHMFL and Department of Physics, Florida State University, Tallahassee, Florida 32310 2 Department of Physics, Stanford University, Stanford, California 94305 (Received 11 August 1998) Elementary, low-energy excitations are examined by bosonization in one-dimensional systems with quasi-long-range order. A new, independently measurable attribute is introduced to describe such excitations. It is defined as a number w which determines how many times the phase of the order parameter winds as an excitation is transposed from far left to far right. The winding number is zero for electrons and holes with conventional quantum numbers, but it acquires a nontrivial value w = 1 for neutral spin-| excitations and for spinless excitations with a unit electron charge. It may even be irrational, if the charge is irrational. Thus, these excitations are topological. [S0031-9007(98)08377-X] PACS numbers: 74.20.-z, 74.25.Jb, 74.80.-g The concept of elementary excitations due to Landau plays a fundamental role in understanding many properties of condensed-matter materials. It essentially takes a view of renormalization-group theory by assuming that a system scales towards a weak-coupling fixed point at low energies and long wavelengths. Even if true microscopic interactions are strong, elementary excitations remain in a one-to-one correspondence with bare excitations—only their properties are renormalized. In spite of the success of the basic concept, it cannot be justified rigorously because of the breakdown of the renormalizing procedure. Indeed, there are many interesting instances where the physical picture of weakly interacting elementary excitations with quantum numbers equal to those of bare ones becomes invalid. Examples of such cases are the quantum Hall effect [1] and quasi-one-dimensional conductors [2] where low-energy excitations have unusual spin-charge relations and are not continuously related to conventional electrons and holes. In this Letter, we examine interacting one-dimensional conductors. This is a widely relevant and frequently studied problem [3,4]. We consider a particular situation in which an extra particle is added into a superconductor. Yet, the consequences of this process illustrate that the elementary, low-energy excitations are more complex than one might have argued. Our most important result is that, in addition to the usual spin and charge quantum numbers, an excitation attaches itself to a kink which can be quantified by another quantum number. It is defined as a number w which determines, in multiples of tr, how many times the phase of the order parameter at a given point winds as the excitation is transposed from one to another end of the system. While the winding number w does not constitute an independent degree of freedom in the sense that it would be unrelated to the spin and the charge of the excitation, it does have physical attributes that make it measurable without any information about them. The winding number would be zero for an electron and a hole with conventional quantum numbers, if they were to exist as elementary excitations, but it ac-

quires a nontrivial value w = 1 for a neutral spin- 5 excitation and for a spinless excitation with a unit electron or hole charge. It may even be irrational, if the charge is irrational. Thus, these excitations are topological objects which can be viewed as composite particles made of spin or charge degrees of freedom and dressed by e'7TWphase kinks in the order parameter. The winding number appears naturally in systems which have continuous symmetry and whose ground states develop quasi-longrange order in one dimension and true long-range order in higher dimensions. In addition to superconductors, kinks must then be introduced in spin- and charge-density-wave systems. Our conclusion complements the earlier observation [5] that soliton excitations have unusual fermion quantum numbers in charge-density-wave systems where massless fermions are coupled to a boson field. Here we demonstrate that also the converse is true: elementary charge and spin excitations always carry kinks in onedimensional systems with quasi-long-range order. As a paradigm for low-energy and long-wavelength excitations in a one-dimensional conductor, consider a Hamiltonian where the energy spectrum is linearized at the Fermi points ±k.F and the fermionic degrees of freedom are expressed by slowly varying left (+) and right (—) moving fields i//tT±(x):

1752

© 1999 The American Physical Society

0031-9007/99/82(8)/1752(4)$15.00

<M*) ~ *„+(*)*""** + ^^{x)eiktX.

(1)

Specifically, let the Hamiltonian be H = vFY,

I dx[^l.{x)ptj/(T-(x)

- il/a+(x)pil/a+{x)~\

+ VtJanMJ^Ml

(2)

where Jan(x) = :^n(jc)<^0-n(jc): (n = -h = ±) are the left- and right-moving currents, p = -ihdx is the momentum operator, vF is the Fermi velocity, and V2 and V4 are the forward-scattering constants. The colons denote normal ordering with respect to the filled Fermi

256 PHYSICAL

VOLUME 82, NUMBER 8

REVIEW

sea of the noninteracting system. In the noninteracting system, the Hei sen berg operators are functions of variables x ± vFt: il/lr±(xj) = <J/,r±(x ± vpt). For simplicity, we consider a limit where the backward and umklapp terms are either zero or scale to zero, so that they remain irrelevant in the sense of scaling and the model is exactly solvable by bosonization. Far away from the second-order commensurability where 4-kF is equal to the reciprocal lattice vector, the umklapp processes are effectively turned off and remain so under the renormalization-group flow. In Abelian bosonization [3,6-8], the left- and rightmoving fermion operators are expressed in terms of boson fields, i/r^x) = ( F ^ / T I T T ) : e * ' * « w : , where Fa is the Klein phase operator which establishes the correct anticommutation relations for different fields [3,8]. For instance, with this definition, the current operator becomes J mix) = (l/27T)dx$>an(x). Because the Hamiltonian is quadratic in the boson fields „.„, dx

/

^{(hvpS^

+ v 4 )[ax*«r«U)][a,* M)1 U)]

+ V2[ax*«n.to][a,*/rtito]}.

(3)

LETTERS

22 FEBRUARY

1999

A t (x)A(y) = Sscfr - y) :eir>A< cosAs:,

(8)

where the prefactor is

e* w =(^) 2 (^r /2

(9)

The operators, A„ = [c-(x). The exponent is defined as /ASC = 1 + V1' where t) = [(1 + g)/(l — #)] 1 / 4 . In the ground state, Gs?(x) = G SC U)—thus, the asymptotic forms of the correlation function are CJJJC(X) X x~^c and Gsl(q = 0, 0) at low energies and small momenta, for g < 0. The effect of excitations on pairing fluctuations can be examined by computing the correlation function in the state |* t > = , where f/CO) = [>/+(()) + (/»!_ (0)]/V2 adds an electron into the system at the origin and time t = 0. First, let the electron be far away from the points x and y where the pairing fields are measured, \r\
it is readily diagonalized by a set of unitary transformations. First, define spin and charge fields O j n U) = [*!„(*) - <J>J„«]/A/2 and C„« = [*,„(*) + <J>l„(;t)]/\/2. Second, define sn(x) and G ^ ( x , y ) = <^ T |A t (x)A(y)|^ T ). (10) cn(x) cosh + $>cn(x) sinh , with g = V2/ -.2iaSvvi/ Using the identity [(pl.±(0),dxcpv,±(x)] (hvF + V4) and tanh2 = g(\g\ < 1). The Hamilton(x2 + a 2 ), it simply becomes ian is transformed into

"= Z *"» ( f ^ K ^ M ] 2 + [Bx9v-(x)f), V — S,C

3/

^

(4) where vs = VF and vc = VF/ cosh 20 are the spin and charge velocities. In the Heisenberg picture,
- if>H(x)4>i-(x)y^2,

(5)

Gsc(x - y) = (OlA+WAtyJlO) (6) probes the ordering fluctuations of interest in the ground state |0). The Heisenberg operators A(JC, t) are denoted by A(x), where x = (x, t). Below, we will always consider cases where x and y are measured at equal times t: x = (x, t) and y = (y, t). Using the identity [9], • e<"
it can be shown that

_ .eia
ps

„2

\Msc/2

G$(x,y) = (-3-) (-^—5)

cos R2 + (ID

when r = 0. For given r, the influence of the electron on the correlation function decays as R~4. This behavior is universal as the scaling exponent is independent of the strength of interactions. The result is consistent with the natural observation that a single electron has no effect on the bulk properties of a superconductor. Second, let x and y be arbitrary. The correlation function can be written in the form G$(x,y)

(277-a)- 1 Gsc('-)J'(x,y),

(12)

where J ( x , y ) = [J+(x,y) + J"-(x,y)]/2 and J ± ( x , y ) = cos[0 B (*, ± ) - 6a(ys±)] X e ±iwl[e.(xc±)-e0{y,±)-]±iwjie.(x^)-e.(y.*)'] (13) We have defined 6a{x) = arctan(x/a), w\ = 77cosh, and W2 = —7/sinh 0 (w\ > W2 > 0, for g < 0). Note that 6a(x) —* Y sgn(*), as \x\/a —* ». This is our principal result, and it can be obtained by noting that [±(x)] = ±2iSv„iaictan(x/a). Initially (t = 0), the function J reduces to JT(x,y) = cos2[0„Gx:) 0a{y)~\- For \r\ •« \R\, we recover Eq. (11). In the 1753

257 PHYSICAL

VOLUME 82, NUMBER 8

REVIEW

scaling limit where r,R —»°o with z = r/2R < 1 fixed, the asymptotic behavior of Gsc is Gl^(x,y) = (2^ar1gSc(r)

-(T)'

2 a

(15) xl + a1 which shows that the electron destroys the order parameter in its vicinity [10] while preserving J > 0. The orderparameter suppression dies out as x~2. However, in the interacting system, the electron decays instantaneously into elementary excitations involving either spin or charge degrees of freedom. The kinky nature of these excitations is revealed, when the system is allowed to evolve in time. For illustrative purposes, consider the ground state with one right-moving electron initially added at the origin |¥T> = ^/-(0) |0>, so that f = T-. The spin and the charge excitations carry kinks in the order parameter, and they both separately contribute to the total phase shift an equal amount of IT, because wi + w2 = 1. Initially, the kinks overlap but then split as the excitations move apart. If only the spin excitation is located between the points x and y at a later time t, the correlation function is negative: Gsc changes sign if either x or y crosses the position of the spin. This shows clearly that the initial state |M>j) of the system must evolve into a state where the superconducting order parameter has an ordinary kink, or an antiphase domain wall, precisely at the point where the spin is located [11]. The winding number of a spin- j particle is 1. In contrast, the splitting of the charge into right and left moving charge components, q[ and q\, leads to two irrational kinks whose winding numbers are w\ and H>2. The winding number of the kink is uniquely related to its charge—in units where the electron charge is one, the relation is particularly simple: w\^ = q*2- The total winding number, as well as the total charge, is conserved. Because the spin and charge velocities differ, the spin and charge kinks also propagate with the different velocities. This suggests a new and potentially attractive way to detect spin-charge separation in a superconductor by using the Josephson effect. It would also serve as evidence of irrational charge. Figure 1 describes schematically the effect of spin and charge on the superconducting order parameter. The above result is generalized in a straightforward manner for N electrons initially injected into the system at positions R* = (/?;,0) (( = ! , . . . , N ) : 1754

22 FEBRUARY 1999

G&'fx.y) = (2na)-Mgsc(r)r[Wx

~

R

* ~

Ri)

•

(16)

(14)

where K/a = 4z 2 /(l - z2). Again, the electron is seen to suppress superconducting correlations in the manner described by a universal scaling exponent. In general, the order parameter A is obtained from
LETTERS

If an infinite number of electrons is randomly distributed with a mean distance (,, the correlation function decays exponentially: G s c (x,y)

-4r(f)/f

&

(17)

for r(t) = |r| + min{0, \_{vs - vc)t - |r|]sin 2 J wi, (vs + vc)t — \r\ — 2vctsm2 j w\) ~» a. In the weakand strong-coupling limits, r(t) — min[|r|, (vs — vc)t]. Thus, for any nonzero concentration of injected electrons, nex = l/i > 0, superconductivity is destroyed. One may also ask whether other quantities than the order parameter show any signatures of kinks. Indirect evidence of kinks could be looked for in the density of states and the conductivity, for example. They are probed in specific heat and optical absorption measurements, (i) That there are new states at low energies ("midgap" states) associated with the kinks is evident from the density of states N(co). In the absence of excitations, at zero temperature, N(a>) <* ai", where a = 5 (1/Vl — 82 ~ 1). Enhanced superconducting fluctuations will always lead to a pseudogap in the density of states, because a > 0, for g < 0. If there are excitations present in the system, the density of states must correspondingly be modified at low energies such that N(a>) oc (co2 + y 2 )"/ 2 + (w2 + y2)a/2,

(18)

2

where y\ = (2/iv c /€)[cosh 0 - sin(-5-cosh2(/>)] and 72 = (2Rvc/()[l - cos(f sinh20)]. Thus, nonzero N(a> —<• 0) implies that kinks give rise to new states at low energies, (ii) The conductivity, however, does not exhibit any unusual behavior due to kinks, because it is sensitive to the concentration of charges and the interactions between them—injected electrons behave undistinguishably from the existing particles, and the

Charge q* FIG. 1. Schematic illustration of propagating kinks in the order parameter A(x) « !F(x,&>) after a right-moving electron has broken up into spin and charge excitations in a superconductor away from half filling; Vi = 0 and V2 < 0. The left-moving charge component is not shown. Note that, in the interacting system, the spin velocity is larger than the charge velocity, vs > vc.

258 VOLUME 82, NUMBER 8

PHYSICAL

RE

charge dynamics remains the same. If kinks and excitations were pinned, a different kind of behavior would be expected. In general, a nonzero backward scattering term which scatters electrons of opposite spin across the Fermi surface in opposite directions must be included. As a result, leftand right-moving spin degrees of freedom are coupled so that the spin will also split into left- and right-moving components. While it appears that the spin by itself cannot have other values than integers and half integers, irrational values are allowed for the average spin and the concomitant kink. The concept of a winding number applies equally to charge-density-wave and spin-density-wave instabilities (at 2kp) that are described by operators p(x) = ^if>t4x)if>tr-(x)ei2k-3c

+ H.c.

(19a)

IEW

LETTERS

22 FEBRUARY 1999

ter. For example, in a superconductor, the Josephson effect could provide a direct method to measure winding numbers associated with injected elementary excitations. Using Hartree-Fock theory, we have already pointed out that spin excitations in superconductors tend to form antiphase domain walls (kinks) in the order parameter [14]. An analogous observation concerning impurity spinons has been made in the context of one-dimensional Kondo lattices [15]. Thus, these results corroborate the conclusion that kinks are generic excitations of superconductors. We are grateful to Steven Kivelson for his insightful comments on the manuscript. M. I. S. is indebted to Alexander Fetter for his hospitality at Stanford University, where this work was completed. The support by the NSF under Grants No. DMR-9527035 and No. DMR-9629987 and by the U.S. Department of Energy under Grant No. DE-FG05-94ER45518 is gratefully acknowledged.

(charge-density-wave) and

(spin-density wave); T" are the three Pauli matrices (a = 1,2,3). The only differences are that the irrational winding numbers w depend in a unique fashion on the nature of quasi-long-range order and whether the excitations require phase or amplitude kinks in the order parameter. In a fractional quantum Hall system, edge excitations form a chiral Luttinger liquid [12]. Their unusual quantum numbers are determined by topological order of the quantum Hall state present in the bulk, and they may manifest themselves indirectly in tunneling and shot-noise measurements, for example. Because edge states lack a clear quasi-long-range order classified by a symmetry group, no conventional order parameter exists and edge excitations cannot carry kinks in the same sense as lowenergy excitations in superconductors do. Whether the concept of kinks can be generalized to include topological orders is an interesting question. In conclusion, elementary excitations in interacting one-dimensional conductors always carry kinks in the order parameter. The winding number characterizing a kink cannot be arbitrary, but is determined by the spin and the charge of the excitation and by quasi-long-range order of the ground state—in other words, by the interactions. This is not an entirely unexpected result, because electrons are interpreted in bosonization as solitons [13]. The novelty of our formulation is that the kink structure of the excitations becomes observable, if the ground state develops quasi-long-range order. This implies a completely new conception of probing unusual quantum numbers of elementary excitations, which manifest themselves through structural and dynamical modulations in the order parame-

[1] The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987). [2] A. J. Heeger, S. A. Kivelson, J. R. Schrieffer, and W.P. Su, Rev. Mod. Phys. 60, 781 (1988). [3] V.J. Emery, in Highly Conducting One-Dimensional Solids, edited by J. Devreese, R. Evrard, and V. van Doren (Plenum, New York, 1979). [4] J. Solyom, Adv. Phys. 28, 201 (1979). [5] See for example, R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976); J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981); S. Kivelson and J.R. Schrieffer, Phys. Rev. B 25, 6447 (1982). [6] D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 (1965). [7] E. Abdalla, M. Abdalla, and K. Rothe, Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (World Scientific, Singapore, 1991). [8] J. von Delft and H. Scoeller, Ann. Phys. (Leipzig) 7, 225 (1998). [9] Using the identity, eia^x) = :(»«'«>«:e-<'2/2, Eq. (7) can be cast into all different equivalent forms needed in this paper. [10] Obviously, the exact form of the short-distance behavior is sensitive to the details of the ultraviolet cutoff, although the relevant length scale is still the Fermi length a. Also, fast oscillations at 2kp have been neglected. [11] In a triplet superconductor, on the other hand, different forms of y~ correspond to different values of the spin projection. The spin excitation has the same kink profile in the Sz = 0 component of the triplet order parameter as in the singlet order parameter, while the Sz = ±1 components have pure phase twists through complex values (cf. charge kinks). [12] X.G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992). [13] S. Mandelstam, Phys. Rev. D 11, 3026 (1975). [14] M. I. Salkola and J. R. Schrieffer, Phys. Rev. B 57, 14433 (1998). [15] O. Zachar, S.A. Kivelson, and V.J. Emery, Phys. Rev. Lett. 77, 1342 (1996).

1755

Journal of Low Temperature Physics, Vol. 122, Nos. 5/6, 2001

Rapid Communication Fractional Electrons in Liquid Helium? R. Jackiw,1 C. Rebbi,2 and J. R. Schrieffer3 Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA E-mail: [email protected] 2 Boston University, Boston, Massachusetts 02215, USA 3 Florida State University, Tallahassee, Florida 32310, USA (Received December 29, 2000)

We argue that electrons in liquid helium bubbles are not fractional, they are in a superposed state. In an analysis of several unexplained observations on the mobility of electron-inhabited "bubbles" in liquid helium, H. Maris has suggested that such bubbles can fission into two smaller daughter bubbles, each containing half of the original electron's wave function and each allowing the detection of fragments of the original electron, which has divided into two pieces that act as though they are fractions of the original particle.1 In this connection it is important to keep in mind two significantly different concepts of fractional fermions. On the one hand, there are physical situations where each individual measurement yields a fractional result. Two examples are the field-theoretic models for fermions propagating across domain walls, as realized experimentally by solitons on poly acetylene; 2 ' 3 and also the excitations in the fractional quantum Hall effect.4 For these the fractional characteristics are sharp observables (quantum mechanical eigenvalues) without dispersion,5 and the phenomena put into evidence previously unsuspected fermion fractionization. Alternatively, the fraction is an expected value, not an eigenvalue. Then repeated measurements always yield either the full quantum numbers of the particle or a null result, and the "fraction" is just the probability of finding the full particle. This is characterized by a nonvanishing dispersion, which remains nonzero no matter how far apart the measurements are performed. The phenomenon discussed by Maris belongs to the second class: fractional quantum numbers do not arise, only fractional expectations with 587 0022-2291/01/0300-0587519.50/0 © 2001 Plenum Publishing Corporation

588

R. Jackiw et al.

nonvanishing dispersion. The reason for this is the following. Let the electron's wave function be presented as \J/+ = (I//J + \l/2)/y/2, where t/^ and i//2 are (normalized) wave functions peaked at the first and second daughter bubbles, respectively. The expected electron number localized around the first bubble is n = ( + \Nf\ + ) = dVf\l/*+ij/ Here / is a sampling function, which localizes the volume integral in the region of the first bubble. It is true that n = 1/2. However, one must also look at the variance {An)2 — < +1 N2 \ + > — n2. We remember that there exists another state i//_ = («Ai — ^ V v ^ that is almost degenerate with the state i{/+. Retaining just these two states, we find: (An)2 = ( + \Nf\ + y< + \Nf\ + y+< + \Nf\-y(-\Nf\ = K+

+

}-n2

\Nf\~y\2 2

= j"dK#*>A_

_ I

~ 4'

Thus no matter how far apart the two bubbles are taken, one cannot isolate a sharp fraction. Indeed the effect is a standard quantum mechanical result for a superposed state. Here the superposition is in location: the electron is either in one bubble or the other, but before the measurement one cannot decide in which bubble it resides. While "half the electron's wave function" is in both bubbles, measurements will find a full electron in half the bubbles. Finally, we observe that (as remarked by K. Canter) inasmuch as the helium bubble is stabilized by Pauli repulsion between the electron in the bubble and the orbital electrons in helium, for a bubble to exist its inhabitant must be identical with the electrons in the atoms, and cannot be just a fraction thereof. Ultimately it is an experimental question whether split bubbles exist and whether they have any role in explaining the mobilities. We note that arguments have been presented that challenge, on dynamical grounds, the splitting.6 While we do not assess the experimental data, we assert that fractional electrons do not belong in a theoretical description. Indeed, as Maris himself states, experiments require that each bubble acts as though it contains a full electron charge.

Fractional Electrons in Liquid Helium?

589

REFERENCES 1. H. Maris, J. Low Temp. Phys. 120, 173 (2000); see also Physics Today 53(11), 9 (2000), and New Scientist 2260, 25 (2000). 2. R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976). 3. W.-P. Su, J. R. Schrieffer, and A. Heeger, Phvs. Rev. Lett. 42, 1698 (1979); R. Jackiw and J. R. Schrieffer, Nucl. Phys. B 190 [FS3], 253 (1981). 4. R. B. Laughlin, H. L. Stormer, and D. C. Tsui, Rev. Mod. Phys. 71, 863 (1999). 5. S. Kivelson and J. R. Schrieffer, Phys. Rev. B 25, 6447 (1982); R. Rajaraman and J. S. Bell, Phys. Lett. B 116, 151 (1982); R. Jackiw, A. Kerman, I. Klebanov, and G. Semenoff, Nucl. Phys. B 225 [FS9], 233 (1983). 6. V. Elser, eprint cond-mat/0012311.

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Ill Quantum Hall Effect

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265 STATISTICAL P H A S E S A N D T H E F R A C T I O N A L Q U A N T U M HALL EFFECT

Steven A. Kivelson Department of Physics, University of California at Los Angeles, Los Angeles, CA 90095, USA

In the mid 1980's, the most exciting development in condensed matter physics was the discovery and elucidation of the basic physics of the fractional quantum Hall effect. Schrieffer's work in this field included the first constructive derivation of the fractional statistics of the Laughlin quasiparticles, and some of the first work to hint at the deep connection between the theory of the quantum Hall effect and the well understood phenomenon of superfluidity. This latter work, which attempted a description of a quantum Hall fluid as a quantum melted Wigner crystal, may indicate a more generally useful approach for the study of highly correlated electron fluids.

3.1

Introduction

Just a year after the discovery of the fractional quantum Hall effect [l], a basic understanding of the entirely new quantum state of matter which gives rise to this effect was achieved, almost magically, through a wave-function ansatz proposed by Laughlin [2]. The incompressible fluid nature of this state, the rigid correlations imposed by a combination of the repulsion between electrons and the lowest Landau level constraint, and perhaps most remarkably of all, the fractional charge of the quasiparticles supported by this state all emerged simply from this ansatz. The only fundamental feature of this state not addressed in this initial work was the statistics of the quasiparticle. Both Haldane [3] and Halperin [4] considered this issue as part of an effort to understand how condensation of a dense fluid of these quasiparticles (defined relative to a Laughlin "parent" state) could lead to a hierarchy of "daughter" quantum Hall fluids. Haldane proposed an ansatz in which the quasiparticles were assumed to be bosons while Halperin asserted they were anyons, but the two constructions lead to the same hierarchy. Since the statistics of a quasiparticle is only well-defined where the mean spacing between quasiparticles is large compared to their size, it was unclear from either of these papers what_the physical statistics truly were.

3.2

Fractional Statistics and the Quantum Hall Effect

This problem was solved definitively and elegantly in the paper of Arovas, Schrieffer, and Wilczek [5] (ASW). What ASW noted was that the Laughlin wave-function for any dilute collection of quasiparticles could be adiabatically deformed in such a way that the quasiparticles could be exchanged along paths in which the quasiparticles were always far apart. Moreover, the Berry's phase could be simply computed. It consists of two pieces: The

266 first is path dependent, and is proportional to the net directed area enclosed by the paths. This term has the interpretation of the effective magnetic flux enclosed by the quasiparticles.a The second is proportional to the "braiding" number of the path, which is to say the effective number of clockwise minus counterclockwise exchanges — this term has the interpretation of a statistical angle. For instance, if just two quasiparticles are exchanged along a path which encloses no other quasiparticles, the statistical phase is 6 if the path goes clockwise and — 6 if it goes counterclockwise. Clearly, if 0 = nir, the phase change is independent of the sense of the exchange path, and indeed if n is even, the quasiparticles are manifestly bosonic, while if n is odd they are fermionic. However, ASW showed that 6 = Tr/m for the Laughlin state at u = 1/m. (Here, v is the electron density per magnetic flux quantum, or "filling factor" of the lowest Landau level.) This construction proved that the quasiparticles are anyons, with statistics equal to those asserted by Halperin. ASW also pointed out that the quasiparticles could be treated as bosons or fermions, but in this case one needs to imagine that, attached to each quasiparticle there is a quantum of fictitious "statistical flux," so that 0 (or 8 — TT) can be thought of as the Aharonov-Bohm phase obtained when a charge circles a flux. This notion, which was further developed in a later paper by Arovas, Schrieffer, Wilczek, and Zee [6], represented the crucial first step to the Chern-Simons field-theoretic description [7; 8; 9; 10; 11] and the composite fermion approaches [12] for quantum Hall systems, both of which are based on a form of flux attachment.

3.3

Cooperative Ring Exchange Theory

SchriefTer's second contribution to the theory of the quantum Hall effect was work carried out in collaboration with Arovas, Kallin, and me [13]. This work was not completely successful, in that the formalism was cumbersome, and aspects of the approach were unsatisfactory. Nevertheless there were important, and influential ideas concerning the physics of the quantum Hall liquid that originate in this work. I also believe that the basic idea — to approach the physics of highly correlated electron liquids as quantum melted solids rather than as strongly interacting gasses — is correct and important and likely to be increasingly central to theoretical approaches to a host of problems in the coming decade. It certainly is the basic idea underlying the approaches to the theory of both high temperature superconductivity [14] and the metal-insulator transition [15] with which I have been involved in recent years, so you could say I have voted with my feet. The starting point of the cooperative ring exchange theory is the classical ground-state of interacting electrons in a high magnetic field, namely the triangular Wigner crystal. We then studied, in the semiclassical limit valid for large magnetic field, B, the effects of quantum fluctuations about this classical ground-state. (Since the magnetic field quenches the kinetic energy, the semiclassical approximation turns out to be an expansion in powers ° In fact, the effective magnetic field seen by the quasiparticles is proportional t o the mean electron density, rather than to the actual magnetic field.

267 of ljr§ = \/2u ~ J B - 1 / 2 , where I = ^hc/eB is the magnetic length and TQ is the mean spacing between electrons; in contrast, for B = 0, the usual expansion parameter is r~l, where the Bohr radius replaces L) There are two distinct classes of fluctuations — small amplitude fluctuations which make analytic contributions to the ground-state energy in powers of £/ro, and instantons which result in the exchange of electrons from different sites of the crystalline lattice. Of these latter, the most important are the ring-exchange processes, in which a set of electrons along a closed ring on the lattice exchange places by moving one position over. Although the contribution to the energy of any one such path is exponentially small in proportion to exp[—aN(ro/£)2], where N is the number of electrons exchanging and a is a number of order 1, the number of such paths also increases exponentially as exp[-ha'N] where a' ~ log(4) « 1.4. Moreover, what is special about the ring exchange paths, as we showed by explicit calculation, is that a is a relatively small number, which we calculated to be a « 1/6. Thus, we noted that if the crystal did not melt first due to a proliferation of small amplitude fluctuations, it would melt via a proliferation of cooperative ring exchanges when the magnetic field is reduced to the point at which (£/ro)2 = a/a'. The other point about the cooperative ring exchanges is that they have a phase factor which is the sum of contributions from the enclosed flux and from the number of particles exchanged. Only for certain commensurate filling factors, v = v*, do all the large ring exchanges add coherently — this commensuration, we showed, could give rise to the fractional quantum Hall effect and the existence of fractionally charged quasiparticles. Indeed, the sum over ring-exchanges was equated to a bosonic ring-exchange theory [16] in an effective magnetic field

Beff = B- mpfo

(1)

in which the Fermi statistics of the electrons account for the magnetic field shift. Here m is an odd integer, p is the areal density of electrons, and (j>o = hc/e is the flux quantum. Thus, when v = \/m {i.e. when B = mpcfio), the ring-exchange sum is the same as would be encountered in a theory of a bosonic supersolid. The phase that results from condensation of ring exchanges is a peculiar one in which crystalline order and quantum Hall fluid behavior coexist. Although such a "Hall crystal" phase can occur under special circumstances [17], generically one expects the Wigner crystal to melt via a first order transition to a translationally invariant fluid state. We argued that even in such a fluid state, the correlations responsible for the fluid behavior are the same as in the Hall crystal, and so can be understood on the basis of cooperative ring exchange theory. Indeed, somewhat later Lee, Baskaran, and I [18] were able to more or less show that the essential (long distance) features of the Laughlin wave-function can be derived by considering cooperative ring exchange processes on a lattice with a slowly fluctuating geometry. Alternatively, one could imagine approaching the quantum Hall liquid in a somewhat different manner: Consider electrons in a fixed magnetic field corresponding to a quantized Hall state, say u = 1/3, in the presence of a periodic potential of strength V with one

268 valley per electron — for example, one with the same triangular lattice structure as the corresponding Wigner crystal. For large V, the electrons are clearly localized on the lattice sites and the system is insulating. For V — 0, the ground-state is the Laughlin liquid. Because the Laughlin state has a gap, the transition between the Laughlin liquid and the insulating state must occur at a non-zero critical value of V = Vc; it is reasonable to suppose that this transition is continuous, and well described by the cooperative ring exchange theory. Assuming there are no further phase transitions as V is reduced from Vc to 0, it is reasonable to conclude that the condensation of cooperative ring exchanges captures the essential features of the quantum Hall liquid. By now, the cooperative ring exchange theory of the fractional quantum Hall effect has been largely superseded by later, simpler approaches which incorporate the valid insights derived from this work. However, I believe that it was an important step in the development of the theory of the quantum Hall effect. This work, along with the early work of Girvin, MacDonald, and Platzman [19], represented the first steps in pinning down and exploiting the deep relation between quantum Hall physics and the properties of charged superfluids [ll]. Indeed, this work was recognized by Girvin and MacDonald [20] as providing key motivations for their seminal paper in which the non-local order-parameter which characterizes the quantum Hall state was first recognized. This paper was also the first place, to the best of my knowledge, in which the thermodynamic properties of the state were related to those of a system in a shifted effective magnetic field, Eq. (1). Unfortunately, we did not realize at the time that this expression can be much more directly obtained by a flux attachment transformation involving the electrons, as was first recognized, in the context of the theory of anyon superconductivity, by Laughlin [21]. Eq. (1) is a key relation which underlies both the composite boson approach of Zhang et al. [7] and the composite fermion approach of Jain [12]. Indeed, the influence of the cooperative ring exchange theory on the development of these more successful approaches was acknowledged in both these papers.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10

D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983). B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984). D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984).* D. P. Arovas, J. R. Schrieffer, F. Wilczek, and A. Zee, Nucl. Phys. B 251[FS13], 117 (1985).* S.-C. Zhang, T. E. Hansson, and S. A. Kivelson, Phys. Rev. Lett. 62, 82 (1989). S. A. Kivelson, D.-H. Lee, and S.-C. Zhang, Phys. Rev. B 46, 2223 (1992). A. Lopez and E. Fradkin, Phys. Rev. B 44, 5246 (1991). B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993). D.-H. Lee and M. P. A. Fisher, Phys. Rev. Lett. 63, 903 (1989).

269 [11] For a non-technical description of the Chern-Simons mapping between the quantum Hall liquid and a charged superfluid, see S. A. Kivelson, D.-H. Lee, and S.-C. Zhang, Scientific American, March, 1996, pg. 86 - 91. [12] J. K. Jain, Phys. Rev. Lett. 63, 199 (1989). [13] S. A. Kivelson, C. Kallin, D. P. Arovas, and J. R. SchriefFer, Phys. Rev. Lett. 56, 873 (1986) and Phys. Rev. B 36, 1620 (1987).* [14] S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998); V. J. Emery, S. A. Kivelson, and J. Tranquada, Proc. Natl. Acad. Sci. 96, 8814-8817 (1999); J. Zaanen, M. L. Horbach, and W. van Saarloos, Phys. Rev. B 53, 8671 (1996). [15] S. Chakravarty, S. A. Kivelson, C. Nayak, and K. Volker, Phil. Mag. 79, 859 (1999). [16] This view of the quantum Hall transition is analogous to the view of the superfluid transition in He discussed in R. P. Feynman, Phys. Rev. 9 1 , 1291 (1953). [17] B. I. Halperin, Z. Tesanovic, and F. Axel, Phys. Rev. Lett. 57, 922 (1986); G. Murthy, Phys. Rev. Lett. 85, 1954 (2000). [18] D.-H. Lee, G. Baskaran, and S. A. Kivelson, Phys. Rev. Lett. 59, 2467 (1987). [19] S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett. 54, 581 (1985). [20] S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett. 58 1252 (1987). [21] R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988). (The symbol * indicates a paper reprinted in this volume.)

270 VOLUME53, NUMBER7

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13 AUGUST 1984

Fractional Statistics and the Quantum Hall Effect Daniel Arovas Department of Physics, University of California, Santa Barbara, California 93106 and J. R. Schrieffer and Frank Wilczek Department of Physics and Institute for Theoretical Physics, University of California, Santa Barbara, California 93106 (Received 18 May 1984) The statistics of quasiparticles entering the quantum Hall effect are deduced from the adiabatic theorem. These excitations are found to obey fractional statistics, a result closely related to their fractional charge. PACS numbers: 73.40.Lq, 05.3O.-d, 72.20.My

Extensive experimental studies have been carried out 1 on semiconducting heterostructures in the quantum limit OIQT » 1, where uio^eB^m is the cyclotron frequency and T is the electronic scattering time. It is found that as the chemical potential /A is varied, the Hall conductance cr^ = IxIEy = ve2lh shows plateaus at v = n / m , where n and m are integers with m being odd. The ground state and excitations of a two-dimensional electron gas in a strong magnetic field B0 have been studied 2 " 4 in relation to these experiments and it has been found that the free energy shows cusps at filling factors v=n/m of the Landau levels. These cusps correspond to the existence of an "incompressible quantum fluid" for given n/m and an energy gap for adding quasiparticles which form an interpenetrating fluid. This quasiparticle fluid in turn condenses to make a new incompressible fluid at the next larger value of n/m, etc. The charge of the quasiparticles was discussed by Laughlin2 by using an argument analogous to that used in deducing the fractional charge of solitons in one-dimensional conductors. 5 He concluded for v" 1/m that quasiholes and quasiparticles have charges ±e*=±e/m. For example, a quasihole is formed in the incompressible fluid by a twodimensional bubble of a size such that 1/m of an electron is removed. Less clear, however, is the statistics which the quasiparticles satisfy; Fermi, Bose, and fractional statistics having all been proposed. In this Letter, we give a direct method for determining the charge and statistics of the quasiparticles. _ In the symmetric gauge A ( F ) = -fB 0 x T we consider the Laughlin ground state with filling factor e = l//n, 2

*.-im-**)"exp(-iXl*/l >. j < k

722

<»

where ZJ=XJ + iyj. A state having a quasihole localized at z 0 is given by +«,°-^+nl-(*i-zo)*«.

(2)

while a quasiparticle at z 0 is described by *«Io-Ar_n*(3/a*i-V«tf>*.».

(3)

:

where 2iraoB0= <po — hc/e is the flux quantum and TV + are normalizing factors. To determine the quasiparticle charge e", we calculate the change of phase -y of i//m ° as r 0 adiabatically moves around a circle of radius R enclosing flux d>. To determine e*, y is set equal to the change of phase, (e*/Kc)0,

(4)

that a quasiparticle of charge e* would gain in moving around this loop. As emphasized recently by Berry6 and by Simon 7 (see also Wilczek and Zee 8 and Schiff9), given a Hamiltonian H(z0) which depends on a parameter z 0 , if z 0 slowly transverses a loop, then in addition to the usual phase / £ ( / ' ) dt\ where E(t') is the adiabatic energy, an extra phase y occurs in
= i(*(t)\d
.

(5)

From Eq. (2),

- ^

-N\XJ-\n\z,-

2o(/)l*.

+,

°,

(6)

so that ^W/Vi(C*°||£ln(r,-z0)|*,;Z°).

(7)

Since the one-electron density in the presence of

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the quasihole is given by

/' , W-<*. + ' , lS«{: ( -r)l*. + , , >.

(8)

we have 4 r - ' ' fdxdyp+z°(z)-%-\nlz-z0(t)], (9) J at at where z=x+/y. We write p °(z) = p 0 + 8p *°(z), with po=vl0. Concerning the p 0 term, if z 0 is integrated in a clockwise sense around a circle of radius R, values of \z\ R contributes zero. Therefore, this contribution to y is given by

= -27T«= -277-^/4.0,

(10)

where (n > R is the mean number of electrons in a circle of radius R. Corrections from 8p vanish as (ao/R)2, where a0=(Kc/eB)l/2 is the magnetic length. This term corresponds to the finite size of the hole. Comparing with Eq. (4), we find e* = vc, in agreement with Laughlin's result. A similar analysis shows that the charge of the quasiparticle *m"2°.s-**. To determine the statistics of the quasiparticles, we consider the state with quasiboles at za and zb,

As above, we adiabatically carry za aroound a closed loop of radius R. If zb is outside the circle \zb\ = R by a distance d » a0, the above analysis for y is unchanged, i.e., y «= - 2irv/(/>0. If zb is inside the loop with \zb\ — R « —a0, the change of (n)R is — v and one finds the extra phase Ay = 27rv. Therefore, when a quasiparticle adiabatically encircles another quasiparticle an extra "statistical phase" Ay = 27r^

(12) 10

is accumulated. For the case v = I, Ay = 2ir, and the phase for interchanging quasiparticles is Ay/2-=7T corresponding to Fermi statistics. For v noninteger, Ay corresponds to fractional statistics, in agreement with the conclusion of Halperin.11 Clearly, when v is noninteger the change of phase Ay when a third quasiparticle is in the vicinity will depend on the adiabatic path taken by the quasiparticles as they are interchanged and the pair permutation definition used for Fermi and Bose statistics no longer suffices.

LETTERS

13 A U G U S T 1984

A convenient method for including the statistical phase Ay is by adding to the actual_vector potential A 0 a "statistical" vector_potential A # which has no independent dynamics. A^ is chosen such that {e'/fic)<$A4,-dT=Ay>=2nv,

(13)

when z„ encirlces zb. One finds this fictious A^ to be A,(F-n)=

0 o f x ( r - Tb) 2ir| r - r

4

(14)

r

if the quasiparticles are treated as bosons and 4>o~' #o(l — Vv) if t n e v are treated as fermions. Thus, the peculiar statistics can be replaced by a more complicated effective Lagrangian describing particles with conventional statistics. 12 Finally, we note that if one pierces the plane with a physical flux tube of magnitude , the above arguments suggest that a charge ve/o is accumulated around the tube, regardless of whether 4>/4>o is equal to the ratio of integers. This work was supported in part by the National Science Foundation through Grant No. DMR8216285 and No. PHY77-27084, supplemented by funds from the National Aeronautics and Space Administration. One of us (D.A.) is grateful for the support of an AT&T Bell Laboratories Scholarship.

'K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,494(1980). 2 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 3 F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983). 4 B. I. Halperin, Institute of Theoretical Physics, University of California, Santa Barbara, Report No. NSF-1TP-83-34, 1983 (to be published). 5 W. P. Su and J. R. Schrieffer, Phys. Rev. Lett. 46, 738 (1981). 6 M. V. Berry, Proc. Roy. Soc. London, Ser. A 392, 45-57 (1984). 7 B. Simon, Phys. Rev. Lett. 51, 2167 (1983). 8 F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984). 9 L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), p. 290. 10 Although
723

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24 FEBRUARY 1986

Cooperative-Ring-Exchange Theory of the Fractional Quantized Hall Effect Steven Kivelson,(,) C. Kallin, Daniel P. Arovas,(b) and J. R. Schrieffer Institute for Theoretical Physics, University of California, Santa Barbara, California 93106 (Received 3 September 1985) A semiclassical path-integral approach is used to calculate the contribution of large-correlatedring exchanges to the energy of a two-dimensional Wigner crystal in a strong magnetic field. This correlation energy Ec(v) shows cusps at fractional fillings vc** n/mof the lowest Landau level. The uniform Wigner crystal is locally unstable for v*vc and the theory predicts the existence of fractionally charged quasiparticles to accommodate the extra density v - vc. PACS numbers: 71.45.Gm, 73.40.Lq

Recent experiments on the low-temperature, largemagnetic-field (B0) conductivity of high-mobility electron layers1 provide evidence of the existence of a family of novel condensed phases of the twodimensional (2D) electron gas at "special" rational values of the dimensionless density v, where v is the mean number of electrons in the area 2ITIQ =4>(/fi0 covered by one flux quantum o~= hc/e. Specifically, (1) the anomalously small value of o-^ suggests2 that there is a gap in the spectrum of current-carrying states, and hence a cusp in the correlation energy Ee(.v) at the special values of v; (2) the fractional quantization of v^ may be evidence3 of the existence of fractionally charged quasiparticles. We have studied this system in the absence of impurities, in the high-magnetic-field limit where the spacing K
We have studied this problem, also beginning from a WC state, but using a systematic, semiclassical approximation.8,9 Our results may be summarized as follows. We find exchange effects in which L electrons in a ring coherently rotate to an equivalent configuration leading to contributions to £c(v) which can be orders of magnitude larger than pair-exchange contributions because of the reduced tunneling barrier. In addition, these contributions exhibit nonanalytic cusplike behavior for certain rational values of v.10 Rings with large L play a dominant role for two reasons. Firstly, although the contribution from any single ring decreases exponentially with L, there are a very large number of rings with large L(~ KL where K is the connectivity). Secondly, they can make a nonanalytic contribution to Ec(v) as a result of interference between different exchange rings. The contribution from each ring contains a phase factor 0**2n XBQA{V)/O (Bohm-Aharonov effect), where A{v) is the enclosed area. For arbitrary v, the contributions from large rings add incoherently. However, because A(v) is always approximately equal to an integer multiple of the area per elementary plaquette of the WC, for certain rational densities vc the different rings add in phase. It is this effect which makes these densities energetically favorable and which leads to cusps in Ec(v) at i/=-!/,. when arbitrarily large rings are included. Our model derives from a LLL path-integral representation for the partition function Z = Tre -fi*N with N

U

]

HN

Pl+

B

27 °

£xr

'

T*). j
The single-particle Hamiltonian admits a continuous representation for LLL eigenstates, <^R(T) — (r|R): (riR) = ( 2 i r ) - 1 / 2 e x p { - | ( r - R ) 2 + | / ( r x R ) - z ) with //ilR) --jtfwjR) and where we set / 0 = 1 . The resolution of the LLL projection operator, P0 -(l/2ir)fd2R\R)(R\, may be used to develop a

© 1986 The American Physical Society

873

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path-integral expression 11 for Z: Pes* where N = vB0/0 and I |r,l) = |r, . . . i>>. This yields

PISW

J"\

(1) where . ^ i s a normalization constant and the boundary conditions require Ry(0) = R P ( j)(/3). The action for continuous paths i s " ' 1 2

S\R\-\fidT+ X^*y-**>

(2)

j*k

where P is the matrix element of the Coulomb potential between coherent states, K ( R ) = y>/7r(e 2 /«) x exp( - ?R2)I0( jR2). We will refer to the integration variable T as the (imaginary) time. The partition function Z is evaluated within the semiclassical approximation. This entails the finding of all paths RC(T) which extremize the action [/?c is a vector function with 2N components R ^ ( r ) ) and then the inclusion of quantum fluctuations by expansion of the action to second order in R — Rc. In this way Z can be expressed as a sum over classical paths, z=

Xc°f*c'e

-SIR')

(3)

LETTERS

24 FEBRUARY

1986

addition, as discussed earlier, we focus on those paths which are most likely to lead to structure in Ec(v) because of their systematic dependence on v; that is, we consider paths, such as those illustrated in Fig. 1, which consist of a cyclic, coherent superposition of nearest-neighbor exchanges. For convenience, we factor out the leading-order contribution that is common to all classical paths by writing Eq. (3) as

Z-ZoXcZc-ZoXc^Rc^~S[Rr\

(5)

where D = D/D0 and S = S— S0. Let us consider the contribution of a single large exchange ring to Z. The real part of the classical action is approximately proportional to the number of electrons in the ring L and the imaginary part is 0= ±27r(#/<£ 0 ) + TT(L - 1 ) , where is the enclosed flux, the ± refers to positive or negative sense of rotation, and the term TT(L — 1) reflects the Fermi statistics. If the classical path exactly followed the straight-line segments between sites (as shown in Fig. 1) then 0 = ± n(v~'— 1 )NA + 7r(mod2ir), where NA is the number of enclosed plaquettes of area TTIQ/V, and we have used the fact that NA is even (odd) when L is even (odd). Similarly, the real part of the action would be a0(v)L, where ao is independent of path. 13 We have argued that the net effect of deviations from such linear paths is to renormalize a 0 . 8 The fluctuation determinant is D[/?<.]= - T f l - ^ r e x p l - A a L + C K l n L ) ] , where T 0 is the cooperative tunneling time. 14 Therefore, we approximate the contribution of a large-ring exchange as AZ C (v) = TQ l dr exp[ - a (v) L + ihNA + O (InZ.)),

C

where D[R ] is the fluctuation determinant. The extremal (classical) paths satisfy the equations of motion

iXj - a Vj/a Yj, iYj - - a Vj/aXj.

(4)

where K, = 2/-y K ( R , - R y ) . These are simply the imaginary-time E x B drift equations. To find solutions to Eq. (4) which satisfy the boundary conditions Ry(0)=R>>(y)(/3), we analytically continue the path integral to complex values of Xj and Jy.12 The path with the smallest action is the triangular WC. This path with pairwise exchange and Gaussian (phonon) fluctuations about it makes the leadingorder contribution to Z, Z 0 = DQe °=exp[— /3 x NE0(v) ], where E0 is the energy per site of the static WC as computed by Maki and Zotos. 5 £o(") is a smooth, monotonic function of v for v < y . Since the shear modulus is negative for v > v + — 0.45, 5 we restrict our analysis to v 1 — v + , the same considerations apply to the hole lattice.) So as not to have prohibitively large action, the important classical paths must resemble the WC at most points in space and time. Therefore, we consider only classical paths whose initial configuration is the static WC. In 874

where a = a0 + Aa and h = 7rd> - 1 — 1). Whether or not large-ring exchanges contribute significantly to Z(v) is determined by the numerical value of a(v). We have estimated a(v) for the simple case in which a single line of electrons exchange,

FIG. 1. Examples of exchange paths and their representation in terms of a configuration of the dual lattice.

274 VOLUME56, NUMBER 8

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so that Xj((3)=Xj(Q) + av, Yj(fi)-

y / 0 ) , where av

= (4TT/V3I>) 1/2 is the lattice constant of the WC. We

assume that only the electrons in this one line move in the background of the static potential of all other electrons. Thus we overestimate a; however, we believe that the relaxation corrections are small. Wefindthat the extremal path corresponds to rigid motion of the line [R y (r) - R / 0 ) = R(r) for yon the line] and that <*(-) — 0.81 (as a function of the density, a is approximately proportional to 1/V). As we shall see, this indicates that large-ring exchanges are important at densities of experimental interest. A general path in Eq. (5) contains many exchange rings and for small a, they are sufficiently dense that the exchange events do not form a dilute gas. Therefore, we include what we believe to be the most important interactions between exchange events that overlap in space and time. The time interval /3 is divided into slices of width r 0 and, because the classical paths are exponentially localized in time, we ignore interactions between exchange events which occur in different time slices; i.e., Zc — lZsUce]fi T°, where ZsUce is the trace over all exchanges in a given time slice. The exchange paths in a given time slice are enumerated in terms of integer-valued spin variables S x ; 5X is defined to be the number of clockwise minus the number of counterclockwise exchange rings that encircle the plaquette A. Hence, A labels a site on the dual (honeycomb) lattice. We associate with each spin configuration an energy

#DG = « X (Sx-V 2 +iV»5X

(6)

where (\,y) denotes nearest-neighbor sites. Then we make the approximation ZsUce = Trexp(-//oG) which is exact for all configurations of isolated rings and includes a repulsion between rings that share one or more nearest-neighbor bonds. Equation (6) is the Hamiltonian of the discrete Gaussian (DG) model in an imaginary field, where a - 1 plays the role of temperature. This model is known to have a phase transition at a critical value of a = a c (/i) 1 5 For h = 2trm [v- l/(2m + l ) ] , ac takes on its maximum value, which we estimate to be a c = 1.1. For a(v) > a c , the system behaves like a classical WC, while for a(v) < ac, the system is highly quantum mechanical and arbitrarily large exchange rings dominate the behavior of the system. To study the a
**-£

Gky

h

where Gky ~ ln\Ry-Ry\ is the (honeycomb) lattice Green's function and
24 FEBRUARY 1986

small-a phase can be analyzed by a study of the ground-state properties of the CG. The h = 2Trm ground state of HQQ has qk^m and zero energy. The ground state for | h — 2nm I s S/j « 2TT has a fraction 8h/2n of sites with charge l — Bh/2ir, which themselves form a Wigner lattice. The remaining sites have charge — bh/2ir. Thus the free energy FCn of the CG at small a is proportional to |8/j|ln|27r/8/j[, which for our problem implies Ec(v) — ISvllnlSi'l for |8i/| « l/(2m + 1). Since dEc/dv diverges as 8v — 0, the uniform WC state is thermodynamically unstable in the open neighborhood of f = l/(2m + l). This motivates the need for quasiparticles discussed below. From studies of Josephson junction arrays in a transverse field,16 FrG is believed to have cusps of the form |8/i|ln|2ir/8/iT at all rational h/2n for a(v) < ac(h(,v)).u From our estimate of a(v) and from Monte Carlo calculations of Shih and Stroud,16 we find that this inequality is most likely to be satisfied by v j , r> f» fi T> f« a n d f» although some of these phases may be unstable with respect to competing phases.18 Since the v = — uniform-density state is much more stable than any other special density, one might also consider a hierarchy of states formed by starting with a WC of quasiparticles and repeating our analysis for ring exchange of quasiparticles with fractional charge and fractional statistics.19,20 In this case, one obtains a sequence of stable densities in agreement with previous hierarchical analyses.20,21 We will discuss the relative stability of these states elsewhere.8 Our model is thus incompressible since it is rigid with respect to uniform dilations. However, there are quasiparticle (qp) excitations of charge Q* which correspond to local dilations of the Wigner crystal by an amount 8-4. As a result of the deformation of the lattice, all rings that enclose the qp acquire an extra phase A0 — 2nBobA/0 relative to those that do not. One can see immediately from the CG representation that this will cost a logarithmically divergent energy unless A0 = 2ir x integer, and hence the elementary excitations have charge Q* = ±ve. We have constructed an approximate qp with a core size -— l0 and estimate its creation energy E^iv) -~0.5v1e2/el0. When v is near a favored density vc, the system can reduce its energy by forming a nonuniform state with density (averaged over a plaquette) equal to vc everywhere except in the core of the qp. The cusps in the energy are thus E(v)—vc~l\v-vc\E<w(vc) for v near vc. (We have not evaluated the creation energies with sufficient accuracy to distinguish between quasiparticle and quasihole energies.) Of great interest is the magnetophonon (mp) spectrum. In the sparse-ring phase, the mp dispersion resembles the standard
275 VOLUME56, NUMBERS

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spectrum.8 This effect and the issue of dissipation are presently under consideration. We conclude with some brief comments on the consistency of our approach. In the small-a phase, the gas of exchange loops is quite dense. Hence, the timeslice path decomposition is not well defined. However, the major role of the time-slice approximation is only to provide an ultraviolet cutoff corresponding to a repulsion between overlapping exchange loops. One particular feature of the dense-ring phase is the proliferation of intersecting rings in the dual spin model. Some of these configurations correspond to highaction paths in real space (e.g., crossings), and should be discouraged by the inclusion of additional shortranged spin-spin interactions.22 We expect that such terms should lead to a renormalization of a, and that they will not change the universality class of the spin model. An interesting open question concerns the existence of long-ranged charge-density-wave (CDW) order. At finite temperature, the magnetophonons will destroy any long-ranged order. At zero temperature, none of the terms that we have computed explicitly destroy the CDW. However, we have yet to establish fully whether or not such order is actually present in this limit, and the relation between our theory and that of Laughlin remains unclear. Nevertheless, it seems likely that our results do not depend essentially on the answer to this question, since the vanishing compressibility at rational v derives from the coherent addition of many large exchange loops, an effect that could persist in the absence of CDW order. We are grateful to A. J. Berlinsky, V. J. Emery, F. D. M. Haldane, T. C. Halsey, D. Stroud, and E. Tosatti for useful conversations. This work was supported in part by the National Science Foundation through Grants No. DMR 82-16285 and No. DMR 83-18051. One of us (D.P.A.) acknowledges support by an AT&T Bell Laboratories Scholarship, the work of another of us (C.K.) is partially supported by a Canadian Natural Sciences and Engineering Research Council Postdoctoral Fellowship, and one of us (S.K.) is the recipient of an Alfred P. Sloan Fellowship.

'"'Permanent address: Department of Physics, State University of New York at Stony Brook, Stony Brook, N.Y. 11794. (b, Permanent address: Department of Physics, University of California, Santa Barbara, Cal. 93106.

876

IEW

LETTERS

24 FEBRUARY 1986

•See, for example, D. C. Tsui, H. L. Stormer. and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982); H. L. Stormer, A. M. Chang, D. C. Tsui, J. C. M. Hwang, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 50, 1953 (1983). 2 R. B. Laughlin, Phys. Rev. B 23, 5632 (1981); B. I. Halperin, Helv. Phys. Acta 56, 75 (1983). 3 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 4 D. Yoshioka and P. A. Lee, Phys. Rev. B 27, 4986 (1983), and references therein. 5 K. Maki and X. Zotos, Phys. Rev. B 28, 4349 (1983). 6 F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 54, 237 (1985); D. Yoshioka, B. I. Halperin, and P. A. Lee, Phys. Rev. Lett. 50, 1219 (1983). 7 S. T. Chui, T. M. Hakim, and K. B. Ma, to be published; see also S. T. Chui, Phys. Rev. B 32, 1436 (1985). 8 S. Kivelson, C. Kallin, D. P. Arovas and J. R. Schrieffer, to be published. *This approximation is justified by the fact that the quantum parameter is small for v < T . See Ref. 8 10 The possibility that large-ring exchanges might lead to a phase transition was discussed in a quite different context by R. P. Feynman, Phys. Rev. 91, 1291 (1953). See also S. Chakravarty and D. B. Stein, Phys. Rev. Lett. 49, 582 (1982). "See, for example, L. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981), Chap. 27. 12 When discontinuous paths are important, one must use the discrete version of Eq. (1). See Ref. 11 and J. R. Klauder, in Path Integrals and Their Applications in Quantum Statistical and Solid State Physics, edited by G. Papadopoulos and J. T. Devreese (Plenum, New York, 1977), p. 5. l3 The effective interaction between particles on the exchange path falls off sufficiently rapidly that a0 is independent of path. See Ref. 8. 14 The fluctuation determinant has been evaluated approximately. It contains a factor - 1 per ring-exchange event. The Tf1 term arises due to the zero mode associated with time translation. 15 S. T. Chui and J. D. Weeks, Phys. Rev. B 14 4978 (1976). 16 W. Y. Shih and D. Stroud, Phys. Rev. B 32, 158 (1985), and references therein. 17 T. C. Halsey, unpublished. 18 Note that the plateau at v = j - does not appear in the usual hierarchy scheme. In this context, we note that a plateau in (TW at v= -j- as well as a broad minimum in va at v= y have been reported by G. Ebert et a/., J. Phys. C 17, L775 0984). "D. P. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984). 20 B. I. Halperin, Phys. Rev. Lett. 52, 1583, 2390(E) (1984). 21 F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983); R. B. Laughlin, Surf. Sci. 142, 163 (1984). 22 At the level of the spin model, several unphysical paths have already been discarded or suppressed. See Ref. 8.

276 PHYSICAL REVIEW B

VOLUME 36, NUMBER 3

15JULY1987-H

Cooperative ring exchange and the fractional quantum Hall effect Steven Kivelson,* C. Kallin, Daniel P. Arovas,* and J. Robert Schrieffer Institute for Theoretical Physics, University of California, Santa Barbara, California 93106 (Received 30 January 1987) We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, v = v,. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of | v—v, | in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.

I. INTRODUCTION The remarkable discovery of anomalies in the transport properties of a two-dimensional (2D) electron or hole gas in a strong magnetic field at integral 1 and fractional 2 filling v of the lowest Landau level has stimulated considerable theoretical activity. In particular, in semiconductor heterojunctions it is observed that the Hall conductivity axy exhibits plateaus at integer filling factors v, = 1,2, . . . , as well as at rational values of v, with odd denominators, e.g., v , = y , y , j , . . . , with axy quantized very accurately to values vje1/h. In addition, the longitudinal resistance pxx is observed to become extremely small near v,. While the integer effect can in essence be accounted for within the context of one-body theory, the fractional quantum Hall effect (FQHE) is fundamentally a consequence of many-body correlations. Most theoretical attempts to account for the F Q H E are based on the following general picture. 3 , 4 Near certain filling factors v, the energy E(v) of the system is particularly low, exhibiting cusplike behavior at each v,. As a consequence, at these cusps the chemical potential [4 = dE /dv is discontinuous, corresponding to a gap 2A = ( e * / e ) ( / i + — /x_) in the spectrum for creating a pair of widely spaced quasiparticle excitations of charge e*. In the presence of defects, added charge Sv = v —v, becomes pinned at the defects, causing axy to be constant near v,. For large 8v the defects are saturated and the unbound added charge enters the condensate increasing its density toward its value at the next plateau. Presumably, the minimum of pxx at v, is also a consequence of the gap 2A, with intrinsic dissipation being thermally activated. On the basis of this picture, a number of approaches have been advanced to account for the origin of the cusps as well as the nature of the low-lying excitations. Early theoretical attempts focused on a Hartree-Fock approach to the 2D electron gas in a strong magnetic field B.5~8 Within this approximation, it was shown that E is a smooth function of v, without cusps. The first success36

ful theory of the F Q H E was proposed by Laughlin. 9 Working in the symmetric gauge A = { B x r , with B in the direction normal to the 2 D (x,y) plane, he assumed the electronic ground state for a fractional filling factor v = 1/m is well approximated by a Jastrow-type function, 2 2 *O=A-0 n * o •

(1-2)

The factor z,•— ZQ raises the angular m o m e n t u m of electron i, measured about z0, by the a m o u n t + 1, leaving a region of reduced electron density at ZQ- Since the mean density per quantum of flux is v, the charge of the quasiparticle is e* =ve, a fractional value. This is the analog of fractionally charged solitons in one-dimensional systems. 12 While a factor (z, — z0)a for noninteger a would give charge ave, only integer a is allowed if one is to remain in the lowest Landau level. For v = y (and a = l ) , Laughlin estimates the quasihole creation energy to be A_=0.026e2/e/. The quasielectron is obtained by lowering the angular momentum of each electron, thereby increasing the electron density near the origin. Similar arguments to those above lead to a charge — ve and a creation energy, which Laughlin estimated to be A + = 0 . 0 3 0 e 2 / e / for v = | . It has been suggested that these excitations obey fractional statistics. 13,14 1620

© 1987 The American Physical Society

277 36

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL . . .

Based on these results, Laughlin argued that the state describes an incompressible system in that v can be changed by a small amount 6v only if the extra charge is added in the form of quasiparticles. Therefore, the energy increase near v, is linear in 8v, bE =N | 8v | A, where A' is the number of flux quanta in the area A of the system. In a compressible system, &E is quadratic in 8v and therefore the system exhibits gapless sound waves or magnetophonons. Extensive numerical calculations"^ 1 7 have been carried out for small numbers of electrons to investigate the ground state and excitations of an interacting electron gas in a strong magnetic field. Haldane and Rezayi 15 have recently studied up to eight electrons in a spherical geometry for filling factor v = \, using a Coulomb potential. Remarkably, for six electrons, the projection of their fully diagonalized state on the Laughlin-Jastrow state is over 9 9 % . To investigate the excitation spectrum Girvin, MacDonald, and Platzman 1 8 used an approach similar to that of F e y n m a n " for 4 He. In this scheme one considers the Fourier component p± of the density operator suitably projected onto the lowest Landau level. When this operator is applied to the ground state, one assumes that most of the weight arises form a single mode. Using the oscillator strength sum rule and the structure factor 5 ( k ) obtained form the Laughlin-Jastrow function, one finds a spectrum which exhibits a gap for all k, reflecting the incompressibility of the system. In addition, these authors find a magnetoroton minimum analogous to the roton minimum in 4 He. The above discussion has been limited to the "fundamental" steps, V = 1 / / T J , where m is an odd integer. To account for steps at densities n /m where n differs from 1 and from m— 1, Haldane, 1 0 Halperin, 14 and Laughlin 2 0 proposed a hierarchy of condensates, each step of the hierarchy being formed from the quasiparticles of the preceeding step. While giving a good qualitative description of the n /m steps, the mean spacing between quasiparticles is comparable to their intrinsic size, making a detailed theory of this approach difficult. Laughlin's ground state appears to be a remarkably good approximation to the exact ground state. However, it is not clear which of the correlations that are built into Laughlin's wave function are essential for the fractional quantum Hall effect, and what their physical origin is. On the one hand, the fact that this wave function keeps the electrons well apart at short distances appears to be very important. On the other hand, Laughlin's wave function builds in strong correlations between the position of one electron and the positions of all other electrons. Moreover, the incompressibility which characterizes the quantum-Hall ground state, is a long-wavelength collective property of the system. In particular, it is an interesting open question whether or not an order parameter which exhibits long-range or quasi-long-range order can be defined for the system. 2 1 , 2 3 By adopting a fresh approach to this problem, we believe that we can shed some light on these issues. The present work has as its point of departure the lowdensity limit investigated by Maki and Zotos. 8 As v—>-0

1621

(or | 1—v| —>0) the electrons lor holes) form a Wigner lattice. Corrections to the zeroth-order Madelung energy arise from lattice vibrations and pairwise exchange. These energies are smooth functions of v. Maki and Zotos calculated the contribution to E from exchanges involving a triangle of three electrons. They treated the problem as two pairwise exchanges and found an energy lowering when the Wigner and magnetic lattice periods are commensurate. Unfortunately, the magnitude of the effect was found to be 4 orders of magnitude too weak and a flat minimum rather than a cusp was found at densities v = l / t t i , where m is an odd integer. It is clear that the Aharonov-Bohm phase is what leads to this effect,24 although it is difficult to see how one can obtain the required cusps or the correct order of magnitude of the strength of the cusps from this argument alone. As we will discuss, the phenomenon is a rather subtle one. It is not even prima facie clear whether exchange processes should be expected to lower the ground-state energy, and hence produce a particularly stable ground state when they add in phase, or in fact raise the energy. In the absence of a magnetic field, the antisymmetric ground state always has higher energy than the unsymmetrized ground state—exchange always raises the energy of a fermion system. This is because exchange builds nodes into the wave function and, hence, increases the ground-state kinetic energy. However, in a strong magnetic field, the kinetic energy is quenched. In this case, the dominant effect of exchange can be to reduce the repulsive interaction between electrons. Hence, exchange can indeed lower the ground-state energy. In a recent paper 24 we proposed an explicit mechanism whereby exchange processes can lead to cusps in the ground-state energy. Furthermore, we showed how the quasiparticle charge is quantized in this approach. The basic idea is that rings of electrons undergo cooperative tunneling processes in which each particle moves to the site initially occupied by one of its neighbors. It is this tunneling current when coupled to the vector potential A, through the Aharonov-Bohm effect, which leads to coherent superposition of the exchange energies from all rings when the flux inside each ring is a multiple of the flux quantum. The sharpness of the cusps arises from the importance of large rings, which are extremely sensitive to changes of density. T h e large overall magnitude of the effect is due to the low tunneling barrier which arises from the fact that the electrons are moving cooperatively along the tunneling path. In essence, as the density increases, the number of ring tunneling events per unit time increases until the system passes through a phase transition, after which the system is filled with rings. These rings enter with arbitrary phase except near critical values of the density given by rational values n /m, where they add in phase, lowering the energy in a cusplike form. Quasiparticles are formed, much as in the Laughlin theory, and have their charge quantized by the fact that charges other than the proper values have infinite energy. Recently, Baskaran 2 5 has argued that the cooperative ring-exchange mechanism need not be tied to the Wigner crystal, but may be defined relative to the typical

278 36

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

1622

configuration of an incompressible fluid. While we are not sure this problem has been completely resolved, we also feel that the crystalline lattice is not an essential feature of the theory described in this paper. We will return to this point in the conclusions. This paper provides a detailed analysis of the theory put forward in our earlier Letter 24 and also includes new results on the excitations. The structure of the paper is as follows. In Sec. II we discuss the path-integral formulation of the problem and the coherent-state basis. The essential role played by cooperative ring-exchange processes is deduced and the contribution from various types of exchange paths is discussed. In Sec. Ill the problem of summing over all cooperative ring-exchange contributions is mapped onto the discrete Gaussian model, and in turn onto the Coulomb gas, whose properties are largely known. This leads to a ground-state energy exhibiting cusps of the form | 8v | In | 6v | , which motivates the need for quasiparticles. Quasiparticle excitations are deduced in Sec. IV where it is shown that the condition that all exchange rings add coherently leads to fractional charge quantization. Boson excitations (collective modes) are also discussed. A summary, remarks and conclusions are given in Sec. V. In addition, there are six appendices. A discussion of how exchange processes can lower the energy of a fermion system in a strong magnetic field is contained in Appendix A. Appendixes B and C contain the formalism necessary

Z(v) = T r e - ' w = - ^ T N\

2

s g n ( c r ) / TJ d2rk{ru

in deriving a coherent-state path-integral expression for the partition function. Appendix D contains some details of the calculation of the classical action, Appendix E reviews some of the statistical mechanics of certain models that we use, and Appendix F contains a lowest-order calculation of the magnetophonon spectrum in the dense exchange phase. II. PATH-INTEGRAL FORMULATION A. Coherent-state path integral One can develop a path-integral formulation for a twodimensional system of electron confined to the lowest Landau level by using a coherent-state representation. 26,27 We use such a formulation to calculate the partition function Z ( v ) for the partially filled Landau level in the steepest-descent or saddle-point approximation. The ground-state energy is then obtained from the partition function as

E(v)=—

limp-,,,, — lnZ(v) ,

where B is the inverse temperature. The partition function for a two-dimensional system of N electrons in a perpendicular magnetic field B = — B0z may be written as

(2.1)

. . . , i> | e " " | rff(1), . . . , !•„,„,>

where the sum is over all permutations a and, in the symmetric gauge, 12

H

2m*

Pi +

X--BozXr, 2c

+ t/,(r,)+

2

Here, K2(r) = e 2 / e r is the Coulomb potential and U\ is the interaction with the neutralizing background. In the limit that the splitting between Landau levels, fkoc=fieB/m*c, is much greater than the Coulomb energy (m*—>-0), we can project the Hamiltonian onto the lowest Landau level. This results in an expression for Z as a path integral over an overcomplete basis of directproduct single-electron coherent states | R ) which span the lowest Landau level. The position-space representation of I R ) is 0 R (r) = < r | R > = ( 2 i r ) - , / 2 e x p

(2.2)

^d-i-rj)

yi < i)


•i-|R,-R2|2

where we set / = V^Sc/eJ? = 1. [Henceforth, we will adopt the notation a A b s ( a X b ) - z for two-dimensional cross products.] The general properties of these coherent states are reviewed in Appendix B. We stress that the state label R is a continuous quantity. The overlap between two such states is given by

iR,AR 2 (2.4)

The derivation of the discrete-time path integral proceeds in the usual way, as discussed in Appendix C, and one obtains Z(v) = M X

sgn(a)/ n

IT dRjinek

-S[R) (2.5)

|r-R|2+^-(rxR)-z (2.3)

+

where ./V is a normalization constant, Ry(«E) labels the center of the coherent state occupied by particle j at (imaginary) time r=ne, E=B/M, and, from the definition of the trace, the R; satisfy the boundary conditions Rj(0) =

Raij)(B)

The action function in Eq. (2.5) is

(2.6)

279 COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

36 M

2 Vjk(nz) -OU2

-j[R,(ne + E)-R,(/iE)HR y (nE)- -iR,(nE)Xz]-fEC/,(ne))-l-e

S[R]= 2

1623

(j


(2.7)

where

VAr)jk


j V2 [ R / T + E J R ^ T + E ) )

(2.8)

y


The discrete-time path integral in Eq. (2.5) is well denned for time step E small but nonzero. The integration over the variables Xj and Yj in each time slice can be performed along any contour in the complex Xj and Yj planes that runs from — oo to + oo. In general, the continuum limit (E—>0) of Eq. (2.5) is fraught with mathematical difficulties because of the existence of discontinuous paths with finite action. One can handle this difficulty either by keeping E nonzero at all intermediate steps of the calculation, or by considering the usual Feynman (Weiner) path integral for the partition function and taking the limit m* —*0 at the end.27 (These subtleties are inconsequential for calculating the classical action, but can be important for calculating the sum over paths in the vicinity of the classical path i.e., the fluctuation determinant.) Despite these difficulties, the continuum version of the path integral can be used to develop a saddle-point approximation for the partition function,27,28 as discussed in the next section. In this case, Eqs. (2.5) and (2.7) become

]J 2)Rj(T)e^s^^

Z(v) = cS2,sgn(a)f

,

(2.9)

J = l

N

S[R] =

fdr

-±RJARJ

+ 2

+ U(RJ)

r(Rj-Rk)

Z =

^e'slRC]D[Rc]

(2.12)

where 5[i? c ] is the action evaluated along the classical path (and includes a phase factor arising from Fermi statistics) and D [R c] is the fluctuation determinant. It follows from the discrete-time equations of motion that the classical (extremal) paths are continuous, except possibly at the endpoints (see Appendix C) and therefore can also be derived from the continuous time action. The classical paths satisfy the equations of motion:

(2.10)

i.k (J < k)

lX

> a r , ' ,Yj

where £/(R)= is the neutralizing background term and V is the matrix element of the Coulomb potential between coherent states,

K(R)=^^V*2/8/0(*V8).

pect, the semiclassical approximation is valid at low electron densities where the ground state is believed to be a Wigner crystal. In fact, we shall see that g/Sv<0.1 for v < | , and therefore this approximation is possibly justified at intermediate densities as well. In the saddle-point approximation the path integral is evaluated by a multidimensional version of the method of steepest descents. Thus our prescription is to first find all those paths RC{T) which extremize the action, oSYoRyfr) \ft=Rc = 0. [Rc is a vector function with IS components R,(T) which we will call a "classical" path. It is important to keep in mind, however, that R , ( T ) labels the center of a coherent state, i.e., it is a guidingcenter coordinate and not the coordinate of a point electron.] The path integral can then be expressed as a sum over saddle-point contributions in which the contribution of paths in the neighborhood of each classical path is evaluated by expanding the action to quadratic order in R —Rc. In this way Z can be expressed in terms of a sum over classical paths (assuming them to be well separated in path space)

(2.11)

B. Saddle-point approximation Having formulated the problem in terms of a path integral, we proceed to evaluate it within the saddle-point (or semiclassical) approximation. For the class of paths which we consider, there is a single quantum parameter, which we will call g/8ir, which is simply proportional to the ratio of the extent of the electron wave packet due to its zero-point motion to the mean area per electron. Hence, g/87r is proportional to v and, as one would ex-

axj '

(2.13)

where Vs 2i
280 KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

1624

boundary conditions. I n this case one obtains an additional contribution to the classical action from the discontinuities at the endpoints. 27 However, for the specific paths which we will consider, the endpoint contributions to the action are negligible. (See Appendix C.) By making the coordinate change R = R —Rc we can express the fluctuation determinant D[RC] as a path integral of the same form as that in Eqs. (2.5)-(2.7) but with the potential V= 2 , < 7 Vy in Eq. (2.7) replaced by the time-dependent quadratic potential: J (XiKijXj + XiLij Yj + YtLtjXj + Y, Wit Y,) (2.14) where X,;(K C ) =

dXidXj

=RC
R

(2.15) VVu

ay, ay, From the relation of Eq. (2.5) to the phase-space path integral it is clear that D [R c ] is the partition function for a set of N coupled harmonic oscillators with timedependent coupling constants: D [ * f ] = T r r [ e x p f-

J"'drH(RC{T))

= ±2lP<

W

'J[R

C)P

i

+piLtAR">XJ

c

c

)Pj

Z=D0e~So2.D[Rc)e-S[R']

,

(2.17)

w h e r e / 5 = D / Z > o and S = S - S 0 C. Cooperative ring-exchange paths We now focus on those paths which are most likely to lead to structure in Ecf.v) because of their systematic dependence on v. That is, we consider paths, such as those illustrated in Fig. 1, which consist of a cyclic coherent superposition of nearest-neighbor exchanges from the initial static Wigner crystal configuration. Each such exchange event can be characterized by a directed path on the Wigner lattice, as shown in the figure, and by the time 0 < f < / 3 at which the event occurs. Paths involving second- or further-neighbor exchanges (see Fig. 2) have larger real parts of the action (not only because the electrons must tunnel larger distances but also because the electrons come closer together along the path and hence must tunnel through larger potential barriers) and are therefore neglected. Let us consider the contribution to the reduced partition function Z from a single large ring exchange involving L electrons. The classical action, from Eq. (2.12), is

s[/r]=~Y 2 /"*•*.;ARj+/*ra-i>

i.j

+ XiLlJ(R

rewrite the sum in Eq. (2.12) in a form that allows us to focus on the difference between a particular classical path and the static Wigner crystal

(2.16a)

where T is the time-ordering operator, H(Rc)

36

+XiKiJ(R )XJ]

(2.16b)

an6[Pj,Xj]=-ibjk. Let us begin to classify the classical paths. The path with the smallest action is a stationary path, the triangular Wigner crystal. This path and Gaussian fluctuations about it make the dominant contribution to the partition function D0e ° = exp[—0NEoM], where E0(v) is the energy per site of the static Wigner crystal, which is essentially the Hartree energy that has been computed by Maki and Zotos. 8 The contribution to this energy from pair exchange can also be calculated by using the discrete-time version of the path integral. E0(v) is a smooth, monotonic function of v for v < T . Maki and Zotos also noted that for v > v ^ ~ 0 45 the Wigner crystal is classically unstable since the shear modulus is negative. Thus we must confine ourselves to v < v + . (For v > 1 — v + , the same considerations apply for the hole lattice.) All other classical paths apparently have action larger than So', however, there are very many of them. We assume that, so as not to have prohibitively large action, the important classical paths resemble the Wigner crystal at most points in space, most of the time. We will return to this assumption later. In this case, it is convenient to

(2.18)

where R) satisfies the equations of motion, Eq. (2.13), with the potential K = £ , - < 7 [ K ( R , - R , ) - K ( R ? - R ? ) ] , and R° = R y ( r = 0 ) is the position of the jth electron in the initial static Wigner crystal configuration. It follows from the equations of motion that along the classical path the potential V is zero. The term iv(L — 1) in Eq. (2.18) reflects the Fermi statistics. If the electrons involved in the ring exchange are labeled from 1 to L, then the Rf satisfy the following boundary conditions:

FIG. 1. An example of a cooperative ring-exchange path.

281 COOPERATIVE RING EXCHANGE AND THE FRACTIONAL . . .

36

,dr . ) e

Z>[tf c ]=-(

1625 (2.22)

To

FIG. 2 A ring-exchange path involving further-neighbor exchanges. This six-particle exchange process is of a substantially smaller amplitude than that depicted in Fig. 1.

R?(j8) = Ri.(0) , RJ + l (0) = RJ(O>, ; = 1 , . . . , 2 , - 1

(2.19)

where A a is a real constant which acts to renormalize ao and ro is the characteristic tunneling time. As in all instanton calculations, by constraining the exchange to occur at time T, we break time-translation invariance. Therefore, in obtaining an expression for D[RC] from Eq. (2.16), the resulting zero mode is suppressed and instead an explicit integral over the flip time is included, as represented by the factor dr/r0. We have not explicitly calculated the prefactor, which is denoted by ( • • • ) in Eq. (2.22), for a general classical path. (The calculation of D for an especially simple path is discussed below in Sec. I I D . ) We can at best estimate the leading order contribution of large cooperative ring exchanges (L —<• oo ) to InZ (v). Corrections that are sublinear in L (i.e., there are lnZ. corrections from the prefactor) are too subtle to be included at the present level of approximation. However, the phase of the prefactor is important and must be correctly calculated. In Appendix A we argue that the fluctuation determinant is real find negative. The results of Eqs. (2.21) and (2.22) can be summarized

RJ(0) = RJ(O), j not on the ring . as T h e real part of the action is approximately proportional to L, since each electron tunnels through a similar barrier, and the imaginary part, from Eqs. (2.18) and (2.19),

dr

5[r]E-*'i=^cxP[ TO

-a(v)L±i2irfNA+0(\nL)]

, (2.23)

0=(2.20) 6=±2irt/0) + ir(L-l)

,

where R j ' is the real part of Kj, /^>0=NA /2v, A^ is the number of enclosed plaquettes of area ftv=ir/2v. In this case, the phase would be

0=±1T — - 1 v

NA -\-TT (mod2ir)

(2.2l)

where we have used the fact that, for the triangular lattice, A ^ is odd (even) when L is odd (even). Similarly, the real part of the action would be ao(v)L, where ao is independent of path. In fact, due to the presence of corners, the classical paths do not follow precisely straight-line segments, ao >s not independent of path, and for any particular single ring exchange there are corrections to the phase expressed in Eq. (2.21). We will argue that the net effect of such corrections is to renormalize ao, but first let us consider the fluctuation determinant. The important contributions to the reduced fluctuation determinant D[RC] come from those regions of space and time where Rc differs appreciably from a Wigner crystal. From Eq. (2.16), this implies that

where a = a 0 - f - A a and / = {[( 1/v)—1], It is apparent from Eq. (2.23) that when v _ 1 is an odd integer, the contribution from different ring exchanges will add in phase. Although the contribution from a given large ring exchange is exponentially small, the number of ring exchanges involving L electrons also grows exponentially. Thus when the tunneling coefficient a ( v ) is sufficiently small, arbitrarily large ring exchanges will contribute to the energy, and when their phases are such that the different contributions add coherently, then a particularly stable ground state is found. This phase relation is also preserved when some further-neighbor exchanges are considered, and we do not expect the inclusion of such exchanges (which we have neglected) to change the results qualitatively. One feature of the classical path which is worth noting is that the characteristic time r 0 over which the electrons are actually moving is determined by local interactions and so is roughly independent of the length of the exchange loop. Moreover, as is typical for tunneling paths, the motion is exponentially localized in time so that interactions between exchange events that are separated by a time interval greater than T 0 are negligible. In deriving Eq. (2.23) we have assumed that the classical paths were constructed from L straight-line segments between the lattice points so that the enclosed area A is an integral multiple of the plaquette area, A =N Ailv, and that the real part of the action is L times the real part of the action for motion along a single line segment, SQ(L,NA) — aoL±i2trfNA. In fact, the action associated with a single ring exchange takes this form only for paths with a very high degree of symmetry, of which the

282 KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

1626

straight-line path shown in Fig. 3(a) and considered in the next section, is a particularly simple example. A general path has many corners near which the electron deviates from straight-line motion. As noted previously, this im-

]

JdS'dA'

(e-S^D\Rc\)L,N=(--)e'

36

plies that the action is not simply So- We can express the effect of these deviations by considering the average contribution to Z from all exchange loops of fixed length L and fixed number of enclosed plaquettes N A :

-S':ti2
•P(S',A,\L,NA)

(2.24)

Here A = NAClv + A' is the enclosed area and P(S',A' \L,N A)dS'dA' is the probability that one of these rings chosen at random will have action S=S0(L,NA ) + S'±i2irfA'/£lv. We expect that in the limit of large L, S' and A' should be approximately independent Gaussian random variables with first moments bs and 5 ^ , and second moments As and AA, respectively. Therefore, in this limit, the effect of deviations from a straight-line-segment path can be expressed as a simple renormalization of the exponent in Eq. (2.23): aL-i2irfNA-+[a

+ &s/L

-As/2L

+

{2Trf)2AA/L]L±i2Trf(NA+SA}

We need to determine the L and A ^ dependence of 6 and A for this expression to be useful. If we assume that the dominant interactions which cause the path to deviate from a straight-line segment are determined by the local configuration of corners of the path, then P is approxi-

(o)

(2.25)

mately independent of NA; 8S, As, and AA are proportional to L, and 5 ^ vanishes like \/L for large L. (In fact, the discreteness of the lattice implies the presence of an order L° term in 6 ^ . We believe this term is an artifact of the crystalline lattice.) From Eq. (2.25), this implies that the net effect of corners on the contribution from large ring exchanges is to renormaiize a to a + a.\. To obtain an approximate upper bound on the magnitude of a\, we have compared the action of extreme case of a zigzag path, holding all nontunneling electrons fixed [see Fig. 3(b)] with that of the straight-line path discussed below. The resulting bound on tt| i s a ^ a . For the typical path in which the density of corners is much lower and relaxation effects are included, a.\ will be much smaller.

D. Calculation of a ( v )

(b)

FIG. 3. (a) A "straight-line exchange" path, (b) Zigzag path.

The numerical value of the tunneling coefficient a ( v ) determines whether cooperative ring exchanges contribute significantly to the partition function at a given density v. We estimate this coefficient for a particularly simple exchange path. Consider the path in which one row of electrons exchanges one step in the x direction, so XjiP)=X:(0) + av and y,(/?)= r / 0 ) , where Q V = ( 4 T T / V / 3 V ) / 2 is the lattice constant of the Wigner crystal. We impose periodic boundary conditions in the x direction, XJ{T)=XJ + L(T). Since this path encloses no area, the action can be chosen to be pure real. To make the calculation tractable, we assume that only the electrons in this one row move, in the background of the static potential of all the other electrons. T h u s we probably overestimate a(v). The classical action, and hence ao(v), can be readily calculated numerically (see Appendix D), but in order to calculate the fluctuation determinant and Aor(v) it is convenient to work with an analytic approximation for the actual potential in the action. We have checked numerically that for | Yj | « a v , the actual potential is well approximated by

283 COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

36 L

1627

Qx

V' = ea v

+

av

+ 2

-^-1[\-cos(2irXJ/av)] (2TT)

Kxik

-j)

Xt

-X,

+

Kyik~j)

(2.26)

For v = | , we found that the best fit to the actual potential was obtained with Qx = 0 . 2 6 0 and Qy = 2 . 4 4 , and that Q and Kij) are only weakly dependent on v. That Qx/Qy = 0 . 1 0 6 7 « 1 reflects the important fact that when the entire row moves coherently in the z direction, each electron is moving toward a site left vacant by its neighbor, and so the potential barrier is very much less than for motion in the y direction. With this form of the action, both the classical action and the fluctuation determinant can be evaluated explicitly. Since S is a quadratic form in Yj, the motion in the y direction can be integrated out exactly. This yield a new, effective action Selr for the x motion, with a quadratic kinetic energy T

= — fp c

dr

2

Jt

[±4>jMij-k)<j>k+2ifKxij- -k)i1-(t>kf}+ 2 a , ( l - C O S * ; ) J

where <j>j=2irXj/av, Mii -j) = the matrix inverse of

w'o-Me,+£*,<*>

i2ir)2a6vWi-j ' and WiJ -KM

-j)

(2.28)

while for a continuum theory with effective elastic constant K, the propagator is 60ik,a>) = [M'co2 + Kikav)2

+ Qx]-i

.

2

[Mii— j) is an exponentially decreasing function of | i -j | , and hence M* = 2 ; Mij) = i2ir)2al/Qy acts as an effective mass.] S e l r is the action for a one-dimensional sine-Gordon chain. The classical path satisfying the boundary conditions, 0,(0) = 0 and 4>ji0) — 2ir, corresponds to the simultaneous coherent motion of all the electrons, i.e., $jiT) = foir). Hence, S [ / ? c ] = a 0 (v)L, where a0iv) = iQx /Qy)W2 iS/V^irfr'1, is independent of K. Note that, as promised, a 0 (v) is rather small due to the softness of the potential in the x direction. The characteristic classical time over which the transition occurs can also be readily calculated and one finds T0M = alAQxQy)w2ie1/el)-i. To evaluate the fluctuation determinant D[RC] we approximate the discrete sine-Gordon chain in Eq. (2.27) by a field theory with a finite ultraviolet cutoff A ~ l / a v and then use known results for the sine-Gordon field theory. 30 Of course the effects of fluctuations with wave number k of order A are treated very crudely in this approximation. However, if the quantum parameter g /8TT (defined below) is small compared to 1, the effect of short-wavelength fluctuations is small, as reflected in the weak A dependence of the results. Usually, to take a continuum limit of Eq. (2.27), (<£,—<#*) is replaced by (j — k)avdx2+K0ikav)2[M\/kay)

(2.27)

+ ±] + Qx\-'

,

For Kk smaller than Qx, the propagator is approximately independent of k, while for large k the propagator is small. Thus, over the relevant range of k, the two propagators are approximately equal for K = * 0 [ T - i n i k e a v ) ] « \ K 0 [ M K 0 / Q X )+ 3 ] , /Qx)ln.

where kQav = iK analysis is

iL&

S"~±fffdTfLdT g

J

0

J

What

emerges

from

this

+ nto[l— cos0]

0

(2.29) where, by rescaling our units of time so that (}'=p/m0T0, we are left with a single dimensionless classical coupling constant mQ = iQx /ic)xna^x and a single quantum parameter g/%ir = {Qy/K)U2

(v/3/8)v«0.24v ,

(2.30)

where G / 8 i r = l marks the point at which the classical ground state becomes unstable. Note that for all relevant values of v, we have that g /Sir « 1. From Eq. (2.16), the effect of Gaussian fluctuations can be readily calculated and the results can be summarized as follows. (1) They produce quantum renormalizations of the bare parameter m0 which enters the action, which simply results in mo being replaced by m=m0(mo/2A)*/8"~/»j0(fit/»c)*/16\ Physically, this reflects the fact that because of the zero-point fluctuations of the field, the effective barrier height between the different minima of the potential is less than its classical value. (2) They shift the soliton creation energy, measured in units of the renormalized frequency m, to

284 KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

1628

Es/m = \—g/iir due to the temporary reduction of the zero-point energy of the field that occurs during the tunneling event (a dynamic Casimir effect). (3) The zero mode produces a prefactor of the form (To)~'dT{S)u2. Combining all these results, we obtain at last the expression Z ) e ~ i = ( T 0 ) ' V r e x p [ - a ( v ) i : +Oi\nL)]

,

(2.31)

where 1/2

a(v) = v ' V37T

Qy

' g/16tr

Q* 4«r

873-

(2.32) and g / 8 j r « 0 . 2 4 v is denned in Eq. (2.30). F o r v = g/Zir~0.0%« l, and a ( j ) = l . O . The bare value a0 1.48 at v = y, so the the quantum renormalization ~ 3 3 % at this density. The tunneling coefficient a ( v ) calculated in Eq. (2.32) is shown in Fig. 6.

{, is is is

III. MAPPING ONTO A DISCRETE MODEL A. Sum over paths Having obtained an approximate expression for the action due to a single isolated ring, we now turn to the task of enumerating all such exchange rings and summing their contributions to the partition function. If a ( v ) is large, so that the system is in the sparse-ring phase, this can be done by making use of the dilute-gas approximation:

\nZ~

2

2ML,NA)

( • • • )e '

Xcos(2TrfNA)(e~'ss)LiyA

(3.1)

where JV(L,NA ) is the number of polygons which can be drawn on a triangular lattice with circumference of length L and with N A enclosed plaquettes. The factor ( / 3 / T 0 ) ( • • • ) comes, as in Eq. (2.22), from the fluctuation determinant, the factor 2 cos(2irfNA), where / = 4-(v _ ' — l), results from the interference between paths in which the exchange occurs in the clockwise and counterclockwise directions, and the term (e~bs)LN is the average over all shape-dependent variations in the classical action for paths of fixed L and NA. For large L, N(L,N A )~!KL, where !H~A is the connectivity of the lattice. It is clear from Eq. (3.1) that the dilute-gas approximation is valid only if a » l n 5 ¥ . For a<\r&{, the entropy associated with large paths outweighs the "energy" a.L. However, in the dense-ring (small-a) phase the dilute-gas approximation breaks down, as signalled by the observation that the free energy computed from Eq. (3.1) is superextensive. For small a, the sum over the important classical paths becomes considerably more complicated. Firstly, the important paths consist of overlapping and intersecting rings of electrons exchanging essentially simultaneously so that

36

the action associated with such a path is not simply the sum of the actions associated with the constituent rings. We will refer to this as interactions between rings. Secondly, we can no longer restrict our attention to rings in which the electrons move only to nearest-neighbor sites; for small a , exchanges involving further neighbors also become important. We have not been able to develop a well-controlled approximation for summing the contributions of all these paths. We have, however, developed an approximate method for summing the contribution of an important subclass of these paths by relating the sum t o the partition function of a two-dimensional classical spin model. T h e equivalence to the spin model is asymptotically correct for large a, and we believe it captures the essential features of the dense-ring phase which occurs at small a. Since the mapping is not exact, we begin with a general discussion of the essential ingredients. (I) We consider the subclass of classical paths which can be constructed as a linear superposition of individual ring-exchange events. Each individual event is characterized by a set of directed segments along the nearest-neighbor bonds which comprise the ring and which specify the direction of the exchange motion, and by an imaginary time which specifies when the exchange event occurs. (2) The real part of the action increases approximately linearly with the net length of the exchange rings. (3) The imaginary part of the action is proportional to the directed area enclosed by the exchange paths, due to the Aharonov-Bohm effect. In addition, we associate an extra contribution to the imaginary part of the action, also proportional to the enclosed area, which incorporates the Fermi statistics. (4) T h e exchange motion occurs on a characteristic (imaginary) time scale of order r 0 . Exchange events that occur at times much greater than T 0 apart are essentially noninteracting. With these points in mind, we construct our spin model in three steps. First, we divide the time interval /? into time slices of approximate duration T 0 . All exchanges that occur in different time slices are assumed to be independent and noninteracting. Thus, Z ~ ( Z s i i c e ) T°, where Z s i ic e >s the sum over all exchanges that occur in a given time slice. This approximation is plausible since the exchange paths are exponentially localized in time. Within a given time slice, we associate with each exchange path a set of integer valued "spins" Sk defined on the (dual) lattice of plaquetts such that S* is equal to the number of clockwise exchange loops minus the number of counterclockwise exchange loops that encircle plaquette k (see Fig. 4). Thus the classical paths are the domain walls of the spin model. To each classical path there corresponds a unique spin configuration on the dual lattice. However, in the presence of overlapping or intersecting loops, the mapping is not one to one; more than one classical path may correspond to each spin configuration. For example, when two intersecting ring exchanges occur in the same time slice, we obtain spin configuration of the sort shown in Fig. 5(a). This spin configuration corresponds to at least two possible exchange paths, depending on whether the ring to the left or the ring to the right exchanges first. There are also paths which overlap on a line segment, as shown in Figs. 5(b) and 5(c). If the com-

285 COOPERATIVE RING EXCHANGE AND THE FRACTIONAL . . .

36

---,%

*

"•

^

^ZnosTre

0 / * 0 ' " 0 / \ 0 / >

; ; o\ . o\ ' o ' ; o > * - 0 / \ 0 / v 1 0 \ ' 0 v / 1 >/ 1

f

•

/ o 4— » 0 / v 0 / \ 0 / \ 0 \ ,' 0 \ ' 0 \ ,' 0 \

x—>----•---

.

' > 0 / - 0 / \ 0 'V1 /' 0 \ / 0 > ,' 0 \ ;< 0 \ / 1 » 0 / \ 0/\-l / \ 0 / \ 0 ' \ 0 \ / 0 \ / - l ^ / - l \ / 0 ^ / 0 \ ,' 0 \ ,' ' \ 0 / \ - l /' \ - 1 / ^ 0 / \ 0 /' \ 0 / v ' 0 \ .' 0 \ ,' -1 \ / 0 \ ,' 0 \ ,' 0 \ / 0 0 / \ 0 \ / 0 \ '' » 4-

-V-

• 0 / \ 0 / , - --*: , * 0- ' v 0 . / \ ' 0 \ / 0 > / 0 \ -' 0 \ /

-»'

.»

U-

*

1629 (3.3)

The mapping to the discrete Gaussian model makes the following assumptions concerning the action of various classical paths. (1) There are various interactions between paths which reflect the fact that multiple classical paths correspond to each spin configuration. (2) There is an effective interaction between loops which share one or more traversed segments in that any segment traversed n times carries with it an action n2a(v). (3) There are no interactions associated with corners or bends in the exchange path; the real part of the action is simply proportional to the length of the path. (4) There are no interactions between disjoint exchange

loops.

¥

FIG. 4. Mapping to a spin model: The integer spin variables assigned to a plaquette reflects the net ring-exchange circulation about that plaquette in a given time slice. mon segment is traversed in the same direction by both paths, as in Fig. 5(b), again there is a degeneracy in the mapping since either the path on the right or the path on the left can exchange first. However, the degeneracy is much more important in the case where the common segment is traversed in opposite directions, as in Fig. 5(c). The resulting spin configuration is the same as if the common segment were not traversed at all. This is not likely to be important since we expect the action associated with the path in which the segment is multiply traversed to be considerably larger than the action associated with the path in which the segment is not traversed at all. Indeed, we have found that if we suppress the contribution of paths in which any segment is multiply traversed during a given time step it does not change our results qualitatively (see below). We now have an approximate representation of the sum over classical paths in terms of a sum over classical spin configurations. The remaining step is to construct a spin HamiUonian such that the "energy" associated with a given spin configuration is equal to the action associated with the corresponding classical path. This can be done to arbitrary levels of complexity, but in light of the fact that the mapping onto the spin model is itself only approximate, we have contented ourselves with studying the simplest model which is consistent with the general features of the classical paths listed above. Thus, we associate with each spin configuration an energy #DG=a 2

(S/.-Sr)2 + 2m^lfkSk ,

(3.2)

where (k,y) denotes a nearest-neighbor pair on the dual lattice, / \ = o-} = ( v - ' - l ) / 2 , and x/0 = l v - 1 is the magnetic flux through plaquette A. in units of the Dirac flux quantum, HOG can D e recognized as the Hamiltonian of the discrete Gaussian (DG) model in an imaginary field.31 The sum over paths within a given time slice can thus be approximated by

FIG. 5. Unfaithful aspects of the mapping, (a) A spin configuration which corresponds to two intersecting exchange paths, (b) A spin configuration corresponding to two exchange paths which overlap additively on a single segment, (c) A spin configuration corresponding to exchange paths which overlap in a canceling sense on a single segment.

286 1630

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

Assumption (1) concerns the fact that the mapping is not one to one as has already been discussed. The most important example of this is that paths in which a given segment is traversed in opposite directions is not included at all in Eq. (3.3); it is as if there were an infinite action associated with such paths. On the other hand, if the segment is traversed more than once in the same direction, a finite action is associated with the path, as stated in assumption (2). In the absence of loop-loop interactions, t h e action associated with an flu m e s - t r a v e r s e d segment, such as the n=2 exchange shown in Fig. 5(b), would be na, whereas in the D G model it is n2a. We choose an interaction of the form (Sk— Sr )2 since the tunneling amplitude in a magnetic field is proportional t o the square of the distance trasversed. However, we note that the solid-on-solid model, 3 2 with an interaction of the form | S^ — S^ | , is believed to lie within the same universality class as the discrete Gaussian model. If we had represented the sum over paths with a solid-on-solid model we would have neglected interactions between overlapping loops (other than those included from mapping different exchange paths onto a single spin configuration). This fact suggests an insensitivity of the results of our analysis to the actual approximation used for loop-loop interactions. In fact, one could restrict neighboring spin variables t o differ by 0 and ± 1 only, thereby eliminating all configurations which correspond to loops that share segments, and still obtain similar results. This case is a special solid-on-solid model which has transition temperature very close to that for the D G model. 3 3 Assumption (3) says that for paths with only singly traversed segments, the real part of the action is assumed to be simply aL, independent of the shape of the exchange path, i.e., how many corners it has, or how many intersections it has (although some interaction effects due to intersections are included as discussed above.) These approximations could, in principle, be improved by including further-neighbor and multispin interactions in the Hamiltonian, i.e., an interaction which counted the number of corners or the number of intersections. However, we do not expect the inclusion of short-ranged spin interactions to change the nature of the model in any essential manner. Interactions between spatially separated segments of the exchange path are not zero [assumption (4)], but they appear to be sufficiently short ranged that they do not have any important qualitative effects. For instance, if we consider two simultaneously exchanging rings of N electrons, each separated by a distance X which is large compared to their radii, we can calculate the interaction energy using a multipole expansion. T h e lowest-order contribution comes from the / =N multipole + 1 and the interaction energy falls off as r0(r0/X)2N , which is small in comparison with the local line energy of the ring. When rings are in close proximity, where the multipole expansion converges poorly, the interaction effects are potentially important. Although we have no proof that such short-ranged interactions are not qualitatively important, we believe that such terms may also be introduced as additional short-ranged spin-spin interactions in the D G Hamiltonian which do not alter the na-

36

ture of the transition. We have discussed the steps which lead from the sum over classical paths t o the discrete Gaussian model, highlighting the approximations that arise at each step. While it is possible to refine the process by introducing additional interactions in the spin model, there are inherent errors associated with the time-slice approximation. Thus the spin model is not reliable on a quantitative level. The value of a which appears in the D G model, for example, should be interpreted as an effective a, which is only roughly related to the value of a appropriate for the motion of a single isolated ring which was calculated in the previous section. However, we also believe that the relation between the sum over classical paths and a spin model with similar long-distance behavior to the discrete Gaussian model is correct despite our inability to establish an exact correspondence. This belief is based on the fact that the discrete Gaussian model incorporates all the features of the sum over classical paths, listed at the beginning of this section, which we believe embody the essential physics of the problem. B, Equivalence with other models The discrete Gaussian Hamiltonian is related to other well-studied models. For example, application of the Poisson summation formula produces an exact equivalence to the spin-wave-vortex-gas system. In addition, the D G model may be mapped onto the planar (XY) model using a duality transformation and the Villain approximation. (In the presence of an imaginary field, t h e D G model maps onto the frustrated XY model.) All these models have received a great deal of attention, both analytical and numerical, and many existing results may be applied directly to our ring-exchange model. In Appendix E we review the equivalence between the D G model of Eq. (3.2), the frustrated XY model, which is described by the Hamiltonian « = T2a

2 [l-cos(fl,-fl,.-^)],

(3.4)



and the vortex-gas model HvG=-^^{mi-f,)Gij(mj-fi)

.

(3.5)

In Eq. (3.4) the dynamical variables are the angles (6,} and the sum is over all nearest-neighbor pairs < r , r ' ) on the triangular lattice JL. This model has been used to describe Josephson-junction arrays in a transverse magnetic field,34 where superconducting islands interact via a proximity coupling. The vector potential Arr' lives on the links of X and is constrained by the requirement that the integral § A-d\ around a plaquette yield the total magnetic flux through that plaquette, which we write as 2irfk in units of the elementary flux quanta: 2p]aqu«te^T' = 2'r/A. with A„=(2e/fie) §' A-rfl. In Eq. (3.5) the dynamical variables are the integer charges m, and the prime on the sum denotes that only neutral configurations contribute to the partition function, i.e., the constraint 2< ("»i— fii — O is imposed. The asymptotic form of the lattice Green's function for large 11 —y | is Sjj ~ In | i —j | . Note that in

287 36

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL . . .

these models, since local integral flux increments are "invisible" (i.e., the system is invariant under fk —*fk + 1), all thermodynamic properties are dependent only on the fractional part of fk. If the flux is uniform (i.e., fk =/" Vfc), an additional invariance, /—* —/, results.

T

1

1

1

1

1

1

[

T

1

1631

1

1

[

F

1

1

1

[

1

1

1

1

-

a{v)

: (a)

C. Critical couplings and commensuration In our model, the coupling a and the flux per plaquette f are not independent. Rather, they both depend on the dimensionless density v, with / ( v ) = (l + v ) / 2 v and a(v) derived earlier. In investigating our one-parameter phase diagram, however, we shall find it useful to consider a and / as independent parameters. There have been several approaches to the statistical mechanics of models such as those defined in Eqs. (3.4) and (3.5), including mean-field theory, 35 Monte Carlo, 3 4 , 3 6 , 3 7 and ansatz ground states. 38 All these studies indicate that the ordered-state properties are very sensitive to the value of/. In general, the critical coupling ac is an increasing function of the "degree of rationality" of / . In other words, writing 3i(f)=p/q, with p and q relatively prime, one finds that ac is largest when the denominator q is smallest. The value of / which leads to the greatest a-c is f=0, i.e., v = l / ( 2 y ' + l). Since for / = 0 all rings add in phase it follows that, independent of / , the 2D electron gas (2DEG) is in the sparse ring phase whenever a(v) is larger than ac a t / = 0 . Since ac is a nonuniversal quantity, one can only obtain an estimate of its value from calculations for the various spin models considered above. On the square lattice, for example, estimates of the transition temperatures give a range of ac from 0.45 to 0.87, where the lower values come from the XY model 39 and the higher values come from the D G or solid-on-solid models. 33 The values of ac are larger for the lattices of interest (namely, the honeycomb lattice for the D G model and the triangular lattice for the XY model) and range between 0.7 and I.3. 40 These differences depend on the details of the models (i.e., the short-distance behavior). Since the mapping to the spin model is not exact, there is no compelling reason to chose any particular one of these values of ac. However, favoring (somewhat arbitrarily) the DG model, since this was our starting point, we use the estimate of ac ~ 1.1 at f=0 for illustrative purposes. A comparison of our calculated function a ( v ) and the corresponding critical coupling a c ( v ) is shown in Fig. 6. At low densities a>ac, and the system is in the sparsering phase. (In the language of the vortex gas, this is the screening phase.) As the density is increased, the curve a(v) dips below ac(v), provided t h a t / ( v ) attains a simple rational value, and a dense-ring phase results. Though the function a c (v) exhibits a local cusps at fillings other than Vj = l/(2j + 1), these higher-order commensurations are probably irrelevant to the fractional quantized Hall effect, as the figure suggests. Using the approximations for a and ac discussed above, the only lowest-order commensurate Hall state that our theory allows is the T state. Hierarchical generalizations of the cooperative ringexchange theory will be discussed in the following section.

)*•••• ) U

1 I

• L__J

L_J

I

I

I

1 L

V OM

1 1 T J I

• ' ' 1 •• I'l-pi-

: (b) 0X6

-

•;

-

{ :

i>.03

^ C M

-

: -j

0.01

' 0.1 '

0.8

' oi' '

0.4

-"-»

FIG. 6. (a) A comparison of the calculated values of a(v) to the estimated critical coupling a c (vMsee discussion in text). The Maki-Zotos value, aMZ(v) = fr/(vv / 3) is much larger than a(v). For v—j, aMz = 5.2. (b) Unrenormalized value of the inverse tunneling time TO ' vs filling factor fraction v.

D. Deviations from perfect commensuration The most straightforward way in which a state might change its density but still retain energy-favoring correlations is via uniform dilation. A central feature of the Laughlin theory is the incompressibility of the ground state, which disallows any such infinite-wavelength charged excitations. This is in great contrast to the normal concept of a crystal, which can dilate, thereby changing its density at very little energy cost. A uniform dilation in our model is described by changing the filling fraction v. It is then natural to ask how the free energy of the system depends on the deviation 8v, with v — vj + 8 v . For ov small, we can take / = — Sv/2vj, and the allowed vortex strengths will be integers minus / ( ! / I « D- Without loss of generality, we consider the c a s e / > 0 , i.e., uniform expansion. The chemical potential for a single vortex of (integer) strength q is Hq—iP'q /la, which strongly discourages excess vorticity. The system also must satisfy the constraint of overall charge neutrality. These conditions require that for small a, the ground state consists of a superlattice of vortices of

288 1632

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

strength 1 — / atop a uniform q — — / state, the superlattice constant being determined by the neutrality constraint. 38 This leads to an energy per plaquette of < S ~ ( i r / 8 a ) / l n ( l / / ) . Allowing for arbitrary sign o f / , one finds

«—-£l/|ln|/|.

8aevG

l/|ln|/|

the small-quasiparticle limit. The excess energy of this configuration is then

£, = -

2

(3.7)

where £VG is the dielectric function of the vortex gas. The physical origin of the cusp in 7 is clear. At slightly incommensurate densities, different rings within the same time slice lose their phase coherence. This decreases the total partition sum, thus increasing the free energy. In the sparse-ring phase, the increase 5 7 is analytic. However, in the dense-ring phase, the exchange condensate induces a rigidity with respect to uniform dilations, as reflected by the cusp in Eq. (3.7). IV. ELEMENTARY EXCITATIONS

2a T0

1

a T0

The fact that d 6/dv is negative for v near vy leads us to consider an inhomogeneous ground state at v, + 5v in which local distortions (analogous to Laughin's quasiparticles) account for the deficit or surplus density. In the dense exchange regime, rings which enclose the distortion will acquire an additional Aharonov-Bohm phase Ad=2-rrB8A /4>O relative to those rings that do not. Thus, the creation energy of such a defect would be logarithmically infinite unless the Aharonov-Bohm phase is an integer multiple of 2TT, that is 6 / = 6 / 4 /2irl2 is an integer, or equivalently, the quasiparticle charge is quantized in units of e * = v e . The quasiparticle creation energy is a sum of a term due to the real (1 / » Coulomb force, which is a decreasing function of the quasiparticle radius, / ? q p , and a ring-exchange term which is an increasing function of/?qp. To calculate the equilibrium quasiparticle radius and energy, ^? q p and Eqp, we consider a trial quasiparticle state in which the area distortion 6^4 is equally shared by a number of plaquettes comprising a roughly circular region in the Wigner lattice. We assume that the quasiparticle is a charge v disturbance atop a primitive v—\/m ground state, in which case the ground-state configuration of the associated vortex-gas system consists of a single vortex f7io = l surrounded by N =vR2/Hv "expanded" plaquettes of area f l v + S f i and vorticity 6 / = l//V. We treat the quasiparticle as a two-dimensional one component plasma, in which the vortex corresponds to an electron, and the expanded plaquettes to the neutralizing background. This assumes that the quasiparticle extends over a reasonably large number of plaquettes, although the conclusions are nonetheless qualitatively valid even in

2

SfjGjk&fk

_ir__

In

ring-exchange

(4.

where we have used the long-distance behavior of the honeycomb-lattice Green's function, lim TrG,y = V 3 \n(Ru/av)

+ ir ,

(4.2)

derived in Appendix F . The approximate Coulomb energy of the distortion is given by Er =

:

(4.3)

eR

where J is a constant, given by (4.4)

2ir

which can be modified to properly account for latticecorrugation effects. Balancing these two energies by setting d{Er+Ec )/dR=0, we find

A. Quasiparticles 1

1-Gokbfk

ar0

(3.6)

and the ground-state energy per plaquette has a cusp at f=0. Of course, what we really need is the free energy per plaquette, J at temperature kBT=l. By considering thermal excitations in the vortex gas, we obtain J7«

36

,2

Rqp = —= ^

V2TT

-(v2ar0)/ ,

el

(4.5)

In

IT

As a function of v, .R qp (v) is monotonically decreasing, which reflects one's intuition that a dense exchange gas prefers a small quasiparticle radius, the logarithmic ringexchange energy overwhelming the lattice Coulomb energy in this limit. For large quasiparticles, the effective potential, o ( r ) = — vG(r)/aT0, should be screened by the vortex-gas dielectric function, e.g., u ( r ) — * u ( r ) / e ( r ) . As one enters into the sparse-ring regime, virtual vortexantivortex excitations completely screen out the strong logarithmic interaction, rendering the potential effectively short ranged. It is known that the large distance limit of e(r) has an infinite discontinuity at the KosterlitzThouless transition point, and that e _ l ( r — o o ) is zero in the sparse-ring regime. 42 Our theory therefore predicts an abrupt vanishing of the charged excitation gap at the transition point. [To be precise, the gap vanishes abruptly as a ( v ) increase above ac with f = {v~'— l ) / 2 held fixed. This may hold some relation to the apparent first-order transition seen by Haldane and Rezayi 1 5 when the pseudopotential components of the interaction potential are varied at fixed filling.] B. Collective modes While Laughlin's ground-state and fractionally charged elementary excitations provided the first microscopic picture of the fractional quantized Hall effect, a detailed understanding of the neutral collective m o d e was not avail-

289 1633

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL . .

36

able until the work of Girvin, MacDonald, and Platzman (GMP),' 8 who generalized the Bijl-Feynman "single-mode approximation," 4 3 originally used in deducing the phonon-roton curve for 4 H e . G M P found that a liquid ground state, such as Laughlin's, would evidence a k —»0 gap in the collective mode spectrum as a generic occurrence. Their magnetophonon-magnetoroton spectrum is also in excellent agreement with the numerical results of Haldane and Rezayi. 1 5 , 1 6 Our state, on the other hand, seems to share a certain kinship with the Wigner crystal, which has gapless magnetophonon excitations and a long-wavelength dispersion resembling cok~kln. These two pictures appear to be quite different, and a detailed investigation of the collective mode within the ring exchange theory is therefore required. Unfortunately, we have made only limited progress in this direction. T h e results are somewhat heartening, but there remain several major unanswered questions. One possible scheme for investigating this issue is to adopt a lattice viewpoint and then work out the renormalized phonon frequencies in the presence of ring exchange. The basic idea is this: long-wavelength phonons oscillate on time scales a> ~~' » T 0 , SO by inverse adiabatically treating the coupled ring-phonon problem, one can trace out over the fast degrees of freedom (the ring exchanges) and arrive at an effective action for the phonons. The simplest such ring-phonon interaction would account for the phase variation incurred by rings which enclose deformed plaquettes. The ring exchange thus couples to the surplus a n d / o r deficit magnetic flux: tf,.ph = 2 m 2 S ;

f+j

(4.6)

with

/y=/-f6/y = | ( v - ' - l ) +

bAj

(4.9)

2W 2

(We have ignored the fj independent term arising from the spin-wave component of the free energy.) One can now expand Eq. (4.8) in terms of the correlation functions of the vortex gas and obtain Hcir— — T 0

Gj,k + — 2 GjiQirGr 2a 2 8/7

8/t

;.*

-0(S/ 3 ) (4.10) Qjk = < (rrtj —/Mm* -f)

>6/=o

Consider for the moment deviations about the odd denominator states, for which / can be taken to be zero. In this case, the vorticity-vorticity correlation -17/& in the dense ring < m ( R ) m ( 0 ) > behaves as -R phase, where a is the bare coupling a renormalized by fluctuations of the vortex gas. 44 F o r our purposes it will be enough to take a~a, in which case the Fourier space version of Eq. (4.10) becomes

Ht!f = rol—

2 8 / ( k ) §(k)-

27T2

S(k)G
x 8 / ( - k ) + 0(5/3) , Q(k)^-Qok{2-v/a)

(4.ll)

with S(k) the k-space lattice Green's function discussed where 8j=BbAj — obAj/2irl2, and BAj is the change earlier. Provided a < i r / 4 , which is the Kosterlitzin area of plaquette j in the presence of the phonon field. Thouless criterion for being in the insulating phase of the We neglect corrections to the real part of the ring action vortex gas, 42 the long-wavelength density fluctuations indue to the electron displacements, as we suspect that such teract via a k"2 coupling, and the plaquette distortions contributions will be both small and smooth. We have couple by a logarithmic interaction in real space. In the also ignored terms proportional to the rate of change of a > i r / 4 region, the real-space interaction is short ranged. &,(&i) although they may be important even in the adi-(When a is large, it is more convenient to expand the abatic limit. We think Hr.vh in Eq. (4.6) is the dominant effective Hamiltonian in terms of the correlation functions term, since it is the phase commensuration, reflected by of the discrete Gaussian model.) the — | 5 / ) In | 8 / | cusp in the free energy of the vortex The consequences of these results are most easily gas, that should discourage long-wavelength density flucworked out within the framework of a continuum elastic tuations and thus raise the frequency of the longitudinal theory (a detailed calculation is given in Appendix F) of a branch of the phonon spectrum. (Actually, in the prescharged membrane in a magnetic field. T h e Lagrangian ence of a strong magnetic field, the low-lying collective then contains four terms: kinetic, electromagnetic, local mode is a longitudinal-transverse hybrid. This will prove elastic, and "long ranged." In the absence of a field, the to be of considerable importance below.) longitudinal and transverse modes have long-wavelength The effective phonon Hamiltonian is obtained from dispersions of the form r

(4.7)

= Tre rings

a>2(k) = C2k2 ,

Using the equivalence of the discrete Gaussian and spinwave-vortex-gas models, we obtain Hcl

-TO 'in T r exp \mk I

a>2(k) = [u(k) + C , ] * 2

2a

^(mj-fj)GJk{mk-fk) J,k

(4.8)

(4.12)

where u(k) is the Fourier component of the long-ranged density-density interaction, and C\ and C j are elastic constants. As alluded to above, the magnetic field strongly mixes these branches, and in the high-field limit one obtains a cyclotron mode with co + (k = 0) = a>c [assuming that lini|t_o' c 4 l | (k) = 0], a " d a low-lying "magnetopho-

290 1634

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

non" branch with
(4.13)

In a two-dimensional system with \/r interactions, v{k) = 2rre2/k, whence tu;(k)<xk l/2 and &>,(k)<xA:. This leads to the well-known result for the conventional 2D Wigner crystal 8 ' 45 of co_(V)ock V 2 . In our model, the presence of a dense exchange gas leads to a logarithmic contribution to u(r), which raises the longitudinal branch, giving o>j(k)ocfc°. The transverse branch is of course unaffected by such an interaction, so the hybrid mode co~ acquires a linear dispersion and only partially reflects the stiffness induced by the logarithmic interaction. The existence of a k=0 gap in the magnetophonon spectrum, as predicted by Girvin et a/., 18 is so well confirmed by numerical simulations that one must conclude that an acoustic spectrum is the result of some faulty reasoning. We feel that there is no essential problem with the physics behind the ring-phonon Hamiltonian of Eq. (4.6), nor with the adiabatic approximation of Eq. (4.7). However, the expansion of H^s in terms of the vortex-gas correlation functions is really a perturbation expansion in the 8/}. The effective Hamiltonian Has[ j 6/}) ] >s manifestly invariant under the local gauge transformation 8/}—•8/}-t-l, and this symmetry is clearly lost if one works to any finite order in perturbation theory. One expects that the vortex gas will respond (over any length scale) to an area deformation of magnitude larger than 2rr/2 by nucleating a vortex charge, and this physics is obscured in the present treatment. The fully nonlinear model is extremely complicated, and we have not yet succeeded in determining what long-wavelength spectrum it reproduces. C. Hierarchy schemes The fractionally charged Coulomb gas has especially stable ground states at all rational values o f / , as can be seen from Fig. 6, with the state at integer / being particularly stable This leads to the possibility of stable incompressible ground states in our original system (the 2DEG in a magnetic field) at various rational densities, with v _ 1 odd being the most likely candidates for special stability. In fact, from our estimation of the tunneling coefficient a(v), it appears that v = y may be the only stable incompressible state. However, the experimental observation of the FQHE at other rational densities 46 leads one to consider other possible CRE-stabilized ground states within our formalism. In this section we will consider two possible scenarios for stabilizing higher commensurability states. [Density v = n/(2m + n ) corresponds to f = m/n or commensurability n.] The first is to consider as our starting configuration a distorted version of the original triangular Wigner crystal (WC) with unequal size plaquettes and hence a larger unit cell. The second is a hierarchy scheme, analogous to that considered previously on the basis of Laughlin's formalism, which constructs the higher-order commensurate states as condensates of a dense gas of the quasiparticles discussed in Sec. I V A . Let us begin by considering the first

36

scenario. The reason the state with integer f = m is so stable is that it is wholly unfrustrated; the ground state has zero vorticity in each plaquette. This reflects that fact that for v = ( 2 m + l ) _ l all rings add in phase. At all other densities the system is frustrated; each plaquette must have nonzero vorticity. For rational f = m/n, supercells can be defined consisting of n plaquettes which have zero net vorticity. This reflects the fact that certain classes of rings add in phase. However, the resulting stabilization energy is a subtle interference effect which would be small even for arbitrarily small a. By distorting the lattice the frustration can be relieved so that all the rings add in phase, even at densities other than v=(2m + 1 ) _ 1 , but at a cost in elastic (Coulomb) energy. We have discussed this competition in the context of the quasiparticle, and similar considerations hold here. If we consider the limit where the exchange energy dominates, then at any density other than v = (2m + 1 ) ~ \ the system distorts to form a mixture of large and small plaquettes, each with a half-integer flux penetrating it, so that the system is again unfrustrated. Even if we do not consider this extreme limit, it is clear that for n > 1, the underlying triangular lattice will distort, since by increasing the size of the WC unit cell the system can reduce its frustration and hence lower its energy significantly. This effect is more important for larger n. Let us consider the effect of lattice distortion on the stability of the high-order commensurate phases. Consider, for example, large n and fj —f«1. Then the ground state of the vortex gas consists of superlattice of vortices of strength 1 — / atop a uniform q = — / state, as described in Sec. III. If the lattice is distorted so that the area of each positively charged (1 —f) plaquette is reduced by an amount 8ft [fj = / - ( l / 2 v ) ( 6 f t / f t v ) ] , then the charge on the positive sites will be reduced to 1 — / — l / 2 v ( 6 f t / f t v ) . Since we consider only total-area preserving distortions, the charge on the negative sites is similarly reduced. Thus there is a reduction in the energy of the vortex gas (exchange energy) which is linear in 8ft. Of course, since the perfect triangular lattice minimizes the electrostatic energy of the system, there is an increase in the direct energy, but this is quadratic in 8ft. For general n, if 8ft is the magnitude of the lattice distortion, we expect for small 8ft that the total energy is of the form

Ef asij j - + -

-FJn)

/

a(v)eVGv

6ft ftv

ln(n)

(4.14)

where Ej- is the energy of the undistorted lattice, F\ and Fz are constants of order 1 which depend on the lattice pattern, and the factor ln(n) is from the mean separation between positively charged sites. This energy is minimized by nonzero 8ft. The magnitude of 8ft tends to be larger for larger n, where the Coulomb gas energy is larger, and for smaller v, where the electrostatic energy is

291 36

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

relatively small. The dependence of the constants / } on the crystal structure (n) is unimportant for large n. However, for small n, which are the commensurabilities of interest, the vortex-gas ground states in general are not known, and in addition the ground-state configurations may not be compatible with the lattice distortions described above. For example, for n = 2, we can find no pattern of lattice distortion which does not result in rather long-range electric fields For instance, one can form a striped phase, in which rows of plaquettes are alternately large and small. This pattern results in a permanent dipole moment in each unit cell, and hence is relatively less favorable energetically. Thus it appears that lattice distortion does not help stabilize the n = 1 state very much. By contrast, for n — 3 one can find a lattice pattern which is greatly stabilized by distortions. For example, by breaking the lattice up into hexagons which consist of six plaquettes, the energy can be reduced by contracting one third of the hexagons and expanding the other two thirds. However, it is not known whether this configuration actually corresponds to the vortex-gas ground state for n = 3. Note also that by minimizing Ej in Eq. (4.14) with respect to 8ft, we find a value of 8 f l ~ f t v . From this we conclude (1) lattice distortion can be a very important source of stabilization of a phase, and (2) the small amplitude analysis of Eq. (4.14) is inadequate for treating the effect; energies must be computed for the distorted lattice itself. The details of this discussion are based on the notion of an underlying lattice. As is the case with our quasiparticles, we feel that the competition between local electrostatic and ring-exchange energies is a mechanism which is not sensitive to any underlying crystalline order. Since the v=y case is so much more stable (at least, in the absence of lattice distortions discussed above) than any other density, it is appealing to consider a hierarchy scheme analogous to that proposed by Haldane, 10 Halperin,14 and Laughlin 20 in connection with Laughlin's ground states. In this scheme, the density is changed from v=\/m by adding fractionally charged quasiparticles. When the density of quasiparticles is sufficient, one finds a new stable incompressible ground state in which the quasiparticles themselves have condensed into a Laughlin-type state. The generalization of this scheme to the cooperative ring-exchange formalism is straightforward. The density is changed from v=\/m by adding quasiparticles and the contribution to the energy from cooperative ring exchanges of quasiparticles is calculated. The analysis proceeds as for the electrons, but with a renormalized tunneling coefficient a' and with fractional statistics, 13,14 appropriate for the fractionally charged quasiparticles. As in the usual hierarchy scheme, the quasiparticles are assumed to behave as particles with fractional charge, interacting via the Coulomb potential. In this case, a' is given by Eq. (2.32) but with v—<-v* = m2(v— 1/m), where v* is the density of quasiparticles per "quasiflux quantum" $J =hc/Q" =m0. As before, there are three contributions to the phase associated with a single cooperative ring exchange; these are the statistics phase, the Aharonov-Bohm phase, and the fluctuation determinant phase, which we will denote by 0S, 6A, and 8D, respectively. The statistics phase is

er.

1635

J_ m u <j)

:-^(£-l)±^JVq]

(4.15)

where A<6,; is the change of the azimuthal angle of particle i relative to particle _/', L is the number of quasiparticles on the ring, Nw is the number of quasiparticles enclosed by the ring and the ± refers to the direction of the exchange path. For the triangular lattice, this phase can be written as m

(4.16)

m

where N4 is the number of enclosed plaquettes. Aharonov-Bohm phase is 6><=±-

ITNA

The

(4.17)

Therefore the contribution to the partition function from a single cooperative ring exchange involving L quasiparticles of charge q * = e /m is dr

exp

-a'(v)L ±iirN

TO

A

1 |Av*

m

±i—+i0D m

(4.18) The requirement that rings of all sizes should add in phase produces an expression for the densities of possible incompressible states:

v.

±—=2n, m

n=±l,±2,.

(4.19)

For example, from the v = j state, one obtains the v=\,\, . . . states. There is, as before, a phase factor per ring of ±ir/m +0D. In the case of fermions, we argued that OD=IT, and hence this factor is trivial. W e do not have an analogous argument for particles with fractional statistics, although a statistics dependent phase 6D is not ruled out. In fact, it has been suggested 47 that in the dense ring phase, a phase factor per loop simply renormalizes (increases) the effective a in the D G model. We comment that it is possible that these two "hierarchy" schemes which we have discussed are equivalent. The distorted lattice in the first scheme looks like a lattice at some other density with quasiparticles on some fraction of the sites, which is precisely the second scheme. The difference between these two schemes though is that the first is an analysis done completely within the electron formalism and concentrates on static distortions of the lattice to relieve the frustration, while the second involves treating the quasiparticles as fundamental particles and looks at dynamic distortions (quasiparticle motions). Looking into the possible equivalence of these two

292 1636

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

schemes could shed some light on why the usual hierarchy scheme appears to be valid even though one is pushing the quasiparticle concept well beyond its usual range of validity. V. CONCLUSIONS We have formulated a theory of ring-exchange condensation in an interacting two-dimensional electron gas at high magnetic field, which describes a transition (at zero temperature) from a low-density, sparse-ring phase to a higher-density ring condensate at a critical value of the Landau-level filling fraction, v c = | . This phase transition is precipitated by a cooperative ring exchange (CRE) instability. A C R E event is one in which a set of electrons in the lattice, forming a ring, executes a cooperative motion, the result of which is a cyclic permutation of that set. Expressing the partition function as a sum over C R E events, we find that the individual terms will in general interfere randomly with one another unless the effective magnetic flux per plaquette, / ( v ) = ( v _ l — 1 ) / 2 , is a simple rational number. This effective flux accounts for both the Aharonov-Bohm phase incurred during a C R E motion and for the permutation signature that arises from Fermi statistics. The rings will contribute to the energy in a singular matter if the tunneling amplitude e ~alv) is sufficiently large, and if the flux / ( v ) is a simple rational value, which allows certain classes of rings to interfere constructively. We find that this exchange condensation, which produces an incompressible ground state and a cusp in the ground-state energy as a function of v, is likely to occur in the vicinity of v = | , and possibly near other odd denominator rational filling fractions. The C R E tunneling parameters are evaluated using a coherent-state path-integral formalism, which restricts the Hilbert space to the lowest Landau level. Unfortunately, there are several technical issues which cloud our theory at present. In particular, we do not yet properly understand the phase of the fluctuation prefactors emerging from the path integral. It is also unclear whether the spin models which we use for our analysis and which only approximately treat interactions between C R E events, retain enough validity in the dense-ring phase, where there is a substantial overlap of rings at any given time. These issues deserve more attention. The C R E theory also implies the existence of fractionally charged quasiparticle excitations, with e* = ± v , e and where v, denotes one of the preferred rational filling fractions. The stability of these quasiparticles vanishes discontinuously as one proceeds into the sparse-ring phase. In addition, we have derived a Hamiltonian, Heg, which describes the coupling of ring-exchange processes and Wigner lattice magnetophonons. A naive expansion of Hclr to lowest order leads to a renormalized magnetophonon spectrum which obeys an acoustic dispersion cot ~ k, at long wavelengths; the unrenormalized Wigner lattice magnetophonons satisfy ca\i~kin. There is good reason to question the validity of the naive expansion, and indeed 18 previous theoretical and numerical results find a gap for the long-wavelength collective mode. We are currently attempting to treat collective excitations without linear-

36

izing the effective potential as in our present approach to these modes. The cooperative ring-exchange approach emphasizes the importance of correlation between large numbers of electrons over large distances, as in critical phenomena. This view is in contrast with the numerical work based on Laughlin's wave function which emphasizes the importance of electron pair correlations at short distance. Nevertheless, the results of these apparently contradictory theories are quite similar. For example, both schemes lead to the incompressibility of the system at fractional densities v , = « / m with m and odd integer, fractionally charged quasiparticles, and a gap 2A in the quasiparticle spectrum. An important question is whether these theories describe the same underlying physics in different languages. While the Laughlin state ^o = I I ( z / - z y ) m e x p [ - 2

!** | 2 / 4 ]

(5.1)


features the pair correlation factor (z, —Zj ) m , the product over i and / leads to correlations between chains or rings of particles; e.g., (z\—z1)m(z-i—z-h)m(z-!,—z\)m describes a three-particle ring correlation. Thus the peculiar properties of the 2D electron gas in a strong magnetic field may ultimately arise as described by the Laughlin state from the existence of such large rings which are implicitly contained in this function. The sharpness of the quantization of the magnetic densities v, = « /m suggests that large distance scales are involved, otherwise finite size fluctuation effects would appear to spoil this precise quantization. Unfortunately, we do not have a direct mathematical link between the Laughlin and C R E approaches, although we believe such a link may well exist. Recently Lee, Baskaran, and Kivelson 4 8 have demonstrated that the long-wavelength properties of Laughlin's ground state can be derived from a generalized C R E approach. If in fact these two theories describe the same or closely related physics in two very different languages, does the C R E scheme give anything new beyond the Laughlin scheme? We believe the answer to this question is yes in that the C R E scheme is constructed to treat a continuous range of densities rather than only v=n/m and its immediate vicinity. Thus, the C R E predicts the energy varies as | &V | In j 6v | for a spatially uniform deviation of v from a magnetic value n /m. Since the chemical potential n = dE/dv diverges for such uniform deviations, it follows that the extra charge can be accommodated at lower energy by clumping it up in quasiparticles of charge Q. The quantization of Q, for example Q = ± v , e near density v, = 1 /m, is a simple consequence of the condition that the energy of a quasiparticle be finite; other values of Q lead to divergent energies. The emphasis on long-range correlations also suggests that finite-size effects might be observed in the F Q H E , as in critical phenomena. Whether periodic boundary conditions in model calculations eliminate such effects remains to be determined. In addition, certain calculations which emphasize long-range correlations may be easier to perform in the C R E language, as is the case using the renormalization-group techniques in critical-point problems. This raises the

293 36

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL . . .

question of whether there exists an effective-field theory which properly handles the C R E effects for longwavelength properties. Indeed Girvin and MacDonald 4 9 have recently proposed a possible order parameter for the F Q H E . Their work which is based on Laughlin's wave function, suggests a topological order parameter analogous to that of the two-dimensional XY model and, hence, further suggests a connection to the C R E theory. While the above discussion of the C R E approach is developed in the context of an underlying Wigner lattice, fluctuation effects arising from C R E processes or other tunneling processes have a tendency to reduce the strength of crystalline order. It seems likely that for densities v larger than the critical density vc for C R E to be effective, the lattice melts and forms a strongly correlated liquid. Whether or not this is the case, we believe that the C R E mechanism is operative and in no way depends on a charge-density wave ground state for its existence. ACKNOWLEDGMENTS We gratefully acknowledge useful conversations with G. Baskaran, A. J. Berlinsky, S. Chakravarty, F. D . M. Haldane, J. K. Jain, D. H. Lee, P. B. Littlewood, A. Luther, L. S. Schulman, D. Stroud, and D . J. Thouless. This work was supported in part by the National Science Foundation (NSF) through Grant Nos. D M R 82-16285 and D M R 83-18051. One of us (S.K.) acknowledges the hospitality of the Institute for Theoretical Physics, University of California, Santa Barbara (ITP-UCSB) where much of this work was carried out, and two of us (S.K. and J.R.S.) acknowledge the hospitality of Nordisk Institut for Teoretisk Atomfysik (NORDITA) where further work was completed. One of us (D.P.A.) acknowledges support by A T & T Bell Laboratories. Another of us (C.K.) was partly supported by the Canadian Natural Sciences and Engineering Research Council. One of us (S.K.) was partially supported by the Alfred P. Sloan Foundation. APPENDIX A: THE PHASE OF THE PREFACTOR In this appendix, we discuss the phase of the fluctuation determinant. It follows from the analysis of Sec. I I C that in the dilute loop gas phase, ring exchanges lower the energy of the system at densities v„ = (1m + 1) ~' only if the fluctuation determinant for a single ring exchange is negative. If the determinant were positive, in the dilute regime, ring exchanges would increase the energy of the system at densities vm. In the dense-ring phase, it has been argued that independent of the sign of the fluctuation determinant, ring exchanges always lower the energy at such densities. 47 However, the critical value of a is smaller if the determinant is positive, and thus the very existence of a dense-ring phase is called into question. It is therefore necessary to determine the phase of the fluctuation determinant for a single ring exchange. In the absence of a magnetic field the fermion ground state for a given real JV-body Hamiltonian will lie at a higher energy than the corresponding unsymmetrized ground state. Furthermore, the sign of the contribution to the energy from a ring exchange involving L fermions

1637

can be shown to be positive for L even and negative for L odd. 5 0 This is a direct consequence of the well-known argument which shows that the (unrestricted) ground-state wave function has no nodes. 4 3 This argument, in turn, depends crucially on the kinetic piece of the Hamiltonian. In essence, any wave function which vanishes can lower its energy by reversing its sign in various regions (and rounding out the resulting cusp) so that it is always positive. By eliminating the nodes of the wave function, the kinetic energy is reduced. Although the potential energy is usually increased by this procedure (i.e., for repulsive interparticle interactions which fall off monotonically with distance, by eliminating modes at r,=rj, which are introduced through the antisymmetrization of the wave function, the potential energy is always increased), the change in the kinetic energy dominates. In the presence of a magnetic field, the Hamiltonian is no longer real, and the usual analysis of the sign of exchange terms cannot be applied. A strong magnetic field has two effects on the sign: it introduces Aharonov-Bohm phases as the particles move around and it quantizes the kinetic energy, or, in the strong-field limit which we are considering, it quenches the kinetic energy. The first effect will obviously enter in the phase associated with a ring exchange. This phase is explicitly included in the action of the classical path and, at densities v m = ( 2 m + 1 ) ~ ' , where there are an odd number of flux quanta per plaquette, it contributes a factor of ( — 1 )L for a ring exchange involving L electrons. T h e form of the kinetic energy also has an important effect on the phase. The zero magnetic-field argument suggests that in a strong field the energy may be lowered by having as many nodes as possible along the lines where particles come together, r , = r ; . Indeed, if the pair interaction is sufficiently short ranged, the Laughlin wave function, which has all nodes along such lines, is the exact, nondegenerate ground state of arbitrary permutation symmetry at density v = l / m . 5 2 If one assumes that it is energetically favorable for the many-particle wave function to vanish when two particles coincide, then one concludes that ring exchanges always lower the energy at densities v m = ( 2 m -+- I P 1 , in other words, that the fluctuation determinant must be negative for fermions. If one calculates the contribution to the energy from pairwise ring exchange using the Maki-Zotos variational wave function, one finds that in the strong-field limit ring exchanges do always lower the energy at densities v m . Although one is not guaranteed of obtaining the correct sign for exchange contributions to the energy from calculating overlap integrals using a variational wave function, since the Maki-Zotos wave function is the starting point for our analysis, it is consistent to use the sign obtained from it. The fluctuation prefactor has been further investigated by Jain and Kivelson, 28 who considered a single-particle model in a strong magnetic field, in which the particle hops between sites equally spaced along a ring of radius r0. By choosing a sufficiently simple model potential, the classical action may be evaluated analytically and the fluctuation prefactor then extracted by comparing the separation of low-lying energy levels (obtained by numerical

294 1638

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

solution of the projected Schrodinger equation) to e~ '. Jain and Kivelson postulated that, in the large r 0 limit, the phase of the prefactor tends to the value e,lr/n, where n is the number of tunneling centers along the ring. Numerical results have thus far confirmed this speculation, although only the cases n = 2 and n = 4 have been studied. As applied to our ring exchange theory, this result seems to imply that a factor of — l per ring arises from the fluctuation prefactor. APPENDIX B: SINGLE-PARTICLE BASIS In this appendix, we discuss a continuous representation for the single-particle basis spanning a given Landau level. The bulk of the discussion relies on many wellknown results, which we now summarize. 1. Cyclotron and guiding center variables The dynamics of a two-dimensional system in the presence of a uniform magnetic field B= — B0z may be separated into two distinct motions, cyclotron and guiding center motion. Working in the symmetric gauge, where the vector potential is /*, = i/Joe,;'/, the cyclotron position and momentum are given by r

Si =

e

T >--Z ijPi

b=(px-ipy)/Vll = v/2/

H0 = -fioic{aia+\)

21

^O0(z,z)= =

.

(m',n'

1 (2ir/ 2

, -zz/4/2

(B6)

(c*)* (bf)m .--. v n ! Vm !

These (independent) pairs satisfy the canonical commutation relations [ir,-,!;] = [*/• Pj)— — '*8y. with no other nonzero commutators. The free-particle Hamiltonian is

(B7b)

27r/ 2 m!

XL„m-"(zz/2l2)e (B2b)

(B7a)

\m,n >=8 m m '8„„' ,

"

(B2a)

2C'J'J

I1

(B5)

The normalized fiducial state, |0,0>, defined by a | 0 , 0 ) = fc | 0 , 0 ) = 0 , possesses the real space representation

while the corresponding guiding center variables are

—

,

2. Basis states

,

F

] = 1 . The Hamiltonian H0 be-

with 0=\/2wl1.

(Bib)

K; =Di

(B4b)

4/

Construction of the usual Fock space of state vectors proceeds by application of the raising operators a and b :

'=Pf+^jJ€OrJ

Pi = ir< + -j oPj >

3+-W 2

which obey [a,a*] — [b,b comes

(Bla)

V

I2€

36

V21 -Tz/*l'

(B7c)

where LjHx) is a Laguerre polynomial. This is the oftenused "angular momentum basis," so named because the elements \m,n) are eigenstates of the angular momentum operator, Lz=€jjriPj

Ha=—

TTiTTj

(B3)

(Throughout the discussion, two-dimensional vectors will be indexed by Roman labels, and etj will denote the antisymmetric tensor of rank two, with e 1 2 = + l.) The magnetic length is / = Vtfc /eB. One may now define the independent oscillator ladder operators in terms of the complex coordinates z —x +iy and z—x —iy:

= «i>ti-ota) .

\Kn)=(°reiRS-RI,WV-2,

= V2/ 3 + — T2Z 4/

| 0,0)

(B9a)

which is radially localized about the guiding center R: PR.n

a=(§x+i£y)/V21

(B8)

Next, we introduce the normalized coherent state,

=

1 (2irl1n\)W2

with R =RX +iRy, given by

z-R

Vll etc.

2 e(zK-IR)/41 e-

\z-R

|2/4/2

(B9W The coherent-state overlap is

(B4a) < R'.n' | R,n ) =6„„. e '*'* - * * '"^e " I * "*'' 2 / 4 ' 2

(B10)

295 COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

36

APPENDIX C: COHERENT-STATE PATH INTEGRATION

We stress that we are dealing with a continuous representation here; the coherent states are eigenstates of the nonHermitian operator b, with b \ R , / i ) = ( . R / V 2 / ) j R,/i >, and thus they are not orthogonal. Nevertheless, the projector P„ onto a given Landau level takes a simple form: />„=

-J

X

\m,n){m,n

The usual procedure by which the imaginary time propagator (x' | e ~eH \x") is transcribed over to a path integral entails dividing the inverse temperature ft into a large number of equal intervals of size E, and subsequently inserting a resolution of unity, I = J " r f j c | j i c X . x | , a t each step. In any physical context, the expression I always represents a projector onto a relevant subspace, the states | x ) likewise so restricted. Using the coherent-state basis and projector discussed in Appendix B, a coherent-state path integral can be derived for the propagator. In this appendix, we shall discuss the essential features of this formalism, including the method of steepest descents approximation (SDA). The treatment follows that of Schulman 26 and Girvin and Jach. 51

|

d2R |R,n>
(BID

In the lowest Landau level (M = 0 ) , the above formulas can be summarized (dropping the n = 0 index):

<6R
1

T^^^ , ' r x R ' i / 2 ' 2 e "' r " R , 2 / 4 ' 2

(B12a)

= e _ i R ' x R ' i / 2 ' J e " < R - R ' , 2 / 4 ' 2

W

(B12b)

1. Single-body path integral

2

dR 2vlz

1639

RXRI

(B12c) The matrix element

-ev R,> =

/

2ir/

/

2

4/

lRi

iiRj

+

2TT/ 2

dRk

f n 5 J t f [R] = - | 5 - 2

d2R M

+

--- rO(e

i-Rj)-Rj{Rj

+

2

(CI)

) ,

i-Rj)\

+ e 2

K(R^, |R,)

y= o

j=0

with

limit of the coherent-state path integral, K(R'!R)=/ = V(R'\R)


.


,-py

R

'>=/jf«o,-*(alR(T,>-

S[R(r)]

(C4)

Rl0) = Rf

As mentioned in the text, the continuum limit of path integral is not well defined. The origin of difficulty is the nonorthogonality of the basis states | Whereas the usual quantum mechanical propagator proaches a 8 function in the zero time limit, e.g., l i m , _ 0 <x'\e ~
this the R). ap-

,

the coherent-state propagator tends to the overlap < R ' | R > =e - • " • * K - / 2 ' 2 ( ? - < R - « o 2 / 4 / >

f

and there is thus a nonzero probability for the particle to undergo an instantaneous hop over an arbitrary distance. This is reflected in the fact that the continuous time limit (e—»0) of the coherent-state action functional, +

+r

\Rj-i),
Rj + 1-Rj

= +2l2c^-V(RJ

+ i\Rj)

,

R/{Rf-R(P))}

2

dr

Rj - * , _ , = - 2 ' 2 E - ^ - y(Rj

oRj

J_i-RiO)) S[RM]=^j[Rl{R 4/

is dominated by discontinuous paths, and indeed the limit is ill defined. In general, one must either work with the discrete-time path integral, or use some other procedure to make (C4) well defined (see below). Fortunately, we are only interested in the semiclassical limit when V(R) is a slowly varying function of its argument over the length scale /, and we can employ the SDA to evaluate the path integral. The path which extremizes the discrete time action satisfies the discrete-time classical equations of motion

4/

(RR-RR)+V(R

\R)

(C3)

is linear in time derivatives, and thus discontinuous paths have finite action. This, in turn, implies that the E—>-0

with jj= 1, . . . , M. From these equations, one s£es that the \Rj] are evolved forward from initial data R0=Ri, while the [ Rj) are evolved backward from final data RM + I=R/', the differences RlDA-Ri and Rff^-R/ are not, in general, infinitesimals. It is therefore convenient to write

296 KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

1640

SM[R-]—-—T[R

o(Ro — R I l + ^M + l l ^ M i l - ^ M )]

+ —j 4/

2

[RjUij-Rj-i)-Rj{Rj +

i-Rj)]

J = ]

+ e 2 V(Ri ; =o

+

i\Rj)

(C6)

.

For sufficiently small E, and whenever the classical path is discontinuous, the discrete-time classical equations of motion can be replaced by their continuous time limit: R =

r. | R) , -2l- d V(R bR 2

R = +2l ~V(R dR

\R)

supplemented by the boundary conditions R(l3) = Rf, R{Q)=R,. The classical paths may be well defined in the small e limit. Caution must be exercised in extending the above formalism toward evaluating the single-body partition function,

I

d2R (Rk-^IR) 2nl2

-[R? f

.i

2. Many-body path integral The derivation of the many-body path integral directly parallels that for the single-body problem. In what follows, we shall denote by R° the guiding center of particle i (1
<5?"|e"'"'|5?*> = / ft f l ^ f e - 5 ' « ( W , + 0(e 2 ) , 217/ Z

(C8)


for the SDA equations may have no continuous solutions.

$*[#]= 2

The trouble arises from the boundary conditions that must be employed when integrating over the endpoints in Eq. (C8). In this case, the periodic boundary conditions R{0) = R{O) and R(P) = R{0) must be applied to the first-order equations of motion in Eqs. (C7a) and (C7b). Such a system need not possess a solution, and the best that one can do is to satisfy the Euler-Lagrange equations locally, while admitting discontinuities at certain points along the path. (As discussed in the body of the paper, these problems do not arise for our ring-exchange paths.) We shall return to the issue of path discontinuities at t h e end of this appendix.

(C7)

,

36

+,

(R°+'-R?)-R?(R!'+i-Rn]

where the discretized action is

+^

2

V2(R°

+

\R? +

(CIO)

'\R°j,R°)

4/dvyi

where the continuous-time action is

The potential is written as a sum of two-body terms:
r.v)=i 2

ViiTi-tj)

,

S[JHT)]=

Urj)

2 -^\R'i[R!-Ri(0)] , = , 4/

+

Rr[Ri'-RdP)]\

(Cll) V2(R',Q'\Q,R)=(R',Q'\

f

/

2|R,Q>/

•

The SDA equations of motion are Rf-R'-^-Uh-9^

f_

(CI 4)

(C12) +

\R'

+

_ X>{mr)]e-s^{Tn

•>Ri{0) = Ri

VjiR^R^R^R,)

i.j

a — 1 D o ~] \

1

\Rj,RV

with q_=l,_L. . ,M, and subject to the boundary conditions R?=R'i and Rf + ' = *,"• In the continuous time limit, the many-body path integral reduces to '

•jjjiRiR,-**,)

J n

V2(R?,R°\Raj-\R',

2

.v Rf = ' - R f = + 2 / 2 e ~ ; 2 VilR°

1

2

+ 1 2

2>R? j = i

(ft"\e-'rv\7l,)=

+ /""

,

(CO)

The SDA equations of motion are then

(CI 5) /J, = + 2 / 2 - ^ < V [ # ] . dRi The iV-fermion partition function, ZN, may now be cast into the form

297 1641

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

36

2

" = 777 2

= -^T

1

sgn(cr)/ n - ^ T < R " U >

sgn(er) f

B[5?(r)]exp

Ra(m\e-0V\Rx,...,RN>

J

n

where the boundary conditions require that RaW(0) = Ri(O) and RaW(0) = Rj(O). Again, the firstorder equations of Eq. (CI5) will in general be incompatible with the imposition of this periodicity constraint. For reference, the diagonal coherent-state matrix elements of the Coulomb potential K2(r) = e 2 /er are V2(R,Q

= =r. VT : e - , « - G ! 2 / 8 ^ / o ( | i { _ c | 2 / 8 / 2 ) ? \Q,R)-2el (C17)

where IQ{X) is the modified Bessel function of the first kind. 3. Discontinuous paths T h e presence of discontinuous paths implies that the naive continuous time limit of the discrete-time coherentstate path integral is not well defined. The discontinuous paths still show up in the continuous-time path integral since the action has only a term linear in the time derivative; there is no kinetic energy term quadratic in the time derivative to force continuity of the paths. However, the weight of discontinuous paths is not necessarily the same in the discrete-time and continuous-time formulations of the problem. This problem is not of central importance to us here since, as discussed previously, in the e—»0 limit, the classical paths tend to a smooth path, with possible discontinuities arising at the endpoints. Thus, the continuoustime formulation can be used to calculate the properties of the classical paths, but the prefactor, which we do not compute explicitly (except in the case of a displaced line), must be computed using the discrete-time formalism. Alternatively, Klauder 2 7 has shown that the coherent-state path integral can be defined by introducing an artificial kinetic energy term in the action, and then taking the limit as m * —»0 at the end of the calculation. Again, this procedure is only necessary when computing the prefactor. Even with one of these prescriptions for computing the prefactor, there remains an ambiguity. The potential term in the continuous-time action is not uniquely specified since V(Jj + l \zj) does not approach any function of z, and Zj alone as e—»0. In Eq. (C7) we made the replacement V(zj + x\zj)—*V(zj\Zj). Since zj + \—Zj is typically of order /, we could equally well have used any function of the form V(z | z)[ 1 + Od2/R2)], where R is the characteristic distance over which V varies. (By definition, l/R « 1 where the semiclassical approximation is valid.) Schulman has suggested that this ambiquity is related to operator order ambiquities in the original formulation. However, Klauder has shown (for polynomial potentials) that if one uses the prescription of taking the m * —»0 lim-

2

^RlR,-RiRl)

+^ 2

,f, 4/2

VI(Ri,Rj\RJ,Ri)

(CI 6)

it, the same propagator is obtained whether one chooses the potential term to be V(R) directly, or, as we have done, its matrix elements K ( R ) =
APPENDIX D: ACTION FOR A DISPLACED LINE Imagine the simplest possible exchange path, namely that of a uniformly shifting line of charges within the Wigner crystal. When the line JL is displaced, we have R, = T , - l - a S / e / , with 8 , e z unity if and only if lattice site i lies on the line in question. The matrix element of the Coulomb potential between coherent states was given in Appendix C:

V(R)=-

/o(KV8/2)

lel

(Dl)

Accordingly, the energy of the displaced line configuration relative t o that of the perfect Wigner crystal is A£ = • I I H R i - R y ) - •K(Ti-Tj-)]

(D2)

The sum is broken up into three terms. The first term includes all pairs (i,j) in which both sites i and j lie off the line. This contribution to A£ is obviously zero. The second term involves all pairs (i,j) where one of the sites, say i, is on the line and the other, j , is off: A£2= 2

[^(Ti + a - T y ) - •K(T,--Ty)] .

(D3)

Clearly the line energy is extensive, hence t h e energy per tunneling electron can be written C/laJsAEz/JVii, c = X

[K(Ty-a)-K(Ty)]

(D4)

where we have chosen the origin to lie on the line. T h e sum over / can in turn be written conveniently as a sum over all lattice sites minus a piece with . / £ - £ . T h e third and final term is, of course, that arising from both i and j on the line. Since the tunneling is cooperative, this contribution to the classical action vanishes. We therefore find l / ( a ) = 6eaii — 8E 0I1 . T h e slowly convergent sum for Sean can be converted into a rapidly convergent sum by means of the Poisson summation formula,

2S(r-T)=-^2 e ' T

and the result is

"" G

(D5)

298 1642

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

1

6e»ii = -

G G*0

1)

(
G/

(D6)

In the above formulas, T and G denote vectors in the direct and reciprocal lattices, respectively; flv — 2irl2/v is the Wigner-Seitz cell area. The identical technique may be applied to the " o n " sum: Simply sum over all direct lattice vectors, but include, in an integral representation, a Kronecker 6 function which enforces the online restriction. One finds 6E„

ve el

u;

dc 2v

G G#0

1 \GI+cgt

Xe -(G+cgr; (e

fcg)a_ j )

(D7)

with g = 2 y / v / 3 a v . While the virtue of Eq. (D7) is its rapid convergence, the integration complicates the numerical evaluation, and in practice it is more convenient to calculate the slowly converging expression 6e on =

VTl lei

wave-vortex-gas system. The frustrated XY model is described by the Hamiltonian H=K

70
2

,

2

v(or-ffr-Ari

(E3)

In this appendix, we review the equivalences between the discrete Gaussian model, the XY model and the spin-

The partition function may then be written as

exp[K(S„.) + i S „ ( < 9 , - 0 r - > * „ . ) ]

(E4)


T o each distinct nearest neighbor pair (r, r ' ) on JL one associates a directed line from r to r'. The bond spin Srr- is then written as the difference of nearest-neighbor-site spins R and R ' on the dual lattice JCD, where the segment from R to R ' intersects that from r to r' such that (r' — r ) X ( R ' — R ) - z > 0 . This assignment satisfies the "zero divergence" constraint that arises from the ( 8 , | integrations. Lastly, the vector potential term on JL gives rise to a uniform imaginary field on JLD, and the duality transformed partition function takes the form

Z=Tre

(E2)

The function V{6) is assumed to be periodic, and may be expressed as a Fourier sum,

APPENDIX E: EQUIVALENT MODELS

r"

(El)

where the dynamical variables are the angles j 6r) and the sum is over all nearest-neighbor pairs < r, r ' ) on some lattice JL. This model has been used to describe Josephsonjunction arrays in a transverse magnetic field,38 where superconducting islands interact via a proximity coupling. The vector potential A„- lives on the links of JL and is constrained by the requirement that the integral § A-dl around a plaquette yield the total magnetic flux through that plaquette, which is written as 2-rrf in units of the superconducting flux quantum: 2Piaqu«tc Arr' = l^f< with A„- = (e*/ik) J'' A-dl, and e* = 2e (the charge of the Cooper pair). Following Jose et al.,** consider a general Hamiltonian of the form

n[\er\}=(D8)

[ 1 -cos(6>, -ffT -A„.)\



70[(T-a)2/8/2]

2

36

-u\\sR\\

(E5)

The value of the normalization constant JV as well as the functional form of the Villain coupling Ky(K) are determined by matching the values of the left- and right-hand sides of the above equation, as well as their second derivatives, at 0 = 0. One finds that for AT > U i r ) ~ ' , Ky~K is a good approximation. The function V[S) then becomes Py{S)= — S2/1KV, which completes the mapping to the D G model. If JL is a triangular lattice, then JLD is a honeycomb lattice. The transformation to the spin-wave-vortex-gas system is even simpler, and follows directly from application of the Poisson summation formula

2 v(SK-sK)+iTrif2,sR ,

R

which is a generalization of the D G model. To recover the D G Hamiltonian, we make use of the following approximation, valid for large K:

>^JV

x

-ll/2)*ri/(8-2irm|-

(E6)

2

6(S-«)=

X

e2v

(E7)

Inserting such a term for each dual lattice spin S,, the trace over all integer 5, can be replaced by a product of integrals:

299 COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

36 ZDO=

2

1643

ex

P [ - « 2 <S,--S,) 2 -2iri2AS*

/"

II ^

II J

2

6<5

/-»;> e x P \-a-2,iS,-Sj)2-Ivi

2/*S k


n • = - oo

= 2 / " nrf5'.-expf-a2's.-g.>,iy+2m'2s*('«*-/*»l =

ZswZ\

(E8)

We have implicitly defined the lattice Green's function Sjj, which is discussed in more detail in Appendix F. Our coupling a is related to the Villain constant by a=\/2Ky. The spin-wave partition function is trivial and results from an unconstrained trace over the Gaussian Hamiltonian Hsw: # s w = a 2 Wi-jr < U,j)

-r

z

f«

n j i

sw= J

(E9)

-Hit*,})

1. Maki-Zotos revisited Maki and Zotos8 found that the Hartree-Fock chargedensity wave calculations could be reproduced accurately by the simple ansatz wave function |*>=(JV!)-1/2 2

\\d4>ke

= exp LNilf

d2k In <2ir)2

tj

\K,(U,*o(2

,IW>> •

(Fl)

a£iN

-TrS(k)

In the above formulas, N represents the number of sites in JLD, and ft is the area of the Wigner-Seitz cell on JL; the trace over £(k) is with respect to internal basis indices, which are present if the lattice is not a Bravais lattice. The spin-wave system does not exhibit critical behavior at any finite value of the coupling a. All the interesting physics lie in the vortex-gas piece, a

possible, a linear spectrum results. In this appendix, we shall discuss the details of this calculation.

-/;> , (E10)

Now Sjj is divergent, and it is customary to identify the finite difference Gij=2viSu— Sy<). The divergence of §u manifests itself in a "neutral vorticity" constraint, as it is easy to show that only neutral configurations (i.e., those which satisfy Q[m]= ^(m,-— / , ) = 0) have nonzero Boltzmann weight in the partition sum. This leads one to the following result:

The Maki-Zotos (MZ) state is just a determinant formed of single-particle coherent states centered on the sites of a triangular lattice R, = T , . The ground-state energy is then evaluated according to 1

-$(" : 2

j
*\/<*|*

\TJk I

2(R,-R*)+

j
2

r 3 (R,,R,,R*)+

i <j
(F2)

Magnetophonons may be investigated by letting the guiding centers deviate from the lattice sites, e.g., R, —>T/ + u , . Working within the harmonic approximation, and in addition neglecting all contributions from three-body terms, etc., the "bare" magnetophonon Hamiltonian becomes

= J 2it< V k > " . < k ) u y ( - k ) , 277-/2

-HVG = - -r— 2 ' ( m < ~fi )Gjj(mj

—fj)

(Ell) -GiGjViO)]

Note that the above derivation allows for nonuniform flux fk per plaquette. APPENDIX F: RING EXCHANGE AND MAGNETOPHONONS As discussed in the body of this chapter, density fluctuations in the a
(F3)

where we have suppressed the label " 2 " on K2, and where subscripts on nonboldface variables index a Cartesian component, and are not to be confused with particle label subscripts. The dynamics behind the Hamiltonian of Eq. (F3) are concealed in the projection to the lowest Landau level in which the cyclotron motion of the u, variables is quenched. The resulting Hamiltonian can be written in terms of the noncommuting guiding center variables. Using complex coordinates r, =u,> +iu,,y,

300 1644

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER 1

Zk-

2ire2l V(q) = ^^-^U(ql)

2£-*'T'r/=/V2(fl*+6+_*

VN

36 .

(F9)

(F4) U 2

z-k-

e _

%=l^2(a*-k

The MZ potential,

+ bk)

K(R) =

one obtains the magnetophonon Hamiltonian H°mp=^[X(k)bfkbk+X(k)btkb_k k

+
(F5)

,

where X(k)=±ll[xx(k) + %y(k)] , (""


H% can be diagonalized by a Bogoliubov transformation with the result that the bare magnetophonon branch is given by

Vire2 2 2 2 2 4el sech(R /8l tfo(R /8t )

differs from the projected Coulomb form of Eq. (Dl) in its short-wavelength behavior—this is a consequence of the exchange corrections introduced by antisymmetrization of the MZ wave function. Maki and Zotos found that, whereas the total energy is adequately represented by the Hartree term throughout the range 0 < v S 0 . 4 , the shear modulus C|, and hence the magnetophonon spectrum, is quite sensitive to the effects of the two-particle antisymmetrization inherent in Eq. (F10). The Fourier components of the Fock-corrected potential are given by 1'

^ ? = [ |*-{k) | 2 - [^(k) | 2 ] '

\[U(kl) +

„(k)

n = 1

(F7)

In the long-wavelength limit, <J>,y(k) tends to the continuum result, .„2

(F10)

,

(FID U„(gl)--

C0]kikj+Ctk%j+O(k3)]\

VT.

Vn(n

2«+1

«(«+!)

g2!2

J&l

X/0

C0 = | 2 [8C/(G/) + 7(G/)£/'(G/) + (G/) 2 tr(G/)]

+ 1 ) exp

2nin+l)

(F8) 2

C, = i 2

2. Ring-exchange effects

[3(G/)t/'(G/) + (G/) tnG/)]

where G is a reciprocal lattice vector, and where we have defined the dimensionless potential Uiql) by

(2irl2r2\8AJk) ^ v ( k ) - —

#„*= —

H

a

We begin with the effective Hamiltonian of Eq. (4.8), obtained by tracing over the ring-exchange variables in the presence of a phonon field:

g^(k)g w .(k)g V v (k) &Av(-k)

.

(F12)

*

In the above expression, the Greek subscripts refer to basis elements of the plaquette lattice (/x=l for upward-pointing triangles, n = 2 for downward-pointing triangles, and summation convention employed for these indices). One finds that the Fourier component of the area fluctuation is given by SA^T)=^Lr2,e*TbAfl(k) 1 VN

2e,l'T[iyk)6k+i?;!(-k)6T:lc]+~-2e "k-k,-T[zyk,k'»T_kA_k.+z>;(-k,-k')6j;*k]

with

and where T,, T 2 , Tlt and T2 are vector and complex representations of the two primitive direct lattice vectors Ti i 0 and T 0 1 , respectively. The triangular lattice Green function is written

fi1(k)=^[r2(e'kT'-i)+rI(i-e"''T2)] ik-T,

ik-T,

B2(k)=^[7V"-T" 2/

Z),(k,k') = ^ - ( l - e 4; „ ,, , ,.

2

2l

,.

D 2 (k,k )=—— (1— e 4i

e

'kT

ik-T,2

'_i) + 7-ie""'(l-e*'2)], ik-T.

(F14) )(l-e

' k T2, . . ,

(F13)

)(1— e

ik'-T

-ik-T,

') ,

-ik'-T,

Me

S(k) =

-ik-T, 2

1+e

'+e

ik-T, 2

1+e '+e {6-2cos(k-Ti)-2cos(k-T2)-2cos[k-(T,-T2)]j"

ik-T, -ik'-T, 2

e

(F15)

301 1645

COOPERATIVE RING EXCHANGE AND THE FRACTIONAL .

36

Now only the lowest-order terms at small k are investigated. This approximation involves neglecting the Q^v(k) contribution to the effective interaction (valid in the longwavelength limit when a is less than ac), and retaining only terms bilinear in the operators b and b : # e i r = £ [Mk)btbk

+

Uk)blkb^k

k

+ tl>(k)bkb-k-+il>'(k)b}b^k]

,

Hk)=-^~{2irl2)-2[BJk)SMJLk)Bt{k) 2ar0

+*;<- - k ) S „

4>(k)--

7T 2

2a TO

(2TT/2

2

[S^(k)^v(k)Bv(-k) + B)i(-k)§)lv(-k)BJk)]

The complete magnetophonon Hamiltonian is then given by the sum of Hctl and # m p . In the long-wavelength limit, where V377A.(k)4ar0 (F17) kx + iky tMk)4ar0 the phonon spectrum takes the acoustic form ak where the "sound" velocity is

-k)flv(-k)] , el

(F16)

"Present address: State University of New York, Stony Brook, NY 11974. 'Present address: McMaster University, Hamilton, Ontario, Canada L8S 4M1. •Present address: James Franck Institute, 5640 S. Ellis Ave., Chicago, IL 60637. 'K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,494 (1980). 2 D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 3 D. Yoshioka, B. I. Halperin, and P. A. Lee, Phys. Rev. Lett. 50, 1219(1983). "B. I. Halperin, Helv. Phys. Acta 56, 75 (1983). 5 H. Fukuyama, P. M. Platzman, and P. W. Anderson, Phys. Rev. B 19, 5211 (1979). 6 D. Yoshioka, Phys. Rev. B 27, 3637 (1983). 7 D. Yoshioka and P. A. Lee, Phys. Rev. B 27, 4986 (1983). 8 Kazumi Maki and Xenophon Zotos, Phys. Rev. B 28, 4349 (1983). 9 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). ,0 F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983). " S . A. Trugman and S. Kivelson, Phys. Rev. B 31, 5280 (1985). 12 W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698(1979). "Daniel Arovas, J. R. Schrieffer, and Frank Wilczek, Phys Rev. Lett. 53, 722 (1984). 14 B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984). I5 F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 54, 237 (1985). ,6 F. D. M. Haldane, Phys. Rev. Lett. 55, 2095 (1985). ,7 Daijiro Yoshioka, Phys. Rev. B 29, 6833 (1984). ,8 S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett. 54, 581 (1985); S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. B 33, 2481 (1986). " R . P. Feynman, Phys. Rev. 91, 1291 (1953); 91, 1301 (1953); R. P. Feynman and M. Cohen, Phys. Rev. 102, 1189 (1956). 20 R. B. Laughlin, Surf. Sci. 142, 163 (1984). 21 S. M. Girvin and A. H. MacDonald (unpublished). 22 G. Baskaran (unpublished). 23 D. J. Thouless, Phys. Rev. B 31, 8305 (1985).

.

24

v/31 ar0

=ck,

(F18)

Steven Kivelson, C. Kallin, Daniel P. Arovas, and J. R. Schrieffer, Phys. Rev. Lett. 56, 873 (1986). 25 G. Baskaran, Phys. Rev. Lett. 56, 2716 (1986). 26 L. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981), Chap. 27. 27 John R. Klauder, in Path Integrals and Their Applications in Quantum, Statistical, and Solid State Physics, Vol. 34 of NATO Advanced Study Institutes. Series B: Physics, edited by George Papadopoulos and J. T. Devreese (Plenum, New York, 1978). 28 Jainendra Jain and Steven Kivelson (unpublished). 29 For a general path, there may also be contributions to the imaginary part of the action from terms such as Rt f\R". These contributions can be treated in the same way as the deviations from straight line segments, and also lead to a renormalization of a 0 . 30 R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1982). 3I S. T. Chui and J. D. Weeks, Phys. Rev. B 14, 4978 (1976). "Robert H. Swendsen, Phys. Rev. Lett. 37, 1478 (1976). 33 W. J. Shugard, J. D. Weeks, and G. H. Gilmer, Phys. Rev. Lett. 41, 1399 (1978). 34 S. Teitel and C. Jayaprakash, Phys. Rev. B 27, 598 (1983). "Wan Y. Shih and D. Stroud, Phys. Rev. B 28, 6575 (1983). 36 S. Teitel and C. Jayaprakash, Phys. Rev. Lett. 51, 1999 (1983). "Wan Y. Shih and S. Stroud, Phys. Rev. B 30, 6774 (1984). "Thomas C. Halsey, Phys. Rev. B 31, 5728 (1985); J. Phys. C 18, 2437 (1985). 39 J. Tobochnik and G. V. Chester, Phys. Rev. B 20, 3761 (1979). 40 See Ref. 37 for transition temperatures on triangular and honeycomb lattices. One can also estimate ac for different lattices by multiplying the calculated value for one lattice by a mean-field-type ratio of lattice coordination numbers. 41 J. M. Caillol, D. Levesque, J. J. Weis, and J. P. Hansen, J. Stat. Phys. 28, 2 (1985). 42 J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). 43 R. P. Feynman, Statistical Mechanics (Benjamin, New York, 1972). ^Jorge V. Jose, Leo P. Kadanoff, Scott Kirkpatrick, and David R. Nelson, Phys. Rev. B 16, 1217 (1977).

302 1646 4s

KIVELSON, KALLIN, AROVAS, AND SCHRIEFFER

Lynn Bonsall and A. A. Maradudin, Phys. Rev. B 15, 1959 11977). 46 H. L. Stormer, A. Chang, D. C. Tsui, J. C. M. Hwang, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 50, 1953 (1983). 47 D. Lee (private communications).

48

36

D. Lee, G. Baskaran, and S. Kivelson (unpublished). S. M. Girvin and A. H. MacDonald (unpublished). D. J. Thouless, Proc. R. Soc. London 86, 893 (1965). 51 S. M. Girvin and Terrence Jach, Phys. Rev. B 29, 5617 (1984).

49

50

IV Surfaces

This page is intentionally left blank

305

SCHRIEFFER'S C O N T R I B U T I O N S TO SURFACE P H Y S I C S

Theodore L. Einstein 1 and James W. Davenport 2 1

Department of Physics University of Maryland College Park, MD 20742-4111,

USA

Center for Data Intensive Computing, Bldg. 463B Brookhaven National Laboratory, Upton, NY 11973-5000,

USA

In this section are the four most influential (or at least the most often cited) of Schrieffer's papers in surface physics. The first paper [l], his first major publication as a graduate student at Illinois, preceded his work on superconductivity and dealt with the effective carrier mobility in the surface space-charge layers of semiconductors. The seminal insight was that when dealing with carriers in this region, in which quantum behavior dominates in the direction normal to the surface, collisions of the carriers with the surface must be included. This "initial attack ... laid the groundwork for most of the later work" [2]. This paper postulated that "the electrons confined in the narrow well of an inversion layer would not behave classically." Clearly if the electron wavelength is comparable to the distance from interface to the classical turning point, then carrier motion perpendicular to interface cannot be treated classically. Instead, one expects discrete levels, leading to a sequence of subbands with perhaps free-electron dispersion in the plane parallel to the surface. The paper starts with the Boltzmann equation for a system with a large space-charge field Ez(z) and small uniform field Ex parallel to the surface. The resulting expression is changed into a first order differential equation by introducing a new variable (the perpendicular component of energy), with the goal of seeking the leading order (in Ex) effect on the distribution. The key assumption of uniform random diffuse scattering at the surface followed ideas of K. Fuchs [3]. Schrieffer further assumed that the scattering time was independent of energy. He analyzed two potential distributions, both of which became infinite in the bulk. In the simpler case, the space-charge region was characterized by a constant field (so a potential linear in z), with clear simple results. He also treated a somewhat more accurate approximation for the potential, albeit still going to infinity in bulk so that the number of carriers per unit area near the surface was finite. He expressed the results in terms of two dimensionless ratios, with graphs. For deep wells, the theory can be used for accumulation as well as inversion layers. The paper stimulated a great deal of theoretical work, initially to develop potentials that did not diverge in the bulk. While there was doubt expressed about whether the discrete quantum levels could be observed in experiment due to broadening by interface scattering, several experimentalists took up the challenge, with fabulous success, leading to decades of progress in studying the two-dimensional electron gas and its role in a slew of devices such as field-effect transistors. Many of the highlights of this work, with some comments on experiments, were republished in the proceedings of a conference [4], which is sometimes cited instead of the original Physical Review paper, e.g. in the authoritative review by T. Ando, A. B. Fowler, and F. Stern [5] and in the Nobel Laureate article by Daniel C. Tsui [6], which credits this

306 paper with stimulating his tunneling experiments on InAs to observe directly the discrete quantized energy levels and Landau levels of a two-dimensional electron gas. The remaining three papers were written at the University of Pennsylvania over a decade and a half later, after Schrieffer's renowned contributions to superconductivity and his first major period of interest in magnetism. At the dawn of the 1970's there was a rush of experimental investigation in surface science driven primarily by the development of ultrahigh vacuum equipment that allowed extended investigation of pristine surfaces. As the first elite theorist to become interested in the subject, Schrieffer had a major impact on this renaissance and offered the hope that the essentials of surface phenomena could be understood in terms of simple models and unifying concepts. His paper on the "Theory of Chemisorption," appeared in the proceedings of the International Conference on Solid Surfaces, held in Boston, Oct. 11-15, 1971 [7] and was associated with a major invited address that still is vivid in the memories of many of the early leaders of the field. Ostensibly a review of contemporary theoretical understanding of chemisorption, it promulgated the conceptual scaffolding for much subsequent thinking about binding on surfaces. Benefiting from work in previous years on magnetic impurities, most of the discussion is couched in terms of the Anderson [local impurity] model [8], which had previously been applied to the chemisorption problem, particularly by D. M. Newns [9]. The hopping term underlying covalent coupling with the substrate is equivalent to the Hiickel picture of quantum chemistry. In contrast to the myriad of terms in that discipline, only one Coulombic integral U is retained, the dominant one in which all four spatial orbitals are the same. In other words, Schrieffer boils the model down to the bare essentials to focus on the relevant physics. The first analysis is by the Hartree-Fock self-consistent-field molecular orbital method, emphasizing the work of Newns [10] and others. Then he turns to the regime of strong U, when electron-electron correlation dominates. After recounting van der Avoird's use of Heitler-London or valencebond (VB) methods to investigate dissociation barriers of adsorbing hydrogen molecules [ll], he presents a preliminary account of work with R. Gomer on the induced covalent bond theory [12]; the challenge they met was to formulate VB for an infinite system rather than just a finite cluster. The non-orthogonality of the adsorbate atomic wavefunction and the substrate metal one-electron wavefunctions leads to an antiferromagnetic exchange of strength J. For weak J (relative to the substrate bandwidth) conventional exchange repulsion is retrieved, but in the opposite limit the adatom and nearby substrate orbitals form a surface complex which then rebonds to the indented solid. This tour-de-force calculation, while leading to deep insights, proved difficult to apply to specific systems. The Anderson model viewpoint, which includes atomic orbitals at the most fundamental level, complements the electron-gas approach developed contemporaneously by Walter Kohn and collaborators [13]. It allows readier access to and tuning of the effective model coupling parameters to investigate generic physical phenomena, but is more difficult to improve systematically to describe specific systems. It is based in part on the recognition that most of the active metals are transition or noble metals, where d-electrons play a central, if not dominant, role. The framework has proved to be of enduring significance in such problems

307

as atom scattering off metal surfaces, for which the atom-substrate hopping and the atomic energy level can be generalized to distance-dependent expressions. The paper [14] on the indirect electronic interaction between adatoms on a metal is the most cited work of the four published here, even without including a subsequent recounting of the work in the 24th Nobel Symposium proceedings [15]. It was motivated largely by an effort to explain the large variety of ordered ("superlattice") patterns of adatoms on metal surfaces at low temperatures, indicative of an anisotropic adatom-adatom interaction that could be attractive or repulsive (in contrast to more conventional electrostatic or elastic repulsions). Grimley had shown a few years earlier [16] that two widely-separated (far enough apart so that their orbital overlap is negligible) adatoms could still interact by coupling to electronic wave functions of a metallic substrate. Schrieffer and Einstein improved upon this work in several ways. Rather than a free-electron substrate with an artificially-created (by the cut-off) surface reactivity, they used the simplest model that would build in the atomicity and anisotropy of the substrate band structure, namely a simple-cubic lattice with a single-orbital tight-binding model. The interaction energy is calculated by computing the shift in one-electron energies using a Green's function method, essentially finding the Fredholm determinant. In contrast to several competing contemporary approaches, this procedure is non-perturbative, including scattering between adatom and substrate to all orders by means of a phase shift resulting from a self-energy of the adsorbate Green's function. Since the adatoms tend to be nearly neutral and well screened by the substrate, there is no need for the subsequently-developed Harris formalism [17]. The results are expressed in terms of the effective parameters of the Anderson model, so that one can compare the energies with other properties that can be calculated. Furthermore, the approach can be readily generalized to multi-adatom interactions, and in fact to the interaction energies of an entire fractional, ordered overlayer (i.e. an adatom superlattice). A pedagogical review of developments (up to the mid-1990's) on indirect interactions between adsorbates is given by T. L. Einstein in [18]. A shortcoming of the approach is that systematic improvement turned out to be challenging. E.g. N. Burke's attempt [19] to characterize self-adsorption on tungsten and O/Ni(100) by including more substrate orbitals was unsuccessful quantitatively (but okay qualitatively). More generally, the complementary density functional approach — the basis for later self-consistent, total-energy calculation — had the advantage, since extending the number of basis states was easier and since correlation effects were relatively unimportant. An heir to Schrieffer's approach is the semiempirical INDO (intermediate neglect of differential overlap), which has contributed to understanding of strongly correlated systems such as the copper oxides of high-temperature superconductivity [20]. The asymptotic regime of the pair interaction is intriguing since it can be described analytically and is analogous to the celebrated RKKY (Ruderman-Kittel-Kasuya-Yosida) interaction between magnetic impurities, so dominated by electrons at the Fermi energy and with an oscillatory term in 2kpR, where R is the distance between adatoms. In contrast to the R~3 of RKKY, the interaction decay envelope varies like R~5 (due to cancellation of the

308 leading term of the Green's function by the image of the adatom). Hence, this interaction is unobservably small in its asymptotic regime. Lau and Kohn [21] noticed that if surface rather than bulk states mediated the interaction, the decay was R~2. Remarkably, over two decades later, such effects have been observed using scanning tunneling microscopy [22; 23]! Furthermore, these effects appear to play a role in the interaction between steps on vicinal surfaces [18; 24]. As surface science matured during the mid 1970's, attention shifted from analytical models to more computationally exacting efforts to model physical behavior. Ever eager to confront the challenges posed by the pressing problems of the day, Schrieffer was ready to turn to large-scale (for that time!) computational investigations when appropriate. Much of the burgeoning interest in surface physics was fueled by the development of various surface sensitive probes, which in turn demanded progressively more detailed interpretation. Of particular interest were photoelectron spectroscopy and electron energy loss spectroscopy. Schrieffer had taken an interest in the multiple-scattering solutions of the Schrodinger equation [25]. While they had been developed to describe the ground states of molecules, he realized that they could be applied to continuum states as well by using formalism worked out by J. L. Dehmer and D. Dill [26]. Since many surface problems involved adsorbed molecules, they would thus provide an important link to the experiments. With (then graduate student) Davenport, the theory was applied to valence state photoemission [27]. Soon after, the approach was extended to electron scattering [28] in collaboration with Davenport (by then a postdoctoral fellow) and with W. Ho (who did his dissertation with E. W. Plummer, stimulated to do the relevant experiments). As "the simplest quantitative formulation of resonance scattering by an isolated, oriented molecule" [29], this paper set the stage for subsequent research on the subject. An important feature of electron scattering from molecules is the appearance of scattering resonances. These are usually simply related to unoccupied molecular orbitals which happen to lie in the continuum. Since molecular orbitals have definite symmetry and since molecules adsorbed on surfaces are usually oriented, there are strong angular-dependent scattering effects which can be observed. In addition, these resonances lead to enhanced vibrational excitations of the adsorbates. The theory required innovatively accurate numerical wave functions for the electrons. The simplest approximation for the vibrational wave functions, viz. in terms of harmonic oscillators, then sufficed. These theoretical studies stimulated a renewed experimental interest, which was rewarded by observations of such resonances in the early 1980's by Demuth, Sanche, and coworkers. These studies have continued to the present [30]. While the impact of this research on theory is most recognized in the treatment of angular-resolved photoemission, it is arguably even greater on atomic scattering. The idea of temporary, adiabatic excitation of atoms to states with different vibrational properties from the ground state has been profitably applied to a broad range of situations [31] and promoted the thinking leading to layer-focused multiple-scattering methods [32]. It became increasingly clear that detailed numerical analysis would be needed to account

309 for the idiosyncrasies of each system. As a leading experimentalist (Peder Estrup) once quipped, "There are no simple systems in surface science." Despite astonishing progress in accounting in detail for the electronic properties of surfaces and in the ability to resolve quickly the inconsistencies arising in experiments, the early hopes that surface science would revolutionize catalysis were only partly realized. As he exited the field of surface physics in pursuit of one-dimensional conductors, Schrieffer summarized much of his assessment of the state of understanding in a review chapter with former students Einstein and J. A. Hertz entitled "Theoretical Issues in Chemisorption" in [33]. Acknowledgments TLE is supported by the NSF through Grant EEC-0095604 and MRSEC Grant DMR 00-80008. JWD is supported by the U.S. Department of Energy under Grant # D E AC0298CH10886. References [1] J. R. Schrieffer, Phys. Rev. 97, 641 (1955).* [2] D. Frankl, Electrical Properties of Semiconductor Surfaces (Pergamon, Oxford, 1967),

[3] [4]

[5] [6] [7 [8 [9 10 11 12 13 14

15 16 17 18 19 20

p. 110. K. Fuchs, Proc. Cambridge Phil. Soc. 34, 100 (1938). Semiconductor Surface Physics, ed. R. H. Kingston (University of Pennsylvania Press, Philadelphia, 1957), p. 55. T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). D. C. Tsui, Rev. Mod. Phys. 71, 891 (1999). J. R. Schrieffer, J. Vac. Sci. Technol. 9, 561 (1972).* P. W. Anderson, Phys. Rev. 124, 41 (1961). D. M. Newns, Phys. Rev. 178, 1123 (1969). D. M. Newns, Phys. Rev. B 1, 3304 (1970). A. van der Avoird, Surface Sci. 18, 159 (1969). R. H. Paulson and J. R. Schrieffer, Surface Sci. 48, 329 (1975). See, for example, N. D. Lang, Solid State Physics 28, 225 (1973). T. L. Einstein and J. R. Schrieffer, Phys. Rev. B 7, 3629 (1973).* J. R. Schrieffer in Collective Properties of Physical Systems, eds. Bengt Lundqvist and Stig Lundqvist (Academic, New York, 1973). T. B. Grimley, Proc. Phys. Soc. (London) 90, 751 (1967); 92, 776 (1967). J. Harris, Phys. Rev. B 31, 1770 (1985). T. L. Einstein in Handbook of Surface Science, vol. 1, ed. W. N. Unertl (Elsevier, Amsterdam, 1996), Ch. 11. N. Burke, Surface Sci. 58, 349 (1976). See, for example, Y. J. Wang, M. D. Newton, and J. W. Davenport, Phys. Rev. B 46, 11935 (1992).

310 K. H. Lau and W. Kohn, Surface Sci. 75, 69 (1978). J. Repp et al., Phys. Rev. Lett. 85, 2981 (2000). N. Knorr et al., Phys. Rev. B 65, 115420 (2002). W. W. Pai et a l , Surface Sci. 307-9, 747 (1994). K. Johnson, Adv. Quantum Chem. 7, 143 (1973). J. L. Dehmer and D. Dill, Phys. Rev. Lett. 35, 213 (1975); D. Dill and J. L. Dehmer, J. Chem. Phys. 65, 5327 (1976). J. W. Davenport, Phys: Rev. Lett. 36, 945 (1976). J. W. Davenport, W. Ho, and J. R. Schrieffer, Phys. Rev. B 17, 3115 (1978).* R. E. Palmer and P. J. Rous, Rev. Mod. Phys. 64, 383 (1992). Q.-B. Lu, A.D. Bass, and L. Sanche, Phys. Rev. Lett. 88, 147601 (2002). See, for example, J. W. Gadzuk, Phys. Rev. B 44, 13466 (1991). P. J. Rous, Comp. Phys. Comm. 137, 33 (2001). T. L. Einstein, J. A. Hertz, and J. R. Schrieffer in Theory of Chemisorption, ed. J. R. Smith (Springer, Berlin, 1980), Ch. 7. (The symbol * indicates a paper reprinted in this volume.)

311 PHYSICAL

REVIEW

VOLUME

97.

NUMBER

3

FEBRUARY

1,

I9SS

Effective Carrier Mobility in Surface-Space Charge Layers* J. R. SCHMEFFER

Physics Department, University of Illinois, Urbana, Illinois (Received August 16, 1954) Carriers held to a region near the surface by the potential well of a space charge layer may have their mobility reduced by surface scattering, if the width of the well is of the order of a mean free path. An effective mobility, which may differ from the bulk mobility by as much as a factor of ten, has been obtained from a solution of the Boltzmann equation. Solutions have been carried out for two types of potential functions: (o) a linear potential corresponding to a constant space-chargefield,and (b) a solution of Poisson's equation including an external bias applied normal to the surface. The results have been used to calculate changes in surface conductance of germanium with changes in surface potential and predict the "field effect" and "channel effect" mobilities.

I. INTRODUCTION

T

HE surface of a semiconductor is the seat of a space-charge double layer produced by a surface charge distribution which is counterbalanced by a space-charge region consisting of holes, electrons, and impurity ions.1,2 The surface charge distribution arises from the trapping of holes or electrons at the surface of the material. There are three obvious types of ^surface traps. First, by adjusting the mathematics of the energy band solution for an ideal infinite crystal to take the surface into account, one may find allowed levels which correspond to states localized near the surface, and which lie in the forbidden gap of the energy level diagram. A second type of surface trap arises from the impurity ions found in increased ^quantity near the surface. A third type is that arising from chemisorbed material on the semiconductor surface. The chemisorption process in general requires a charge transfer from the body to the surface of the semiconductor. The chemisorption traps are often separated from the bulk material by an oxide layer. By changing the gaseous ambient surrounding the sample, one can change the density of chemisorption traps and therefore alter the magnitude of the space-charge double layer. Morrison8 has used this technique on a free germanium surface to displace the energy bands near the surface relative to the Fermi level. The conductance of the sample is changed by such a displacement because the carrier concentrations in the space-charge region are different from the bulk concentrations. Figure 1(a) shows a ^-type semiconductor with a large positive charge on the surface causing an »-type surface layer, or "inversion layer," to be formed. Similarly, a large negative surface charge would cause the bands to rise and a strongly p-type layer would exist at the surface. The intermediate case is obtained when a small positive * This work was supported by a grant from Motorola, Inc. ' J. Bardeen, Phys. Rev. 71, 717 (1947). ' W. H. Brattain and J. Bardeen, Bell System Tech. J. 32, 1 (1953). ' S. R. Morrison, Tech. Report No. 2, Electrical Engineering Research Laboratory, University of Illinois, and private communication.

charge exists on the surface which is counterbalanced primarily by a space-charge of impurity ions, termed an "exhaustion layer." Since the hole concentration is reduced under such conditions while the electron concentration is still relatively small, the conductivity in an exhaustion layer will be smaller than that of the bulk semiconductor. If an alternating field is applied normal to the surface, by arranging the semiconductor to be one plate of a condenser, an effective mobility of the carriers introduced by this field can be determined from the change in conductance of the sample. This type of effective mobility will be called the "field effect" mobility. Unlike the bulk carrier mobilities, the field effect mobility can change its sign, the positive value corresponding to electrons. Morrison finds in general that the sign of the field effect mobility changes when the ambient causes the conductance to go through a minimum value.* Brown8 and Kingston6 have measured the conductance of a "channel" or »-type surface layer on a normally p-type base region of an n-p-n junction transistor, as a function of applied reverse bias. By fitting theoretical curves to experimental data, Brown estimates the mobility of electrons in the channel to be one-fifth to one-tenth the bulk mobility. Kingston carried out measurements in a water vapor atmosphere for several values of vapor pressure. The water vapor tends to form a positive surface charge distribution on the ^-type base and creates an «-type inversion layer as described above. Kingston finds the surface conductance appears to vary inversely with applied bias, and has proposed a theory to explain this effect by taking into account the reduction in mobility of the channel electrons. To estimate the change in conductance of a semiconductor due to a space-charge layer existing at the surface, one must consider at least two effects. The first is simply the change in the number of holes and electrons in the space-charge region. The second is the

641

* W. Shockley and G. L. Pearson, Phys. Rev. 74, 232 (1948). »W. L. Brown, Phys. Rev. 91, 518 (1953). • R. H. Kingston, Phys. Rev. 93, 346 (1954).

312 642

SCH R I E F F E R

N-Type Surface Layer

the space-charge potential. T h e general solution for the surface conductance is given in Part I I for diffuse surface scattering and applied to (a) a linear spacecharge potential corresponding to a constant electric field, and (b) a solution of Poisson's equation including an externally applied potential, Va, across t h e surface. The change in the number of holes and electrons in the space-charge layer as a function of ^»o (see Fig. 1), has been computed for impurity densities from intrinsic u p to 1018/cm3.9 These results are combined with the carrier mobility obtained from the surface scattering considerations to discuss Morrison's work a n d predict the carrier mobility in the channel effect. n. GENERAL THEORY We consider a volume extending inward from a unit surface area of a semiconductor, subjected to a n electric field Ex, parallel to the surface, and a field E, along the inward normal to the surface, due to the space-charge layer. The carriers are regarded as free in t h e sense that the energy depends upon the absolute square of the wave vector only. Under steady state conditions, the distribution function for t h e holes or electrons is determined by the Boltzmann equation, 10 v • g r a d r / + a • grad, / = - ( / - / 0 ) / T ,

FIG. 1. (a) Energy level diagram of an »-type inversion layer existing at the free surface of a p-type material, (b) A voltage V, applied across the surface shown in (a). reduction of the carrier mobility by surface scattering if the width of the space-charge potential well is comparable to a mean free path or less. For example, if electrons in an n-type inversion layer are held near the surface by the potential well, the surface may scatter the electrons more frequently than the conventional bulk scattering mechanism, therefore reducing the electronic mobility appreciably. This reduction may be quite important in the channel effect where widths of the order of several hundred angstroms are attained. If a large dipole exists at the free surface of a semiconconductor, surface scattering will play a role in determining both the total change in conductance due to ambient and the results of the field effect measurement. The considerations presented here primarily deal with the influence of surface scattering on such measurements. The increase in resistance, of thin metallic films from a decrease in electronic mobility, in the absence of a magnetic field, has been discussed by Fuchs 7 and a generalization of the problem to include magnetic effects was carried out by Sondheimer. 8 The theory presented here in general follows these analyses, the essential difference arising from the spatial dependence of the unperturbed carrier distribution function due to ' K. Fuchs, Proc. Cambridge Phil. Soc. 34, 100 (1938). " E. H. Sondheimer, Phys. Rev. 80, 401 (1950).

(1)

where v and a are the velocity and acceleration of a carrier, r the relaxation time, and f=fo+fi(v,z), /o being taken as the Maxwell-Boltzmann distribution function and / i a small perturbing function. Thus,

/„=C exp[- («»*/2+#)/*r],

(2)

where \f> is the potential associated with Ez, m the effective mass and q the charge of the carrier. When products of / i and Ex are neglected in (1), fi is found to satisfy the equation: d/i

qE.df!

dz

m dv,

v—+

/,

qVzEJo

T

kT

+-=

.

(3)

By introducing an energy parameter, e,=$miv ! +?W'-'A»),

(4)

where ty, is the value of the potential at the surface, Eq. (3) is reduced to qE, dfi m du,

/i

qvxExf0

T

kT

(5)

If the boundary condition of random scattering a t the surface is imposed by making / i vanish at v!=v„, where v„ is the positive z component of velocity asso• This calculation has been carried out independently by R. H. Kingston (private communication). A somewhat similar calculation has been carried out by C. G. B. Garrett. 10 See A. H. Wilson, The Theory of Metals (Cambridge University Press, London, 1953), Chap. VIII.

313 EFFECTIVE

CARRIER

dated with a carrier at the surface with energy parameter t«, Eq. (5) has the solution, mvJSxf0 ,_ ^ f-'expfXfo'.e,)] dv,', •expC-R(v„e.)2 I » E,(v,',e.) kT

h

and where m

f

q ^o

dv," rE.(v.",e,)

(7)

The current density can now be calculated as ix=q I dv^lvydv,vxfi.

\-qf J / kT

(8) 2 i r « V kT /2trkT\

dz

(14)

qE„

By combining Eqs. (12), (13), and (14) and writing Mbuik=9T-/»», one obtains *W«>,iik= 1 —exp(a2)(l —erf a).

2vqEJOkT /•"

f dv.f

(13)

where N is the total number of either holes or electrons in the well, depending upon whether a ^-type or w-type surface is under consideration. If the zero of potential is taken at the surface, N is given by N=C fdvjvydvrdz exp - (

Introducing the distribution function, Eq. (6), performing the vx and vv integrations and integrating the current density over z to obtain the total current in the potential well for a unit surface area, we find I,=

The reduction in the conductivity of the carriers in the well due to surface scattering can be taken into account by defining an effective mobility such that 7»=iV9M.n-E.,

(6)

K(>.',«.) = -

643

MOBILITY

(15)

Figure 2 shows this ratio plotted as a function of a. For the limit of large E,„ the expression reduces to

Xexpjj - ( — + q ^ / k T \ - K

Meff/Vbuik=2a/7r!.

(16)

Thus the effective mobility is inversely proportional to the space charge field for large constants fields. For *.'. (°) X small fields, the effective mobility reduces to the bulk £.' value as expected. (b) The second case considered is that in which E, The z integration has been carried to infinity assuming the form of \fi insures a negligible contribution from the is obtained from a solution of Poisson's equation for the region beneath the space charge layer. Since space-charge layer. For definiteness, the problem will be solved for a p-type material with an n-type inversion dK/dv,=m/qrE„ layer existing at the surface and an external bias, VG, applied in the z direction such that the energy bands Eq. (9) reduces to are depressed further at the surface as shown in Fig. EjCkTr' /•" 1(b). The results are valid for an «-type material with 2 / , = 2irff — exp[- qi>./kT~\ I du a p-type inversion layer with an appropriate change in the definition of quantities involved. Xexpl-t!/kTj.exp2K(t,)-2K((,)-l']=DA, (10) ' exp(tf')

where K(c,) is evaluated from.Eq. (7), the upper limit being (2t,/m)i = v„, and D the group of constants appearing before the integral. To evaluate the current explicitly, some form of the space-charge potential must be assumed. (a) The first case considered is E,=E,„ a constant. The relaxation time is assumed constant and the surface scattering random. By introducing Eq. (7) into Eq. (10) and defining « = (qE,.T)-i(2mkT)i,

(11)

one obtains Ix=irWakT£l - exp(a2) (1 - erfa)} 1

C exp(— x>)dx.

erfa=—2 I

IT* •'o

(12)

•^rr

FJG. 2. The effective mobility for the constant field case, as a function of the parameter a.

314 644

J . R.

SCHRIEFFER

We define $ to be the mid-gap potential relative to the mid-gap potential deep within the sample. ^„ and vp are the quasi Fermi potentials and assumed constant over the region of interest. From Fig. 1(b), we note

However, to evaluate K, Eq. (7), E. must be expressed as a function of v, and t, by using the defining relationship for t„ Eq. (4). The expression for K becomes M

(-

#=

er\SrNakT.

The charge density is taken as p^JL-N.+fr

e x p ( - ^ / * r ) - « o exp(^/*r)],

)TVI

»o

kT

N,

exp(ef/kT)

+

[—lT-("--)/kT

(17)

where

P« +— exp(-ef/kT)\\

>]!'•,

»,=«o exp[.a/',/kT2= electron density at the surface, where

»o=«< exp[(A£f—eVJ/kT'], pt=tnexpZ-AE,/kT],

e^.

(21)

«^=e^.+ 2w«'«>—e«-

(18)

We now make the approximation that the bracketed portion in the denominator oi Eq. (21) is negligible Hi is the intrinsic carrier density, N. the acceptor compared to the terms retained, over the region of major contribution to the current integral. This apdensity, and K the dielectric constant. proximation is equivalent to setting the field due to Poisson's equation becomes the impurity ions equal to a constant value, given by its value at the surface, and neglecting the contribution dfy Aire of holes to the space charge. If of/, is greater than —2= [-iV0+/>o e x p ( - ^ / * r ) - » , > exp(e^/*r)]. several kT, the population of states in the region where dz K i r l f f l i / is comparable to ep,, is very small compared (19) to the population at the bottom of the well. The bracketed expression also indicates that the maximum Let dtff/dz=y('=0 at some point within the sample as possible hole contribution to the denominator correan origin for integration. Multiplying Eq. (19) by sponds to a decrease of e^, by kT. Thus for most dfy/dz and integrating over z, we obtain the electric problems of interest, the above approximation appears field as a function of the potential. valid and K reduces to

E*-

SVeA^r

kT\/p<,\

[*+—j (— )(exp[-^/*r)-l)

V(«./*r) K= - (2/7T)*0 f •'n

+ (—Vexp|>/*r]-l)j]. (20)

dy 5+exp[-(€./*r)+y»:

(22)

where 0 = - __ e x p ( - ^ . A D = - — erL8»o J tTi8n,i If. <*. B=

Na e+.

(23b)

exp(-e&/W)= »o kT

(23a)

n, kT

\

...

V"

......

-lit"'

/ oi

at

-o) * • oi-ee«r«>.i

FIG. 3. The effective mobility corresponding to the potential obtained from a solution of Poisson's equation for the space charge layer, plotted as a function of the parameter 0 for several values of the parameter B.

«' Fro. 4. The effective mobility plotted as a function of the depression_of the energy bands at the surface.

315 EFFECTIVE

CARRIER

645

MOBILITY

-

ae

f.. -8

-'

'[,

6

Ma liT

L>„.

FIG. 5. The change in surface conductivity as a function of the depression of the energy bands at the surface.

FIG. 6. The "field effect" mobility as a function of the depression of the energy bands at the surface, assuming the charge associated with surface traps is independent of the applified field. where

The current is given by substituting Eq. (22) into Eq. (10) and after some calculation gives h=DkT

2VZ/S[(2}+1)»-.B1]

+

x f

f

exp

-Z-2(2/T)»/S

(24)

1

—-=1+ 2v2/3[(B+l)*-5»] X

\f

OJ-X-2(2/TW

x f •'o

(.B+exp(-x+y*))-*dy\dx-l

. J

(25)

i

This ratio is plotted in Fig. 3 as a function of |8 for several values of B. Figure 4 shows fieii/nbuik for intrinsic germanium. III. APPLICATIONS The results of Sec. I I may be applied to the problem of determining the position of the energy bands at the free surface of a semiconductor if the conductance of the sample is known relative to the conductance at a definite band position. The conductance due to the space-charge layer is given by Aa=e\ji

Cn).uAn+n

L

J0 L

An effective mobility is again denned by Eq. (13) and proceeding exactly as in the constant field case we obtain the expression,

m»ik

o

*«>rexpW'/fc:r)-i d^/dz

Ap = pA

[B+exp(-*+y)T-«yL*-l|.

Hat!

An=no I

(„>,«&p~],

(26)

dHdz

]#, k, \

(27a)

(27b)

and n 0 and p<> are defined as in Eqs. (18) by setting F „ = 0 . The effective mobility is to be considered different from the bulk mobility when the carrier is constrained to conduct in the potential well, thus neglecting the small correction due to some carriers of opposite sign scattering from the potential barrier rather than the surface. Since all quantities in E q . (26) are expressible as functions of ^,o, this relationship and the sign of the field effect give the band position. There is some question as to what values to assume for the effective mass and relaxation time for both holes and electrons. Assuming m„ = mp=Q.25m„ we have plotted Aa in Fig. 5 for intrinsic, »-type and ^>-type germanium. The relaxation times have been determined from the relationship Mbuik=eT/m. I t should be noted that the conductance decreases and goes through a minimum as the surface conductance tends to become inverted. For the p-type material this is due to the hole density near the surface decreasing to a small value before the electron density has increased appreciably. The conductance minimum can be used to establish the definite band position needed above. Changes in conductance may be interpreted as moving along this conductance curve and the energy band position is read directly once the minimum has been established. The magnitude of the field effect can be estimated if one assumes the charge associated with surface traps is unchanged when the field is applied. Since there can exist no net field deep within the semiconductor, Gauss's law applied to the free surface gives the total charge, Q, in the space charge region by the value of the field just inside the surface, Eq. (20). Defining a

316 646

J.

R.

SCHRIEFFER

Ul

0.50 "~-S~_

0.35

FIG. 7. The theoretical effective mobility corresponding to Kingston's data on the channel effect, for ambient water vapor pressure (a) 19.8 mm of Hg, (i) 14.5 mm, and (c) 4.6 mm.

approach the bulk values indicated by the dotted lines. Morrison's observation that the sign on the field effect changes when the ambient forces the conductance to go through a minimum is explained by realizing the field effect essentially senses the slope of the n-type curve in Fig. 5. By applying the external field, the bands are either lowered or raised a t the surface, depending upon the sign of the applied field. If the sample is a t a conductance minimum, the conductance will increase in either case however the sign of the induced charge is opposite, thus accounting for the observed effect. An estimate of the effective mobility to be expected in the channel effect can be derived from Kingston's data of channel conductance versus reverse bias on n-p-n junction transistors by extrapolating his conductance curves to zero bias. Since Kingston's data give directly the conductance due to the existence of the channel, the depression of the bands at zero bias is given by the value of \t s0 corresponding to the intercept. The values obtained in this way for several values of ambient water vapor pressure are

field-effect mobility, MF.E., one finds Vapor pressure

MF .E. = dAo•/'dQ

19.8 mm of Hg 14.5 mm of Hg 8.6 mm of Hg 4.6 mm of Hg

= Ai ( „)etf»o[exp(e^,/*r) - 1 ] +M(p)eii/'o[exp(-eiA./*r)-1]

(d+

oV(n)eff

+ \dz / UXAn [iV„+no exp(et,/kT)-p0

oV(j>)e(f~| 1

\-Apexp(-efo/kT)l,

(28)

where {d\j//dz)+, is given in Eq. (20), the derivatives of the effective mobility are obtained from Eq. (25), and An and Ap as defined above have been tabulated as functions of eif',/kT. The first two terms in the numerator of Eq. (28) are due to the number of electrons and holes in the space-charge layer changing, while the last terms arise from the fact that the effective mobility of the carriers already present in the space charge layer changes. In general both types of terms must be considered. The field-effect mobility is shown in Fig. 6 for intrinsic germanium. If surface scattering were not included, the mobility for a large surface dipole would

Ao-(Fa-O)

26 micromhos 10 micromhos 6.6 micromhos 4.0 micromhos

^•B

0.25 volt 0.22 volt 0.21 volt 0.20 volt

If one assumes the Fermi level at the surface is stabilized relative to the energy bands, as Kingston has proposed, 11 the effective mobility is then given by Eq. (25), where /S assumes a fixed value for each value of ^so and B depends linearly upon Va. Figure 7 indicates the mobility is reduced to about one third the bulk mobility for a reverse bias of several volts. ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Professor John Bardeen who suggested t h e problem and made many valuable suggestions during the course of the work. I am indebted to Dr. S. R. Morrison and Professor Harry Letaw, Jr. for several informative discussions and their critical reading of the manuscript. Thanks are also due to Mr. Walter Helly for carrying out the numerical integrations on the Illiac. 11 R. H. Kingston, Phys. Rev. 94, 1416 (1954).

317

Theory of Chemisorption" J.

R. Schrieffer

Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received 26 July 1971) The basic concepts in the theory of chemisorption of atoms on metals will be reviewed. The sharp atomic levels of the free atom are broadened into virtual levels by electron tunneling between the atom and the solid. For rapid tunneling, correlation effects on the adatom are weak and a selfconsistent field molecular orbital approach is appropriate. The adatom density of states can vary from a Lorentzian level to states split above and below the valence band of the solid, corresponding to the bonding and antibonding orbitals of a surface complex formed between the adatom and its surroundings. The Anderson model treats only Coulomb interactions on the adatom, while the CNDO method of quantum chemistry has been recently employed to include Coulomb interactions in the solid. For strong intra-atomic Coulomb interaction on the adatom, correlation effects are important and the localized magnetic moment on the adatom as predicted by the unrestricted Hartree—Fock approximation is quenched. Correlations have been treated by extended valence bond methods, multiple scattering theory, and configuration mixing techniques. Semiquantitative agreement with experiment is obtained in a number of systems.

I. Introduction The theory of chemisorption has long remained a challenging theoretical problem.1 The problem is complicated by the wide variation in the nature and strength of the adsorbate-substrate interaction as the properties of the free surface and the adsorbed species vary. In this review we concentrate on the adsorption of simple atoms on metal surfaces. Even in these systems, current theoretical treatments are a t best semiquantitative and many basic questions remain to be answered. 2- ' 2 To fix ideas, consider a neutral atom far from a clean metal surface, as shown in Fig. 1. If the ionization energy I of the atom is less than the work function $ of the metal, an electron will be transferred from the atom to the solid. Alternatively, if the affinity energy A of the atom is greater than 4>, an electron will be transferred from the metal to the atom. In these cases, the chemisorption bond will tend to be ionic, although level shift effects enter when the adatom is in contact with the surface which can alter this conclusion. If I> and A <<£, the atom is stable in its neutral state and the bond will tend to be neutral. For an open shell atom, the difference between the ionization and affinity energies is due to the Coulomb repulsion U between the valence electrons and the added electron. For hydrogen, * This work was supported in part by the National Science Foundation and the Advanced Research Projects Agency through the Laboratory for Research on the Structure of Matter, University of Pennsylvania. Reprinted

1 = 13.6 and A =0.7 eV so that C/= 12.9 eV. Since 0 is typically 4-6 eV, one expects that hydrogen would be adsorbed in the neutral state. However, when alkali atoms are adsorbed on tungsten, they tend to loose their valence electron to the solid, since the work function of W is 5.85 eV and / varies from 5.36 eV for lithium down to 3.87 eV for cesium. When the adatom is brought toward the surface so that it interacts weakly with the metal, the atomic states are no longer exact energy eigenstates of the system. since electrons can tunnel between the adatom and the solid, broadening the adatom states.13 Thus, an affinity level initially above the Fermi level E? will develop a tail that extends below £ ? and becomes partially occupied as shown in Fig. 2(a). On the other hand, an ionization level initially below E? will have a tail extending above EF and will be partially empty as in Fig. 2(b). Thus the total density of states on the adatom is the sum of these two broadened levels one centered at — I and the other at — I+U= -A. In the Hartree-Fock (HF) approximation, the oneelectron wave functions \j/n extend over the entire system, including the adatom, with the electron density on the adatom being composed of the sum of many small contributions from each of the occupied single-particle states. Since the HF approximation

FIGURE 1. Electronic

levels for atom from surface.

from: THE JOURNAL OF VACUUM SCIENCE AND TECHNOLOGY

far

\$~f

i ^ T

£F >/////////A %%%%?

\IV If"

VOL. 9 NO 2

561

318 56i

H. Schrieiiei

FIGURE

2.

(a)

The

adatom

virtual level centered above the Fermi level, (b) The adatom virtual level centered below the Fermi level,

treats only the average charge density in the system the mechanism which gives rise to the two levels in the atomic limit discussed above, namely different discrete charge states of the adatom, is absent in the (restricted) Hartree-Fock approximation. The unrestricted HF approximation allows for different H F potentials V, for up and down spin a. In the limit that the adatom is far from the surface the up spin level will be at — / and the down spin level at — A, or vice versa, as in the atomic description. More complete analysis shows that due to interactions neglected within Hartree-Fock, the localized spin on the adatom is flipped rapidly by the solid and a spin-zero-state results. This dynamical quenching of the localized spin is the analog of the Kondo effect12-14 which occurs in bulk alloys and is related to the zero spin of a conventional electron pair bond in the limit that the interaction with the solid is strong. We first discuss the HF approximation of chemisorp tion and then consider the effects of correlation and spin quenching on the picture. II. SCF-MO Approach The success of molecular-orbital schemes in describing the structure and binding energy of molecules suggests that these schemes would also be useful in the theory of chemisorption. Coulomb forces between electrons play a very important role in the problem since they lead to large shifts of the atomic energy levels as the occupancy of these levels change. Thus, a self-consistent (Hartree-Fock) treatment of these interactions is essential. The simplest formal model to treat in this framework is that of Anderson15 which was originally introduced to describe dilute alloys and has been extended to chemisorbed atoms. 4 ' 8,910 The model assumes that the isolated solid with its clean surface

can be described in the one-electron approximation The single-particle states ^t(r) of the substrate behave as Bloch states deep in the bulk and have decaying tails extending into the vacuum. For simplicity, the" adsorbate is taken to have a single orbital


# H F = E «*«*» + E «»»»»«++ E (VkpCk,+cf. + Vkv*cv,+ck,) he

-U{n¥t){n.i),

(2.1)

¥

where cia and c,> create and destroy, respectively, electrons in the tth orbital with spin a, and is the corresponding occupation number operator. The states k and

(2.2) and the averages (w,„) are to be calculated selfconsistently. In the restricted HF approximation (nr,) is forced to be independent of spin a- while in the (unrestricted) HF approximation, solutions occur for large U in which ( n , t ) ^ < « ^ ) . The potential V is the HF potential for the substrate and V„,

-f

(2.3)

\pk* Vipdr.

Hav is diagonalized by the HF eigenfunctions !/•„, which are linear combinations of the \pk and


E t(0Cfl)

I (*.„*,) I

/_

A„(E)dE

= (nr.)

(2.4)

319 Theory or Chemiaorpiion

where A^iE), the local density of states or spectral weight function for the adatom orbital, is denned by AV.(E)=

£ |(*„,#>.)|«(£-0-

(2-S)

A<E:

riei

FIGURE 3. The level width r (£) and level shift A (£) functions for a symmetric band of width W centered at «..

563

n

By using.a. Green's function approach, one can readily show that for an assumed value of (nv,), A^, is given b y15

If the h states are expressed in terms of localized orbits j centered on the substrate atoms then

T(E)/ir A„.(E)--

[£-e„,-A(£)?+r<(£) The "level-width" function T(E) is r ( £ ) = x l \Vltf\H(E~ek)=Tr\Vkv\jLSN(E),

(2.7)

where N(E) is the density of k states of energy £ in the substrate. The "level-shift" function A(£) is the Hilbert transform of V(E), P /•- r ( £ ' ) A(£) = - / -dE',

2r=s*/r-2r|K*,U,W(«„)-

(2.9)

Alternatively, if Vtr is sufficiently strong that r ( £ ) is large compared with both W and the difference between «,. and tc, the level shift function A (E) is very large and has the general form sketched in Fig. 3. We see that for E above the top of the band, the level shift is positive while for E below the bottom of the band A(£) is negative. This feature leads to two sharp peaks of A »«(£), one below the bottom of the band and one above the top. Since r ( £ ' ) = 0 for | £ ' — e„| >\W, one can approximate £ ' in the denominator of (2.8) by tc and write

r(E')dE' 1 £-6,

L k

Ft I8

(2.11)

i

where the sum arises primarily from orbitals on the atoms in the immediate vicinity of the adatom. Since T (£) = 0 for £ outside the band, we see that A has delta-function peaks at energies £ given by E— e. • C l / ( £ - « . ) ] E \(J\V\
C E - e , , ) ( £ - * „ ) - - £ \(j\V\v)l*~0.

(2.121 (2.13:

(2.8)

where P denotes that the principle part of the integral is to be taken at the pole. For given (n^,_„) one can qualitatively understand the local density of states by studying two limiting cases. Suppose the solid has a single energy band of width W and centered at tc, which participates in the chemisorption bond. If Vkv is sufficiently weak, then r ( £ ) and A(£) are small compared with W and A has a narrow peak near «,, which is well approximated by evaluating V and A at E = t^,. If t^, is outside the band, we see from (2.7) that T(««,,) = 0 and Af,(E) is a delta function, i.e., a sharp bound state occurs at an energy slightly shifted from <»,. If tt. is within the band, A is a Lorentzian function of half-width r («,,„) centered at e*,+A(^«), i.e., a virtual bond state or a resonance level occurs. The width of the level 2r agrees with the uncertainty principle and the golden rule for a decaying state of lifetime T,

A(£) =

A ( £ ) = [ l / ( £ - e . ) ] E \{J\V\
(2.6)

(2.10)

This is just the secular equation for the bonding and antibonding levels arising from the adatom orbital mixing with a linear combination of localized orbitals which have been broken off from the solid. Thus, one has a surface complex formed from the adatom and a cluster of surface atoms with the bonding state of the complex below the band and the antibonding state above the band. 9 The self-consistency condition (2.4) together with the expressions (2.2) and (2.6)-(2.8) allow one to solve for A*, and (n^). It is easily seen that there always exists a nonmagnetic solution (n,t) = (»M) to these equations. If U is greater than a critical value Uc = l/Ar(EF), where Ar{Ep) is the density of states at the Fermi surface for the nonmagnetic solution. then the nonmagnetic solution is unstable: In this case an additional pair of solutions enter which are stable having (nvt)>(nri) or (»^>>(»,t). These magnetic solutions reduce to the correct atomic limit of a single electron in the

Ue. The chemisorption energy is given within the HF approximation by" 1 /•"•

A £ = E - /

• T J _«,

tan-'

T

r(«)

_

1

Lis— «»,. — A(e)J Xde-e¥.

<»„},

(2.14.)

where — r < t a n _ l < 0 . If bound states of energy e(. are split off below the band, then £ » *i« should be added to (2.14) and the inverse tangent is then defined by 0 < t a n - l < i r . This expression includes the shift in

320 564

J. R. Schneffer

L_J

1

i

1

1

6

L__

-^ >

1 /I Cr / l r Ti /Cu /NI

-

/

4 Cr v

UJ

-

I-Ti/

< TK 2

Cr /

/N>

-

Ni

0

i 2

Ti

1 4

'

1

6

|<5'|(eV)

FIGURE 4. The chemisorption energy AE, calculated in the Hartree-Fock approximation by Newns for the metals noted, plotted as a function of the hopping integral |/3' |.

energy due to the density of states of both the adatom and the solid changing when the atom is adsorbed. Frequently, it is assumed that the change of density of states in the solid is negligible (the compensation theorem 15 ). If N(e) varies appreciably on the scale of r, a significant fraction of the chemisorption energy arises from this change. Newns9 has carried out self-consistent solutions of AE as a function of 2Z* I ^*»>l2i f° r a density of states at the surface of the perfect crystal of the form given by a one-dimensional chain iV(<) = DV(0)Ao](eo2-*2)* =0

M<6° \*\>f.

(2.15)

The bandwidth 2e0, the Fermi level and the work function were chosen to represent the substrates Ti, Cr, Ni, and Cu. V was chosen to be 12.9 eV and «,"=—12.9 eV. Newns' results are shown in Fig. 4. Since | V^l1 is difficult to estimate accurately from first principles particularly when d orbitals are involved, Newns used the observed AE to deduce the interaction strength £ » | T/^l'sfl' 2 . The value of 0' is between 3.7 and 4.2 eV. The self consistent solutions turn out to be nonmagnetic ((»»t) = (w^)) for the observed values of AE, although Cu is close to the magnetic limit. For /3' large compared with the bandwidth of the solid, AE is given by the surface complex limit in which a surface atom or cluster of atoms is split off from the solid and bonded to the adatom. In this region AE= — c+2f}', where c is the energy required to break the surface atom with one electron free from the solid and 2/3' is the molecular binding energy resulting from two electrons occupying the bonding orbital. One sees that the surface molecule limit is not an unreasonable approximation even for A E ~ 3 eV. Since A£ is fit to agree with experiment, the only independent check of the theory is through the value of the charge q on the adatom. The change

in work function A on chemisorption is related to the charge transfer q-e by the dipole length d. Since the calculated excess of electrons on the adatom is of order 0.2 to 0.3, and A is observed to be 0.3 to 0.S V, one finds an effective dipole length of approximately 0.1 A, an order of magnitude smaller than a H-metal bond length. I t is possible that a major fraction of the charge is transferred parallel to the surface if the H is adsorbed at a concave site, although this appears unlikely for closely packed planes such as the (110) surface of the bee lattice and the (111) surface of the fee lattice. There are several obvious shortcomings of the H F approximation to the Anderson model which could account for at least part of this discrepancy. First, electron-electron correlation effects will reduce the energy of the system by keeping electrons apart and thereby reduce the value of $' required to obtain agreement with the observed A£. Nonorthogonality of the adatom and solid wave functions will lead to the same effect. The decrease of /9' tends to make the system magnetic, a regime in which the H F approximation is not valid. An additional difficulty is the neglect of Coulomb forces other than on the adatom eliminates image potential effects, an important effect for alkali atom adsorption. Image potential effects in the presence of virtual levels have been discussed by Gadzuk and others. 1017-21 It appears that the Gomer and Swanson expression ZV/4(r-f-&, _1 ) is a reasonable approximation for the image potential experienced by a fixed charge Ze a distance r from the surface where k,~L is of order the Thomas-Fermi screening length. In practice one must take account of the fact that the virtual level, within the HF approximation, changes its width and its mean energy as the adatom is brought toward the surface so that the charge Ze on the adatom varies as a function of distance from the surface.10 Bennett and Falicov" have also treated the virtual level problem with image forces. They assume that the overlap of t//k and p comes primarily from the region outside the metal. The classical image potential is used and the energy of the adatom level is set by Slater's semiempirical rules for the atomic energy levels as a function of charge in the various atomic states. A self-consistent solution of the charge on the adatom is then carried out. Estimates of the virtual level width were made for K, Rb, and Cs adsorbed on a free electron metal with an abrupt jump in the potential at the surface. Similar estimates have been made by Gadzuk10 and it is found that A £ ~ l . 5 - 3 . 0 eV for the metal-adatom separation being of order 2.5-3.0 A. Recently, Bennett, McCarroll, and Messmer23 have investigated the chemisorption of H, C, N, O, and F on graphite by using the extended Hiickel theory

321 Theory of Chemisorption

(EHT) and the CNDO (complete neglect of differential overlap) methods. An advantage of such calculations is that the detailed geometric arrangement of the surface can be handled naturally, rather than using one-dimensional models,9 parametrized wave functions or matrix elements,4 etc., as was done in most earlier calculations. Earlier calculations of Coulson and Blyholder8 on short chains of atoms showed that the chemisorption energy is not very sensitive to the length of the chain. The CNDO approach as opposed to the EHT explicitly includes Coulomb interactions between electrons and between cores thereby stabilizing the charge transfer effects which lead to the absence of a minimum in some of the EHT calculations. Periodic boundary conditions were imposed on an 18 atom carbon layer. While the absolute magnitude of the calculated binding energies are much too large (by a factor of 3 or more), it is argued that locations of stable adsorption sites, the relative binding energies and charge transfer for a series of atomic adsorbates can be predicted. Another approach is through a moment method applied to the tight-binding model of the metal and die adatom, as Cryot-Lachmann and Ducastelle24 have discussed. The moment method is based on the identity

/;

« v . (•)«*««<« I » • ! « > .

(2.16)

relating the local density of states on atom a to the diagonal matrix element of the tight-binding Hamiltonian taken with respect to the ath localized orbital. By summing over atoms with and without the adatom present, one can approximately deduce the shift of the density of states and energy. This technique has been applied to the adsorption of transition metal atoms on tungsten, as studied experimentally by Plummer and Rhodin." The theory predicts that the binding energy varies as a quadratic function of the valence difference ZA.—ZU between the adatom and the substrate atoms, in rough agreement with experiment. While the method allows one to include crystalline structure and the degeneracy of the d bands, it does not include the Coulomb potential in a consistent manner. The theoretical curve has a considerably broader maximum than the experimental curve, as shown in Fig. 5, a result arising from the neglect of correlation effects on the adatom, as Newns 2 ' has discussed. Allan and Lenglart27 have studied the binding energy of an atom adsorbed on the (100) surface of a crystal described by a tightt binding s band scaled by a factor of five to represent d bands. They include Coulomb interactions on the adatom by using the Friedel sum rule to obtain self-consistency. As for the moment calculations, a rounded curve is obtained for A£ vs ZA.— Zu-

565

FIGURE 5. The chemisorption energy for transition series atoms adsorbed on W. The theoretical curves are the results of the Cyrot-Lachmann and Ducastelle moment analysis.

III. Electron-Electron Correlation Effects The SCF molecular orbital schemes, while giving a reasonable zero-order picture of the chemisorption bond when the bond is very strong so that T>U, overestimate the importance of Coulomb interaction between electrons when the bond is weak. There is a competition between the one-body terms in the Hamiltonian which tend to delocalize the electrons, and the Coulomb repulsion between electrons which tends to localize the electrons by keeping them far apart. The H 2 molecule is a well-known example of this competition. The importance of correlation effects in bonding is measured by the ratio of the effective Coulomb interaction U (of order the difference of the nearest-neighbor interatomic and the intraatomic interactions) and the splitting energy Ae between the bonding and antibonding molecular orbitals. For U/Ae>l correlation effects are important. In the chemisorption problem, the width of the virtual level T plays the role of the molecular-orbital splitting energy and correlation effects are important for E7/nT> 1, particularly if there is of order one or more electrons on the adatom which are participating in the bonding. A number of techniques have been advanced for including correlation effects in the theory, several of which we review below. Van der Avoird11 has developed a scheme which treats both the substrate atoms and the adatom in the Heitler-London or valence bond scheme. Following the early work of Eisenschitz and London,28 van der Avoird" has developed a technique based on t h e valence-bond scheme which takes proper account of the symmetry of the wave function in each order of perturbation theory. This description is best suited to adsorption on insulators in which localized states can be properly used to describe the solid. For metals, the valence-bond method is questionable since the electrons are delocalized and the bandwidths are generally larger than the chemisorption energy and

322

"HH

(O

Another approach to strong electron-electron correlations has been given by Wojciechowski.7 He treats the exchange repulsion between the adatom and the doubly occupied band states of the metal in the same manner as Pollard,30 by taking account of the nonorthogonality of the adatom and metal states and using first-order perturbation theory. Wojciechowski assumes that states with two electrons on the adatom are very high in energy and therefore he admixes only ionic states with an electron transferred from the adatom to the solid in states k above the Fermi surface. The wave function of the system is than given by

O>

FIGURE 6. Contour map of the interaction energy (in atomic units, 27.2 eV) of the HjPtj system studied by van der Avoird.

cannot be neglected. In first order, the interaction energy is the result of a conventional valence-bond calculation, •,-<*,|il7|#,>/<#.|il|*,>,

(3.1)

where A is an operator which projects onto states of the proper symmetry and V=H—H0. H0 describes the system of noninteracting atoms. In second order, if one uses the Unsold average energy denominator, 2 ' one has *=s-(l/A£*„)C((#o| VAV\*,)/(*,\A

|*„»-eiJ]. (3.2)

This result corresponds to the van der Waals or dispersion interaction plus an exchange polarization energy in which the atoms are virtually excited by an exchange of electrons. Using this formalism van der Avoird has treated the problem of Hj interacting with two Pt atoms as shown in Fig. 6. He finds that there is no activation energy required to dissociate Hi when it is roughly 0.75 A above the P t atoms, in agreement with experiment. For simplicity, the Pt d electrons are replaced by an effective s orbital containing one electron. The adsorption energy arising from these unpaired spins is 75 kcal/mol while the experimental value is 28 kcal/mol. A repulsive contribution arising from the bonding d orbitals and the conduction electrons lowers the adsorption energy somewhat, a rough estimate of the net binding energy being 45 kcal/mol. A more complete analysis would include a larger number of atoms representing the solid, and the explicit treatment of energy band effects with these orbitals. The present treatment does show that a large fraction of the interaction energy comes from three and four body forces when the hydrogen atoms are separated from the Pt atoms by less than 1 A. Thus, it appears that pair-wise chemical bonds are not sufficient to describe this system in the interesting range of parameters.

¥ =¥„ +

E a***.-

(3.3)

Since the metal electrons are not excited in these configurations, the adatom electron is simply trans'ferred back and forth to the metal without spin flip. Thus, the scheme is similar to MuUiken's theory of donor-acceptor complexes with no electron pair bond being formed. The approach appears to overestimate the charge transfer and gives too weak a binding because of the lack of mixing of


* = * , . + E «***,+ E 6***..

(3.4)

where **,_, for kr so that adion states have small weight, they obtain an antiferromagnetic exchange interaction ff«xoh = N ' E

Jkk'Cv,+Ck,C„+Cr,

(3.5)

between the adatom spin and the spin density in the metal (iV is the number of atoms in the substrate). If the width Wb of the conduction band is large compared to J one can treat H„cj, by first-order perturbation theory and the conventional exchange

323 Theory of Chemfsorptlon

FIGURE 7. The interaction energy &E divided by the energy of the free surface complex Wm, plotted as a function of the bandwidth divided by W„. The weak and strong-coupling perturbation-theory results and an interpolation between these results for intermediate coupling are sketched for the induced covalent bond theory.

567

AE Wm

Wb

repulsion is obtained. For Wb of order J, a delocalized spin polarization is induced in the solid surrounding the adatom. The induced spin arises from both the nonspin flip (s = W\,) limits and an interpolation is made for intermediate coupling strength. A sketch of the results of Paulson is shown in Fig. 7 for a hydrogen atom adsorbed on the (100) surface of a simple cubic metal directly above a surface atom. Newns" has recently treated the problem of two transition metal atoms, including correlation effects within the basis set of the d shells on the two atoms. One atom was taken to have five d electrons representing tungsten and the binding energy was studied as a function of the number of d electrons on the other atom for a range of values of the intra-atomic Coulomb interaction. It was found that correlation effects were most important for two half-filled d shells, with the H F solution being approached when one or nine electrons are in the d shells. Correlations tend to make the adsorption energy vary as the triangular shaped curve A£ = £ 0 | 5 - » | for n electrons on the second atom as is observed experimentally," rather

than the bell-shaped curve given by HF theory, as shown in Fig. 8. One can interpret Newn's result as arising from a bond energy for each of the 15-n \ unpaired spins. Recently, Gadzuk, Hartman, and Rhodin10 have treated correlation effects in the problem of alkali HARTHEE-FCCK

FIGURE 8. Plot of the theoretical binding energy AE for two transition metal atoms as a function of the filling of the valence shell of one of the atoms. The second atom is taken to have a half-filled valence shell. The curves are numbered by values of U/ff where 0 is the hopping integral. Relative experimental binding energies are given by broken lines as a function of the number of d electrons N for N<5 and by dash-dot lines as a function (10 -N) for N>S. Experimental AE values (Ref. 24) have been scaled to pass through the maxima of the lower two curves. Arrows indicate the magnetic transition points, with the magnetic region to their left. Horizontal bars give the experimental AE, that for Ti being an upper limit.

324 568

J . R- Schrieffer

atom adsorption. They employed an effective Coulomb potential V,u which was derived by Schrieffer and Mattis" for the low-density limit, i.e., the number of electrons or holes on the adatom involved in the bonding is small compared to unity. For U large compared to the level width T, if the center of the virtual level is above the Fermi level and only a small tail is below Ey, correlation effects are unimportant since there are rarely two electrons simultaneously in the virtual level even if correlations are neglected. As the center of the virtual level sinks toward Ep, a larger probability of two electron occupying the virtual level occurs and U,u becomes important. One finds that regardless how large U is, U.H is always such that no localized moment occurs on the adatom, in agreement with the Heitler-London-like theories which hold for approximately one electron on the adatom. Gadzuk and coworkers obtain reasonably good agreement with experiment using a plausible adatom surface spacing.

IV. Conclusion While considerable advance has been made recently in the theory of chemisorption, including the selfconsistent treatment of Coulomb interactions between electrons, the effects of electron-electron correlations and the explicit treatment of surface geometry, the calculation of the magnitude of binding energies is still far from being quantitative. The alkali adsorbates are the most readily handled in a quantitative manner.10 It appears that the HF approximation is invalid for H adsorption on metals although it may be appropriate for the chalcogenide adsorbates since the intra-atomic Coulomb interaction is smaller for these heavier atoms. In the future, a more detailed treatment of the influence of surface geometry on chemisorption may be expected, as well as the inclusion of multiple bands in the solid and several orbitals on the adsorbate. The inclusion of configuration interaction in the HF scheme and ionic effects in the valence bond approach will give a more reliable theory in intermediate coupling cases vV^iU. The theory of interactions between adatoms is also progressing so that hopefully one will be able to account for the structures observed in adsorbed layers. 3 *" In addition, work is beginning on the kinetics of desorption." The theory of chemisorption should remain a fertile area for many years ahead.

References 1. For a review of the literature prior to 1967 see L. W. Swanson, A. E. Bell, C. H. Hinrichs, L. C. Crouser, and B. E. Evans, "Vol. II, Literature Review of Adsorption on Metal Surfaces," Final, .NASA Rep., Contract NAS-8910, July 1967; H. Moesta, Chemisorption und Ionization in Metall-Meta.ilSystemm (Springer-Verlag, Berlin, 1968). 2. G. R. Baldock, Proc. Camb. Phil. Soc. 48, 457 (1952). 3. T. Toya, J. Res. Inst. Catalysis (Japan) 6, 308 (1958); 8, 209 (1960). 4. T. B. Grimley, Proc. Phys. Soc. (London) 72, 103 (1958); 90, 751 (1967); 92, 776 (1967); Advan. Catalysis 12, 1 (1960); J. Am. Chem. Soc. 90, 3016 (1968); Molecular Processes on Solid Surfaces (McGraw Hill, New York, 1968), p. 299; J. Physique 31, Cl-85 (1970). 5. J. Koutecky, Trans. Faraday Soc. 54, 1038 (1958); Surface Sci. 1, 280 (1964). 6. A. J. Bennett and L. M. Falicov, Phys. Rev. 151, 512 (1966). 7. K. F. Wojciechowski, Proc. Phys. Soc. (London) 87, 583 (1966); Acta Phys. Polon. 29, 119 (1966); 33, 363 (1968). 8. C. A. Coulson and G. Blyholder, Trans. Faraday Soc. 63, 1782 (1967). 9. D. M. Edwards and D. M. Newns, Phys. Lett. 24A, 236 (1967); D. M. Newns, Phys. Rev. 178, 1123 (1969). 10. J. W. Gadzuk, Surface Sci. 6, 133 (1967); 6, 159 (1967) and in The Structure and Chemistry of Solid Surfaces, edited by G. A. Soraarjai (Wiley, New York, 1969); J. W. Gadzuk, J. K. Hartman, and T. N. Rhodin (to be published); G. Allan and P. Lenglart, J. Physique Suppi. 31, Cl-93 (1970). 11. A. van der Avoird, Surface Sci. 18, 159 (1969). 12. 1. R. Schrieffer and R. Gomer, Surface Sci. 25, 315 (1971). 13. R. W. Gurney, Phys. Rev. 47, 479 (1935). 14. J. Kondo, Solid State Physics (Academic, New York, 1969) Vol. 23, p. 183. 15. P. W. Anderson, Phys. Rev. 124, 41 (1961). 16. T. B. Grimley, J. Phys. C 3, 1934 (1970). 17. J. H. de Boer, Advances in Catalysis (Academic, New York. 1956), Vol. III. 18. N. S. Rasor and C. Warner III, J. Appl. Phys. 35, 2589 (1946.) 19. J. R. MacDonald and C. D. Barlow, Jr., J. Chem. Phys. 40. 1535 (1964); 43, 2575 (1965); 44, 202 (1966). 20. L. D. Schmidt and R. Gomer, J. Chem. Phys. 42, 3573 (1965). 21. R. Gomer and L. W. Swanson, J. Chem. Phys. 38, 1613 (1963). 22. D. M. Newns, J. Chem. Phys. 50, 4S72 (1969); Phys. Rev. B 1, 3304 (1970). 23. A. J. Bennett, B. McCarrolI, and R. P. Messmer, Phys. Rev. B 3, 1397 (1971); Surface Sci. 24, 191 (1971). 24. F. Cyrot-Lachmann and F. Ducastelle, Phys. Rev. (to be published). 25. E. W. Plummer and T. N. Rhodin, J. Chem. Phys. 49, 3479 (1968). 26. D. M. Newns, Phys. Rev. 25, 1575 (1970). 27. G. Allan and P. Lenglart, J. Phys. (Paris) Suppi. CI, 93 (1970). 28. R. Eisenshitz and F. London, Z. Physik 60, 491 (1930). 29. A. Unsflld, Z. Physik 42, 563 (1927). 30. W. G. Pollard, Phys. Rev. 60, 578 (1941). 31. J. R. Schrieffer, J. Appl. Phys. 38, 1143 (1967). 32. J. R. Schrieffer and D. C. Mattis, Phys. Rev. 140, A1412 (1965). 33. J. Koutecky, Trans. Faraday Soc. 54, 1038 (1958). 34. T. B. Grimley, Proc. Phys. Soc. (London) 90, 751 (1967); 92, 776 (1967), T. B. Grimley and S. M. Walker, Surface Sci. 14, 395 (1969). 35. H. Suhl, J. H. Smith, and P. Kumar, Phys. Rev. Lett. ZS, 1442 (1970).

325 PHYSICAL REVIEW B

VOLUME 7 , NUMBER 8

15 A P R I L 1973

Indirect Interaction between Adatoms on a Tight-Binding Solid* T. L. Einstein and J . R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania (Received 24 July 1972)

19104

The indirect interaction between adatom pairs on the (100) surface of a simple-cubic tightbinding solid is investigated within a molecular-orbital approach. A general scheme for calculating the surface-denslty-of-states change and the Interaction energy of one and two singlelevel adatoms is presented, and contact (and a correction) is made with Grimley's formulation. The method permits binding above surface atoms, at bridge sites, or at centered positions, and yields interaction energy as a function of band filling, adatom energy level, and a general hopping potential V between an adatom and the nearest surface atomfe). Calculations have been carried out for V/Wb in the range 1/12-1/2, the upper limit giving split-off states (W6= bandwidth). The single-atom interaction shows little dependence on binding type, in all three cases being most attractive when the Fermi energy equals the noninteracting adatom level, with a strongly V-dependent strength. For the pair interaction, one finds a strength at nearest-neighbor separation of about an order of magnitude smaller than the absorption energy of a single adatom. This interaction has an exponentiallike dropoff and sign alternations as one moves along the (10> direction. Under reasonable conditions, the nearest-neighbor interaction is often repulsive while the next nearest, third nearest, or fourth nearest is attractive, suggesting the patterns c(2 x 2), (2 x 2), and c(4x 2), respectively, which are frequently observed in the adsorption of simple gases on the (100) surfaces of transition metals. On the basis of two-dimensional Ising-model calculations including second-neighbor interactions, one can estimate the strength of V from the observed disordering temperature of the adatom lattice; the result is similar to that obtained from estimates based on the heat of adsorption.

I. INTRODUCTION An intriguing aspect of the chemisorption of simple g a s e s onto transition m e t a l s i s the variety of surface s t r u c t u r e s observed. Below a critical temperature, the pattern within surface adlayer islands i s highly dependent on the substrate, but not always a r e these a r r a y s identical in structure to the surface layer of the adsorbate. The n e a r e s t site occupied (relative to some occupied site) i s often the second, third, o r even fourth n e a r e s t neighbor r a t h e r than the (first) nearest neighbor. l Over a decade ago, Koutecky 2 deduced that these adatoms interact (share electrons) via the subs t r a t e - m e t a l electron band and therefore exhibit oscillatory behavior reminiscent R u d e r m a n Kittel-Kasuya-Yosida (RKKY) i n t e r a c t i o n s . ' In the last s e v e r a l y e a r s , Grimley 4 - 6 and Newns 7 have formalized this "indirect interaction" on the basis of the analogy to the Anderson model for dilute magnetic i m p u r i t i e s . 8 However, the interaction energy has only been evaluated in the asymptotic regime a n d / o r in the simple (and often invalid) approximation that a virtual level could describe the interaction. This paper p r e s e n t s a first attempt ?.t the theoretical evaluation of the two-adatom indirect i n t e r action at nearby binding s i t e s . As such, we have allowed ourselves a number of simplifying a s s u m p tions. We deal with the (100) surface of a simple cubic semi-infinite lattice, using the formalism 7

and p r o g r a m of Kalkstein and Soven 8 to obtain s u r face Green's functions. Since we a r e i n t e r e s t e d in covalent binding, we work with t r a n s i t i o n - m e t a l s u b s t r a t e s , as these provide the strongest bond. A tight-binding model replicates the d band, and the sp band is neglected, a s in Grimley 4 and N e w n s . ' Thus the possibility of hybridization as an explanation for H on a W (100) surface, s u g gested by Tamm and Schmidt, 1 0 is excluded from the beginning. The adatoms a r e assumed to have a single energy level, a reasonable approximation for a simple gas atom o r an atom with a partially filled d shell. Correlation effects a r e not considered explicitly, but can be roughly accommodated by self-consistently adjusting the p a r a m e t e r s of the problem, such a s the adatom level. This a p proximation i s reasonable in s y s t e m s where t h e r e is only a weak tendency toward localized-moment formation on the adsorbate, i . e . , U/vT
326 3630

T. 32

31

•

•

Zl

ZZ

•

•

L.

•

34

33 Z3

24

•

•

•

•

• •

•

II

12

13

14

•

•

•

•

•

•

•

X

•

EINSTEIN

•

AND J .

• X

•

•

•

• X

•

X

•

•

•

•

FIG. 1. Charts indicating the labeling convention for the binding of adatom pairs on the (100) surface of a simple cubic crystal, and the four possible types of binding. The first diagram shows atop W) binding. One adatom is always taken to bind to site (11); the second can sit at any other side, and that site then labels the interaction. Thus, the pair-interaction energy and the Green's function corresponding to this chart are referred to as AWf3 and Gf3, respectively. The other charts describe the 23 interactions for centered (C), bridge(B), and bridge-perpendicular (BP) binding.

need only consider the one- and two-body t e r m s . The calculations distinguish binding directly above the surface (called "atop" o r A here, "and o n - s i t e " or "linear" sometimes elsewhere), at a centered site (symmetrically between four surface atoms; called C), o r at a bridge site (symmetrically b e tween two; called B). In the last case, we d i s tinguish bridge B and bridge-perpendicular BP binding in the pair problem: In the former case, a line joining the two surface neighbors of an adatom tends to be parallel to a line joining the two adatoms (i. e . , the angle between them i s l e s s than 45°); in the latter case, the lines a r e m o r e nearly perpendicular (angle g r e a t e r than 45°). If the two adatoms sit diagonally, the two cases become identical. Figure 1 i l l u s t r a t e s the four binding s y m m e t r i e s , and also d e s c r i b e s our labeling convention for surface s i t e s : We label s i t e s by the subscripts {ij) of a square lattice, so that (11) i s the "origin" and i, j a r e generally positive int e g e r s — s y m m e t r i e s in the interaction energy, and ultimately the surface G r e e n ' s functions, allow all discussions to be framed in t e r m s of one quadrant of the plane. In dealing with the problem of adatom p a i r s , we will denote the location of one adatom by (11) [i. e . , the lattice site (11) o r the associated symmetry site, a s illustrated in Fig. 1] and the other adatom at some other {ij). Since the pair interaction tends to be weaker than the variation in single-adatom adsorption energy with binding symmetry, we calculate h e r e only the case that both adatoms select the same binding symmetry. It i s straightforward to generalize the calculations to a more general situation. The pair interaction then determines the configuration within an island

R.

SCHRIEFFER

of adatoms: After t h e f i r s t adatom b o n d s , we visualize a second adatom approaching the surface nearby. Since any site (with the s a m e binding symmetry a s the first) will provide the s a m e onebody energy lowering, it i s the p a i r interaction which d e t e r m i n e s the most favorable position for this second adatom. Correspondingly, a third adatom will bind with the same site s y m m e t r y a s the first two at a relative position that i s most favorable in t e r m s of the sum of its two-body interactions with them. We view this p r o c e s s as continuing a s m o r e adatoms join the domain. Thus, from a chart of the p a i r interaction it is generally easy to determine the s t r u c t u r e of an island. Section n d i s c u s s e s in detail the analytic p r o p e r t i e s of the solution. Contact i s m a d e with Grimley's w o r k , 4 ' 6 and many of his approximations and derivations (that p a r a l l e l ours) a r e d i s c u s s e d . The r e a d e r who is principally i n t e r e s t e d in c a l culational r e s u l t s should r e f e r directly to Sec. Ill, where computations for the interaction energy for a single adsorbed atom and between two nearby adatoms a r e presented. General c h a r a c t e r i s t i c s for a large range of input p a r a m e t e r s a r e d e scribed, and experimental information (binding energy, s u r f a c e - a r r a y c r i t i c a l t e m p e r a t u r e for disordering, probability of occurrence of p a r t i c u l a r adlayer patterns) i s used to evaluate the adatom-surface hopping p a r a m e t e r and to check that the calculational r e s u l t s a r e r e a s o n a b l e . Section IV s u m m a r i z e s the investigation and d i s c u s s e s possible extensions. II. MODEL A. Green's Functions In our model we consider a perturbed H a m i l tonian of the form X = X0 + V. 3C0 i s the H a m i l tonian for a semi-infinite solid, with e i g e n s t a t e s \k) and eigenvalues e„ plus the Hamiltoman for the isolated adatoms relevant to the problem, one o r two h e r e , with eigenvalue Ea. The potential V connects the adatom state la) with a binding site l i ) , allowing hopping between them. In this paper we r e s t r i c t ourselves for simplicity to t h e (100) face of a simple cubic c r y s t a l . \ij) d e n o t e s the Wannier function associated with site {ij). F o r binding directly above a surface atom. I t ) = 111); for binding at a bridge site, |f> = ( | l l > + | i a »

[or | 0 = (|11> + | 2 1 » ] ;

and for binding at a centered site, |0-i(|u>+|i2>+|2i>+|22>)

.

The semi-infinite metal XQ we use h e r e i s that of Kalkstein and Soven, 9 since we will u s e t h e i r Green's-function calculational technique in comput-

7

INDIRECT INTERACTION

BETWEEN ADATOMS

ing the interaction energy. Using linear-combination-of-atomic-orbitals (LCAO) o r tight-binding formalism, the most realistic simple approximation for rf-band metals, they start out with a p e r fect-crystal Hamiltonian Kj with only one-center and nearest-neighbor two-center r e a l - s p a c e m a t r i x elements. The one-center element, i . e . , the atomic-orbital self-energy, implicitly defines their energy z e r o . We take their isotropic twocenter element £ x to be - T, where T i s a positive number. In three dimensions, these choices lead to an energy band of width Wb= 12T, which is centered at the energy z e r o . Unless otherwise stated, we will m e a s u r e energy in units of IT (or equivalently \V/^, and will thus drop 2T when clarity does not require it. Kalkstein and Soven then add a potential to cancel interactions a c r o s s the cleavage line (which divides the metal into semi-infinite halves), and a perturbation V to a c count for surface effects such a s electron r e d i s t r i bution. Since we a r e not putting in Coulomb c o r relations, we neglect V. Invoking periodicity in the two directions parallel to the surface, one can show that the one-electron G r e e n ' s function for an atom on the perfect surface i s G u (£,k l ,) = ( l / 2 r 8 ) [ + ( i J + t ( 4 r 8 - l l . 1 ! ) 1 / 2 ] ,

(2.1)

where for our simple surface co = £ + 2T(coskxa0

+ cos*,,ao) ,

(2.2)

a 0 being the lattice p a r a m e t e r and the root is + i s g n o ) ( u ) 2 - 4 T 2 ) 1 / 2 for o> 2 >4T 2 . To obtain this "atop" surface Green's function, one must perform a sum over the surface Brillouin zone: Gft.«"Gfi(*)«

S

G 11 (B,k,)e 1I . , <«ir 1I u> •

tv.
(23)

Since the first index of G will generally be 11, one can usually omit it, as indicated, with no loss of clarity. The following r e m a r k s concerning the Green's functions calculated by the Kalkstein-Soven program a r e helpful in the subsequent analysis, (i) The atop diagonal G r e e n ' s function can be r a t h e r well approximated by a triangular density of states (with E in units of ZT)—although this simplification was not employed in the present calculations: lmG&{E)

J i i r U -i\E\),

\E\ <3 \E\ >3

3+ £

3-E

The sharp c o r n e r s of the triangle obviously a r e smoothed in the actual result. This gives a b e t t e r fit than the semi-elliptical density of states of the one-dimensional chain 7 :

2 u SH9 •E ) \ 1°.

3631

l£l<3 \E\ >3 (2.5)

ReG4(£)-^[£:p(£i!_9)1/2]>

| £ | >

3 and E £ 0.

(ii) As suggested in (i), the imaginary p a r t of G*X{E) i s s y m m e t r i c about E = Q, the c e n t e r of the band, while ReG^tE) i s a n t i s y m m e t r i c . T h i s r e sult a r i s e s a s follows. G n ( £ , k(l) is s y m m e t r i c in k,„ i . e . , kx and ky: Gu(E, £„) = G u ( u ) l E , k j ) , w h e r e ai = £ + 27'(cosfe J a 0 + cos£j,a 0 ), and coska0 is s y m metric in k. Hence the sum over - u/a < k„ kv< Tt/a can be reduced to four t i m e s the s u m o v e r a quadrant, say, 0 ) = -ReG u (co) and ImG u ( - c j ) = ImG u (co). Since this is true for all *,„ and hence for all k„ in the quadrant, the s y m m e t r i e s of ReG5\(£) and ImGfttE) follow. T h e fact that the r e a l and imaginary p a r t s of G have o p posite symmetry can be viewed as a result of the fact that they satisfy a K r a m e r s - K r o n i g r e l a t i o n . (iii) The symmetry of Giltij(E) a l t e r n a t e s with each lattice spacing of separation, i . e . , ImGf,(- E) = ( - l)"""ImGfj(£), (2.6) ReGf,(-E) = - ( - i r * " R e G f , t E )

,

where m-i-\ and n = j - 1. By using the s y m metry in k„ which p e r m i t s the reduction of the k„ sum to a_quadrant, we see that the contribution of g)!„-iR u -R u ) i s j U S t cosTOfe.OQeosnfejaD. In p e r forming the a) inversion of (ii), i. e . , kx- n/a - kx, e t c . , we use the fact that cosmkxaa = ( - l) m cosradr - kxa) and similarly for ky, and the above s y m m e t r y i s obtained. (iv) Since, as we saw in (iii), the distance factor e n t e r s only through the factor cosmkxa0cosnkJ,a0, we can write down various m i r r o r equalities [where the 2's a r e required by our choice of (11) rather than (00) as the origin]: Gt,,j{E)

(2.4)

ReG^)^£lnl(3+£»3-£)L|ln

ImG>,(£)

ON A . . .

= GtJ(E) = Glz.J(E)

(2.7)

to account for G r e e n ' s functions outside the quadrant i, j i l . Moreover, by the symmetry in kx and ky in the summation, we find G*S(E) -G^,(E). With the aid of these t h r e e equalities, we can write any G r e e n ' s function Gulk,,.(E) as Gfj(£), w h e r e ; i s the l a r g e r of \k -k'\ + 1 and I f - 7 1 + 1 , and i i s the s m a l l e r ; that i s , we can reduce all formulas in GA into expressions of G A 's in the octant b e tween (10) and (11), inclusive.

328 3632

T.

L. E I N S T E I N AND J .

(v) It is easy to check that JiaGfjiE) vanishes outside the band, i. e., for I El -3. Then to2 is always 11 (i. e., 4T2), so that there is no imaginary part in GniE, k„). For E just above the bottom of the band, u>2< 1 only for kx and ks nearly zero, so that coskxa0 + coskya0~ 2. For such E, the leading factors cosm£xa0cos»£,,a0 will be near unity, or at least positive. Hence we have the general result that for any ij, ImG*t(E) increases from zero as one initially increases energy from the bottom of the band: the extremum nearest the bottom is positive. By similar but more tortuous reasoning, or more simply by looking at the Kramers-Kronig relation

ReGft(£) = - - < p J

E,^£

'dE' ,

one verifies that ReG^tE) is negative at the bottom of the band, for any ij. A more detailed look shows that ImGlE) is initially proportional to the f power of the magnitude of energy difference from the upper- or lower-band edge. (vi) The number of extrema in the parts of G^ increases rapidly with i and j . For i, j f 3, there are i+j-l (rn+n+ 1) extrema in ImG^lE), or equivalently, there are tn+n internal zeros, or i+j zeros including endpoints. If i" or j 2 4, there are more than m + n + 1 extrema, although no simple empirical formula seems to fit the computational result. For use in Sec. Ill, we point out in particular that (ij) = (14), (15), and (16) have 4, 9, and 10 extrema in lmGfj(E), respectively. From (iii), we find that if one of the indices is raised by one, the number of extrema must increase by an odd number. With the Kramers-Kronig relation one can demonstrate the observed result that ReG?,(E) has one extremum more than imGft(E). To determine the Green's function for other binding positions, we note that

G )=

n

(2 8)

'^ ^I^CT| )

-

formally. To express bridge binding of the same separation, a0[(i - 1)2 + {j - l) 2 ] 1 ' 2 , with the adatom's surface neighbors both in the same row (toriij), we substitute GfJ(E) = Gtj(E) + i[GtiJ.1iE)

+

Gt,Jtl(E)]

.

R. S C H R I E F F E R

substitute |lD8P = i(|H>+|21» and Bp
G^(E) = GJ1(E) + \[Gtu,^)*Gtui^)] • ( 2 - 1Q ) The assumption liiZj is convenient, and possible, as elaborated in remark (iv). Moreover, should an index not be positive in a general formula, remark (iv) indicates how to reduce the errant GA to the selected octant. Thus, for example, Gft = Gfj + Gi2. Note also that Gf,p = Gf(. Finally, for centered binding, one makes substitutions of the form | y ) c = » ( | y > + \i+i,j)+

|»,j+i>+ | i + i , j + i »

for \ij) to find, for example, that Gfj = Gfj + 2G*2 + G&, using the two mirror symmetries of (iv). Figure 1 illustrates the different types of binding. We present two additional remarks concerning the surface Green's functions with any of the four binding position symmetries. (vii) ImCf^E) and ImGftlE) are both positive throughout the band. Recall ImGftCE) = ImG?1(£) + ImG£(£) =£

(l + cosA;xa0)ImGCE, kh) ,

where ImG(E, k„) £ 0. But 1 + coskxa0 > 0. Similarly, for ImG&lE), the factor is (1 + cosftjOo + cosfcj,a0 + coskxaB coskya0) = (l + coskxa0)(l + coskya0)Z0 By the symmetry in kx and ky, one might write (1 + 2 c08kxaB + cos&jOo cosfej,a0) in actually performing the sum. Figure 3, which reproduces the calculated ImGfi, confirms the claim of this remark. (viii) ImG^Us) 2 I lmGfj(E) I, i . e . , ImG£(£) =F JmGfjiE) 2 0. This difference (sum) is just the sum over the positive Gn{E, k„) times the factor

(2.9)

The choice of the symmetric combination of perturbed atomic orbitals indicates the selection of the bonding state, i . e . , the combination lowering the metallic energy distribution Apf/E) - (l/jr)ImGfy(2?). Section HI elaborates on this manifestation of the importance of the symmetry of the orbitals, describing the skewing of Ap*(£) and its effect on the binding energy of adatoms. Alternatively, to have the antidomain, with the bridge neighbors in the same column, we would

7

1 Tcos{mkxa(l)cos(nky a0) ^ 0 . More generally, we can apply the Schwarz inequality since ImGfy is a Hermitian inner product on a Hilbert space:

Thus

|imGf,te)| = |<*x>! 2[(x<0|,|0>x)(xx>] , / 1

.

329 INDIRECT

*

/ ,

1

•

1

INTERACTION

•

I *

BETWEEN

I r--3»

K=X0+ V (3C and XQ have eigenvalues E, and ilt respectively) can be derived a s follows. We o b s e r v e that

Det

h = a0 A/2 FIG. 2. Diagram illustrating a possible system in which V is the same for atop (A), bridge (B), and centered (C) binding: for approximately spherically symmetric orbitals this statement about the potential is true if the adatom is the same distance h from its nearest substrate neighbor(B) regardless of binding type. The minimum value for h is a 0 /V2, for which a "centered" adatom lies in the surface plane. Although convenient, the assumption of the same V for all binding types is not at all necessary in our analysis.

= |lmGf : (£)|=ImGf 1 (E)

,

(2.11)

where the last step follows from (vii), and the p r e ceding step from surface translational lattice symmetry. The potential V r e p r e s e n t s the entire interaction of an atom with the bulk; we assume no overlap with nearby adatoms. V i s nondiagonal. It has the form Val, or Vu- V*tl, w h e r e small italic l e t t e r s denote an adatom's noninteracting level and n u m b e r s signify a surface atom. We define V= 1 Va, I. F o r atop binding, an adatom electron can hop only onto the bulk atom just below it. F o r bridge (center) binding, it can hop to any of the two (four) n e a r e s t metal a t o m s . This potential p a r a m e t e r can be scaled separately for each of the three binding positions to fit single-adatom binding e n e r g i e s . F o r the sake of comparison, it is convenient to assume that r e g a r d l e s s of binding type, the hopping potential has the same strength: Vt„ = Val if adatom i and surface atom n a r e n e a r e s t neighbors, and 0 otherwise. This assumption would certainly be justified in the case of spherical orbitals and s i m i l a r a d a t o m - s u r f a c e - a t o m distances for the three types of binding. If we take the adatom to be in the surface plane for centered binding, then this distance h i s a0/ V"2, where a0 is the lattice constant. An atop adatom would be assumed to be this far above the surface, while for bridge binding it i s \a0 above the surface (45° angle) (cf. Fig. 2). This s c e nario i s not at all unreasonable as a f i r s t approximation, and we shall see that for single-atom binding, one gets very s i m i l a r binding energies for the three types at fixed V. B. Interaction Energy /. Basic Formation A general method to calculate the change in the density of s t a t e s of a perturbed Hamiltonian

3633

A D A T O M S ON A .

(Fr^r«)^-K-")=n E

-Cj-i6 (2.12)

Thus we can w r i t e

i

- Im -— lndetl(

=

^

I m

j \E-E,-i6

Y ){E~X-i6) ~

= S [S(E-Ej)-6(E-ej)]

E-fj-it) = Ap .

(2.13)

i

If we define G° = (E - K „ - i6)m\

d

then

K g -^-J U5 ~ 3C -' 5) =det(1- GV)

so that Ap = i Im ^ r In detU - G°£) 7T

(2.14)

or.

We select the representation with the e i g e n s t a t e s of 3C,j a s basis v e c t o r s , so that G° is diagonal, and the mn matrix element has the form 6m>„ - G° ,mVm„. If V has m a t r i x elements only between a set of s t a t e s 1, . . . , n, then outside the » x » submatrix in the upper left-hand c o r n e r , one will have only l ' s along the diagonal and 0 off it. Thus, the determinant of the m a t r i x is that of the submatrix. As Grimley 4 d i s c u s s e s , conservation of e l e c t r o n s implies rEr

.EF

J_^ ApdE

(2.15)

&EF Po(EF)

where A p = p - p 0 .

Hence

AW= 2( / _ / EpdE - Jf = 2{l*"{EAp

+

EpadE) EFAEFp0{Er)]dE}

= 2 / _ / {E-EF)&pdE

.

(2.16)

The factor of 2 accounts for electron-spin degene r a c y . This proof w o r k s for infinitesimal AEF, which is the case for a very large s y s t e m . Henceforth, we can let EF = EF. Inserting our e x p r e s sion (2.14) for Ap and integrating by p a r t s , we find AW= - (2/jr) J_B/ Im l n d e t ( l - G°V)dE.

(2.17)

Let us investigate this expression for adsorption of one and two a t o m s . F o r the single atom, P connects only the s t a t e s a and 1 (where 1 may be

330 3634

T.

L . E I N S T E I N AND J .

a symmetric combination ofstates).

Thus

R.

SCHRIEFFER

define (?*„ a s G o a /(l - GaoGf, | VU | 2 ) = [E - Ea - ^ G * (£)]"' ,

18) then and

det(l-C°K)Mlr = l-(G-*)2(Gf2)V AW:slnffl*

f J..

\

E-E„-i6/

dE .

(2.19) The form of the determinant indicates that we a r e calculating the interaction for an electron hopping from the adatom to the n e a r e s t solid atom, interacting with the metallic d band, and then hopping back to the adatom orbital. G^CE) i s shorthand for Gu,n(E), as discussed above, where X = A, B, BP, and C. Recalling that Im In i s just the a r c tangent, we note that for small V, AW<* V2, the perturbation-theory prediction. F o r V very large (relative to the band width Wb), the integrand a p proaches tan' 1 (V 2 ImGf 1 /V 2 ReGf 1 ). Hence the integral over the band approaches a constant (with respect to V). In addition, there i s a pole term arising from a split-off state. As discussed later in this section, the split-off contribution, and hence AW, i s linear in V for large potential. In this regime, the adatom and nearby surface atoms form a surface molecule, which bonds to the indented solid. u In the two-adatom problem, we find that detU - G°V) 1 = det

- Gaof ol 0 0

-GxnVu 0 - G, 2 v 2i 1 0 0 1 -Gf,Vl0 - GjjVaj 0 - Gb6Vt2 1

^l-Ca.GSIv^Ml-C^G*!^! 2 ) •C^G&G^G?,! v j

2

(2. 20)

|

the final expression being an algebraic r e a r r a n g e ment after expansion of the determinant by m i n o r s . By symmetry, Vib=Vu, Gbb = Gaa, Gn = Gn, and G 2 i - G 1 2 . G12 i s shorthand for Gf l l ( J (E), i . e . , 2 denotes the second position ij. Our formulation allows an adatom electron to hop to the nearest solid atom, propagate to the surface atom nearest the second adatom, hop out to this second level, interact, hop back to the solid, propagate, and hop onto the original adatom. We find det(l - G°V) = [(1 - G^Gf: | Vla \2) - G„Gf21 Vu|2] x[(l • G „ G £ | f i . | 2 ) + G M C f 2 | K l a H

.

(2.21)

Since we a r e seeking the pair interaction, we must subtract from this expression, which applies to two adatoms on a surface, the det(l - G°v) for two single noninteracting ( i . e . , infinitely separated) absorbed atoms. We recall that V= I V., I. If we

,

so that l m l n [ l - ( G * ( E ) ) 2 (Gf 2 (E)) 2 V4 }dE

AVrtr= - M J-BO

2 £j B

~,)m

f \ , /, V'&UE)? \ ., Imln 1 ( -[^-^-^1(g)f)rfg(2. 22) 2. Commentary

Equation (2. 22) is essentially identical to G r i m l e y ' s . 4 , s Our G"oa corresponds to h i s G A „ if one r e s c a l e s our Ea by Una o r a s s u m e s U=0. In Grimley and Walker 6 it is assumed that «„ i s r e placed by n„, the adatom occupation n u m b e r for single-particle adsorption. Thus, they neglect Grimley's Anderson-model correlation t e r m AW2 = - ZU(nz - n l ) , * a s we do in not treating Coulomb interaction effects explicitly. T h e i r qA c o r r e s p o n d s to our l^Gu while qAB, qae, o r q^ goes o v e r to o u r V*G18) i . e . , ValG12 V 26 . As the m a t r i c e s in (2.18) and (2.20) i l l u s t r a t e , o u r perturbation also has no diagonal p a r t . Since 1 - GA^q\t i s equivalent to o u r 1 - GlfiuV*, we s e e that our r e s u l t s for Ap and AW Bllr agree with the r e s u l t s of Grimley and Walker 6 up to a minus sign, which a r i s e s from o u r use of an imaginary infinitesimal of opposite sign to conform to the convention of Kalkstein and Soven. l l This choice of sign m e a n s that the i m aginary p a r t s of o u r Green's functions will have the opposite sign from t h e i r s . Since "Im In" i s just the arctangent of the imaginary p a r t of the argument over the r e a l part, our integrand will have the opposite sign from t h e i r s , canceling the sign discrepancy. In Grimley and W a l k e r ' s derivation 6 of the p a i r interaction energy (called 0 m i l ) in their Appendix, they suggest that the final expression r e q u i r e s the approximation (d/dE)GA„ = - G 2 ,., at least in the region where the integrand i s l a r g e . In o u r language, this approximation becomes V21 (d/dE)Gn I « 1. F o r physical potentials V, such an approximation i s not valid, and our derivation h a s not r e quired it. In fact, Grimley and W a l k e r ' s formulas do not require it either, as one might s u s p e c t since our answers a g r e e . Their use of the approximation in the appendix cancels an implicit u s e of it in the text: In their equation (29), (1/q^ )(BqaB /Be) should be replaced by (l/qttfi){aqaB/de) + (1/GAJ x(dGAJae) + GA„. It should be noted that the approximation that T^GntJ?) can be replaced by a complex constant

331 I N D I R E C T I N T E R A C T I O N B E T W E E N A D A T O M S ON A appears in most applications of the Anderson model. In particular, it is used in the original paper 6 and most subsequent treatments of two bulk impurities. 1 2 ' " In fact, there is evidence that the approximation i s not always valid in the bulk-impurity problem. 1 4 Except for weak binding7 the approximation is invalid for the problem considered here. In the localized moment or divacancy problem, one assumes that the free-electron-like s bands are dominant. In chemisorption, the interaction between the "impurities, " i. e . , adatoms, i s mediated by d-like electrons, which are better represented by a tight-binding model. The treatments of two close bulk impurities 12 further a s sume that the interaction propagating term V*G1Z(E) is a complex constant or, equivalently, that it is overshadowed by a constant direct interaction term. Such an assumption would clearly be unreasonable in the present study, except in c e r tain cases of unphysically weak binding. It is usually a reasonable approximation (although we shall not need to make it in our actual calculations) to take I Gl„G\2V* I « 1 and hence write

A

^=!C I m (i^fe r f £ - ( 2 - 2 3 >

This expression is analogous to the first-order expression derived by Kim and Nagaoka15 for two bulk impurities. If we assume V*Gn(,E) = a + iT and VlGli{E) = |3 + iA to be independent of energy (an extremely poor approximation for us), as in Grimley (Sees. 3 and 4) 4 (except that our r and A have the opposite sign from his, so that r is always positive), then AW =

v^

llraTf

=

-*lm(EF

(2.24)

-?T

In the case (Grimley's Sec. 4.1 4 ) \EF~Ea

-a\

= \Er-Ev\«T«Ey

,

2 /3s-A2 •n

2(AS-/3V irr

2 ir

fl?-A8 T

,

4PAE„

(2.26)

since \Er-Er\/T i s small. Indeed, Grimley's expression, which i s proportional to (4E V - 2EF), is physically unreasonable since it depends on the zero of energy. In the case (Grimley's Sec. 4. 24) that \EV-ET\ »V«\EY\, our simple expansion suggests AW=-

4 ir

|3A EF-E

while Grimley's expansion gives (after a bit of algebra) AW=-

0A

It

7T Ep — E-t

Ey which essentially agree for \Er-Ey\ «Er. Only in this limit does Grimley's expression satisfy independence of the energy origin, a general consequence of electron conservation. As Grimley and Walker (Appendix I)6 note, the interaction energy goes to zero as the Fermi energy approaches + °°. (If there are no split-off states, then it vanishes for a filled band.) This very general result is one of the few checks one has when performing actual numerical calculations. In essence, it a r i s e s from the fact that our perturbing potential i s purely off diagonal in the site representation. Hence TrK = TrOC^; that i s , there is no interaction energy (difference in energy from the unperturbed case) if all band levels are o c cupied, even though individual levels may shift. Since this result i s of such importance and utility, we will present short explicit proofs for the singleand pair-adatom c a s e s . Both rely on the fact that all the poles of the Green's functions are in the upper half of the complex energy plane. For single-atom adsorption, we consider the integral (2.19),

where G0„(E) = ( £ - £ „ - i S ) "

1

,

and

T

which differs from Grimley's 4 Eq. (34) for this case, with [7=0. Careful expansion of Grimley's functions / shows that AW

AW = AW, + AW, = ^ | £ - (2Er - 2Er)

AW= - (2/ir) J_l Im ln[l - Gaa(E)G?1(E)V*]dE

we see that &W--

3635

2(A2-^)

+ [higher-order terms in T/Ev or

(Ev-EF)/r]; (2.25) in Grimley's paper, 8 incorrectly appears instead of 4. Hence, with U= 0,

where Ift)are the eigenstates and the eigenvalues of the unperturbed Hamiltonian, and 11) is the binding-site vector (possibly a symmetrical combination of a few surface positions). An integration by parts gives AW= - - £ I m l n [ l - G„(£)Gf,(E)V«]| : .

-NX

V'gfitg) (E-.E~-i6)*

332 3636

T. + V*G„VB) S

L.

E I N S T E I N AND J .

I<1!*>I (E-EkiIE?)

dE i-Gaa(E)Gf1(E)vz

(2. 27)

The surface term vanishes since for any energy outside the band, the argument of the log i s pure r e a l . Hence the term vanishes for large but finite E, and therefore also in the limit E— ±°°. The second term can be made into a contour integral and closed in the lower half-plane of the energy plane, where there a r e no poles. [Note (1 - V^G^Gn)'1 essentially h a s the properties of (E-EaV^Gfj)"1, and I m G ^ 1 0 by r e m a r k (vii), so the poles a r e also in the upper half-plane. ] Since the integrand approaches E"1 asymptotically, the s e m i c i r c u l a r path vanishes, and the integral must be z e r o . Hence AW=0. Similarly for the two-atom c a s e , we integrate AW=by p a r t s . so that

(2/jr) / * Im InU - G^^V*)

AW-

Elm-

(2.28) Again it is easy to show that the derivative t e r m is analytic in the lower half-plane. Also,

SCHRIEFFER

i5, we have that positive imaginary G n i m p l i e s that ReG^CE) is positive above the band and negative below. Now below the band E-Ea is negative, p r e s u m ing Ea to be within the band. The above o b s e r v a tions indicate that - ^ReG^OE), i . e . , K 2 fReGf,<E)l, d e c r e a s e s as E becomes m o r e negative. Hence /(£)=£-.E,-V*ReCf1(E) is a decreasing function below the band a s the energy E becomes m o r e negative. F o r s m a l l V, f(E) will be negative at the band edge, and m e r e l y increase in magnitude a s E s e p a r a t e s f a r t h e r from the bottom of the band. F o r large potential, however, it will be positive at the lower edge. As E approaches - •*>, f(E) approaches E and i s negative. Since f{E) i s continuous and monotonically d e c r e a s ing, it must have a single z e r o below the band, which we denote by Es. Thus, Es-Et-ViReGfliEs)

dE

The surface term vanishes as above,

R.

= 0 , Es<-iwb

.

(2. 30) Similarly, there can be at most one such split-off energy above the band. This discussion p a r a l l e l s Newns's 7 graphical analysis and applies to any case where the change in density of s t a t e s (i. e . , the spectral-weight function) is positive throughout the band. In determining the contribution of a split-off state to AWf la ,,„ it is convenient to write (2.19)

1-GlGljV1 [E-Ea-

rtGf, + Gf,)p-B. (E-E. GW

fjJ

- V(Of, - Gf,)l

(2. 29) By r e m a r k s (viii) and (vii), the positive ImGfjtE) dominate ImGf/E), so that its z e r o s a r e above the real axis. Again, then, we can close below to obtain the vanishing of the integral. 3. Analysis of Split-Off States

Finally, we come to split-off s t a t e s , sharply localized energy states lying above or below the band. In the case of single-adatom adsorption, one is manifested by a z e r o of E -Ea -V*ReGu(E) outside the band. In the pair problem, £ - E„ - V2 x(ReGf, ±ReG?y) vanishes outside the band. The following observations will aid our analysis: Rem a r k (vii) indicates that for any binding position the imaginary part of the "self," o r 11, Green's function is positive throughout the band [and vanishes outside, by r e m a r k (v)]. Hence, since the Green's function satisfies a K r a m e r s - K r o n i g relation [written down in r e m a r k (v); the range of integration need only extend over the band to pick up all contributions to ReG], we find that IReGftfE)! must be monotonically decreasing outside the band a s \E\ i n c r e a s e s . Moreover, with our choice of sign for the imaginary infinitesimal

A

< ' « " =(2/"){ / - '

I m l n (E

~£° -

- jJlmln[E-Ea-VzG^E)-i6]dE}

t6)dE

,

(2.31)

which is the form in which the actual computer computation is made. Within the band, the z'6 of the second t e r m i s unimportant since ImGf^E) i s positive. Below the band, id gives the only imaginary part, so that the integrand of either t e r m is - 7T (0) if the r e a l p a r t of the argument of the natural logarithm is negative (positive). F o r all E below the band, the integrand of the first t e r m i s - Tt. F o r E sufficiently below the band, E -Ea - V^ReGf^E) will also be negative, so that the two integrals cancel for £ sufficiently negative. This cancellation eliminates any difficulty at - ° ° . If there is a split-off s t a t e , then between Es and E0, where EB is the energy of the bottom of the band (i. e . , - 5 Wb), the integrands do not cancel, and we find a contribution of (2/TT){ — ir)(E0-Es), or 2lE s -E0), to AW*. Equation (2.19) can be written A
(2.32)

333 7

I N D I R E C T I N T E R A C T I O N B E T W E E N A D A T O M S ON A . . .

if we set Es = £ 0 when no split-off state exists below the band. As was noted above, the integral term of (2.32) approaches a constant in V. Hence as the potential becomes very large AJ* r J 1 „„~2£ s (V). In this region \ES\ » \E0\, so that we are in the asymptotic region of the Green's function, where ReGf,(E) ~ (1/TT£) J." 0 ' ImGf,(E')dE'=l/£ , where the equality holds for any binding type and reflects the addition of the single-adatom electron to the system. In this region, (2. 30) reduces to a quadratic equation with the solution £ s = i [ £ „ - ( £ * +4V 2 ) 1 ' 2 ]

.

(2.33)

Thus, as V grows very large, Es approaches - V, so that AW(V)~-2V. A state split off above the band can be treated in similar fashion. Such a state must be included if the trace theorem proved just above that AW = 0 as ET grows large (or merely at £ , = 1£0I = | Wt if there i s no split-off state above the band). Since we restrict the Fermi energy to be within the band, this upper state i s unimportant in our computation. In the case of the pair interactions, the inclusion of split-off states i s more complicated, but not more difficult. As suggested by (2.29), it is convenient to decompose (2. 22) as AWfj = - (2/TT) / . * ' dE(hnln{£-

£„ -

^[G^iE)

+ Gf,(E)]-»o} + I m l n { £ - £ „ - V^Gf^B)- G f , ( E ) ] - » } -2Imln{£-£(I-Vi!Gf1(£)-i6})

.

(2.34)

Again, the imaginary infinitesimals are important only outside the band. By remark (viii), ImGuU?) T ImGf/E) is non-negative throughout the band. Hence, as in the single-adatom problem, the Kramers-Kronig relation indicates that ReGfi(E) =F ReGf/£) is positive above the band, negative below it, and monotonically decreasing in magnitude with increasing l £ l (outside the band). For sufficiently large l £ l , the real parts of all three arguments will be negative. Each will contribute an integrand of - ir, leading to cancellation. For sufficiently strong potential, however, the real parts will become positive below the lower band edge. Let Et be defined by £ ± - £ „ - V 2 [ReGf 1 (£ i )±ReGf/£ i )] = 0 ,

£,<£„. (2.35) Then it is clear from the single-adatom problem that the contribution of split-off states to the pairinteraction energy i s - (2/ff)(ir)[(E0 - B.) + (£ 0 - E.) - 2(£ 0

-Es)]

when Er i s within the band. Thus we can rewrite (2.22) as

3637

AWfj = 2(£. + £ . - 2 £ s )

(2.36) again with the understanding that if there i s no zero below the band in any one of the three real parts in the arguments in (2. 34), the corresponding Et or Es i s taken to be £ 0 , so that in the event of no splitting off, the first term of (2. 36) vanishes. Recall from remark (v) that ReGfjQ?,,) i s negative. Hence, as potential decreases, E. will be the first split-off state to disappear (enter the band), followed by £ s , and finally Et. Finally, we examine the asymptotic form of AWf/K). As suggested by the factorization of (2. 29), as V becomes very large, the integral term of (2.36) becomes independent of V, with the value

To analyze the split-off-state term, we must write ReG in terms of a moment expansion:

**G~±t

£• for|£|>|£ 0 |

where Mn = < l A ) / J £ ° '

(ET]mG(E')dE'

i s called the nth moment. Now for atop binding (X=A), n 0 i s finite (and equal to unity) only for (y) = ( l l ) . Since Gf^Gft + G^ and Gfl = G * + 2 G ^ + G22, n0 i s unity for all X when (y) = (11). For (y) not equal to (11), the zeroth moment of ImGf, vanishes in general. There are three exceptions to this statement: Gf2, Gf8, and G£2. The source of the exception i s that the two adatoms share the surface atom between them as a nearest neighbor. In the formalism, this is manifested by the p r e s ence of Gf, as a component, e. g., Gf2 = G^ + 2 (Gfi + Gf,). In these special cases we need not go to higher moment to find the contribution of the split-off states. Solving (2. 30) and (2.35) for large V, we easily find that 2(£, + £ _ - 2 £ s ) * - 27[(1 + ^ ) 1 / 2 + (1 - /I) 1 ' 2 - 2 ] , where £ is the zeroth moment of ImGf>( or equivalent^ the coefficient G^, e . g . , £ = i for Gfs. The physical ramification of this special case i s the growth of a surface macromolecule rather than merely an island composed of dimers. However, in this case the direct interaction may also be i m portant and must be added on in any detailed calculation of the energy of the configuration. In general, however, AWfj(V) is not asymptotically linear in V. In fact, AW(K) approaches the constant determined by the limit of the integral term in (2.36)! The split-off states give a net

334 T.

3638

L.

E I N S T E I N AND J .

FIG. 3. Imaginary part of the "self" (11) Green's function, or n times the change in the density of states, at a binding site for atop (A), bridge (B), and centered (C) binding. The curves verify the downward shift in energy of the symmetrized surface Wannler states The implicit energy unit used throughout is twice the tight-binding hopping matrix strength, i . e . , 27\ or equivalent one-sixth the bandwidth .

0.8 CD

e 0.4-

contribution of o r d e r 1 / 7 in this region if ImGf, has a nonvanishing first moment (x t . [That i s , E, + E.-ZEs~-\(\i\/V).\ If ImG*(£) is even in E, then the split-off contribution i s even higher order in l/V. HI. CALCULATION RESULTS A. Single Atom Adsorption As noted e a r l i e r , the implicit energy unit of our calculation i s 2T, i . e . , j - Wb> the natural unit of the tight-binding model We evaluate numerically the integral AW

SCHRIEFFER

•-H'--('-iW« J2(ES-Ea) \

(3.1)

°

where X=A (atop), B (bridge), o r C (centered); BP (bridge-perpendicular) is redundant to B. The second t e r m i s nonzero when a split-off state of energy Es lies below the lower band edge E0s~i Wb; Es i s determined by E q . (2. 30). F o r homonuclear diatomic adsorbates, the singleparticle interaction energy AW i s minus one-half of the sum of the heat of adsorption and the d i s sociation energy of the molecule. We note that Gf1U?) = Gii(B) + G1A8(£) and G&U?) = GfttE) + 2Gi*2(£) + G&CE)

(3.2)

From r e m a r k s (ii), (iii), (v), and (vi) concerning the Gfy(JE), o r from direct computation, we have (a) lmG^E) i s s y m m e t r i c and positive, with a single extremum in the middle of the band; (b) ImG^CE) i s antisymmetric, with i t s positive peak

in the lower half of the band; (c) ImG^(E) i s s y m metric, with a negative bulge around the c e n t e r of the band, and a positive extremum toward each end. Thus, Ap B = (l/ir)ImGi\(.E) is skewed into the lower half of the band, lowering the average e l e c tron energy, a s one would expect of a bonding state. This effect i s even more pronounced for Ap c , where all three addends combine c o n s t r u c tively n e a r the bottom of the band, while 2Gf2 competes with G^ + G^j n e a r the top (see Fig. 3). In essence we a r e saying that p peaks at lower e n ergy for non-atop binding because the a d a t o m ' s electron mixes with a suitably phased combination of orbitals on the n e a r e s t neighbors in the s u b s t r a t e surface, resulting in only plus signs in Eq. (3.2). We compute AW for Ea, the adatom noninteracting level, at and near the c e n t e r of the band, and at i Wb from each edge. The F e r m i energy sweeps through the band in i n c r e m e n t s of 0 . 1 . We have c a r r i e d out computations on a d a t o m - s u r f a c e couplings of the following strengths: V/T= 1, 2, 3, 4, 5, 6; that is, V/W„ in the range ^ -•§ . F o r V/T = 5 or 6, and often for weaker potential (V/T = A), when E„ i s far from the band center, one finds split-off states of energy Es. Our calculation of AWX revealed the general structure (cf. Fig. 4) of an inverted triangle (the base being the EF axis), smoothed at the base edges. Starting at z e r o (when there i s no split-off state) at the lower edge, AW begins falling with increasing Ef (within a range of about -fc Wb, d e pending on V), soon becoming linear until it reaches its largest (negative) value when Ep = Ea. In e s sence, I AW I is maximum when EF ^ Ea because in this case the maximum number of e l e c t r o n s have their energies lowered with few e l e c t r o n s having their energies raised ( i . e . , the lower half of the

335 INDIRECT

INTERACTION BETWEEN

A D A T O M S ON A . .

3639

FIG. 4. Interaction energy for the adsorption of a single atom. Ea, denoted by the small circle on the abscissa, is - 0 , 3 . A, B, and C denote atop, bridge, and centered binding, re spectively.

-1.0 -

-2.0-

virtual level i s occupied, the upper half i s empty). Alternatively, the adatom level i s broadened by the interaction with the substrate; when it is half-filled, electrons take optimal advantage of this broadening, IAWI then d e c r e a s e s linearly and smoothly vanishes n e a r the top of the band. There a r e no p o s i tive (repulsive) peaks and no noticeable s u b s t r u c t u r e . If one a s s u m e s V i s the same for all binding types, then invariably AW* begins falling first, followed by Aff J , and AW"4 trailing; similarly, they follow the same o r d e r in rising toward the axis. Thus, when EF i s not too near Ea one has I AW* I > !AW B | > I AW* I for EF<E<1 and I AH* I < I &WB I < IAW*I for Er > £„. This is in accord with the observation that Ap for centered binding i s most strongly skewed downward in energy, with the change in density of states for bridge binding skewed moderately, and the atop Ap not skewed

at all. When a state has split off below o r above the band, the interaction energy curve i s negative at the appropriate band edge r a t h e r than going to z e r o there (cf. F i g . 5). The other qualitative feat u r e s of the curve remain the s a m e . Hybridization skewing dictates that a state splits off below the band for C binding first (as V i n c r e a s e s ) , then B, and finally A, assuming the same V for each binding type. At a given potential, the magnitude of the split-off energy, and hence the absolute value of AW at the lower edge, will be g r e a t e s t for C, weakest for A. F o r s t a t e s split off above the band, the symmetry o r d e r naturally r e v e r s e s : A splits off first and most strongly, and so on. Recently, M e s s m e r and Bennett 16 have proposed an extension of the Woodward-Hoffman s y m m e t r y rules of reaction chemistry to the problem of

Aw -2.0 —I

O

-1.0 1

00 H

1.0 1

/ -2.0|-\\\

/

'/S

; ° \

• ' /

>.\ \ -4.0

2.0 v-" > - "

V/T = 5

^

3.0 1

FIG. 5. Interaction energy for the adsorption of a single adatom. Ea~ —1.5. A (doubly occupied) state has split off below the band for each of the three binding symmetries.

336 3640

T.

L.

EINSTEIN

AND J .

chemisorption. They argue that the binding energy i s determined primarily by the matching of the symmetry ot the adsorbate orbital and the substrate orbitals of energy equal to EF. Our r e s u l t s suggest a m o r e involved p i c t u r e in that the binding comes from states throughout the band, with states near EF and/or £ , playing a dominant role only when Ea^Er and V is small. Hybridizing also accounts for the following phenomena: For Ea near the center of the band, the spread among the three i f f 1 (EF-Ea) relative to their average is quite small, on the o r d e r of a few percent. (This percentage d e c r e a s e s a s V i n c r e a s e s . ) F o r £ , = - 1 . 5 , \AW° ( £ , = £ , ) I > lAWfll > I AWAI, the relative spread ranging from 35 to 13% (as V i n c r e a s e s ) . Correspondingly, for £,, = + 1.5, \AWA{EF = Ea)\ > \&WB\ > \AW°\, with an even g r e a t e r spread (50-24%). F o r £ , = - 1. 5, - 0 . 3 , and 0. 0, the average maximum binding energy, i [AW A (£,) + AW B (£,) + AW°{Ea)] is about the same, to within a few percent, with £ , = - 0.3 consistently highest and £ , = - 1 . 5 least in magnitude. The average binding at EF = £ , = + 1 . 5 i s only about 7 - ? the strength at the other three adatomlevel values. F o r most of our range of hopping potential, the interaction lies well between the perturbation regime (AWX Vs) and the surface-molecule limit (AW<* V). If we t r y to fit the data with a relation AW(£,)<* (V/T)", we see that a i n c r e a s e s (within the range 1-2) as V d e c r e a s e s or as £ , moves into a region of relatively weaker binding. In the c a s e of largest V/T, we find ourselves quite n e a r the surface-molecule extreme a = 1. We also find h e r e the familiar result that the interaction energy i n c r e a s e s a s the band n a r r o w s : Since both AW and V a r e scaled by T, and thus W„, AW* W\'u. In the surface-molecule limit, AW becomes independent of the bandwidth, so that the band appears to the adatom a s essentially a single energy level. The maximum binding energy (averaged over X-A, B, C) at £„ = 0.0, - 0 . 3 , o r - 1 . 5 , is roughly i , i , I . o r |- the bandwidth for V/T = 3, 4, 5, or 6, respectively. The data in Table I suggest a ratio of binding energy to bandwidth of about y - f . Thus, within our model, we obtain agreement with experiment when the hopping matrix element between adatom and substrate atom is about three to six t i m e s that between substrate atoms. B. Pair Interaction

To obtain the pair-interaction energy, i . e . , the energy difference between two atoms adsorbed at nearby s i t e s and two adatom s infinitely separated, (Table II), we m u s t compute the integral AW

*,

=~ ~ \

Im l«[l

-Vl(G^(E)f(Gfj(E)f]dE

R. S C H R I E F F E R

7

+ j2(£,.£.-2£ s )

,

(3.3)

where B?.(3) = GJE)/[1

- V*GJE)G?AE)1

•

The second t e r m i s nonzero when t h e r e a r e splitoff states below the band. Es is given by Eq. (2. 30), E± by Eq. (2.35). When a p a r t i c u l a r state has not split off, it is replaced by E0 in Eq. (3.3). We find a s t r u c t u r e much r i c h e r than for s i n g l e atom adsorption. Again we compute with the par a m e t e r s £ , = - 1. 5, - 0 . 3 , 0.0, a n d + 1.5 o r + 2.0; V/T=l, 2, 3, 4, 5; and EF in s t e p s of 0 . 1 . The interaction is computed for eight values of (y): (12), (22), (13), (33), (14), (15), and (16). With the aid of the s y m m e t r i e s of r e m a r k (iv), we know the interaction strength at the 24 lattice points (excluding the origin) within o r on the bounda r y of a square of side 5a 0 centered at the origin, i. e . , at points l e s s than 2.9a 0 away from the origin point. In a (10) direction we know AW for points a s far a s 5a 0 . In o r d e r to improve accuracy, we also linearly interpolate Green's functions and then calculate the integral in steps of 0. 05 r a t h e r than 0 . 1 . F o r the calculation of split-off s t a t e s , s t e p s of 0.025 were used within 0. 2 of the band e d g e s , since when a state i s about to split off from the band, violent oscillations often occur t h e r e . Moreover, for l a r g e V it proved advisable to linearly approximate the G r e e n ' s functions and to thereby evaluate an exact integral over a m e s h spacing r a t h e r than performing a t r a p e z o i d a l - r u l e summation. The sum rule given at the end of Sec. IIB i s generally reasonably well satisfied, except for the m o r e distant points and stronger potentials. In the l a t t e r circumstance, often s t a t e s a r e splitting off only below the band, and the m o s t rapid oscillations occur n e a r the lower edge. H e r e it becomes very desirable to integrate down from the top of the band, as permitted by the sum r u l e . In this calculation, split-off states occur for weaker potentials than for the single-atom c a s e , as in suggested by the factorization 1 -V*(G^(E)f

(G*(£)) s

_ (E - Ea - V^Gfj - X^G^HE - £ - V^G^ + V^G?,) (3.4) where the first factor in the numerator r e p r e s e n t s the downward-shifted state and the second t e r m the upward-shifted one (cf. Grimley 4 ). F o r £ , = - 1 . 5 , splitting off o c c u r s for V/T2 3. F o r £ , = + 1 . 5 , it Just begins to happen at V/T = 3. With £ , n e a r the center of the band, split-off s t a t e s a p p e a r when V/T=4. In the p a i r problem we find a more d r a m a t i c

7

INDIRECT

INTERACTION

BETWEEN

d e p e n d e n c e on p o t e n t i a l than in t h e s i n g l e - a t o m c a s e . F o r V / T = l , the s t r o n g e s t i n t e r a c t i o n o c c u r s for EF=Eat with a s p r e a d of about \-Wt in e a c h d i r e c t i o n ( l e s s f o r E,= - 1 . 5). It i s c h a r a c t e r i z e d by one o r two p r o n o u n c e d c r e s t s and t r o u g h s in AW*, ( s e e F i g . 6 ) . ForV/T=Z, the i n t e r a c t i o n i s s c a r c e l y l o c a l i z e d , but s t i l l l a r g e s t n e a r EF = Ea. (With Ea = - 1 . 5, the s t r o n g v a r i a t i o n of the i n t e r a c t i o n e n e r g y i s s t i l l l a r g e l y confined to the l o w e r half of the band; by Er = 1 . 0 , the i n t e r a c t i o n f a d e s a w a y . ) W h e n V/T= 3 t h e r e a p p e a r s

ADATOMS

ON

A

...

3641

to b e n o l o c a l i z a t i o n w i t h i n t h e e n e r g y band, a n d one b e g i n s to s e e the p r e c u r s o r ( s ) of s p l i t - o f f s t a t e s (cf. F i g . 7 ) . Such p r e c u r s o r s t a t e s a r e c h a r a c t e r i z e d by a n a r r o w p r o n o u n c e d o s c i l l a t i o n of the i n t e r a c t i o n - e n e r g y c u r v e n e a r a band e d g e . F o r V/T = 4, s p l i t t i n g off h a s begun ( i . e . , f o r Ea in the l o w e r half o r m i d d l e of the band, a t l e a s t the s t a t e c o r r e s p o n d i n g to Gft + G*, h a s f a l l e n b e l o w t h e l o w e r band e d g e ; for Ea w e l l i n t o the u p p e r half, a s i m i l a r s t a t e h a s s p l i t f r o m the u p p e r e d g e ) . F o r V/T = 5 o r 6, s p l i t t i n g off i s g e n e r a l l y c o m -

TABLE I. Tabulation of data used in estimating the adatom-surface atom hopping parameter V. The energy of binding a single adatom to a surface, — AW, is obtained readily from experiment. The [d-1 band width of the substrate can be extracted crudely from band-structure computations. By comparing the ratio of — AW/Wt thus obtained with calculations based on the present model, we fix V/T-3-5 to describe chemisorption. Substrate W

Crystal structure bee

bec

W„ <eV)» 10.5(14.1) c

9.2°

Cr

bec

6.9C

Fe

bec

6.0"

Ni

Rh

Pd

Pt

fee

fee

fee

fee

4.6e

6.44'

5.08*

7.16 1

Adsorbate

- Alt' <eV)*

H

3.2

O

6.8

N

7.0

CO

3.6

H

3.1

O

6.3

-AW/Wj

3.2 H

3.0

N

6.4

CO

2.0

H

2.9

CO

1.8

H

2.9

O

5.1

H

2.8

O

4.0

O

4.1

1 7

"These bandwidth values are based on augmented-plane-wave calculations. For bee substrates, the d-band width is taken to be the energy difference £(#25') - £ ( f f 1 2 ) ; for fee substrates, it is estimated by £(L£) -EiLf). Since the r e p r e sentation of an actual d-band by a single tight-binding band is very approximate, these values should be considered to give the bandwidth to one significant figure at best. "Experimental heats of chemisorption for diatomic gases from D. O. Hayward and B. M. W. Trapnell, Chemisorption 2nd ed. (Butterworth, London, 1964), pp. 2 0 3 - 4 . Dissociation energies for H2, 0 2 , and N2 from Handbook of Chemistry and Physics, 44th e d . , edited by Charles D. Hodgman et al. (Chemical Rubber Publishing Co., Cleveland, 1962), p. 3519. C L. F . Mattheiss, Phys. Rev. 139, A1893 (1965). For tungsten, two results are given, corresponding to slightly different crystal potentials It is suggested that the first smaller value is the more accurate prediction of the actual 5d bandwidth of W. d J. H. Wood, Phys. Rev. 126_, 517 (1962). e L. Hodges, H. Ehrenreich, andN. D. Lang, Phys. Rev. 152, 505 (1966), citing J. G. Hanus, MIT Solid State and Molecular Theory Group Quarterly Progress Report No. 44, 29 (1962) (unpublished). r O. Krogh Andersen, Phys. Rev. B 2 , 883 (1970), a relativtstic calculation.

338 T.

3642

L.

E I N S T E I N AND J .

R.

SCHRIEFFER

10* AW,*2

FIG. 6. Nearest-neighbor pair-interaction energy for weak potential (equal to the tight-binding hopping constant); Ea = - 0.3. In this and subsequent figures, A, C,B, andBPdenote atop, centered, bridge, and bridgeperpendicular, respectively. -1.5 -

-3.0

pleted from one edge ( e . g . , even the Gf x -G* t state has separated), and p r e c u r s o r s or actual splitting off is beginning at the other band edge, as in Fig. 8. F o r the stronger potentials (V/T= 3 or 4, o r more), the curving of AW*j i s no longer so s e n s i tive to V, especially away from the edges; inc r e a s e d potential tends to scale up (and possibly slightly shift) the c u r v e s . This discussion is capsulized in Fig. 9, which shows the effects of increasing V while holding the other p a r a m e t e r s fixed: we see (a) state split off below the band and (b) little difference (for stronger V) in the shape of the curves in the band interior. Also unlike the single-atom case, we find many smoothly peaked extrema of AW as EF v a r i e s over the band. The number of extrema of AWfj is g r e a t e r (often by as much as a factor of 2) than the number of extrema of ImGy^, as given in r e m a r k

(vi), except apparently for some types of binding at the distant sites—(ij) = (15) o r (16), a distance > 3. 5a 0 from the origin—although in t h e s e c a s e s t h e r e a r e so many bumps of small magnitude that some may be smothered by the finite m e s h s i z e . Almost always, atop binding has the g r e a t e s t numb e r of e x t r e m a in the energy band, while centered binding has the fewest, the difference being of the o r d e r of a q u a r t e r of the total number. C o n v e r s e ly, the centered e x t r e m a a r e usually g r e a t e s t in magnitude while the atop a r e s m a l l e s t . Bridge binding tends to t r a c k - c e n t e r e d binding, especially for near sites toward the (10) direction—(ij) = (12), (13), and (23), a distance < 2. 5a 0 from the origin; also in this region, bridge perpendicular t r a c k s atop. By tracking, we mean that two c u r v e s have the same, o r nearly the s a m e , number of e x t r e m a of comparable relative height, but often displaced

I 0 2 AW

V/T = 3 -2.0 -

FIG. 7. Pair-interaction energy for fourth-nearest neighbors with a potential not quite strong enough to split off states; £ a = 0 . 0 . Note that (i) the interaction is much less localized within the band than in Fig. 6; (il) the energy scale of the ordinate is similar to that in Fig. 6, indicating that the effect on I AW I of the increase in potential is compensated by the increase in interaction distance; and (iii) the precursors of split-off states are beginning to emerge near the band edges.

INDIRECT

INTERACTION

BETWEEN

A D A T O M S ON A

3643

FIG. 8. Pair interaction for third-nearest neighbors with a potential in the split-off state regimes Ea = - 0.3. For all but atop binding, splitting off below the band has been completed i. e., Et, Es, and £ . are all less than £ 0 . Near the upper band edge, precursor states appear.

by up to ^ Wb. In the (11) direction bridge and bridge perpendicular a r e identical, and their curve lies between centered and atop binding. The tracking relations begin to fail at the intermediate sites—(ij) = (33) and (14), a distance between 2.5a 0 and 3. 5a0 from the origin—and collapse at the d i s tant s i t e s . At distant s i t e s , most of the variation of AWfj i s concentrated in a sharp center peak for EF n e a r the bottom of the band for strong V and near Ea for weak V. On the other hand, a s the potential i n c r e a s e s , the tracking relations improve in the sense that the c u r v e s grow remarkably s i m i l a r in amplitude and in the position of extrema and nodes. F o r the strongest potentials, in fact, all four binding types t r a c k to a degree: While the number of nodes might differ, the over-all general shapes have s i m i l a r p a t t e r n s . The difference in number of extrema in the band for different s i t e s and binding types leads to a multiplicity of patterns as the F e r m i level r i s e s , with some sites a t t r a c -

tive and o t h e r s repulsive in a r a t h e r unpredictable fashion. Obviously, since t h e r e a r e fewer o s c i l l a tions for the n e a r e s t s i t e s , they a r e the m o s t stable to sign change a s the band fills. Only n e a r the bottom of the band (and when t h e r e a r e no states split off below it) do all eight independent s i t e s ( i . e . , all 36 sites, by s y m m e t r i e s ) have the same sign: AW^(EF>-iWb)<0

.

This property is a consequence of r e m a r k s (v), (vii), and (viii), and can be verified by r e w r i t i n g the integrand of (3.3) in the form of (3.4). Taking the logarithm b r e a k s the expression into t h r e e summands. The imaginary p a r t of the log, the arctangent, can be expanded to first o r d e r in the imaginary part over the r e a l p a r t of each of the three arguments, since ImG — 0* at the l o w e r edge and the denominator i s finite and negative when there a r e no split-off s t a t e s . A bit of a l g e b r a and

10 AW 8.0 R /•

4.0 -f-

-4.0

-8.0

L

FIG. 9. Pair interaction for third-nearest neighbors, with bridge binding and Ea = - 0.3, illustrating the effect of increasing the potential from V / T = l - 5 in integer steps. For stronger potentials, the curves in the interior of the band merely increase in amplitude with increasing potential, with little change in shape. At the lower band edge, one sees the evolution of a splitoff state: for V/T= 3, there is a precursor; for V/T=5, splitting off is completed.

340 T.

3644

L.

E I N S T E I N AND J .

the fact that ReGf/ - l w , ) < 0 shows the integrand to be positive, and hence AW to be negative initiallyThe l a r g e number of s i t e s along the (10) d i r e c tion for which AWvtiT i s calculated enables us to say something meaningful about the d e c r e a s e in the interaction energy with distance in the local (nonasymptotic) regime for potentials sufficiently weak so that no split-off states occur. F o r V / T £ 4 , numerical difficulties preclude any quantitative discussion of the distant s i t e s ; hence the s u b sequent paragraph is r e s t r i c e d to V/Ti 3. We find an exponentiallike dropoff with roughly the form

|AW f i j=ye-*'- I , =ye**'' , o ,

(3.5)

where y is of o r d e r unity for V/T= 3. The best value for j3 i s about f- o r 2, depending on how one m e a s u r e s the amplitude of AW (J . If one c h a r a c t e r i z e s the interaction by the largest absolute m a g nitude of AWft{jEF) for the F e r m i energy at any place within the band (as one must for localized binding), then the lower value of 0 applies, giving a dropoff of about f (though ranging from ^, to i ) in the pair interaction a s the separation between the adatoms i n c r e a s e s by one lattice constant.

R.

SCHRIEFFER

This method i s used in F i g . 10, which substantiates the claim of exponential falloff with increasing p a i r separation. A second method c h a r a c t e r i z e s AWfj by an envelope containing most e x t r e m a but e x cluding unusually l a r g e isolated peaks; t h i s viewpoint gives a range of ^ to i for the falloff ratio p e r lattice constant. If we examine the actual falloff ratios a s a function of separation, we find a nonmonotonic behavior that i s distinctly not inv e r s e powerlike in the range of pair d i s t a n c e s considered. Our p r e l i m i n a r y analysis s u g g e s t s that the asymptotic form of the interaction energy goes as an inverse power, but a s fl"5 r a t h e r than a s the familiar R"3 which o c c u r s for bulk i m p u r i t i e s . In any case, a virtual-level approximation, a s enunciated by Grimley and W a l k e r , 6 marked by inv e r s e - p o w e r behavior even for small s e p a r a t i o n s between adatoms, s e e m s quite poor for the range of V considered h e r e . The G r e e n ' s functions a r e highly energy dependent, and of magnitude comparable to the energy p a r a m e t e r . Again, we attempt to fit the dependence on p o tential to a relation of the form AW^ 1 * (V/T)a, where ar ranges from 4 in the perturbation limit down to 0 in the surface-molecule r e g i m e (1 in

TABLE II. Display of the pair-interaction energy suggesting the sensitivity of adatom arrays to changes in the Fermi level, the hopping potential, the adatom noninteracting level, and the binding type. One adatom sits at the origin (11) (denoted "0"); the pair energy for a second adatom at each of the nearby sites is indicated by the number at the site. The magnitude of the number given is 10 plus the common logarithm of the magnitude of the interaction. A plus (minus) sign means that the interaction is repulsive (attractive). Thus, AW= —2.7 x 10"4 would be represented by - 6 . 4 in the table. Each chart is labeled by the symmetric surface array predicted. Since iffy, j=: 4, is unimportant for this determination, only X = C is shown. AWfj

X=

v=

c

A

31 21 11

32 22 12

33 23 13

31 21 11

£ , = 1.2 - 7 . 7 E„ = - 0 . 3 + 8.9 0 V/T = 3

-7.5 +7.5 -8.1 -7.5 + 8.9 - 7 . 7 c(2x2)

-8.3 -9.0

£ , = 1.2 - 7 . 8 - £ „ = - 0 . C + 8.8

-6.7 +7.4 -8.4 -6.7 + 8.8 - 7 . 8 C(2X2)

-8.6 -9.4

£ , = 1.2 - 7 . 7 £ , = - 0 . 3 + 8.6 — V/T=4 0

+ 7.7 +6.9 -8.6 +7.7 + 8.6 - 7 . 7 c(2x2)

+ 8.5 -9.3

— £ , = 0.9 - 7 . 5 £„ = - 0 . 3 + 8.5 V/T = 3 0

+ 7.7 - 7 . 0 -8.5 +7.7 + 8.5 - 7 . 5 c(2x2)

+ 8.1 -9.3

— £ , = 1.5 - 6 . 7 £„ = - 0 . 3 + 8.7 0 V/T = 3

-7.8 +4.4 + 8.3 - 7 . 8 + 8.7 - 6 . 7 c(4x2)

-7.3 + 6.2

v/r=3

0

0

0

0

0

0

32 22 12

BP

B

33 23 13

+ 7.1 - 7 . 4 -8.0 +7.1 -9.0 -8.3 (1X1) -6.2 -7.8 -8.6 -6.2 -9.4 -8.6 (1X1) -7.5 +7.8 -8.4 -7.5 -9.3 +8.5 (lxl) -7.9 +7.7 -8.4 -7.9 -9.3 +8.1 (1X1) + 5.8 - 6 . 8 -7.2 +5.8 + 6.2 - 7 . 3 c(2x2)

31 21 14

15

16

11

+ 8.4 -9.3 -6.6

+ 6.1

+ 4.6

0

+ 8.7 -9.3 + 7.7

-5.8

+ 5.3

0

+ 8.5 -9.1 + 7.6

-6.1

+ 4.9

0

+ 8.4 -9.2 + 7.5

-6.3

+ 4.7

0

-8.4 -8.9 -6.3

+ 5.5

+ 4.5

0

32 22 12

33 23 13

31 21 11

+ 8.3 - 6 . 1 -8.4 +8.3 -9.3 +8.4 (lxl)

-7.9 + 8.2

+ 8.1 - 7 . 6 -8.3 +8.1 -9.3 +8.7 (lxl)

+ 7.6 + 9.0

-7.9 -7.8 + 7.5 - 7 . 9 -9.1 +8.5 (1X1)

-7.8 + 9.1

-7.7 -7.7 -7.1 -7.7 -9.2 +8.4 (1X1)

+ 6.3 + 9.0

-7.7 +7.0 + 6.2 -7.7 -8.9 -8.4 (1X1)

+ 6.7 -8.7

0

0

0

0

0

32 22 12

33 23 13

-7.4 -8.4 + 8.2 c(2 x

-6.1 -7.4 -7.9 2)

+ 7.0 - 7 . 6 -8.3 +7.0 + 9.0 +7.6 c(2x2) + 6.9 - 7 . 8 + 7.5 +6.9 + 9.1 - 7 . 8 (2x2) + 7.8 - 7 . 7 -7.1 +7.8 + 9.0 +6.3 C (2X2) -7.0 +7.0 + 6.2 - 7 . 0 -8.7 +6.7 (lxl)

I N D I R E C T I N T E R A C T I O N B E T W E E N ADATOMS ON A

3645

(5.3)' AW 2.0X10

3.0 +

-2.0

FIG. 10. Pair interaction, with bridge binding and Ea = — 0.3, of adatoms separated by 1-5 lattice constants in the <10) direction. The curves are scaled by (5.3) raised to the separation minus one (in units of «o), to show the exponential character of the decrease in interaction with increasing separation.

V/T =3

the special case of surface macromolecules mentioned at the end of Sec. IIB 3). For peak-to-peak measurements with Ea near the band center, a averages 1.1±0.7 in going from V/T = l to 2, and 3.4± 1.1 in going from 2 to 3. For envelope measurements between V/T= 2 and 3, a is 2.6 ± ~ 1.0 and 2.4±0. 5. We can alternatively consider the various y. F o r V / T = l , y«0. 2; for V/T = 2, y = 0.4 or 0. 38 (method one or two). Hence, in going from 1 to 2, a ~ 1, while going from 2 to 3. a ~ 2. For stronger potentials, the features of the interaction curve in the interior of the band are relatively insensitive to variations of V; we can therefore focus attention on a particular interior extremum and compare its value for V/T ranging 3-5. We find that in general a is a fraction (between 0 and 1), and that a determined from V/T = 3 - 4 is usually greater than a determined from V/T=
longer so. In this range the interaction-energy curve (for EF not near a band edge) is still basically just shifted along the Er axis, but by an amount that is only a small fraction of AE. Figure 11 illustrates this behavior with AE = £Wb. AWnir cannot reasonably be fit by a function of Er - E„ even to lowest order. Moreover, it is not hard to show that in the unphysical limit of very strong potential (the asymptotic regime discussed at the end of Sec. n B 3), the dependence on Ea vanishes entirely. The weak dependence of AWfy on E„ (compared with EF) in the physical range (of V) is fortunate if one is trying to find the appropriate calculated curve for experimentally determined values of the input parameters E„ Er, Wb, and V (from heat of adsorption) since Ea is the most difficult of these four to determine: We recall from the discussion at the beginning of Sec. IIB 2 that our E„ is in fact a rather phenomenological parameter which should be rescaled to give at least a Hartree-Fock account of the (nonmagnetic) Coulomb interaction V between an up- and a downspin electron on the adatom. Clearly, in this approximation Ea is increased by U times the occupation number of the adatom for either spin direction; to lowest order this occupation number can be replaced by that for single-particle adsorption. 6 C. Surface Arrays Armed with a general understanding of the parametric dependence of the pair-interaction energy, we are ready to consider the surface arrays suggested by our results. In about half of the cases, the nearest-neighbor site is attractive (AWlB< 0) and is the most attractive of any site. In these cases, the model suggests that a ( l x l ) adsorbate pattern form. However, in this case one can in general no longer legitimately neglect overlap effects—that is, the direct interaction between

342 3646

L. E I N S T E I N AND J .

R.

SCHRIEFFER

8.0

4.0

FIG. 11. Pair interaction for next-nearest neighbors, with on-site binding, for Ea = - 2.0, -1.0, 0.0, 1.0, and 2.0. The curves verify the relative intensitivity of AWvair to changes in Ea for physical potentials.

- 4 . 0 -i

-6.0 4 adatoms—so that the treatment here is unreliable for any details. Moreover, the model tends to overestimate the number of ( l x l ) arrays. Since all AW are negative near the lower band edge, and since AW12 has the fewest oscillations of all (y) as a function of EF, a disproportionate number of these ( l x l ) patterns fall near the edges (especially for weaker and intermediate potentials V), where not much dissociative chemisorption takes place. (Recall from Sec. in A that the binding energy is smallest near the band edges.) In general our model can suggest adlayers of the form (fexfe) or c(2Ax2Z), k and I integers. Specifically, if the next-nearest-neighbor site, (22), is the most attractive, c(2x2) is favored; if (23) has AW most negative, c(4x2) has an energetic advantage; if (33) is most attractive, c(4x4) results; and if (13) is most negative, (2x2) should prevail [unless (22) is also attractive, in which case after some (2x2) growth the centered site will be filled, giving
mentioned above and because the table underestimates the frequency of ( l x l ) patterns in nature 1 — our calculation predicts remarkably similar ratios. In the band range - 1 . 8 * £ , £ 1 . 8 and for the gamut of parameters without or with split-off states, assuming uniform weighting of parameters, we find roughly the same number of (2x2) and c(2x2) (slightly more of the lattice), and about k—\ as much e(4x2) patterns. As forecast by the preceding discussion of general properties, the viability of a particular pattern for a particular binding type is quite dependent on the parameters, particularly for weak potential. A change of Ea or especially EF by ^Wb within the band interior will destroy a particular non-(lxl) array's energy advantage about a half to a third of the time—for stronger V, the latter factor being much more typical. Changing the potential from weak to intermediate has a more potent effect, altering structure in about i of the instances. On the other hand, going from intermediate to strong potential, there is remarkably little change in structure. In going from V/T = 4 5, with all other parameters fixed, the pattern changes significantly for only about {• of the "samples." Finally, changing binding symmetry has a far more profound effect than varying the parameters listed above: It is extremely rare that for the same EF, E„ and V, more than two of the four types will exhibit the same pattern. This is to be expected from Figs. 6-8. We do have one reliable test for the interaction energy of a c(2x2) pattern. To a fairly good approximation one can view this structure as a twodimensional square-lattice gas with repulsive nearest-neighbor interaction and attractive nextnearest interaction (and neglect more distant interactions). Recalling that a lattice gas is equivalent to the Ising model, we can use the results of

1

INDIRECT INTERACTION

BEETTW E E N

Fan and Wu for the Ising model with second-neighbor interactions. 1 B Since electrons have spin i , to transfer between the two models we make the substitutions wx = HWu{Er)

=- 4J

and

where J and J' a r e the Ising nearest and nextnearest exchange constants, and wj and u>2 a r e both positive. The c ( 2 x 2 ) structure falls into the domain called antiferromagnetic, for which the approximate solution for Te, e 8u, 1 / s = V 2 - e - « „ 2 / 2 + e ^«, 2

(,3=i/fe7c)

t

(3. 6 )

applies. This equation for the ordered lattice critical temperature Tc is accurate to within a few percent for toj 2 w2. Most of our p a t t e r n s fall into this range, with w% ranging from about wx to an order of magnitude s m a l l e r . F o r Wi = wz, (3.6) becomes a cubic equation with the solution e"*"2 /2 = 0.68946 . . . (as compared with a Pade-approximant value of 0. 6 8 3 7 . . . ) ; for MJJ « 7.83M» 2 , one can show by direct substitution that e~***n = 0. 909. Consequently, ^wt ranges from 0. 747 to 0.648. Experimentally, the c r i t i c a l t e m p e r a t u r e i s on the order of ^ eV, " so that wt ~ 0 . 0 3 7 - 0 . 032 eV. In our units o f \ W t , » i ~ 0 . 0 4 for Wb = 5 eV and 10,-0.02 for W6 = 10 eV. These values a r e in r e a sonable agreement with calculated a^ for V / T = 3 and Ea near the band center. IV. CONCLUSION Within the context of a tight-binding model, we have c a r r i e d out a calculation of the indirect interaction energy AW between a pair of atoms adsorbed on the (100) surface of a simple cubic solid. To AW must be added the direct interaction between the adsorbate atoms. The model involves four p a r a m e t e r s , the bandwidth Wb and the F e r m i energy EF of the bulk solid, the atomic energy Ea of the free-adatom orbital <pa and the hopping m a t r i x element V which mixes <pa and the orbitals
A D A T O M S ON A . . .

3647

c r e t e surface unit cell indices of the two a d s o r bates i and j . Our work i s related to that of Grimley, who c a l culated AW in the asymptotic limit of large s p a c ing between adsorbates for a surface c h a r a c t e r i z e d by a p a r a m e t e r which allowed for v a r i a b l e r e a c tivity with the a d s o r b a t e s . The essential d i s t i n c tion between G r i m l e y ' s work and the p r e s e n t analysis is that our model includes c r y s t a l s t r u c t u r e effects and calculations a r e c a r r i e d out numerically so that AW can be calculated a s a function of adsorbate separation, even for closely spaced adsorbates. These differences a r e of considerable importance since we find the interaction i s quite sensitive to the binding type and the i n t e r action i s of significant strength only for i n t e r adatom spacing of a few substrate lattice spacings o r l e s s , the interaction falling off in scale roughly exponentially for the first s e v e r a l lattice spacings, whereas Grimley found a power law for large spacings. F u r t h e r m o r e , the virtual level approximation of Grimley i s probably not valid for p a r a m e t e r s appropriate to observed binding e n e r g i e s , since split-off states occur in this range of coupling, i . e . , a tendency for surface complexes to be formed, which then couple to the indented solid. Nevertheless, the p r e s e n t model i s sufficiently crude that the calculations a r e p r i m a r i l y of qualitative significance, i. e . , it is not possible to this stage to draw comparisons with specific s y s t e m s . We note that the interaction at small spacing v a r i e s from attractive to repulsive, generally on the scale of the lattice spacing, with the sign of AW for a particular spacing varying moderately with Er, and significantly l e s s so with E3 and V for p o tentials of physical i n t e r e s t . It is gratifying that one finds for a reasonable range of p a r a m e t e r s that many of the o v e r l a y e r s t r u c t u r e s observed in low-energy-electron-diffraction (LEED) data appear to be the stable s t r u c t u r e s based on the calculated AW c u r v e s . In addition, the rough agreement between the melting temperature for the c(2x2) lattice of H on (100) W and that calculated from AW fit to the observed chemisorption energy shows that the scale of AW is c o r r e c t . E s t i m a t e s indicate that explicit 3, 4 . . . body forces a r e negligible compared to the pair interaction even at high coverage. A distinctive feature of this work is the strong dependence of AW on the symmetry of the a d s o r p tion site (relative to the s u b s t r a t e ) . Unfortunately, our present analysis does not p e r m i t the hazarding of predictions of which adsorption-site symmetry exists in p a r t i c u l a r experimental s y s t e m s . Seve r a l new approaches to analyzing experimental data should, however, provide information on this subject: Park 2 0 has commented on effects of the antiphase relationship between surface subdo-

344 3648

E I N S T E I N AND J .

mains—certain LEED b e a m s a r e broadened while others remain unchanged. In some c a s e s , it b e comes possible to distinguish bonding at fourfold symmetry s i t e s (atop or centered) from that at twofold symmetry sites (bridge o r bridge perpendicular). Andersson and Pendry 2 1 a r e using intensity v e r s u s energy LEED spectra to investigate the structure of the surface unit cell, in p a r t i c u l a r to deduce the spacing between adatoms and their substrate neighbors. Webb and associates 2 2 a r e seeking s i m i l a r information from the kinematic (single-scattering) intensity, which they extract from (multiple-scattering) LEED data by averaging at constant momentum t r a n s f e r .

*Work supported in part by the National Science Foundation and the Laboratory for Research on the Structure o£ Matter, University of Pennsylvania, of the Advanced Research Projects Agency. *G. A. Somorjal and F. Z. Szalkowski, J. Chem. Phys. 54, 389 (1971), who reference most of the current experimental observations of patterns and offer a lucid discussion of nomenclature. This listing, however, suggests that the (1 x 1) pattern occurs roughly as often as c(2x 2) or (2x2). Infact, (1 x l ) is the most frequently achieved pattern, by a substantial margin. It is however, far more difficult to detect from LEED patterns and also less interesting to deal with; Peder J. Estrup, private communication. 2 J. KouteckJ?, Trans. Faraday Soc. 54, 1038 (1958). S C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963), p. 360, who references the pioneering RKKY papers. 4 T. B. Grimley, Proc. Phys. Soc. (London) 90_, 751 (1967). 5 T. B. Grimley, Proc. Phys. Soc. (London) 92, 776 (1967). e T. B. Grimley and S. M. Walker, Surface Science 14_, 395 (1969). 7 D. M. Newns, Phys. Rev. 178, 1123 (1969). 8 P. W. Anderson, Phys. Rev. 124, 41 (1961). 9 David Kalkstein and Paul Soven, Surface Science 2£, 85 (1971). 10 P. W. Tamm and L. D. Schmidt, J. Chem. Phys. 5J,, 5352 (1969). " T . B. Grimley, Proc. Phys. Soc. 72, 103 (1958). ,2 S. Alexander and P. W. Anderson, Phys. Rev. 133,

R.

SCHRIEFFER

The p r e s e n t analysis should be generalized in a number of directions. A more r e a l i s t i c s u b s t r a t e having d orbitals and self-consistent potential effects a r e clearly of importance. Clearly, the d i r e c t interaction between a d s o r b a t e s m u s t be added to AW in calculating the p r o p e r t i e s of tightly packed o v e r l a y e r s . ACKNOWLEDGMENTS We would like to thank P r o f e s s o r Paul Soven and P r o f e s s o r P e d e r J . Estrup, a s well as D . Kalkstein and R. H. Paulson, for many helpful discussions regarding the above p r o b l e m s .

A1594 (1964); T. Morlya, Progr. Theoret. Phys. (Kyoto) 33, 157 (1965); S. H. Liu, Phys. Rev. 163, 472 (1967). ,3 B. Caroli, J . Phys. Chem. Solids, 28, 1427 (1967). M D. E. Eastman, J . K. Cashion, and C. A. Switendick, Phys. Rev. Lett. 27, 35 (1971). 1B D. J . Kim and Y. Nagaoka, Progr. Theoret. Phys. (Kyoto) 30, 743 (1963). ,e R. P. Messmer and A. J. Bennett, Phys. Rev. B 6_, 633 (1972). "it is interesting to ask which of these cases have small enough work-function changes to indicate neutral rather than ionic chemisorption. Unfortunately, only half the references given by Somorjai and Szalkowski include information on A (1/4) eV, we find that just less than half of those specifying Aip correspond to neutral adsorption, so our sample is hardly statistical. In this select group, we find a single case each of (l x l), (2x2), and c(4x 2), and two examples of e(2 x 2), in reasonable agreement with predictions. 18 Chungpeng Fan and F. Y. Wu, Phys. Rev. 179. 560 (1969). 19 For example, Peder J. Estrup, in The Structure and Chemistry of Solid Surfaces, edited by Gabor A. Somorjai (Wiley, New York, 1969), Chap. 28. 20 Robert L. Park, in Ref. 19, Chap. 28. 21 S. Andersson and J. B. Pendry, J. Phys. C 5 , L41 (1972). 22 Max G. Lagally, Tran C. Ngoc, and M. B. Webb, Phys. Rev. Lett. 2£, 1557 (1971); J. Vac. Sci. and Tech. 9, 645 (1972).

345

ERRATA FOR P H Y S . REV. B 7, 3629 (1973) p. 3633, left column: Let z be the coordination number of an adatom (so z=2 or 4 for bonding onto B or C sites, respectively). Then V = \Vai\ x y/z and V*n = V^i = V/y/z.

p. 3647, left column: (3wi ranges from 0.747 to 1.495, so that wx ~ 0.037 - 0.075 eV.

346 PHYSICAL

REVIEW

B

VOLUME

17, NUMBER

8

15 A P R I L

1978

Theory of vibrationaUy inelastic electron scattering from oriented molecules J. W. Davenport,* W. Ho, and J. R. Schriefler Department of Physics. University of Pennsylvania, Philadelphia, Pennsylvania 19104

(Received 17 October 1977) The electron scattering cross sections, both elastic and vibrationaUy inelastic, have been calculated using the Jfa multiple-scattering method for H2, N2, and CO. The accuracy of the calculational scheme is tested by comparing to data from gas-phase measurements. Good agreement is found between theory and experiment. The formalism is then applied to molecules with fixed orientation by freezing out the rotational motion. The differential inelastic scattering cross sections for vibrational excitation exhibit a variety of angular patterns depending upon the molecule and the energy and direction of the exciting electrons. In all cases which have been calculated, the cross section for vibrational excitation is dominated by negative-ion resonances. The angular distribution patterns reflect the symmetry of these ionic states. These calculations indicate the possibilities of using the characteristics of the inelastic-cross-section patterns as a function of the exciting electrons' energy and direction to study molecules adsorbed on a surface.

1. INTRODUCTION Dill and Dehmer 1 have shown that the multiplescattering formalism developed by Johnson and c o w o r k e r s 2 (for bound states) can be adapted easily to problems involving continuum s t a t e s . This t h e oretical technique has been applied successfully to photoionization from both core 3 and valence 4 levels for randomly oriented gas-phase molecules 3 ' 4 as well a s molecules with fixed orientation. 4 ' 5 In this paper, we apply this technique to elastic 6 and v i brationaUy inelastic electron scattering from m o lecules, another problem involving continuum states. 6 The motivation for this work is the growth in the study of the bonding configuration of molecules adsorbed on a surface by inelastic l o w - e n e r g y - e l e c tron scattering. 7 " 9 The objective is to illustrate the different angular effects that can be observed in the vibrationaUy inelastic scattering from a m o lecule whose orientation has been fixed by a s u r face, as a function of the identity of the molecule and the energy and direction of the incident e l e c tron beam. Our approach to this problem i s b a s i c ally the same a s it was to the problem of differential photoionization from a molecule of fixed o r ientation on a surface. 4 ' 1 0 We assume that if the adsorbed molecule maintains its molecular identity when adsorbed on a surface, the microscopic p r o p e r t i e s of the scattering p r o c e s s will be given by p r o p e r t i e s of the gas-phase molecule. This hypothesis has been thoroughly checked for photoemission from molecularly adsorbed CO. 10 For example, if we were investigating the inelastic s c a t t e r ing from CO adsorbed on a surface, caused by the excitation of the CO vibrational mode, we would first calculate the scattering pattern for an i s o lated molecule with fixed orientation. The surface would only play a role in fixing the orientation and 17

modifying the incoming- and outgoing-wave fields, a s well a s screening the long-range p a r t of the p o tential. In the c a s e of photoemission, t h e s e modifications were treated in a m a c r o s c o p i c c a l c u l a tion. 1 0 ' 3 3 In a more refined calculation, o t h e r effects due to bonding, such a s vibrational frequency shifts and damping effects, can be included, although these a r e small for weak bonding (see Sec. V). One of the most d r a m a t i c features o b s e r v e d experimentally and theoretically in photoemission from molecules i s associated with r e s o n a n t effects in the continuum. 3 , 4 '" ! These r e s o n a n c e s occur both for molecules in the gas phase o r a d s o r b e d on a surface. Resonant effects a r e known to dominate e l e c t r o n - s c a t t e r i n g c r o s s sections in t h e 1-10eV range. Therefore, we will spend c o n s i d e r a b l e time discussing the effects of these r e s o n a n c e s on the differential s c a t t e r i n g c r o s s section, especially for molecules of fixed orientation. In e l e c t r o n scattering these r e s o n a n c e s a r e negative-ion r e s onances. The incoming electron is t e m p o r a r i l y trapped in a virtual bound state of the m o l e c u l e . For example, H2 has a
© 1978 The American Physical Society

3116

J. W. D A V E N P O R T , W. H O , AND J. R. S C H R I E F F E R

number n- 1 up to 8 a r e observed. 1 2 Many of these phenomena a r e expected to be found for electron scattering from adsorbed molecules to the extent that they retain their gas-phase c h a r a c t e r i s t i c s . In Sec. U we describe the details of the f o r m a l i s m . The calculated gas-phase c r o s s sections a r e compared to experimental data in Sec. III. This section i l l u s t r a t e s what a major role the negativeion resonances play in both the magnitude and angular distributions of the scattered electrons. The good agreement between theory and experiment shown in this section is used a s justification for the calculations for molecules of fixed orientation shown in Sec. IV. This section illustrates the effects upon the inelastic-scattering pattern by changing the direction and energy of the incident electron beam. Finally, in the concluding Sec. V, we comment about the applicability of these r e sults in inelastic electron scattering from m o l e cules adsorbed on surfaces. II. FORMALISM The full Hamiltonian contains electron-electron, electron-nuclear, and internuclear interactions. We write it a s H = HM + T,+V(r,R),

(2.1)

where H M r e p r e s e n t s the target molecule, T, the kinetic energy of the scattering electron, and V the interaction potential between the scattering e l e c tron and the target molecule. In the multiple-scattering method, this interaction potential i s approximated in two ways. F i r s t the effects of exchange and correlation a r e taken into account by a t e r m proportional to the one-third power of the charge density. Then V(r,R) = V„(r,R) + V B (r) + K I C (r),

(2.2)

where V„ is the nuclear attraction with R r e p r e senting the nuclear and r the electronic coordina t e s . The static electron-electron interaction or the H a r t r e e t e r m is given by

where n is the electron density and Vsc i s the exchange-correlation potential which we take to be the usual Xa form given by V«(r) = - 3 a e 2 ( 3 « / 8 ^ ) l / 3 , (2.4) where a is a p a r a m e t e r of order 0.7. The second approximation i s that the potential is spherically and volume average to the "muffin-tin" form. That is, nonoverlapping spheres a r e constructed about each nucleus and an outer sphere placed around the entire molecule. Within the

1_7

atomic s p h e r e s (region I) and outside the outer sphere (region III) the potential is spherically a v eraged and in between (region II), it i s volume a v eraged to a constant. The exact p r o c e d u r e h a s been given by Danese and Connally." We r e c a l l that an exact solution to the s c a t t e r i n g problem could be obtained if VIC w e r e r e p l a c e d by E ( r , r ' ) , the electron self-energy, or the optical potential, and this has been c a r r i e d out for c e r t a i n model s y s t e m s and for a few atoms. The s e l f - e n ergy i s nonlocal and energy dependent and not easy to compute except in certain l i m i t s . F o r example, at large distances and low energies, t approaches the classical polarization p o t e n t i a l " i:--
(2.5)

where a ^ is the polarizability. In the Xa approximation, V« = ! a S ° H F >

(2.6)

where £^,F is the H a r t r e e - F o c k self-energy of a uniform electron gas of density «(f) in the limit as £ - 0 . For a = | , the Kohn-Sham value, Vte = 7:°HT. At high kinetic e n e r g i e s the effects of exchange a r e cut off. This has been t r e a t e d by Lee and Beni 15 in the context of extended x - r a y absorption fine s t r u c t u r e (EXAFS). The same type of cutoff o c c u r s for fixed energy a s the density d e c r e a s e s . However, it is known, at least for the electron gas, that these cutoffs a r e not nearly s o rapid a s a r e given by H a r t r e e - F o c k . Indeed, in the r a n dom-phase approximation (RPA) the self-energy is nearly constant and somewhat l a r g e r than the Kohn-Sham value up to p ~ 2kr,u where p i s the T h o m a s - F e r m i local momentum, and kp the local F e r m i momentum. Consequently, the Xa potential i s adequate in the interior of the molecule where the density is high. However, in the outersphere region, the Xa potential should be cut off and matched onto the polarization potential. We have chosen an exponential cutoff because, for e x ample, the H a r t r e e - F o c k self-energy goes like £HF--2*eV/>2

(2.7)

for p/kr large and n is dropping off exponentially with a decay length of o r d e r 2. So we take the potential in region III to be V(r,R) = - a p o l / r 4

+

yoe-2',

(2.8)

and have chosen V0 so that V matches the Xa potential at the starting point of the radial mesh (which is just inside the outer sphere). The c r o s s sections a r e not very sensitive to the form of the cutoff. It would be useful to study the effects of h i g h e r - o r d e r multipoles in describing the longrange part of the potential, since the polarizibility i s in general anisotropic.

348 17

THEORY

OF V I B R A T I O N A L L Y

INELASTIC ELECTRON

Finally, the density which determines the potential in the Xa method is the total density for the system, target plus incident electron. Ordinarily the ground-state density of the target is a good approximation to the total and this approximation works well for H2 where the a„ resonance is fairly broad. F o r the narrow resonances in N2 and CO we have found that the density of the scattering electron must be included and we have done this approximately a s discussed in Sec. III. This makes sense for narrow resonances because the density due to the scattering electron in the region of the molecule would be expected to be large. Given the Hamiltonian in Eq. (2.1), let * * be the target wave functions which a r e eigenfunctions of H„. The full transition amplitude between target s t a t e s M and M' and electron state k" and k' is given by f$M-*'M,)=(-M/2irtP)(e,Xf'~*ll\V\i<x,VC))

,

(2.9)

SCATTERING.

frequency given by the observed excitation energy. It is natural to expand the scattering amplitude in partial waves

/n<= E

YLik)fLL'WY,.-fr),

(2.10)

where *„(?, R) is a solution of (2.11)

He = Te + V ,

for fixed R. F u r t h e r , we take this solution to have the asymptotic form * , ( r , R ) - e ' t - T + / r c ( R ) e""/r

,

(2.12)

so that / is the e l a s t i c - s c a t t e r i n g amplitude at fixed R. Then within the adiabatic approximation and neglecting the difference between the magnitudes k and k' the transition amplitude is given by 16 /(kM-k'M') = /d3fl*J-(R)/i;f**(5)-

(2.13)

The adiabatic approximation is justified on physical grounds because the interaction time (crudely, the time it takes for an electron to t r a v e r s e the molecule, 10" 16 sec) is small compared with the vibrational period (10" 14 sec) or the rotational period (10~ u sec). In fact, for the narrow r e s o nances observed in N2 and CO, the c r o s s sections contain fine structure which has been attributed to nonadiabatic p r o c e s s e s since adiabatic theories do not reproduce it. (See Refs. 11 and 17 and Figs. 2 and 6.) However, this fine s t r u c t u r e i s clearly of secondary importance and we have not treated it. Since we a r e interested in vibrational excitations of the target, we take *j,(R)=X„(R), s a

(2.14)

where xv > vibrational wave function assumed to be that of a harmonic oscillator with vibrational

(2.15)

where the Y's a r e the r e a l spherical harmonics and L is a double index (I, m). Since the YL form a complete set, this i s general. As shown in the A p pendix fLL- (which is proportional to the T matrix) is given by

• £

/*.

l

k

(l*iK)l\KAL.

(2.16)

where if is a r e a l s y m m e t r i c m a t r i x , k2 = E-kinetic energy, and A = (A, n). For s p h e r i c a l potentials iC x . A =-tanr ) l 6, x 5„ | J , where r\, is the p h a s e shift.

(2.17) Then

where * is an eigenfunction of the full Hamiltonian. In fact, we shall make the adiabatic approximation that *(rJ5) = *J,(R)*e(r,R),

311"

(2.18) the well-known e x p r e s s i o n for the s c a t t e r i n g a m plitude. Since our molecular potentials a r e not spherically s y m m e t r i c over all space, the full K matrix must be computed. However, we can use symmetry to block diagonalize K and for linear molecules find p a r t i a l amplitudes for a, it, 6, e t c . s c a t t e r i n g s . Then, the differential c r o s s section for an oriented molecule is given by da. dfi'

E

YL(k){v'\fLL.(R)\v)YL.(k')\

(2.19)

LL'

For the gas phase the amplitude should be t r a n s formed to the laboratory frame and m a t r i x e l e ments taken between molecular rotational s t a t e s . In this paper we wish to compare our c a l culations with g a s - p h a s e data in which rotational excitations w e r e not resolved. Assuming a thermal distribution of initial rotational s t a t e s and summing over all final rotational s t a t e s i s equivalent to averaging Eq. (2.19) over all orientations of the molecule. The dependence on orientation comes from k and k', which a r e defined in the molecular f r a m e . The total g a s - p h a s e c r o s s section can be obtained directly by integrating over £ ' and averaging over k to give

— zKv\f

LL.\v)\*

(2.20)

4* LL'

The differential c r o s s section is given in Appendix B and a g r e e s with Eqs. (46) and (47) of Ref. 1. III. GAS-PHASE CROSS SECTION In this section we present r e s u l t s for g a s - p h a s e molecules using the technique developed in Sec. II.

J. W. D A V E N P O R T , W. HO, AND J. R.

3118

TABLE I. Experimental parameters used in the calculation.

H2 N2 CO

Equilibrium spacing1

Mean polarizability*

Vibrational energy'

fao>

(fl?) 5.33 11.88 13.16

(meV) 545 293 269

1.41 2.08 2.13

TABLE II. Five internuclear spacings used to calculate the angular independent scattering amplitude f^. .*

Internuclear spacings H2 N2 CO a

1.11 1.22 1.41 1.88 1.98 2.08 1.93 2.03 2.13

1.56 1.71 2.18 2.28 2.23 2.33

All distances in atomic units (a„

Root-mean-square displacement of a harmonic oscillator 0.28 0.18 0.19


b>=4)

1

1

1

H "" ° ~ ~ - ^

12.6 00 °
17

-O•o'

o^

10.0

I

1

2

_

""-o~

V 7.5

2 5

Our purpose is to show that the Xa-SW method agrees well enough with the gas-phase data so that it can be used to predict the differential cross s e c tions for the oriented molecules where no angular distribution data exist at the present time. A more complete description of the gas-phase calculation i s presented in another paper where simultaneous rotational and vibrational transitions are also d i s cussed for H2, N2, and CO. In Table I we present the experimental parameters used throughout the calculation. The angular independent scattering amplitude flLi was calculated at five internuclear spacings shown in Table II. For N 2 and CO, the extrema were chosen to correspond approximately to the root-mean-square (rms) displacement of a harmonic oscillator in the « = 4 state. For H2, the rms displacement for the n = 4 state from the equilibrium spacing proved to be too large and the end points were chosen to be somewhat larger than the rms displacement in the n = 1 state. Therefore, only transitions to the first excited vibrational state were calculated for Hj. The vibrational wave functions were taken to be those of a harmonic oscillator. The scattering amplitude fLL< was fitted to a fourth-order polynomial and (for numerical stability) the vibrational matrix element, e.g., in Eq. (2.20), was integrated numerically using the trapezoidal rule with roughly 150 points. Using Eq. (2.20), we have computed the angle-averaged total cross sections for H2, N a , and CO. The result for H2 is shown in Fig. 1. We found a c u resonance which leads to a broad peak in the c r o s s section centered at 4 eV compared

J

i

' 15.0

50

"Reference 19. "Reference 20.

SCHRIEFFER

-

""

L

1

"0

1

1

1

1

1

1

I

INCIDENT ENERGY (eV) FIG. 1. Elastic scattering cross section for H2 as a function of the incident energy. The data in dashed curve are from Linder and Schmidt (Ref. 21). with the experimental value of 3 eV. 21 The absolute c r o s s section agrees to better than 50%. Closer agreement has been obtained with other methods," , 2 3 but our results should be adequate in interpreting oriented molecule effects. We have found that the potentials constructed from the self-consistent ground-state density do not display the IT-type resonances found experimentally for N2 and CO. The reason i s that the resonances are narrow and the density of the s c a t tering electron makes a substantial contribution to the self-energy. The potentials for the ground state are too attractive (representing a s they do the neutral system). Indeed, there are bound states with v symmetry in both molecules about 2.7 eV below vacuum. If we occupy the N 2 level with one electron and compute a new potential (non-self-consistently), we find a IT, resonance in the scattering c r o s s section at 9 eV. Clearly what is needed is a potential intermediate between the ground state and the negative ion. To produce such a potential, we have extended Slater's transitionstate concept and occupied the ? level with half an electron. 2 4 This puts the resonance for N2 at 3.5 eV (experimental is 2.3 eV) and CO at 4.0 eV (experimental is 1.7 eV). We do not expect this model to produce much better agreement with the data than this although a self-consistent treatment of the resonance state is possible. In fact, if this potential were slightly more attractive, N2 and CO would have an electron affinity which would be given approximately within the local density theory by the transition-state potential. Therefore, it is entirely reasonable that such a potential should yield an approximate resonance energy. The total cross sections computed in this way for N 2 and CO are shown in Figs. 2 and 3(a). In the case of N 2 , the c r o s s section shows a strong irt

350 THEORY

1

I

OF

1

1

VIBRATIONALLY

1

1

1

INELASTIC

1

1 30 -

30 l«l

/\

;y\

20 L \ 1

-

o

o

5 i

1

1

1

1

1

1

1

i

1

i

i

I

3119 i

1

(a) CO

-

20

-

15

-

-

10

-

""

5

i5r 0

1

SCATTERING.

25

N2

i 7 V 1 / \

10

ELECTRON

1

i

,

i

i

1

2

3

4

1

5

INCIDENT

INCIDENT ENERGY ( eV )

1

6

1

7

1

8

1

1

9

10

ENERGY ( eV )

FIG. 2. Elastic scattering cross section tor N2 as a function of the incident energy. Data from Golden (Ref. 25) (dashed curve) and Srivastava, Chutjian and Trajmar (Ref. 26) p ) . resonance and good agreement is obtained with m e a s u r e m e n t s by Golden 25 and by Srivastava et a/., 26 except lor the fine s t r u c t u r e . The r i s e in calculated c r o s s section below 1 eV is due to the increasing a, contribution at low energy, which would diminish if the potential was made more attractive, simultaneously pulling down the r e s o nance. To our knowledge, there is no experimental data for the absolute elastic total c r o s s s e c tion of CO, only the relative differential elastic c r o s s section of Ehrhardt et al." In Fig. 3(b), comparison i s made between the calculation and the data normalized to the calculated peak of 90°. Good agreement is obtained for the shape and the relative intensities of the resonances. Schulz 28 has also given the maximum in the elastic total c r o s s section to be 24.0 A 2 , which i s close to the calculated value of 29.2 A2. All gas-phase m e a s u r e ments indicate that CO and N2 a r e very s i m i l a r , both in the elastic and inelastic scattering c r o s s sections. 2 7 ' 2 8 Both show a strong v resonance with the s t r u c t u r e in the c r o s s section l e s s pronounced in CO. The width of the resonance is l a r g e r for CO than N2, which has been attributed to the mixture of p and d waves in the resonance, whereas in the u, resonance of N2 there is no p wave by symmetry. The experimental fact that the s c a t t e r ing c r o s s sections from N2 and CO a r e similar indicates that dipole scattering is not the important p r o c e s s , at least in this energy range. A comparison of the calculated and measured elastic differential scattering c r o s s sections i s a more stringent test of this calculational scheme than is the comparison above of integrated c r o s s sections. We show in Fig. 4(a) the calculated and measured differential elastic c r o s s section for N 2 , 2e and the corrresponding curves for CO in Fig.

O

t o

CO

tfs

I

2

3

4

INCIDENT

5

6

7

8

ENERGY

9

10

(eV)

FIG. 3. Elastic scattering cross section from CO as a function of the Incident energy, (a) Calculated total scattering cross section, (b) Differential scattering cross section. Data from Ehrhardt et al. (Ref. 27) (dashed curves) are normalized to the calculated peak at 90" scattering. 4(b). The experimental data for N2 a r e for an incident electron energy of 5 eV. Since our total c r o s s section is shifted to higher energy by a p proximately 1 eV, the calculated curve i s for an incident energy of 6 eV. F o r CO the m e a s u r e m e n t is at 2 eV, 27 and the calculation at 4 eV. For N2 we have the absolute data of Srivastava et al.2B with which to c o m p a r e . Our calculation is uniformly about 50% higher than the experimental data. Since the data for CO from E h r h a r d t et al."

351 J.

:i 120

4 0

~i

W. D A V E N P O R T ,

W.

HO,

AND

J.

R.

SCHRIEFKER

17

r

(a) O

I

O

If)

«N '0

20 40

60

"0

80 100 120 140 160 180

SCATTERING ANGLE 8

(DEGREES)

20

40

50

80

100 120 140 160 180

SCATTERING ANGLE 8

(DEGREES)

FIG. 4. Differential clastic cross section as a function of the scattering angle for: (a) N2 at incident energy of 6 eV. The absolute measurement (C ) at incident energy of 5 eV is from Srivastava, Chutjian, and Trajmar (Ref. 26). The difference in 1 eV compensates the shift in the peak of the resonance between theory and experiment for the total scattering cross section; and (b) CO at incident energy of 4 cV compared to the data of Ehrhardt et al. (Ref. 27) (O) at 2-eV Incident energy, normalized to the calculation at 90°. i s presented in a r b i t r a r y units, we have n o r m a l ized their values to our calculation at 90 c . Experimentally the minimum in the CO c r o s s section o c c u r s at a l a r g e r scattering angle than it does for N 2 . This is also predicted by our calculation. In general, the agreement between theory and e x p e r i ment for the differential elastic c r o s s section is very good for N2 and CO. We now turn to the inelastic c r o s s section. The effect of the negative-ion resonances is much more pronounced in the inelastic c r o s s section than it was in the elastic c r o s s section. Let us first comp a r e the absolute total c r o s s section for exciting a diatomic molecule from the vibrational ground state to the nth state as a function of the energy of the exciting electron. Figure 5 compares our calculated inelastic c r o s s sections with the m e a s u r e ment of Ehrhardt et al." for CO. The m e a s u r e d c r o s s sections for both the 0 - 1 and 0—2 v i b r a tional transitions a r e dramatically enhanced at an incident energy corresponding to the negative-ion resonant state. Our calculation, a s usual, does not place this resonance at the c o r r e c t energy, but it does reproduce the magnitude extremely well. Experimentally, the cross section for exciting the first vibrational mode falls by an order of magnitude a s the incident electron energy is increased from approximately 1.8 to 3.3 eV. Theoretically, this c r o s s section d e c r e a s e s by a factor of 40 from 3.5 to 10 eV. Figure 5 also illustrates the enhanced excitation of h i g h e r - o r d e r vibrational s t a t e s of the molecule when the excitation energy is near the resonance.

For example, the ratio of the 0—2 to 0— 1 c r o s s section at an exciting energy of 3.5 eV i s 43%, while it d e c r e a s e s to 7% for a 10-eV incident e l e c tron beam. Ehrhardt et al.21 have shown e x p e r i mentally that near the resonance the 0— 7th mode 15

1

1

1

1

1

1

1

1

1

i>j! 1 0

CO v =0-2

1

-

0 5

,/ ^/^ , v

0 35 —

i

i

i

i

-

- A

3.0-

° <

— 2 5-~ b

-

2.01-

' i i

r\

i

i W

i I

\ W

v =0 — 1

\

; \

\ \

15 10 0.5

J AV

n—4—J

i,

O t H <2 3 1 4 1 51 6 7 8 9 10 INCIDENT ENERGY (eV) FIG. 5. Total cross section for the 0 — 1 and 0—2 vibrational transition of CO as a function of the incident energy. The dashed curves are the measurement by Ehrhardt et al. (Ref. 27) with the absolute cross sections obtained from integrating the differential scattering cross sections and normalizing to absolute elastic cross sections.

352 T H E O R Y OF V I B R A T I O N A L L Y

INELASTIC ELECTRON

can be seen in CO with ~6% of the amplitude of the 0 - 1 transition. General agreement has also been obtained b e tween the calculation and the measurement for the total c r o s s sections of the 0— 1 and 0—2 vibrational excitation of N2 and the 0— 1 excitation of H 2 , as a function of the incident energy. The resonance in N 2 is m o r e narrow than CO, both in the theory and the experiment. 2 8 The calculated peak of the resonance is approximately 2-eV higher than the experiment. However, the relative intensities of the 0 - 2 to the 0 - 1 vibrational excitation a r e in good agreement. In H 2 , a broad resonance cent e r e d around 4 eV is obtained from the theory and The absolute magnitude of the the experiment. total c r o s s sections differ by a factor of 2, with the calculation uniformly l a r g e r . However, the g e n e r al shape i s reproduced very well. The differential inelastic c r o s s sections for the 0— 1 vibrational excitation of N2 and CO a r e shown in Fig. 6. The calculation and measurement a r e for an exciting electron energy corresponding to the maximum in the total c r o s s section (see Fig. 5). This is 1.9 eV for N2 (Ref. 31) and 1.83 eV for CO (Ref. 27) experimentally, and 4 eV theoretically for both molecules. The data a r e not absolute so they a r e normalized at 9 = 90°. In N2 there is a m o r e pronounced peak in both the experiment and theory than for CO. This is due to pure d-wave scattering in the n, resonance of N 2 . For CO the

SCATTERING.

90° peak i s much s m a l l e r or nonexisting ( e x p e r i mentally). This is a result of a mixture of p- and d-wave scattering in the ir resonance in CO. F o r a pure £-wave scattering, there would be a m i n i mum in the c r o s s section at 90°. Section IV will illustrate how this p-wzve mixture in the CO s c a t tering can produce differential inelastic s c a t t e r i n g c r o s s sections for a molecule of fixed orientation which a r e quite different from the equivalent s i t u ation in N 2 . IV. ORIENTED MOLECULES Section III demonstrated that the theoretical scheme presented in Sec. II can reproduce r e a s o n ably well the differential elastic and vibrationally inelastic electron scattering c r o s s sections for randomly oriented diatomic molecules. Improved a g r e e m e n t can be achieved, e.g., in the positions of the s c a t t e r i n g r e s o n a n c e s , if a modified m u l t i p l e - s c a t t e r i n g potential i s used. To lower the r e s onance energy for e-N 2 elastic scattering, Dill and Dehmer 6 have used an Xa potential with a = 1 for low kinetic energy and obtained improved a g r e e ment with experiment. More generally, the scheme should be generalized to include nonlocal effects which automatically generate such a v e l ocity-dependent potential, weakening the exchangecorrelation potential at high energy. In this s e c tion we apply the theory to diatomic molecules with 0.6

in o<

¥ 100 120 140 160

SCATTERING ANGLE 8 (DEGREES)

3121

100 120 140 160 180

SCATTERING ANGLE 8 (DEGREES)

FIG. 6. Differential scattering cross section for the 0— 1 vibrational transition as a function of the scattering angle for: (a) Nj at 4-eV incident energy. Measurement by Ehrhardt and Willmann (Bef. 31) P ) is at 1.9 eVj and (b) CO at 4-eV incident energy. Measurement by Ehrhardt et al. (Kef. 27) p ) is at 1.83 eV. Both data are normalized to the calculation at 90°.

353 J . W. D A V E N P O R T ,

3122

(a)

W. H O , A N D J . R .

SCHRJEFFER

17

H,

f ^

.y

\$JU^

»v

FIG. 7. Polar plots for the angular distribution from oriented H2 at 45° incidence, (a) Elastic scattering, (b) 0 —1 vibrational excitation. The incident energy is chosen close to the peak in the total vibrational excitation in the gas phase. Note the difference in the scale. fixed orientation. The scattering process will still be dominated, as it was in the gas phase, by resonances. When the molecule has a fixed orientation, the negative-ion resonance should have two quite separate effects upon the inelastically scattered angular distribution patterns. First, the probability for forming the negative-ion virtual bound state (resonance) should depend not only upon the incident electron energy, but also upon its direction. This is simply a consequence of the symmetry of the negative-ion state. The second effect is in the angular distribution patterns which again reflect the symmetry of the resonant state. For example, if the resonant state is totally d wave (p wave) as it is in K, (H2), then the angular distribution would look like a d wave (/> wave) independent of the angle of incidence of the exciting beam. This angle would only change the amplitude. However, if the resonant inelastic scattering is a mixture of two waves, as it is in CO, we will have angular distribution patterns which change not only in amplitude, but in structure, as the incident direction is changed. This effect results from the angle of incidence dependence of the different partial-wave components in the inelastic cross section. The first example we present is for oriented H2. Figure 7 displays the calculated elastic and inelastic differential cross section for an incident

energy of 4 eV. Although the hydrogen molecule is not very interesting as a surface adsorbate, the case is chosen to illustrate the variety of angular distribution patterns that are possible. The resonance in H2 is distinctly different from the resonances in N2 and CO. It has
354 THEORY

OK

VIBRATION ALLY

INELASTIC

(b) (a)

ELECTRON

SCATTERI.NC.

ISI23

N

N

FIG. 8. Polar plots for the angular distribution from oriented N2 at 45° incidence, (a) Elastic scattering, (b) 0 — 1 vibrational excitation. The incident energy is 4 eV, close to the peak in the total vibrational excitation in the gas phase. Note the difference in the scale.

(H2) = ( - I J cos 2 0 ( | | 2 c o s X where 6OM is the collection angle [Fig. 7(a)], 8, is the incident angle, and fp is the angle-independent scattering amplitude [see Eq. (2.19), fp = / „ ] . The inelastic electron scattering pattern from H2 shown in Fig. 7(b) is now quite simple to understand. The dependence is given solely by the pz wave and the amplitude v a r i e s with the incident electron direction, a s the cosine of the angle with respect to the molecular a x i s . This latter effect is just the probability of forming the p, component of the negative-ion resonance. Our calculations indicate that even off the resonance one obtains an angular distribution s i m i l a r to that of Fig. 7(b). Oriented N2, which is shown in Fig. 8, is nearly a s straightforward a s H 2 , except that the resonant contribution is in the d waves. The geometry is the same a s it was for H2 in Fig. 7(a). The elastic scattering c r o s s section shows a maximum in the forward direction a s it did for H 2) but the rf-wave contributions a r e substantial at right angles to the incident beam. The n, resonance completely dominates the 0— 1 inelastic vibrational transition. Due to the symmetry of this vt state, only the d wave contributes to the scattering and the p wave is excluded. Therefore, to very high accuracy the c r o s s section can be obtained by neglecting all but

the d wave of v s y m m e t r y , i.e., the 1 = 2, m = ±l. This effect is shown by the inelastic s c a t t e r i n g p a t tern in Fig. 8(b). Therefore, we again have a s i t u ation where the shape of the angular distribution is independent of the incident direction. The intensity maximum occurs at 45° incidence and v a n i s h e s with either on axis or perpendicular to the axis CO E = 4V 6, * 45° v = 0 —0

A _v

FIG. 9. Polar plot for the elastic scattering angular distribution from oriented CO.

355 J. W. D A V E N P O R T , W. HO, AND J. R.

3124

incidence. Off resonance, the overall pattern is preserved, with a decrease in intensity and a skewing of the lobes resulting from other I -wave admixtures. The final example we present i s an oriented CO molecule. The previous two examples illustrated a c and n resonance with the inelastic scattering completely dominated by a single partial wave. CO again has a ir resonance, but due to the lower symmetry than N 2 both the pn and d-x components contribute. The elastic differential scattering c r o s s section is shown in Fig. 9. It looks similar to the elastic scattering pattern for N 2 shown in Fig. 8(a), except that the lobes perpendicular to those along the incident beam direction are larger for CO then for N2. This reflects the p-d coupling. The difference in the character of the r e s o -

ld) « • INCIDENCE

SCHRIEFFER

nances in N2 and CO is clearly illustrated by the angular distribution of the 0—1 vibrational transition, shown in Figs. 8(b) and 10(a) for a 45" incident beam. For CO the inelastic c r o s s section is not well approximated by the d-wave or the pwave contribution alone. The two waves mix and interfere, and one needs to include both to account for the angular distribution. Due to the interference between the two waves, the angular patterns and intensities are strong functions of the d i r e c tion of incidence. Figures 10(a)-(c) show the angular distribution for the 0— 1 vibrational excitation for three different incident directions. The c r o s s section for incidence along the axis of the m o l e cule is so small [Fig. 10(c)] that it has to be blown up by a factor of 103 in Fig. 10(d). This decrease of three orders of magnitude is solely a result of being unable to form the negative-ion resonance with electrons incident parallel to the molecular axis. The p-d mixing allows the incident electron to couple to the resonance when it i s incident p e r pendicular to the axis [Fig. 10(b)], in contrast to N2. Inspection of the angular pattern in Fig. 10(b) indicates that the dominant term in the inelastic scattering is the outgoing d wave, while the incident direction can only couple to the pit resonance. Therefore, this geometry i s predominantly p in and d out. Figure 11 is a semilogarithmic plot of the 0—1 vibrational-excitation c r o s s section for CO as a function of the energy for two different collection angles. The solid line i s for a 50" collection angle which is nearly perpendicular to the incident di10.0

Ul II INCIDENCE

i i

: "

i

1

1

1—

1

K" 1

':

w

•

~ •

50"

0.001.

1 : '

140°

: .

vs.

•

1

o.oi r

; •

: :

-<w

1

-°\-°

1

45°

"eouf —s\ \\ — \\ 1 Ii II V 1 \ II1 \\ \\

1.0 F

0.1

r

CO

9;-

t o

FIG. 10. Polar plots for the 0—1 vibrational excitation angular distribution from oriented CO. (a) 45° incidence, (b) Perpendicular to the molecular axis incidence, (c) Parallel to the molecular axis incidence, (d) Part (c) blown up 1000 times In scale.

17

^

r/

i

1

1

1

1

i

i

I

to INCIDENT ENERGY ( e V )

FIG. 11. Intensity at 50° (solid line) and 140° (dashed line) scattering angles (both at 0 = 90°) for CO as a function of the Incident energy for 45° Incident angle.

356 17

THEORY OF V I B R A T I O N A L L Y

INK L A S T I C E L E C T R O N

rection. The dashed curve i s 140c collection, which i s almost forward scattering. The full width at hali-maximum of both curves is approximately 2 eV. At the peak in the resonance, the perpendicular scattering is about 50% l a r g e r than the forward scattering. At an incident energy of 10 eV, the forward scattering i s about 40% l a r g e r than the perpendicular scattering, with both c r o s s sections decreased by nearly a factor of 200.

SCATTERING...

:!I25

s t r a t e has reflectivity R. Then the e l a s t i c c u r r e n t in the specular direction is given by R t i m e s the incident c u r r e n t /„. The inelastic c u r r e n t i s given by

where n is the number of s c a t t e r e r s p e r unit s u r face a r e a . The experiment was for one half a monolayer of CO on Ni(100), 9 with the solid angle

V. MOLECULES ADSORBED ON SURFACES Section III compared the calculated elastic and inelastic c r o s s sections for several diatomic m o l ecules with the appropriate gas-phase data. The agreement between theory and experiment was sufficiently good that we predicted the elastic and inelastic scattering c r o s s sections for molecules of fixed orientation in Sec. IV. In this section, we would like to speculate about the implications of the r e s u l t s presented in Sec. IV on inelastic e l e c tron scattering from a molecule adsorbed on a surface. 7 " 9 The p r e s e n c e of the surface modifies the ideal scattering p r o c e s s presented in Sec. IV in s e v e r a l ways. 34~38 The one which has received the most attention is the reflectivity of the substrate which allows the e l e c t r o n s which have scattered in the forward direction from the molecule to be r e flected into the detector. Forward scattering is dominated by the long-range part of the electronmolecule interaction potential which is in turn given by the molecular dipole moment (actually only the change in dipole moment with internuclear spacing is important for vibrational excitation). For this reason, current theories have treated the scattering from an oriented dipole, including the classical images, using f i r s t - o r d e r perturbation theory. The effect of the image is to cancel the interaction with modes which oscillate parallel to the surface and to quadruple the interaction with perpendicular modes leading to the "selection rule" that only perpendicular modes a r e observed. 3 9 In contrast, the s h o r t - r a n g e interaction can excite modes parallel and perpendicular to the surface with nearly equal intensity. To compare the relative sizes of the long- and s h o r t - r a n g e p a r t s , we note that Persson 3 6 using the dipole model applied to Andersson's experiment (approximately 45 c in and 45° out with a beam halfwidth of 3 D ) has given a vibrational differential c r o s s section of 1 A 2 / s r . This takes no account of substrate reflectivity which could lower this figure by a factor of 5-10. On the other hand, the s h o r t range mechanism gives c r o s s sections of the same size without any reflections at all (cf. Fig. 10). To compare with the data we a s s u m e the s u b -

nsl0"2sr.

(5.2) 2

Taking a c r o s s section of 1 A s r ~ / / « / „ = 7x lO""//? .

l

(5.3)

This is in the s a m e range a s the m e a s u r e d r a t i o (10~ 3 ). An important difference is that the dipole scattering gives a ratio which is b a s i c a l l y independent of R, while the s h o r t - r a n g e s c a t t e r i n g depends on it. This feature could be checked with many existing s p e c t r o m e t e r s . The surface may also modify or even d e s t r o y the resonances which a r e so c h a r a c t e r i s t i c of the gas phase. In photoemission, spatially localized shape resonances have been found to be e s s e n t i a l l y u n perturbed for CO adsorbed on nickel. Though s i m ilar to the negative-ion r e s o n a n c e s d i s c u s s e d h e r e , t h e r e is still no guarantee that electron s c a t t e r i n g resonances exist for adsorbed m o l e c u l e s . The discussion we have given i l l u s t r a t e s that if they exist the inelastic scattering c r o s s section will be a s large or l a r g e r than the best e s t i m a t e s of the dipolar c r o s s sections. The i s s u e can probably be resolved only by experiment. F r o m an experimental point of view t h e r e a r e several advantages to observing the s h o r t - r a n g e scattering. F i r s t there a r e no quasi selection r u l e s , so the complete s y m m e t r y of the surface molecule may be observed. F o r e x a m p l e , Wong and Schulz 40 have shown that for inelastic s c a t t e r ing from g a s - p h a s e C 6 H 6 , it is possible to couple to different symmetry vibrational m o d e s by tuning the incident energy to different r e s o n a n c e s . Another major advantage is that the e x p e r i m e n t does not have to be p e r f o r m e d with the s p e c u l a r beam. This means that one does not have to find the Inelastic signal in the tail of the elastic s i g n a l , nor does one have to worry about " g h o s t s " in the e l e c tron energy analyzer. Finally, it should be obvious from the variation of the angular distributions in the inelastic scattering p r e s e n t e d in Sec. IV that the calculated p a t t e r n s can be used to d e t e r mine the bonding geometry of the a d s o r b e d m o l e cules. This can be accomplished by changing the energy and direction of the incident b e a m a s well a s the direction of detection. In the c a s e of CO, all the r e s u l t s have been calculated with the inci-

357 J. W. D A V E N P O R T , W. H O , AND J . R.

3126

dent beam coming in from the oxygen end of the molecule (see Fig. 9). We have found that the intensities of the angular distributions d e c r e a s e uniformly if the direction of incidence is from the carbon end of the molecule. For example, for 135" incidence, the intensity is d e c r e a s e d to 65& of that for 45c incidence, with the overall p a t t e r n remaining the s a m e . Therefore, it would be i m possible to discriminate between these two bonding configurations from the electron scattering angular distributions. However, by coupling to angular r e solved photoemission m e a s u r e m e n t , the absolute bonding configuration of molecules on surface can in principle be elucidated. The possibilities s e e m quite endless given that the negative-ion resonances can be documented experimentally.

The authors a r e much indebted to E. W. P l u m m e r in the preparation of this manuscript. In p a r ticular, one of us (WH) is grateful for his guidance from the initial stage of this project. Additional discussions with J. K. Kirtley, P. Soven, and S. Lundqvist have been very useful. This work is supported by the NSF Grant Nos. DMR-77-10137, DMR 75-09491, and DMR 73-07682-A03. APPENDIX A: MULTIPLE-SCATTERING EQUATIONS The multiple-scattering theory has been treated in detail previously 1 so we merely sketch the r e sults h e r e . The wave function in the atomic s p h e r e s can be written *(f,)=E^t/{(rjrt(fJ),

(Al)

where / is the r e g u l a r solution of the radial Schrodinger equation, and YL is a r e a l spherical h a r monic. In the outer sphere

(A4) t.L'

In this equation, (A5) with the functions and their derivatives evaluated on the sphere boundary. The diagonal p a r t s of G a r e given by the negative of the cotangent of the sphere phase shifts ^i(

+ Bigltrj]

Y^r,),

(/!,n,)

,

(A6)

t*i.

for the atomic s p h e r e s and by u "

(gl> h) 6 Igi.n,) ° " ' '

'

•_i 1 '

(A7)

for the outer sphere. The off-diagonal p a r t s of G a r e the s t r u c t u r e constants which a r e the coefficients in the expansion of the f r e e - p a r t i c l e Green's function about two different s i t e s and a r e given by Gi)i.= E 4 ^ ' " + , - ' ' / s ( L ' , L ' ' , L ) L"

xn,..(kRtl)YL.{ftu),

(A8)

and if i or j = l, then n is to be replaced by j . Here 1R is the r e a l Gaunt integral IR(L,L',L")=

j

YL(r)YL.(r)YL.(r)dSl

.

^L=H{fvir^LL^KLL,g\.(ri)}YL,(rl),

(A10)

L'

which defines the r e a l s y m m e t r i c K m a t r i x . K is given by (g),n,)^

hL

'

<*/.,»,.)

~

(All)

L

*(r)=

£ +

Cin.ikr^Y^r,) T,ClJ,(krl)YL(rl),

(A3)

where j and n a r e the usual spherical Bessel functions. Matching the functions and their derivatives

Then

(gi.n.)

(A2)

which contains both regular and i r r e g u l a r solutions. The wave function in the constant potential region is obtained using Green's theorem and is given by

(A9)

As discussed elsewhere it is convenient to find s o lutions of the Schrddinger equation having the form

*"'" * ( ? J = Z [AlfUrJ

17

a t all the sphere boundaries leads to the inhomogeneous equation

r G

ACKNOWLEDGMENTS

SCHRIEFFER

To solve the scattering problem we take a linear combination of the * L and choose the coefficients so that the asymptotic form of the wave function is given by Eq. (2.12). This leads directly to the r e sult for the scattering amplitude given in Eq. (2.16). APPENDIX B: GAS-PHASE DIFFERENTIAL CROSS SECTIONS To average the differential c r o s s section over molecular orientations we write Eq. (3.19) a s

358 17

THEORY

O F VIB R A T I O N A L L Y I N E L A S T I C E L E C T R O N

4£ = E YLlk)Yl.Oi)/iJl-A.YA(k')Yti,(k)

,

(Bl)

AA'

w h e r e / d e n o t e s the v i b r a t i o n a l m a t r i x e l e m e n t . T h e F's c a n b e c o m b i n e d u s i n g the Gaunt i n t e g r a l (Eq. A 9 ) , ^~=

E

I R&L'L'')

I R(AA'A")

ft

Where a

> P>

SCATTERING...

3127

and y a r e the E u l e r a n g l e s Which

s p e c i f y the m o l e c u l a r o r i e n t a t i o n . Then, u s i n g the o r t h o g o n a l i t y of the D's and the a d d i t i o n t h e o r e m for s p h e r i c a l h a r m o n i c s w e obtain

-S: = Ls ail in

A,..P,»(cose),

(B4)

JL-Swhere

AA'A"

xKr(t}rr(i'). N e x t r o t a t e the Y's into t h e lab f r a m e u s i n g

(B2) 18

AA'

(B5)

ri.-ElVJiJalSy),

(B3)

•Present address: Institute of Theoretical Physics, Fack, S-402 20 Goteborg, Sweden. 'D. Dill and J. L. Dehmer, J. Chem. Phys. 61, 692 (1974). 2 K. H. Johnson, Adv. Quantum Chem. 7, 143 (1973). 3 J. L. Dehmer and D. Dill, Phys. Rev. Lett. J35, 213 (1975); J. Chem. Phys. 65, 5327 (1976). *J. W. Davenport, Phys. Rev. Lett. 36, 945 (1976). 5 S. Wallace, D. Dill, and J. L. Dehmer (unpublished). 6 D. Dill and J. L. Dehmer, Phys. Rev. A 16, 1423 (1977). 'The first high-resolution experiment on low-energy electron scattering from surface adsorbates was reported by F. M. Propst and T. C. Piper, J. Vac. Sci. Technol. 4 , 53 (1967). 8 H. Ibach, Surf.~Sci. 66, 56 (1977); H. Froltzheim, H. Ibach, and S. Lehwald, Phys. Rev. B 14, 1362 (1976). 9 S. Andersson, Solid State Commun. 20, 229 (1976); 21, 75 (1977). "E. W. Plummer and T. Gustafsson, Science 199, 165 (1977). n S e e review article by G. J. Schulz, Rev. Mod. Phys. 45, 423 (1973). 12 M. J. W. Boness and G. J. Schulz, Phys. Rev. A 8, 2883 (1973). 13 J. B. Danese and J. W. D. Connolly, J. Chem. Phys. 61, 3063 (1974). lf i7. Hedin and S. Lundqvtst, Solid State Phys. 2_3, 1 (1969). is-P. A. Lee and G. Beni, Phys. Rev. B 15, 2862 (1977). l6 D. M. Chase, Phys. Rev. 104, 838 (1956). "A. Temkin, Comments At. Mol. Phys. 5, 129 (1976). 18 M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). 19 G. Herzberg, Molecular spectra and Molecular Structure, Spectra of Diatomic Molecules (D. Van Nostrand,

and P{x) i s the u s u a l L e g e n d r e p o l y n o m i a l .

New York, 1950), Vol. I. 'J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Cases and Liquids (Wiley, New York, 1954). 2l F . Linder and H. Schmidt, Z. Naturforsch. A £ 6 , 1603 (1971). " S . Hara, J. Phys. Soc. Jpn. 27, 1009 (1969). 23 R. J. W. Henry and N. F. Lane, Phys. Rev. LS3, 221 (1969). u 3. C. Slater, Adv. Quantum Chem. 6, 1 (1972). 25 D. E. Golden, Phys. Rev. Lett. 17, 847 (1966). 26 S. K. Srivastava, H. Chutjtan, and s . Trajmar, J . Chem. Phys. 64, 1340 (1976). 2, H. Ehrhardt, L. Langhans, F . Linder, and H. S. Taylor, Phys. Rev. m , 222 (1968). M G. J. Schulz, Phys. Rev. 135, A988 (1964). 29 F. Linder and H. Schmidt, Z. Naturforsch. A J26, 1603 (1971). M A. G. Englehardt and A. V. Phelps, Phys. Rev. 131, 2115 (1963). 31 H. Ehrhardt and K. Willmann, Z. Phys. 204, 462 (1967). 32 R. J. W. Henry and E. S. Chang, Phys. Rev. A 5, 276 (1972). 33 S. P. Weeks and E. W. Plummer, Solid State Commun. 21, 695 (1977). " E " Evans and D. L. Mills, Phys. Rev. B 5, 4126 (1972). 35 D. M. Newns, Phys. Lett. 60A, 461 (1977). ,S B. N. J. Persson, Solid State Commun. 24, 573 (1977). m F . Delanaye, A. Lucas, and G. D. Mahan (unpublished). •"D. Sokcevic', Z. Lenac, R. Brako, and M. Sunjic' Z. Phys. B 28, 273 (1977). "These selection rules are weakly broken if the m o l e cule does not lie precisely in the image plane. 40 S. F. Wong, and G. J. Schulz, Phys. Rev. Lett. ^ 5 , 1429 (1975). 20

V Magnetism and Magnetic Impurities

This page is intentionally left blank

361 THE SCHRIEFFER-WOLFF T R A N S F O R M A T I O N

J o h n W . Wilkins Department

of Physics,

Ohio State

University,

Columbus,

OH 43210,

USA

Historically, interest in magnetic impurities was nurtured by bi-yearly reviews at the International Conference on Low Temperature Physics. Specifically, the curious resistivity anomaly seen in nominally pure transition metals — the resistance increased as the temperature decreased — was eventually traced to dilute impurities such as iron and cobalt. This was clarified in the late 1950's by the groups of Matthias and of Friedel. In 1961 Anderson [l] developed the model that bears his name for a transition metal impurity in a nonmagnetic metal. As the simplest model that can develop a magnetic moment from an unmagnetized impurity, it has attracted great attention and elaboration. What wasn't clear at the time was how the model could explain the resistance anomaly. Interestingly Anderson does not mention or relate his work to a model for scattering between conduction-electron spins and the electronic magnetic moment of an impurity. This s-d model had been around for a long time." The s-d model is often called an exchangemodel because its interaction term is proportional to the scalar product of the conductionelectron spin with the impurity spin, with a constant of proportionality called J, the same symbol as is often used in the Heisenberg exchange model. In 1964 at the 9th International Conference on Low Temperature Physics in Columbus, Kondo reported a calculation of the conduction-electron scattering to third-order in J. If J corresponded to an antiferromagnetic interaction then the resistivity increased at low temperature with a logarithmic dependence. Conventionally the source for Kondo's 1964 result is in Progress in Theoretical Physics [3]. But, in fact, the Columbus talk caused all the excitement. I personally observed Schrieffer's when he returned from the meeting and told me about it. I wish I could remember if the result was already being called the Kondo problem at that time, as that is the name that was attached and has stuck. The "problem" was that the third-order perturbation result couldn't be the end of the story as it would diverge at zero temperature — inconsistent with the experimental resistance becoming constant as the temperature lowers.6 But to anyone interested in magnetism there was a more interesting problem: what did the s-d model have to do with Anderson's model? The former couldn't really arise from exchange. The latter offered a physical, believable description for how a local moment could arise in a metal from a non-magnetic impurity. a

A. C. Hewson, in [2], gives credit to Zener in 1951. See Hewson's book [2] for some history of how the Kondo problem got solved and why it took 20 years. In time so many people worked on the s-d model that "problem" came to stand for any concern. Indeed few worked on the resistivity, it was just too hard, and there were few useful methods. See Hewson also for a discussion of other important papers, such as the Coqblin-Schrieffer Model, not reviewed in this brief space. 6

362 In the 1966 paper "Relation between the Anderson and Kondo Hamiltonians" [4] Schrieffer and Peter Wolff offered an explanation that has stood the test of time. Indeed, while the paper is still being cited about twenty times a year, it has become such a part of our common understanding that many don't reference the two-page paper, assuming all will know the Schrieffer-Wolff transformation. 0 To partially explain the achievement requires some details on the models and symbols. The s-d model has only one parameter J that characterizes the strength of the scattering between conduction-electron spins and magnetic moments.'* In contrast, the Anderson model has two independent parameters. 6 One is the rate Y for scattering between conduction electrons and localized impurity orbitals; the latter are preferentially spin occupied when the magnetic moment develops. The other is the energy cost U when an orbital with a single electron is doubly occupied by a conduction-electron scattering into it. Schrieffer-Wolff relates the s-d and Anderson model parameters by

That J is negative means the interaction is antiferromagnetic as needed to explain the resistivity anomaly. What simplicity! Connecting two models, especially one ad hoc model to another microscopic model is always difficult and is seldom achieved. When it is, there are often many assumptions and restrictions. Schrieffer and Wolff struggled to keep these to a minimum but faced up to those that worried them. At the time, they were concerned with the size of J and felt that the transformation would work only for J much smaller than unity. It would take fourteen years before we knew that it works better than anyone could have expected. Laborious numerical calculations-^ confirm even for essentially infinite coupling - | J\ nearly unity - the connection is accurate. References [1] P. W. Anderson, Phys. Rev. 124, 41 (1961). [2] [3] [4] [5]

A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge, 1993). J. Kondo, Prog. Theor. Phys. 32, 37 (1964). J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).* H. R. Krishna-murthy et al., Phys. Rev. B 2 1 , 1003 (1980). (The symbol * indicates a paper reprinted in this volume.)

c

Continuing the rich tradition of misleading names, e.g., exchange and problem, "transformation" stresses the method first used to derive the "relation" — the authors' suggestion — between the s-d and the Anderson model. d Any feature of the conduction electrons, such as the density of states, is buried in J so it is dimensionless. e Actually there are three parameters. But above we consider only the unrealistic symmetric model in which a singly occupied impurity orbital lies as far below the Fermi energy as does a doubly occupied one above. f See Table VI in [5] where the relation is shown to hold over several decades of \J\ up to one as large as 0.8.

A 1412

PHYSICAL

REVIEW

VOLUME

140.

NUMBER

4A

15

NOVEMBER

1965

Localized Magnetic Moments in Dilute Metallic Alloys: Correlation Effects* J . R . SCHRIEFFER

Department of Physics, University of Pennsylvania, Philadelphia,

Pennsylvania

AND D . C . MATTIsf

IBM Watson Research Center, Yorktown Heights, New York (Received H June 1965) We discuss qualitatively the importance of the correlation energy in determining the ground state of a metal with an impurity atom. For a single, partly occupied impurity d-state orbital, the correlation energy acts to prevent the appearance of a nonvanishing ground-state spin, so that this simple nondegenerate model actually has a complicated structure. In one dimension, we show that this model of an impurity can never lead to a localized moment. In three dimensions, if we take linear combinations of Bloch functions transforming according to the irreducible representations o{ the point group of the impurity-f-crystal, we find that most of the new wave functions are entirely decoupled from the impurity, and only a small subset interacts with it. The noninteractmg majority of states determine the Fermi level, which we therefore take to be fixed. The ground state of the band states interacting with the impurity states depends on the two-body Coulomb repulsion U, and we find that for sufficiently small U the ground state has an even number of electrons with total spin 5 = 0. As V is increased above a certain critical value, the ground state of the interacting subsystem changes to an odd number of electrons, having total spin S = i, and a localized moment is said to exist. The introduction of orbital degeneracy for the impurity d state, and of Hund's rule matrix elements, makes the localized moment much stabler. The results are obtained by a combination of exact energy-level ordering theorems and a Green's-function calculation in the (-matrix approximation.

I. I N T R O D U C T I O N A N D

THEOREMS

T

H E c o n d i t i o n s u n d e r w h i c h a localized m a g n e t i c m o m e n t is a s s o c i a t e d w i t h a s o l u t e a t o m i n a dilute alloy h a v e been investigated within t h e frame-

* Part of this work is a contribution of the Laboratory for Research on the Structure of Matter, University of Pennsylvania, covering research sponsored by the Advanced Research Projects Agency. t Now at the Belfer Graduate School of Science, Yeshiva University, New York, New York.

w o r k of t h e H a r t r e e - F o c k ( H F ) a p p r o x i m a t i o n . , _ * I t is well k n o w n t h a t in m e t a l s t h e H F a p p r o x i m a t i o n o v e r e s t i m a t e s t h e s t r e n g t h of t h e effective e x c h a n g e i n t e r a c t i o n t h r o u g h t h e n e g l e c t of c o r r e l a t i o n s b e t w e e n elect r o n s of o p p o s i t e s p i n o r i e n t a t i o n s (i.e., t h e " c o r r e l a t i o n h o l e " ) . A s a r e s u l t , t h e H F t h e o r y of f e r r o m a g n e t i s m 1 8

J. Friedel, Nuovo Cimento Suppl. 7, 287 (1958) P. W. Anderson, Phys. Rev. 124, 41 (1961). ' P. A. Wolff, Phys. Rev. 124, 1030 (1961)

364 LOCALIZED

MAGNETIC

M O M E N T S IN

in the transition metals is qualitatively in error, as discussed most recently by Kanamori.4 The question arises whether a similar error occurs in the present localizedmoment problem. The theory discussed below shows that at least under certain circumstances this is the case. Specifically, we shall see that a well-known simplified model of an impurity, for which a magnetic moment had been calculated to exist for sufficiently strong Coulomb repulsion among the electrons, does not exhibit a moment when a more accurate calculation is performed. One may then well wonder what would be a satisfactory model of a magnetic impurity in a nonmagnetic host metal, such as a manganese atom in copper. First, we may discard the effects of the electrostatic potential of the impurity on the conduction electrons, as these are the same for electrons of spin "up" as for spin "down." On the other hand, the electrontransfer matrix elements (whereby the conduction electrons of the host metal can hop in and out of the localized, partly occupied orbital states of the impurity) and the Heisenberg exchange forces (in particular, the so-called s-d exchange interaction) do both provide mechanisms whereby the conduction electrons can have their spins polarized by the spin of the impurity. But for practical purposes these mechanisms will be useless if the impurity does not possess a net spin in the first place, so we must first ask when a net spin is energetically favorable. A single hydrogen atom has a single electron, hence a magnetic moment of one Bohr magneton. If it is dissolved in a metal one of three things may happen: (a) The electron may ionize, or (b) at the opposite extreme, a second electron may become bound to the proton, depending on the position of the Fermi level, the dielectric constant, etc. Both of these cases, H + and H~, are nonmagnetic, (c) The third possibility is that the impurity remains in the chemical state H° because the Fermi level, dielectric constant, etc., of the host metal permit the proton to retain its electron, but the electronelectron Coulomb repulsion keeps a second electron away. Case (c) is the magnetic prototype stressed by Anderson,2 except that he found it physically more meaningful to discuss a tightly bound orbital, such as a partly occupied d orbital of one of the transition elements, instead of a hydrogen orbital. Another possible model of magnetic impurities, briefly discussed by Anderson in the Appendix to his paper,2 has two or more degenerate, localized d orbitals on the impurity with a significant probability that these levels will be occupied by two or more electrons (or holes) in the ground state. Then, according to Hund's rules, the state of maximum multiplicity will lie lowest and there will be a net spin of two or more Bohr magnetons. It is possible that nickel and chromium, which normally sustain a magnetic moment not exceeding one Bohr magneton, are examples of the first species, whereas 4

J. Kanamori, Progr. Theoret. Phys. (Kyoto) 30, 275 (1963).

DILUTE

METALLIC

ALLOYS

A1413

iron, manganese, and the rare earths are examples of the stabler Hund's-rule-type impurities. Whether there exist two types of magnetic impurities is a question that has not yet been well answered, and which is outside the scope of the present analysis. The question to which we address ourselves, here is more modest, viz., What are the properties of Ike hypothetical single d-orbital impurity? Is it always magnetic if the Coulomb repulsion is sufficiently strong? Can we describe paramagnetic impurities, such as manganese in copper, by such a simplified model? These questions have already a definite answer in the Hartree-Fock theories,2 •* but we shall find that as soon as the correlation energy is included this answer must be considerably modified. The Hartree-Fock theory, as usual, greatly exaggerates the magnetic state and also misses some of the structure of the ground state. For example, we shall find that for a chain of 2 or more atoms, with the magnetic impurity at one end, it is possible to prove that the ground state has total spin S=0. What this signifies is that even if there is one Bohr magneton on the impurity, say of spin "up," the correlation energy favors having a spin "down" near this impurity and to exchange them so as to form a singlet state of total-spin zero. As this state lies below a localized doublet or triplet state by a finite energy, it will be impossible to measure the localized moment on the impurity: an electron which is part of a singlet state has a spin, all components of which average out to zero, and which therefore cannot be measured by a static magnetic field, nuclear hyperfine splitting, or any other method. The hydrogen molecule is a good example. To be quite specific now, we study the "extra orbital" model of Anderson.2 In this model one considers band states of momentum k which are the Bloch functions of the pure metal and a solitary added localized orbital labeled d. The band states and the localized orbital are assumed to be mixed by a one-body potential V. All two-body (Coulomb) interactions are neglected except those between opposite-spin electrons occupying the localized orbital. Thus, the Hamiltonian of the model is 3C = £ «*nt,+2Z

t4"*°+lI.(ViliCk,,+cd,+H.c.) + UniA.ni,-.

(1.1)

Here, ek and nk:, = ck,,+Ck,. are the energy and number operator for the band state of momentum k and spin s, «i and nd., = Cd,i+Cd,a are the energy and number operator for the localized d state, Vkd is the matrix element mixing the band and localized-orbital states, and U is the Coulomb matrix element between opposite-spin d states. The Fermi operators c*,+ and as+ create electrons in the band state k and localized orbital d state, respectively, with azimuthal spin quantum number J = ± $ . Note that tk, td, U, and NV kl? are all of the order of 1, in a suitable system of units.

365 A 1414

J.

R.

SCHR1EFFER

In order to make plausible the nonmagnetic character of this Hamiltonian, which is what we want to show, we shall first consider a linear chain in which Anderson's impurity atoms are added at either—or both—ends of the chain. We shall then prove a theorem (1.8) that the ground state of this chain belongs to total spin S=0, and is nondegenerate with the lowest state of total spin 5 = 1 or higher. Anderson's impurity is thus markedly unlike an atom of manganese; for if we put a manganese atom at the end of a chain of copper atoms we may properly expect it (and the chain) to have a net spin just as when we imbed it in a three-dimensional solid, in contrast to the situation we now analyze. First consider the case when the impurity is at the farther end of a chain of n atoms. In terms of localized Wannier operators, the Hamiltonian takes the form (assuming only nearest-neighbor overlap) 3Ci= — e £

£

t=i

(c>..*Ci+i..+H.c.)+e,,£fi 1 i,, •

I=~K-

U - VZ(«...*e*.+H.c.)+-(Z: ««,.)2. (1.2) 8

2

8

AND

D.

C.

MATTIS

indices commute with one another (regardless of whether one or both are *), b u t for equal subscripts the rules are i<.2=(ia*)2=0,

ba*ba+baba*=

1,

(1.6)

just as for fermions or Pauli spin matrices. Because the b's all commute, the various configurations can be specified merely by specifying which states are occupied and which are not. T h e order in which the b's appear is immaterial, and the sign of each configuration making up the Hilbert space can unambiguously be taken to be positive. Substitution for the c's in the Hamiltonian now yields n—1

3Ci=-e £

I1(Ji.,\i,.+H.c.)+(.E1ti<+i,1

i—i

-KE,(J,,,%1„+H.c.)+|[/(I1^1,1)!,

(1.7)

an expression involving the commuting operators only. A straightforward application of the method of proof of Ref. 6 leads to the following inequality: Denning Ec(S) to be the lowest energy eigenvalue belonging to total spin S, we have E<,(S)E0(S£

+ 1).

(1.8)

Note that e, c<j, U, and V are all of the order of 1. Some discussion may be useful: We have assumed that « and V are real and positive, for if they are otherwise, only a trivial phase change is required to make them so. We have also included such additional one-body terms as %U(nd,,)2 = %Und,., to complete the square in the interaction terms and make it clear t h a t the interaction is of the electrostatic, spin-independent type (unlike exchange). As for the Bloch energies which appear in (1.1), they are here given by

0 C 2 = - 6 £ £ ( 6 < . . * & i + l . . + H . C . ) + £ i £ ( » _ 1 , . + »n + l..)

e*= — 2ecos£,

-VZ.(6_i..*Jo..+H.c.+6n..*6-+i..+H.c.)

(1.3)

so that the parameter e is one-half the bandwidth. The anticommuting Fermion operators c and c* will now be transformed to Pauli-type operators b and b* by means of the following rules 6 - 6 : 6j,_= exp{tV £ &i, + =exp{ttr

«j,_}ci,_, £

% , - + » Z "),+}'!,+

»< n+l

]
/ = 1 , •••,n+l. (1.4) Here we have relabeled ca,, as Cn+i,« for convenience. The operators b* are obtained from the above by Hermitian conjugation. As for the occupation-number operators, they are clearly the same in the b and c languages: ni,,= bi,.*bi,,=ci,,*c,,,, (1.5)

The same inequality can also be proved if we add a second impurity atom at the other end of t h e chain, merely by labeling the new d-state operators c_i.„ and £_i,„* and the new band-state operators Co,, and Co,,*, so that the ordering sequence (1.4) can be used without modification. The Hamiltonian for this case then reads «-i t—0

«

a

+ * f f [ ( E . • L I . J M - ( £ « «»+i..) 2 ],

(1-9)

and the method of proof leading to the inequality (1.8) can be used also without modification, so t h a t the addition of a second impurity changes nothing. [It might be argued that if we added some electrostatic repulsions among the conduction-band electrons the situation might be changed, but this argument may also be disposed of. For if we add to the above an arbitrary interaction Hamiltonian

xr-'LVijG:

*..)<£,*,..),

(I.IO)

the inequality above may be proved again without modification."} Without repeating the proof in Ref. 6, we may briefly note t h a t the "kinetic-energy" operators involving e and V are the nondiagonal operators in the therefore, the transformation (1.4) is easily inverted to present representation, and because of their negative give the c's as functions of the b's. Two b's with different signs one may invoke variational arguments to prove t h a t the various configurations must enter into the • D. Mattis, Theory of Magnetism (Harper and Row, New York, ground state all with the same sign, regardless of the 1965), Chaps. 4 and 7. •E.Lieb and D. Mattis, Phys. Rev. 125,164 (1962), Appendix. magnitude of U. The nonmagnetic character of the inter-

366 LOCALIZED

MAGNETIC

MOMENTS

IN

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A 1415

acting system is thus caused by the desire of the elec:- parameters, a localized moment begins to appear, i.e., trons to minimize the kinetic energy just as is the casee ground state goes from (a) to (b) above), x(0) will befor noninteracting electrons; this they must do by cor-- come singular. fThis follows since if a localized d-morelations beyond the H F theory. We may also ask howv ment exists, x(T) for the impurity+interacting band much energy it would cost to magnetize the impurity,-, states will exhibit a Curie law varying as T-1 so t h a t i.e., to create a doublet or triplet state in the vicinityY x ( 0 ) = » . ] T o calculate this we add to 3C the Zeeman of the impurity atom. This is different from £ 0 (l)-£o(0)) term which is an energy of the order of 1/JV, and which is thee 3Cz=BZ.s(ni,+Zknt.), (1.12) excitation energy for a state in which two Bohr magne. where B=y.$h is the Zeeman energy of an up-spin electons are more or less uniformly distributed throughout the chain, so t h a t the spin on the impurity is also of the tron in the externally applied magnetic field h and n$ is order of \/N. T o achieve a spin of the order of 1 in the the Bohr magneton. T h e susceptibility is given by vicinity of the impurity, it is easy to see (variationally)) X=WWdB)(ni++Eknk+)\B^ (1.13) that one must in effect break the bond n —* («— 1) or r n—* (w-f 1), and so the energy required will be ~ e orr with the expectation value taken with respect t o t h e ~ V, whichever is smaller. Both these energies are of thee ground state of 3C+3Cz. Clearly it is sufficient to study order of 1, and therefore the nonmagnetic state of thee Xd— 2ve*{d(nd+)/dB) to determine the existence of a impurity lies below the localized magnetic states by a2 localized moment. If this quantity is finite, the ground finite amount. state can only belong to 5 = 0. If it is infinite, t h e ground If the impurity is in the exact center of the chain, state of the impurity system has captured one Bohr half the Bloch waves (those corresponding to the sinfe'v magneton, two or more Bohr magnetons being excluded functions) will have a node at the impurity and so will2 by (1.11). Indeed, Xd, the susceptibility of die d orbital, not interact with it. Similarly for an impurity in thee is more suitable for study than x, the total susceptibility, center of a three-dimensional crystal, only the functionss because in this way we avoid mixing in questions of the which have the full point symmetry of the impurityf Pauli-spin susceptibility of the whole material, and of -f crystal will have a nonvanishing amplitude at thee whether the total number of electrons is even or odd. Xd site of the impurity, and there are only ~i\f1'8 suchi. is a reasonable fraction of x, a n d when either is finite or functions out of a total of N. T h u s in general, there aree infinite, so is the other. two sets of band states: those which interact with the H. ONE-PARTICLE GREEN'S FUNCTION impurity and those which do not, the latter fixing theI Fermi level a t some value t F . By a simple extension off To treat correlation effects it is convenient t o s t u d y the method of proof given above, we can prove (1.8) forr the one-particle Green's function for the localized interacting subsystem in three dimensions as well as ina orbital: one, as discussed in the Appendix. Thus we find, nowv G.(0 = - i < r W . ( 0 * < / . + ( 0 ) } ) , (2.1a) without restriction as to the number of dimensions, defined in the presence of the externally applied mag£o(0)<£0(l)<£o(2)<etc. (1.11a)) netic field, where or ci,{C) = eiw+xz)tCit(o)e-iw.+x.z)t_ (2.1b) £„(l/2) <£0(3/2) <£„(5/2) <etc. (1.11b) T o simplify the dynamics of the problem we eliminate for the interacting subsystem. The actual values of e,'-, the band states from the problem by defining t h e zeroV, U, £ F , and id will determine whether the number of>f order Green's function electrons in the ground state of the interacting subsett Gll.(l)=-i(T{ci.«»(t)c6.w+(0)})<>, (2.2a) is even as in (a) above or odd as in (b) and so whether, associated with the impurity, we find one BohrmagnetonII where the time development of the operators is deteror not. mined by In no case can the ground state of the interacting g ci.v»(l) = eiX"ct.(n(.0)e-<x't (2.2b) subset belong to spin S> 1, i.e., have two or more Bohr magnetons, so we see t h a t this sort of impurity has a a with 3Co the Hamiltonian for noninteracting particles: very tenuous sort of magnetism indeed. I n calculating the ground-state energy of the im3Co=3C+DCz— Unt+ni(2.2c) p u r i t y + m e t a l we shall now adopt an approach based^ and ( ) represents the expectation value in t h e ground 0 on the many-body methods of quantum field theory,'' state of 3C0. I t is readily seen t h a t the Fourier transform the Green's function formalism, which is well adapted " G(,.(t) is given by to the study of the impurity problem, and which was ,s also used in the H F analyses. We shall calculate thee Got(&>) — [ai— td.~ S,()] -, J (2.3a) zero-temperature susceptibility x(0) of the system as aa where function of the system parameters. If, as we change these ;e tdB=td+sB, tk,— tk+sB (2.3b)

367 A 1416

J.

R. S C H R I E F F E R

G+toi)

FIG. 1. Diagrammatic representation of x*.

zfy

A N D D. C.

The diagrammatic representation of (2.8) is shown in Fig. 1. i n . THE VERTEX FUNCTION

G+(u))

and S.(«)-E— , ( s = 0 + , s g n a , = 7 T , (2.3c) * a— ei,+i5 sgnu \ |c<)|/ all energies being measured relative to the Fermi energy. As in Anderson's analysis 2 we neglect the real part of the one-body self-energy S(u) since this is expected primarily to lead to a level shift which can be absorbed into the definition of id- Thus, Go, reduces to Go.(w) = [ w - € ( i , - r - i r s g n w ] - 1 ,

(2.4a)

r=7riV(o)|y w | A v :! .

(2.4b)

where

MATTIS

We have assumed that the density of band states N(u>) varies slowly over a level width r about the Fermi surface and we have replaced N(u) by A^O). If we formally carry out a perturbation expansion of G,(w) in powers of the perturbations V and U, it is seen that by using propagators Go* we automatically include to all orders the one-body mixing potential V. Furthermore, since the Coulomb interaction Und+ntdoes not involve band states, the band states are thereby eliminated from the problem of determining G,(ui). Thus, the perturbation series for G, involves frequency b u t not momentum integrations, a major simplification over the corresponding problem in the many-body band theory of ferromagnetism. I t is clear that the diagrammatics for determining G, with the propagators Go, are the familiar rules of many-body theory with a 2-body interaction U except for the absence of momentum indices and momentum sums. By introducing the proper self-energy 2 , (u>) through Dyson's equation G,(w) = [w— td.—2,(u)+ir

sgnu]" 1 ,

T o determine the vertex function A,(w), we construct an integral equation for 2,(a>). Before proceeding t o the correlation effects we wish to discuss, we will retrieve Anderson's result by noting that within t h e H F a p proximation 2, Hr (o>) is given by dj

—G-. H S V)

as shown in Fig. 2. Since the right-hand side of (3.1) is independent of a, 2 H F is a real constant. From Dyson's equation (2.5) and from (3.1) one immediately finds 2 , H r = (U/ir) c o t - ' [ ( € , _ . + 2 _ . H F ) / r ] .

To find t h e conditions under which a localized moment begins to appear in the H F approximation, suppose we are in a range of parameters id, T, and U (for B=Q) such that there exists no localized moment, i.e., 2 + | B _ O = 2 _ | B _ O - To determine A, Hr (a>), we differentiate (3.2a) with respect to B and find 32. s A,HP=1+JdB

T/T

•1+U-

= 2rf

Here, as before, the limit B —> 0 is understood. Using the fact that when there is no localized moment in t h e absence of B (3.4)

J —v , 2iri

dB

IB—O

A.HF=[1-IM(0)}-1.

(2.6)

A{Q)--

|ImGHF(0)| = (r/7r)[(e„+2HI7+r2:i-1.

(3.5b)

One can interpret A (i) as the density of states in energy of the virtual d level. Notice that A, (and hence x) is singular as UA (0) approaches unity. Thus, the Hartree-

(2.7) (2.8)

(3.5a)

Here A (0) is the spectral weight function of G, with B = 0 evaluated a t the Fermi level

so that /•" da Xrf-W/ rrG+»(»)A+(«).

(3.3)

it follows from (3.3) that

where *-N signifies that the contour along the real axis (— oo, co) is to be closed in the upper half-plane and the limit B —» 0 is understood. I t is convenient to define the vertex function A,(OJ) by the Ward-like relation A.(fc>) = H-id2.(o,)/eJS| B _ 0 >

c)2-,"F\ ). dB J

(2.5)

r da / —G+(o>) dBJr-^2m

G + »(l+—~\—,

(e d _„-t-2_ B H F ) 2 +r 2 :( i - 5 \

dB d

(3.2a)

This is identical to Anderson's result if one uses t h e fact that 2, H F =l/(nd,_,>HF. (3.2b)

we find d Xd=2nll*—(nd+)=2rf— SB

(3-D

JK- > 2ni

I

G6-%''

(
FIG. 2. Hartree-Fock self-energy.

368 LOCALIZED

MAGNETIC

LOW DEI r 4PPR3X VA"3N

MOMENTS

* 3FN ilTY AT™ KM ,TION

IN

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METALLIC

ALLOYS

A 1417

This approximation for 2 , is shown in Fig. 4. From (3.9) we find tM=U/tl+U(v)l, (3.10) where r dx * « = / —G+(*+x)G_(-z). (3.11)

*

By differentiating (3.8) with respect to B, we find 3 2 , (co) FIG. 3. Regions of validity of low-density theory. H F theory predicts magnetism in shaded and cross-hatched areas.

dB

- / * ( o > + o / ) G V ) s+ JJ \\

( 3 - 12)

— . /2wi

dB dB

where we have used Eq. (3.4) and the relations Fock approximation predicts that a localized moment occurs when UA(0)>1,

( H F criterion).

I t is interesting to note t h a t this result is identical in form to the H F band theory of ferromagnetism in metals with a short range potential U, namely, the H F theory predicts that a metal will become ferromagnetic when UN(0)> 1, (3.7) where U is the matrix element of the two-body Coulomb repulsion potential between Bloch states near the Fermi surface and iV(O) is the density of states at the Fermi surface. Since (3.7) overestimates the role of exchange in the band problem it is not surprising that (3.6) overestimates the polarization tendency of the impurity. In order to make progress in including correlation effects, in the integral equation for 2,(w) we restrict the discussion to those regions of td, V, U space in which the average number of electrons (or holes) in the d orbital is small compared to unity. This simplification corresponds in a Fermi-gas problem to working with a low-density gas with short-range forces. This is a definite limitation, but one which allows us to verify the Hartree-Fock theory in a region where the latter definitely predicts the occurrence of a localized moment. This is illustrated in Fig. 3. T h e low-density approximation may be presumed valid in the range <«<;)<0.3 or 1 — («d)5; 0.3, in which range we know that the particleparticle (hole-hole) i-matrix graphs dominate the summation 7 to each order in U. Within this approximation, £,(
/

da' —<(o,-fV)G_.(c/),

7

iU

dx f ' dx —G+ (z+*)G-(-*)<(»-)• J lit

(3.13a) (3.13b)

G-, (u) = G,(oi)=G(o>),

which hold for B —» 0 in the case considered here, i.e., with no actual localized moment being present in the absence of B. We now determine under what conditions x —* °° as we vary the system parameters. The singular behavior of x arises from a singular solution to (3.12) as in t h e corresponding Hartree-Fock case, (3.3) and (3.5). Equivalently, X will be singular if the homogeneous integral equation da'

/(<.) = - / <(«+«% V)/(<->'>

(3.14) 2-iri

has a solution. I n principle, both G and t should be determined self-consistently by solving (3.8) and (3.9). In the absence of the detailed solutions of these equations we can make a reasonable estimate of the correlation effects as follows. The main effect of 2 in determining G and / is to shift the virtual level relative to the Fermi surface. The level width T due to the one-particle potential should primarily determine the form of t h e spectral weight function A (w). Thus, we assume t h a t the spectral weight is adequately represented by the Lorentzian function A ( « ) : A («) = 7r- 1 ]ImG(«) | = ( r / i r ) [ ( ( D - e ^ + r 2 ] - 1 ,

(3.15)

(.3. 8)

where e, rather than td, is now considered to be a parameter of the theory, i.e., the parameters are e, V, and U. With this approximation in (3.11), the real and imagi-

(3.9)

FIG. 4. Low-density approximation of the self-energy.

-N 2iri where the / matrix satisfies l(v)=V+i

= 0, dB

(3.6)

V. M. Galitskii, Zh. Eksperim. i Teor. Fiz. 34, 151 (1958) [English transl.: Soviet Phys.—JETP 7, 104 (1958)].

Zi-

6 P f?.

+

• ••

369 A1418

J.

R. SCHRIEFKER

nary parts of 0(c) are given by

-l * 1 = -

r

s

r

r /(c-w)H-r \

2

(co-2e) +(2r) 2

2(a,-2e) +(2r) _] T(u—2t) 1

2

« 2 +r 2 t 4r ]

tan tan~ -H T (w-2e)

2r

/ c d2= 1 tan -1 —tan"" 2 2 r T (a)~2«) +(2r) \

+ ,>-2*

L

€2+r2 J /

r

1

(3.16a)

(3.16b)

AND D. C.

MATTIS

tion (3.17) we see that a localized moment does not occur even as U —*<*> in the limit where the number of electrons (or holes) occupying the virtual level is small compared to unity, in contrast to the Hartree-Fock result. This shows that the energy of the system is a local minimum at zero magnetic moment. The question naturally arises whether there might not be a secondary minimum with increasing moment, a minimum which might even lie lower than the energy at M—Q. Kjollestrom, Scalapino, and Schrieffer9 have recently shown that the low-density theory does not allow any such subsidiary minimum and that within the limitations of this theory, the ground state definitely belongs to zero moment. IV. DEGENERATE ORBITALS

Here « > 0 and 0(wO). As in Kanamori's treatment of correlation effects in 3-d band metals, we expected the dominant contribution to the magnetization criterion to come from states near the Fermi surface, i.e., 6^iV~0. If t(u>+a') were the difference between the Landau Fermi liquid functions /+_(w,w') —/++(ia),u') for antiparallel and parallel spin electrons in the virtual d state, it would be clear that only states u and « ' in the immediate vicinity of the Fermi energy could enter the instability criterion. This is, in fact, the case within the Hartree-Fock approximation, where l-

u=f+--f+

One can carry out a similar analysis for Ul degenerate d orbitals, 31 being of order 2 or 3 for real d orbitals with crystal field splitting, but as large as 7 for rare earths. Again we take Anderson's model with the two-body interaction 3C' = S

E Uiiitittij,— £ Y.

t < ; «.a'

Jijni.iij,,

(4.1)

31).

(4.2)

i< i M

where J«=

U

(*, .7=1,2,

The criterion for magnetism corresponding to (3.17) When correlations are included, / plays the role of the becomes cI) "irreducible particle-hole interaction" r in the nota3117 (3l-l)(I/-/)tion of Abrikosov, Gorkov, and Dzyaloshinski,8 rather U(0)£1, (4.3) than r 1 " 0 , the limit oi the full particle-hole vertex funcA+Ud> !+(£/-/)« tion which determines the Landau parameters / „ ' . It appears that T m is a good approximation for T ("> in the where low-density limit, a question which we are currently 0 = (1/TT.E) tan-'CE/T). (4.4) investigating. Thus as a rough approximation we have For 31= 1, we retrieve (3.17), while for 7 « £ 7 one finds the criterion for the existence of a localized moment, (3l-l)/-i U UMA(0)>1, (3.17) A(0)21, (4.5) -+L1+U4 {1+Ud>yj where the effective potential Utn is given by where the Hartree-Fock theory gives U (3.18) Uels=t{0) = [17+(3l-l)7]4(0)^l. (4.6) l+{V/irt)ta.rr1(e/r) The magnetism of degenerate impurities arises from Now Ueii increases monotonically as U varies from the fact that if two or more electrons are constrained to 0—>«>, so that remain on the impurity atom, the exchange splitting / I/„H
' B. Kjollerstrom, D . J. Scalapino, and J. R. Schriefier (private communication).

370 LOCALIZED

MAGN'ETIC

M O M E N T S IN

preceding sections, which owes its magnetism—which surely occurs only in the high-density limit(re,j,)>0.3— to the almost accidental fact that a single electron has a magnetic moment.] Anyhow, it is noteworthy that in the case of degenerate orbitals, we find that magnetism is possible even in the low-density limit. V. CONCLUSION

DILUTE

METALLIC

A L L O Y S A1419

orbitals to be occupied. This serves to explain qualitatively why magnetism is almost never observed except in situations where orbital degeneracy plays an important role, and it points out that crystal field effects— which tend to lower the orbital degeneracies—will be important even in metals. ACKNOWLEDGMENTS

Physically, it is suggestive that nickel, which has a The authors wish to express their gratitude to Profesfraction of a hole to be shared among its d orbitals, sor D. J. Scalapino for his interest and active support rarely exhibits a paramagnetic moment in a nonmagnetic during the course of this work. host metal, whereas iron, manganese and the rare earths, with a considerable fraction of the localized orbitals APPENDIX occupied, manage to maintain their moments quite constant in a wide range of materials. This is surely By choosing linear combinations of band functions no accident, but is indicative of what the properties of a adapted to the point group symmetry of the impurity correct solution of Schrodinger's equation should reveal. +crystal, we can in many cases decouple the vast Mathematically, we found that the nondegenerate majority of band states which have vanishing amplitude orbital is never magnetic if it is on the last atom of a at the site of the impurity, and which do not interact chain molecule, and most likely it is also nonmagnetic with it except as a "reservoir" of electrons which mainon any other site of the chain molecule. The reason is tain the Fermi level fixed. It is possible to illustrate this that in one dimension an attractive potential is always simply, by making a few plausible approximations, capable of binding, and we visualize the one-dimensional e.g., by appropriate special choices of tk and VtdWe replace the Brillouin zone by an equivalent sphere, situation as the impurities' electron, say of spin up, binding to its immediate vicinity a conduction-band i.e., we will assume that the Bloch energies «* are given polarization of spin down, which exchange and combine b y n=-2ecos\k\a (Al) to give a nonmagnetic singlet ground state. To produce a net localized moment, we have argued that an energy and that likewise Vu depends only on the magnitude 71 3 0(1) is required, and this lack of magnetism is therefore of the wavevector \k\. This quantity can assume A ' a stable feature. In three dimensions an attractive closely spaced values ranging from 0 to w/a. By Fourier potential does not always have a bound state. As a transforming all the operators in (1.1), the Wannier consequence, the spin up is not necessarily able to operators now referring to spherical shells at distances "capture" a spin down in its vicinity, and may give up R from the origin, we may obtain precisely (1.2) in the this electron to the vast majority of band states which Wannier representation. The «th Wannier operator do not connect to the impurity. This occurs in what we refers to the j-wave component oi a spherical shell at have called the "high-density limit," i.e., the number of J?„=»o, and the impurity is at the origin. The inelectrons on the spin-up d orbital is ~ 1 , and is correctly equality (1.8) is then provable as before, except that predicted by the HF theory. In the low-density limit the number of electrons in the one-dimensional manifold the HF theory still predicts a net moment over a range is not fixed a priori although the Fermi level is. We of parameters, but we have given arguments that this therefore do not know if there are an even number of is incorrect. The "binding," and the correlations which electrons and the ground state belongs to 5 = 0 , or if we have found, are outside the framework of the earlier the number is odd and 5 = J; however, when U—*0 theories, and it is therefore not surprising to find a the number is clearly even and at U —>» it is clearly odd, therefore we know there occurs this level crossing result which is different. In addition, we have found that magnetism is possible at some intermediate "critical" value. These arguments even in the low-density limit, provided there are degenerateserve also to establish Eqs. (1.11).

371 PHYSICAL

REVIEW

VOLUME

140,

N i; M R K R

2

16

S E P T E M B E R

19 6 6

Relation between the Anderson and Kondo Hamiltonians J . R . SCHRIEFFER*

Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania AND P. A. WOLFF

Bell Telephone Laboratories, Murray Hill, New Jersey (Received 24 March 1966) A canonical transformation is used to relate the Anderson model of a localized magnetic moment in a dilute alloy to that of Kondo. In the limit of small s-d mixing, which is the most favorable case for the occurrence of a moment, the two models are shown to be equivalent. The Anderson model thus has low-temperature anomalies similar to those previously discussed for the Kondo model.

W

E have investigated the Anderson model of a localized magnetic moment in a dilute alloy 1 for the limiting case which is most favorable for a localized moment to occur, namely small s-d mixing. We find that Anderson's Hamiltonian can be transformed to a form similar to that of the s-d exchange model used by Kondo, 2 with an energy-dependent antiferromagnetic exchange interaction Jkk-- Since the Kondo effect apparently leads, at sufficiently low temperature, to a condensation in which a localized conduction-electron spin polarization compensates the impurity moment,' we conclude that the Anderson model does not lead to a localized magnetic moment at zero temperature. For temperatures high compared to the condensation temperature Te the impurity moment presumably breaks free from the conduction-electron polarization cloud and a localized moment appears. Thus, the existence of a localized moment in the Anderson model for a given temperature range depends critically on the strength of the effective exchange interaction /**.. T h e Anderson Hamiltonian for a single localized orbital "d" is kg

s

+ Z {v td c k ; Cd ,+ Vu*ctSck.), (t) k,

where tk and td are the one-electron energies of the conduction and localized orbitals, measured relative to the Fermi energy. The d and k states are mixed by the potential V; U is the Coulomb repulsion between opposite-spin electrons located on the d orbital. The model can be characterized by two dimensionless ratios (2a)

r±=Tj\t±\,

Here N{ta) is the density of band states in t h e perfect crystal at energy t„ and the matrix elements are averaged over k states of this energy. If t + > 0 and e_<0, then for Vkd~*0 the ground state is given by the filled Fermi sea and a single electron occupying the d orbital. Since the states with (/-electron spin | and 1 are degenerate, a localized moment occurs even at zero temperature in this case. For small but finite Vkd, i-e. r „ « l , these two spin states are mixed by electrons hopping on and off the d orbital, due to V. Under what conditions can this hopping quench the localized moment? Unfortunately, this question cannot be answered b y treating V directly by perturbation theory, since arbitrarily small energy denominators tk—e*'=^0 enter in fourth and higher orders in V. However, one can isolate those interactions which dominate the dynamics of the system for r„
(3)

have no terms which are first order in V. If we denote the first three terms in H by Ho and the term involving V by Hi, then by choosing 5 to be first order in V, one has [F0,5]=H,, (4) and

(5) From (4) one finds the generator 5 is given by 4

where ta=td+U, = td,

a= + ; a=

5-E

(2b)

*««

—;

and (2c) r^TriVteJlFnlW. * This work was supported in part by the National Science Foundation. 1 P. W. Anderson, Phys. Rev. 124, 41 (1961). ' J . Kondo, Progr. Theoret. Phys. (Kyoto) 32, 37 (1964). > H. Suhl, Phys. Rev. 138, A51S (1965); Y. Nagaoka, ibid. 138, A1112 (1965); A. A. Abrikosov, Physics 2, 5 (1965); 2, 61 (1965). 149

-nd,-~e°ck+cds—H.C.,

(6)

tk—ta

* It would appear that the singularity of S for states with tif^ia leads to difficulties. By carrying out a similar analysis in a Green's-function scheme one finds that tk is replaced by a frequency variable oi, with the behavior near the pole being given by the analytic properties of the Green's functions. The situation is analogous to the Bardeen-Pines versus the Eliashberg elimination of the phonons in superconductivity. See J. R. Schrieffer, Theory of Superconductivity (W. A. Benjamin and Company, Inc., New York, 1964). 491

372 492

R

S C H R I K 1" F E R A N D P

where the projection operators tid.~,° are defined by a

«d,-. =«•<;.-»,

a=+;

(7)

— t— rid,-, > « = — .

While the transformed Hamiltonian (5) is complicated, we will see that in the limit r „ « l , 11 is well approximated by ff0+ff2, where fl2=^[5Jfl13=ff«+flrdi,+ffo'+^ch.

(8)

These four terms can be expressed in terms of the field operators

A

149

W OLFF

that ffdir reduces to a one-body potential which can be eliminated by transforming from the k states to a set of one-electron conduction states which include this direct scattering term. For r a « l , the resultant shift of the conduction-electron wave functions and energies is negligible. Thus, H2 reduces to the s-d exchange interaction (9). For k and k'^^kr, Jtv is given by U J,kpkp= *J»=2\V, kpd\ (13) -<0.

u{u+U)

This coupling is antiferromagnetic, as was previously recognized.5 Were it not for the Kondo effect, H 2 could be treated \tkJ \Ct\l by perturbation theory. As Kondo, Suhl, and Nagaoka3 have pointed out, there is another dimensionless pa(a) an s-d exchange interaction, rameter KmN(0)J0\n(D/kBT) which enters if Jk1c. is H„=-Z A-iC^S**) • (¥/S¥„), (9a) approximated by a constant in an energy interval J 0 ( ~ O about the Fermi surface and zero outside this region. For T 0 , «_<0, we restrict the discussion to this case. 124 (1961); J. Kondo, Progr. Theoret. Phys. (Kyoto) 28, 846 Since all terms in Hi conserve the number of d electrons (1962). * This result is complementary to that of Schrieffer and Mattis, except Heb, which changes the number by two, it found that in the limit of a small fraction of an electron or a follows that H2 does not connect the part of Hilbert who hole on the impurity, on the average, the ground state exhibits space having one d electron (the case of interest) with no localized moment. See J. R. Schrieffer and D. C. Mattis, Phys. Rev. 140, A1412 (1965). As in the present analysis, they the remainder of Hilbert space, i.e., zero or two d found correlation effects suppress the moment predicted by the electron states. Therefore Hch can be neglected. Further- Hartree-Fock approximation. < H r more, in the one-rf-electron subspace, ( &d frd) = l so B. Muhlschlegel and J. R. Schrieffer (private communication). (Ck\\

*»=-(

ICi\\

and * * - (

:

PHYSICAL

REVIEW

VOLUME

185,

N U M B KR

2

10

SEPTEMBER

19 6 9

Exchange Interaction in Alloys with Cerium Impurities* B . COQBLlNf AND J. R . SCHRIEPFEK

Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received 4 March 1969) Starting with the Anderson model for the 4/ 1 configuration of cerium, the transformation of Schrieffer and Wolff is performed, taking into account combined spin and orbit exchange scattering. The resultant interaction Hamiltonian differs qualitatively from the conventional s-f exchange interaction. The Kondo effect, the spin-disorder resistivity, the Ruderman-Kittel interaction, and the depression of the superconducting transition temperature with impurity concentration are worked out for alloys containing cerium impurities on the basis of this new interaction. 1. INTRODUCTION

T

H E occurrence of a resistivity minimum at low temperatures—or the Kondo effect—has been extensively studied for transitional alloys; magnetic alloys with transition impurities show a Kondo effect, while nonmagnetic alloys do not exhibit a Kondo effect. On the other hand, in the series of dilute alloys with rare-earth impurities in lanthanum or yttrium, the alloys with cerium impurities are the only ones t h a t show a resistivity minimum at low temperatures; all the other rare-earth alloys, though magnetic, do not show a resistivity minimum. 1 - 7 The anomalous behavior of cerium metal and alloys has been recently studied in detail. 8 In these cases the 4 / level is very close to the Fermi level, and resonant scattering theory explains their properties. I n the other rare earths, the 4 / levels are generally far from the Fermi level, and the ionic model is valid. To explain the Kondo effect in magnetic dilute alloys with transition impurities, two models are generally considered, the s-d exchange model and the Anderson model, the latter describing the mixing between conduction electrons and localized electrons. Schrieffer and Wolff9 have shown that, in the limit of small mixing, the Anderson Hamiltonian leads to an exchangetype Hamiltonian. To explain the experiments of the Kondo effect in rare-earth alloys, the s-f exchange Hamiltonian is generally used. 1,2 Since the orbital angular momentum * Work supported in part by the National Science Foundation and the Advanced Research Projects Agency. t Permanent address: Laboratoire de Physique des Solides, 1 Associe au CNRS, Faculte des Sciences, Orsay, France. 1 T. Sugawara, J. Phys. Soc. Japan 20, 2252 (1965). ! T. Sugawara and H. Eguchi, J. Phys. Soc. Japan 21, 725 (1966). 1 T. Sugawara, 1. Yamase, and R. Soga, J. Phys. Soc. Japan 20,4 618 (1965). T. Sugawara and S. Yoshida, J. Phys. Soc. Japan 24, 1399 (1968). S H. Nagasawa, S. Yoshida, and T. Sugawara, Phys. Letters 26A, 561 (1968). «T. F. Smith, Phys. Rev. Letters 17, 386 (1966). 7 B. Coqblin and C. F. Ratto, Phys. Rev. Letters 21, 1065 (1968). B B. Coqblin, thesis, Orsay, France, 1967 (unpublished); B. Coqblin and A. Blandin, Advan. Phys. 17, 281 (1968); B. Coqblin, in Proceedings of tie Seventh Rare-Earth Conference, Coronado, Calif., 1968 (unpublished). »J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149,491 (1966).

185

is unquenched and the spin-orbit coupling is large, the spin-spin (s-f) exchange Hamiltonian is conventionally written as H=-2T(g-l)sj, (1) where r is the interaction constant, g is the Lande g factor, s is the conduction-electron spin density at the impurity site, and j is the total angular momentum of the rare-earth impurity. The form of (1) leads to a rather puzzling result, namely, in the case of cerium alloys, g — \ is negative, so that there would be a Kondo effect only if Y were positive, in contrast to transition alloys. Since it is presumed t h a t the strong s-f hybridization in cerium is responsible for the Kondo effect in dilute cerium alloys, while one knows that hybridization exchange leads to negative T, at least for j-state ions, the origin of the Kondo effect in cerium alloys is unclear. To clarify the situation, we consider an Andersontype model for the 4 / 1 configuration of cerium and derive the effective exchange interaction between the conduction electrons and the impurity moment, taking into account combined spin and orbit exchange scattering. Our results differ sharply from the conventional form (1). The spin-disorder resistivity, the Kondo effect, the Ruderman-Kittel interaction, and the depression of the superconducting transition temperature with impurity concentration are worked out for alloys containing cerium impurities on the basis of this new interaction. 2. EXCHANGE INTERACTION HAMILTONIAN For a cerium atom, the large spin-orbit coupling leads to a ground state of total angular momentum y = | in which the orbital angular momentum (/ = 3) and spin of the / electron are antiparallel; the j=l multiplet is widely separated from the ground state and is not of interest here. T o treat the mixing of the conduction and impurity wave functions, we work with conduction states that are partial wave states about the impurity. Since we are only interested in the l = i states of the impurity, and since the conductionimpurity mixing potential is predominantly spherically symmetric, only 1=3 conduction-electron partial-wave states will enter the problem. Furthermore, it is con847

374 848

B . C O Q B L I N A N D J.

venient to make up spin-orbit eigenstates from these partial-wave states so that we work only with states of definite total angular momentum. Let CkM1 create a conduction electron in a state of wave number k, total angular momentum f, and z component M ( = ± § , ± f , ± 5 ) , while £«•* creates an electron on the impurity with j=i and z component M'. Then the Anderson Hamiltonian 10 is H = Ho+Hu (2a) where Ht>=Y. (kfikM+Eo^nM+kU k.M M

12 nMnM', M,M'

(2c)

+ \t„ — €e

Y (3) tb—tJ

where a and b label the initial and final states, respectively. For the 4 / 1 configuration of cerium, these states are of the form

|o> = c * * W | 0 > ,

|&> = c*.*<W|0),

(4)

and the two possible intermediate states for the above states (4) are \c1)=ckMick'M^\0),

|c2> = c * W | 0 ) .

(5)

In this way, we obtain Hi=—

2^ Jkk'Ck'M' CM CM'CkM , kk'MM'

(6)

where Jkr.kr^lVk^U/EoiEo+U).

(7)

While Jkk' is independent of M and M', it depends on k and k' and is roughly constant so long as E0< (e t , (k-)<Eo+U.3 In the following we take Jkk' to be a constant with a cutoff so that 7 = 0 if | ek\ or | £*-1 >D, where D is of order ] £ 0 ' . The Hamiltonian (6) describes spin and orbit exchange scattering, and, in contrast to the s- j exchange model, the change AM — M'—M in the magnetic quantum numbers is not limited to ± 1 or 0, as it is for the s - j interaction. The form of (6) shows that the 10

P. W. Anderson, Phys. Rev. 126, 41 0961). » J. R. Schriefier, J. Appl. Phys. 38, 1143 (1967).

185

magnetic quantum numbers of the conduction and localized electrons are in fact exchanged in t h e scattering process. We note that, for a scattering process in which M=M', the average value of (6) over all possible values of tiM=CMtCM is not zero, in contrast to the s-S or s-j interactions. As a consequence, t h e interaction (6) actually contains both direct and exchange scattering. For many purposes, it is convenient to remove the direct scattering by adding to (6) the direct potential

/

#3 =

tt is the energy of a conduction electron of wave number k, and £0 is the energy of the localized state, both being measured relative to the Fermi energy EF. We neglect here the multiplet splittings when there are two or more electrons on the impurity, i.e., we neglect atomic exchange integrals compared to the atomic Coulomb integral U. We follow the method used previously 11 with the same notation was used there. The canonical transformation replaces Hi by an interaction Hi which is given by

"be

SCHRIEFFER

(2b)

£fi= L (VkCkMicM + Vk*CMfCkM). k,M

Hi^-£(b\Hi\c)(c\H1\a)(

R.

T, ck'M'fCkM'nM 2J-T-1 kMk'M'

(8)

in order t h a t H2 and H3 contain only exchange scattering. Thus, we must subtract from the normal oneelectron (direct) potential the term H3. Thus, the total Hamiltonian is H — ^ ektlkM—J 23 Ck'M'^CkM kM kk'MM'

(

&M.M'

\

CM*CM Z)»«"), W 2j+1 M" J where the k sums are restricted to |«il
(10)

The sum in (10) is over the different angles of k a t | i | fixed and over the two spin directions a; | k a )

375 185

EXCHANGE

INTERACTION

IN

ALLOYS

849

3. RESISTIVITY AND THE KONDO EFFECT denotes a plane wave of wave vector k and spin
(

where

CM1CM

E

2 ; + l M" '=J(kcr\kM)(k'M'\k'
nM"

),

3k.k's

(11)

(12)

Let us compute the value of $k«v,'MM'. For an 1 = 3 and j=\ state, the partial wave \kM) is given as a function of the partial waves \k,l,m,o), where 1 — 3, m is the z component of the angular momentum, and a is the spin \kM)=ctM\k,3,M+h,-\)+PM\k,3,M-\,h),

fiM

^kk'M = 2-, \Ck'M'Cit'

= l(7-2M)/U]*.

(14)

(r,e,v\k,l,m,v)=V(4")jt(kr)Yr(e,
»*'= /

|k
(16)

1-0 = - !

= 4 T T / [ > M F 3 " + S (£2,) i ^ + j B * Y ^ - i

X{a^[F3"'+*(M]%W +/^[F3*''-»(^)]*V.j} •

(17)

If there are n impurities located at positions Rn, the total Hamiltonian is

tf=E

«*»*.k
£

5*.*<.'MMV(k-k'>-R''c,,v.tck,

kok'j'-M.Wn

2 / + 1 M"

(18) - ) •

where c « t ( n > corresponds to the n t h atom. Thus we have derived an exchange-type Hamiltonian taking into account combined spin and orbit exchange scattering, and the forms (9), (11), and (18) are very useful for the following.

r

4/(j+i)-i

L

(2i+l) 2 J ;

+j\(2j+l)nt

two

| Z I W = 0. (21)

The solution of Eq. (21) has the same form as for the s = \ spin, and we can deduce the following results: (1) There is a Kondo effect for cerium alloys, because J is negative. The Kondo temperature is given b y kMT&d)

(0*)*M]

C - 2 l m £ rkk-Ai(.o>)lf(fi>)dw ,

where / ( « ) is the Fermi function, we obtain Nagaoka equations 12

Thus, the calculation of g gives '

[ — 2 lm]CGu.j,(w)]f(u)4i,

mk' •= f

in order to make contact with the i = | case for the Hamiltonian (11). Thus, the plane wave | k
5*.* v

2;+l

(20)

(15)

l

(19)

and the averages

T h e \k,l,m,a) wave function is proportional to ji{kr)Yim{8,
MM

1

CM ;CkM ) -

(13)

where the two Clebsh-Gordan coefficients ecu and (3.« are ccM = lO+2M)/Uji\

Gkk'M — {Ck'M I CkM^) ,

)

MM

e x p C - 1 / ( 2 . 7 + 1 ) \J\n{E,X\,

(22)

where n(Er) is the density of states of the conduction band at Fermi level for one spin direction. We note that in (22) the coefficient in the exponential is 2_/+l and not 2, as for the s-S or s- j interactions. In fact, there are 2 j + l channels for changing the quantum number M on the impurity (instead of 2, as for the s-S and s-j interactions), and these channels add independently to each other. T h e same result occurs obviously in the Kondo resistivity. In (22) the cutoff D is not the width of the conduction band, b u t rather the distance £ 0 from the 4 / localized level to the Fermi level, as in superconductivity theory. W i t h £o of the order of 0.1 eV and 2 j + l = 6, Tk for cerium alloys is of the order of several degrees Kelvin to some 10°, roughly of the same order of magnitude as t h e experimental values for Y-Ce and La-Ce alloys. 4 ' 613 u U

Y. Nagaoka, Phys. Rev. 138, All 12 (1965). S. A. Edelstein, Phys. Letters 27A, 614 (1968); Phys. Rev. Letters 20, 1348 (1968).

376 850

COQBI. IN

AND

J.

(2) Above Tk, there is spin and orbital magnetism. Below 1\, the preceding analysis suggests that there is, in cerium alloys, a compensation of the total angular momentum, i.e., a compensation of both spin and orbital momentum. This has not been investigated at present in detail. However, recent magnetic susceptibility experiments on La-Ce alloys at low temperatures indicate that the effective magnetic moment is 0.5/iB and is a decreasing function of temperature. Hence the orbital moment has been greatly reduced. So in spite of possible crystalline field arguments, it seems to be the first evidence of combined spin and orbital compensation.14 (3) The spin-disorder resistivity is R,=

2vm*n{EF) z#h

4/(/+l) cP-

(2j+l)2

(23)

m* is the effective mass and z the number of conduction electrons per unit volume. We will return to this point later. (4) The exchange scattering resistivity for T> Tt is R=Rll+\J\

(2j+l)n(EF)

ln(0.77Z)/*,r)].

(24)

There should be a resistivity minimum above Tk and a plateau below Tk. These results are in agreement with recent experiments on Y-Ce alloys.*

R.

SCHRIEFFER

can be written as \Ei,MM'(R)

Hu(R)= Z MM- L

( (

(kV|# i |k
\

27 + 1 M"

)

&MM-

\~|

(26)

Z

WW-2

^

*»*'»'

^

tk—tk-

X]^^.MM']2cos(k-k')-R,

(27)

where fk is the Fermi-Dirac function for the energy tk. We see that the terms in the parentheses are identical by interchange of k and k', so that the terms in fkfkvanish, since the denominators are of the opposite sign. Hence we have /* £V""(.R) = 2 Z •—— *»*'»' tk — t^

We have seen, in the preceding section, that the results derived on the basis of the new exchange-type interaction are different from the results obtained with the s-S or s • j interactions. It is interesting to look at other properties of magnetic dilute alloys, in order to see if there are other differences. The first interesting property is the Ruderman-Kittel16 interaction between two magnetic cerium impurities, i.e., the indirect interaction of two cerium impurities via the exchange interaction of their 4 / shells with the conduction electrons. Let us consider two impurities 1 and 2 at a distance R. The total second-order interaction between the two magnetic impurities is given by Z

$M,M-

CMimcM'a) Z nM-m ) . Ei2MM'(R) is the interaction2j+l energy JW"for an individual /J change from M to M' on 1 and the corresponding change from M' to M on 2. Using the fact that (3M.-MM')* = g>k-«-k.M'M, we obtain the value of E12MM,(R) from

4. RUDERMAN-KITTEL INTERACTION

B»(R)=

185

(25)

X|^.*vMJ,,Tcos(k-k')-R.

(28)

We use the form (17) of g and we expand the plane waves in partial waves. For this we take the z axis along the line connecting the impurities 1 and 2. Hence we have ««k-k').* = f; i,r-o

(2l+l)(2V+l)jl{kR)j,.(k'R) XPi(cosek)Pi(cosdk').

(29)

We separate the integration in (28) into an integration over the magnitudes |k| and |k'| and an integration over the angles of k and k'. At last, we obtain Pm*

Z (2/+l)(2/'+l)

''(X)'

Hi is the interaction Hamiltonian given by the second XBl(M)B,.(M')Iii'(R), (30) term of the expression (18). The sum in (25) is for all where the filled k values and all the empty k' vaiues. r The expression (25) corresponds to all the changes Bl(M)=aM2 j dQkPi(cos0k)\ F3M+i(S2*)|2 from M to M' on the impurity 1 and the corresponding J changes from M' to M on the impurity 2. Thus H\%{R) 2 M 2 +/3AM jdClk Pi(cos6k)| Yz ~i(iW | , (31) "A. S. Edelstein (private communication). ** J. C. Slater, Quantum Theory of Atomic Structure (McGrawHill Book Co., New York, I960), Vol. II, Appendix 20.

Jo

'

Jo fc2-*'2

377 J85

EXCHANGE

INTERACTION

T h e value of Bi(M) is easily obtained as a function of the coefficients Ci(3,m;3,m) introduced by Slater 16 for the addition of three angular momenta

IN

ALLOYS

851

Thus, there remain only the six coefficients (,(1,1) = 4 0 0 / 4 9 ,

C ( J , | ) = 16/49,

G(J,1) = 2 5 / 4 9 ,

G ( M ) = - 8 0 / 4 9 , G ( i , « - 1 0 0 / 4 9 , G(f,4) — 2 0 / 4 9 .

Bi(M) = a ^ 2 C ' ( 3 , M+h; 3, M+\) +p^Cl(3,M-\;3,M-h).

T h e average value of G(M,M') is equal to 1. T h e form (37) of the Ruderman-Kittel interaction is T h e only nonzero C (3,m; 3,m) coefficients correspond strongly anisotropic, owing to the different values of to 1 = 0, 2, 4, 6. I n particular, C°(3,m; 3,m) is always the G{M,M') coefficients. T o our knowledge this has equal to 1. never been looked for experimentally. T h e analysis of On the other hand, the integrals Iu-(R) are not in the experiments involving the Ruderman-Kittel intergeneral easy to compute. T h e first one 70o for 1 = 1' = 0 action in alloys with cerium impurities and in cerium can be computed exactly and gives the classical compounds should be repeated on the basis of this new Ruderman-Kittel interaction 16 anisotropic interaction. (32)

l

h„(R)=TkF*F(2kFR), with F (x) = (x cos*—sin*)/* 4 .

(33)

Thus, the first term gives the classical RudermanKittel interaction for a J = 5 spin and is independent of M and M'. However, the terms other than l=l' = 0 in the expression (30), which depend on the M and M' values, are not small compared to the first term and have to be taken into account. T h e dependence of these terms on M and M' gives an anisotropic RudermanKittel interaction. Fortunately, it is easy to compute the asymptotic form of Iw (R) when kFR —> 00; which is generally the only term considered in the Runderman-Kittel interaction for experimental purpose. T h e asymptotic form of the Bessel function is

5. DEPRESSION OF SUPERCONDUCTING TRANSITION TEMPERATURE The superconducting transition temperature Tc for solid solutions of cerium in lanthanum has been measured both at normal pressure 2 and as a function of pressure. 6 An explanation has been presented by Sugawara, 2 by use of s • j or s • S interactions, for the normal pressure experiments. A recent analysis of the pressure-dependent experiments has been m a d e by use of the s-S interaction. 7 I t is obviously interesting to check this analysis on the basis of the new Hamiltonian. The decrease ATC of the superconducting transition temperature with concentration C of cerium impurities is given by 17 (kff|/?,|kV)|2»,

ATc = -Wln(Er)/kBlC((\

(39)

ji(kR)3£wa(kR-\h)/kR, £->=» (34) where ( ( • • • ) ) means the average over t h e solid angle between k and k', over the spins a and a', and over the so that the asymptotic form of Iw (R) for kFR —> 00 is orientations of the impurity magnetic moment. We use the form (11) for the interaction Hamiltonian I„.(R)~Kkr' cosl2kFR-iw(l+r)y{2kFRY. (35) Hi. The average over the angles of k and k ' and the Since / + / ' is always even, the phase of the cosine in the spins a and a' is easily carried out, and we h a v e expression (35) gives only a factor (~l)i(l+l'). If we «||2» call F(M) =

(-1)"2(2'+1)

£

J

&MM-

I

\

1—0,2,4,6

X[a^C'(3,M+i;3,M+i) +pM20(3,M-i;3,M-m,

X (c^CM. \

(36)

and G(M,M')=F(M)F(M'), the asymptotic form of the interaction RUMM' EnMM''(R)c^(m*kFlP/w3)G(M,M') Xcos(2kFR)/(2kFR)3,

(37)

and the total interaction is given b y (26). The coefficients G(M,M') are simple to compute, and there are relations between t h e m : G(M,M')=G(M',M)

= G(\M\,\M'\).

(40)

where (• • •) denotes the average only over t h e orientation of the impurity magnetic moment. Therefore we have

is

R->*>

— £ nu.. ) \ , 2.7+I M" / /

(38)

16 M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954); K. Yosida, ibid. 106, 893 (1957).

« | (k«r|F,ikV)|2)) = 4 / 2 i O - + l ) / ( 2 y + l ) 2 ,

(41)

and one finds that 7r2n(£V)

dT.

=

4J0+1)

P

. (42) dC 2kB (2/+1)2 17 A. A. Abrikosov and L. P. Gor'kov, Zh. Eksperim. i Teor. Hz. 39, 1781 (1960) [English transl.: Soviet Phys.—JETP 12, 1243 (1961)].

378 852

B.

COQBLIN

AND

T h u s , t h e d e c r e a s e of Tc w i t h t h e c o n c e n t r a t i o n C of c e r i u m i m p u r i t i e s h a s t h e s a m e f o r m as t h a t o b t a i n e d w i t h t h e s - S or s- j H a m i l t o n i a n , b u t w i t h a different coefficient. W e will r e t u r n t o t h i s p o i n t in t h e n e x t section. 6. C O M P A R I S O N W I T H

EXPERIMENTS

I n t h i s s e c t i o n , w e p o i n t o u t t h e differences b e t w e e n this new Hamiltonian a n d the two normal s - S and s- j H a m i l t o n i a n s . 1 8 W e d e d u c e t h e v a l u e s of t h e i n t e r a c t i o n c o n s t a n t / for c e r i u m a l l o y s a n d t h e c o r r e s p o n d i n g v a l u e E0 of t h e s e p a r a t i o n b e t w e e n t h e 4 / localized level a n d t h e F e r m i level. Table I summarizes the results obtained with the t h r e e H a m i l t o n i a n s for c e r i u m alloys for w h i c h J<0. F o r c e r i u m i m p u r i t i e s , w e h a v e - $ = 7 , j=%, g— 1 = —rT h e m a i n p h y s i c a l difference b e t w e e n t h e t w o conv e n t i o n a l H a m i l t o n i a n s H, a n d Hj a n d t h i s n e w H a m i l t o n i a n H are t h e following: (1) F o r t h e s p i n d i s o r d e r r e s i s t i v i t y R, a n d for t h e d e c r e a s e of t h e s u p e r c o n d u c t i n g t e m p e r a t u r e Tc w i t h c o n c e n t r a t i o n C, t h e r e s u l t s a r e q u a l i t a t i v e l y t h e s a m e for t h e t h r e e H a m i l t o n i a n s . T h e r e is o n l y a c h a n g e in t h e coefficient c o m i n g f r o m t h e m a g n e t i c m o m e n t of t h e i m p u r i t y . F o r t h e v a r i o u s H a m i l t o n i a n s t h i s coeffic i e n t is

Bj-*(g-iyj(j+l)=5/28, H^>

4/(y+l)/(2./+l)2=35/36.

F o r n u m e r i c a l a p p l i c a t i o n s , H, a n d H s a m e r e s u l t s , w h i l e Hj gives different (2) F o r t h e a s y m p t o t i c f o r m of K i t t e l i n t e r a c t i o n EnMM'{R) between

give almost the results. the Rudermantwo impurities,

J.

R.

S C H R I E F F E R

185

t h e r e s u l t is n o t e v e n q u a l i t a t i v e l y t h e s a m e . H, a n d Hj give a n i s o t r o p i c i n t e r a c t i o n , w h i l e H g i v e s a s t r o n g l y a n i s o t r o p i c i n t e r a c t i o n . I n f a c t , t h e s e differences c a n b e easily u n d e r s t o o d , b e c a u s e H, a n d Hj a r e b o t h b u i l t w i t h / = 0 t y p e s t a t e s w h i l e H is b u i l t w i t h r e a l 1 = 3 states. (3) T h e m a i n difference c o m e s in t h e K o n d o effect. W i t h a n e g a t i v e v a l u e for / , Hj gives n o K o n d o effect for c e r i u m a l l o y s , i n c o n t r a s t t o t h e e x p e r i m e n t a l r e s u l t s . 1 ' 2 H, a n d H b o t h g i v e t h e K o n d o effect w i t h different coefficients. T h e s e c o n d p o i n t w e c o n s i d e r is t h e d e r i v a t i o n of J a n d Eo f r o m t h e e x p e r i m e n t s . T h e R u d e r m a n - K i t t e l i n t e r a c t i o n is a t p r e s e n t n o t useful for t h i s p u r p o s e b e c a u s e t h e r e a r e n o t e x p e r i m e n t s r e l a t i n g t o i t in c e r i u m alloys. T h e K o n d o p r o p e r t i e s a r e n o t p r e c i s e e n o u g h t o g i v e t h e a b s o l u t e v a l u e of / , b u t t h e y can g i v e a definite r e s u l t for t h e sign of J. J is n e g a t i v e in Y - C e a n d L a - C e alloys because t h e y present a K o n d o effect. 1 ' 2 T h e n w e c a n u s e t h e s p i n d i s o r d e r r e s i s t i v i t y a n d t h e d e p r e s s i o n of t h e s u p e r c o n d u c t i n g t e m p e r a t u r e in o r d e r t o o b t a i n t h e v a l u e of 7 . 1 , 2 ' 6 T h e v a l u e of t h e d e n s i t y of s t a t e s of t h e c o n d u c t i o n b a n d for p u r e l a n t h a n u m a n d p u r e y t t r i u m h a s b e e n d e d u c e d from specific h e a t d a t a . 1 9 T h e d e n s i t y of s t a t e s c a n b e t a k e n a s n(EF) — 2.2 s t a t e s / e V a t . a n d t h e effective m a s s m* = 3 f o r t h e t w o l a n t h a n u m a n d y t t r i u m h o s t s . 1 8 B u t , i n f a c t , t h e c o n d u c t i o n b a n d is c o m p o s e d of b o t h a 6s b a n d a n d a n a r r o w 5d b a n d . B a n d calculations on yttrium20 have shown t h a t the c o n t r i b u t i o n of t h e 6s e l e c t r o n s a t t h e F e r m i level is small c o m p a r e d t o t h e c o n t r i b u t i o n of t h e 5d e l e c t r o n s . W e e s t i m a t e t h e c o n t r i b u t i o n of t h e 6s e l e c t r o n t o b e n(EF) —0.5 s t a t e s / e V a t . a n d m* = l.

TABLE I. Comparisons among the three Hamiltonians for cerium alloys for which 7 < 0 . Hamiltonian Spin-disorder resistivity R,

//,= -2/8.-S

Uj=-2J{g-i)».-i

i

2irm*n(EF)cJ s(s+l)

2wm*n (EF)cJ* Cg - DVO'+1)

kk'MM'

2*m*n(EF)cJ*4JU+l)

Depression of the superconducting temperature dTJdC Asymptotic form of the Ruderman-Kittel interaction Eu"K'(R) Kondo temperature Tt

Kondo resistivity R

n(EF) -W 4

Jhn*tf

J2s(s+1)

cos2kFR (2SI-SI)MM-

TT*

(2k

(2i+l)2

«"*

ze*fi

FR)'

= Z)exp • \_2\J\n(EF)} 0.77D1 l+2|/|«(£,)lnkBT J

n(Er) -W

•Pfe-DW+i)

Jhn*kF> cos2kFR -(«-l)H2J1-J2)„«. jr» {2kFRy

No Kondo effect

No Kondo effect

-i*>~

n (EF)P kB

iJij+1) (2j + l)»

Jm*kF< co&2kFR -G(M,M') ** (2kFR)" 7"* = Z>exp

L(2i+l)|/|«(£riJ T RoA l + (2j+l)Jn(EF)\n

L

18 P. G. de Gennes, J. Phys. Radium 23, 510 (1962). >"K. Andres, Phys. Rev. 168, 708 (1968); T. Satoh and T. Ohtsuka, J. Phys. Soc. Japan 23, 9 (1967). " T. L. Loucks, Phys. Rev. 144, 504 (1966).

0.77ZT]

kBT }

379 185

EXCHANGE

INTER

Table II gives the values of 27 computed in the two limit cases, namely, «(£p)=2.2 states/eV at. and m*=3, which gives a lower limit for | 7 \, and n (£*•) =0.5 states/eV at. and m*= 1, which gives an upper limit for | 7 | . Table II gives also the corresponding values of E0. Because the Coulomb integral U is much larger than Eo in cerium alloys, 7 reduces to the simple form J~\VkF\yEo.

(43)

We can compute £ 0 either as a function of the HartreeFock half-width of the level, as previously done,7 or directly, by taking a reasonable value for the mixing parameter Vty We cannot hope for better than a rough magnitude agreement, and we use a value VkF=0.07 eV in the calculations of Table II, The values of | £o | are slightly overestimated because we have not considered here the normal Heisenberg exchange interaction.7 The results of Table II must be analyzed carefully. In a resistivity experiment, for example, a current is given almost exclusively by 6s electrons, so that the values of 27 and £ 0 which we deduce from this experiment correspond roughly to the 6s band parameters n(Ep) =0.5 states/eV at. and m* = 1. On the other hand, both 6s and 5i electrons contribute to the superconductivity mechanism, and the values of 27 and £o will be rather close to the values corresponding to the total conduction-band parameters n (EF) =2.2 states/eV at. and m*=3. Therefore, this analysis gives the following results: (1) In Y-Ce alloys, the value of 27 is of order —0.4 eV and the 4 / level lies an order of 0.03 eV below the Fermi level. The effect of pressure will be particularly large in this case, because the 4 / level is extremely close to the Fermi level. Because of the extreme smallness of £o in Y-Ce alloys, the second-order formula (43) is relatively dubious at normal pressure and cannot give better than an order of magnitude for £ 0 . Moreover, this formula should not be valid for high-pressure experiments, because E<, tends to zero. (2) In La-Ce alloys, the value of 27 is smaller, of order —0.1 eV, and the 4 / level lies roughly 0.1 eV below the Fermi level. The effect of pressure on the superconducting transition temperature has been already studied and gives an important increase of 171 (by \ under a 10 kbar pressure) and a corresponding important decrease of £ 0 . 6 ' 7 In the above discussion, we have neglected the Heisenberg exchange, although, as we mentioned, it is no doubt dominated by the strong hybridization exchange in Ce and should not modify our conclusion. In addition, we have studied only the 4/ 1 configuration and have not treated rare earths other than Ce. For the 4/" configurations («> 1), the angular momentum algebra is more involved and has been treated in a different model by Watson, Koide, Peter, and Freeman.21 Since »R. E. Watson, S. Koide. M. Peter, and A. J. Freeman, Phys. Rev. 139, A167 (1965).

CTION

IN A L L O Y S

853

TABLE II. Values of 2/ and Ej for La-Ce and Y-Ce alloys. m»-l

m*-i

Y-Ce alloys (from spin-disorder resistivity)

27 = - 0 . 4 3 eV Eo = -0.02S eV

2 7 = - 0 . 1 2 eV Eo - - 0 . 0 8 e V

La-Ce alloys (from spin-disorder resistivity)

27--0.14eV Eo - - 0 . 0 7 cV

27 = - 0 . 0 4 eV Eo - - 0 . 2 5 e V

La-Ce alloys (from superconducting temperature)

27 = - 0 . 1 8 eV Ei - - 0 . 0 5 S eV

27 = -O.0S eV E.= -0.12eV

the hybridization exchange is no doubt less important in these other rare earths, they are of less interest from the above point of view. We note that the 4/ 1 ' configuration (one 4 / hole) can be treated as above by interchanging holes and electrons. Thus, the preceding results can be applied to magnetic ytterbium alloys, with a total angular momentum j=%. In this regard, a resistance minimum has been recently reported in some silver-gold alloys containing ytterbium impurities.22 The preceding analysis suggests several types of experiments: (1) Look in detail for a spin and orbit compensated state at temperatures below Tk in Ce alloys, such as La-Ce and Y-Ce, by methods used for transition impurities. (2) Look for changes of Tt and other Kondo properties with pressure, since £o is likely to be a strong function of pressure.7,8 Perhaps one can see the disappearance of magnetism'and the Kondo effect at very high pressure in La-Ce and Y-Ce alloys. (3) By comparing various transport and magnetic properties, establish the validity of the new exchange interaction (6) relative to the conventional s-f interactions. 7. CONCLUSION Thus, we have derived an exchange Hamiltonian and have accounted for various properties for the 4 / 1 configuration of cerium. This analysis gives a consistent description of the resistance minimum and resolves the difficulties encountered by the conventional s-f exchange model. New results for the spin-disorder resistivity and the depression of the superconducting temperature versus impurity concentration have been derived, and a strongly anisotropic Ruderman-Kittel interaction has been obtained. ACKNOWLEDGMENTS We gratefully acknowledge discussions with Professor A. Blandin, Professor B. Muhlschlegel, and Professor M. Peter. • J. Boes, A. J. Van Dam, and J. Bijovet, Phys. Letters 28A, 101 (1968).

380 VOLUME 23, NUMBER 2

PHYSICAL REVIEW

LETTERS

14 JULY 1969

THEORY OF ITINERANT FERROMAGNETS EXHIBITING LOCALIZED-MOMENT BEHAVIOR ABOVE THE CURIE POINT* S. Q. Wang, W. E. Evenson,t and J. R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received 5 June 1969) By using a functional integral formulation of the theory of itinerant ferromagnets above the Curie point, we show that for strong Coulomb interaction U, there are localized moments exhibiting a characteristic Curie-law susceptibility with the correct free spin-i limiting value of the Curie constant. For weak U the same formulation gives a Paulilike susceptibility, again with the proper limit, while for intermediate values the theory gives a smooth interpolation between the extreme cases. The presence of local-moment aspects in band ferromagnetism has long been a baffling problem. The most striking example of this is iron. The high-temperature susceptibility, neutron scattering, alloy experiments, etc., all point to the presence of localized moments, while transport properties show unambiguously the itinerant character of the d electrons. 1 We report here on the first results of a new theoretical approach to such systems. The theory is based on an exact transformation of Stratonovich2 and Hubbard3 which eliminates the two-body interaction in favor of a Gaussian average over fluctuating onebody potentials. We concentrate here on the paramagnetic phase, leaving cooperative effects for future publication. Since there is little short-range order at high 92

temperatures, we expect the problem to be equivalent to an aggregation of one-center problems. The one-center problem can be represented by an Anderson model4 of an "impurity" atom immersed in an effective band. While orbital degeneracy (Hund's rule) is important in practice, most of the essential features are already contained in the nondegenerate orbital model treated here. The Hamiltonian is H0 +Hlt where if a = ±l, H0= Z/«*o"Ao+Z/erfoB
o

+ E l > * c * 0 t c d 0 + H.c.],

(la)

H^Undindi =-i£/("rft-"rff)2 +jU{ndi

+ndi).

(lb)

381 PHYSICAL

VOLUME 23, NUMBER 2

REVIEW

In Eq. (lb) we have used ndo* = n da. The second t e r m of Hl can be absorbed into H0 by defining edo=eda+?U. Following Hubbard, Mflhlschlegel 5 has shown that the partition function can be written as the Gaussian functional average over an effective "magnetic" field £(T) of the partition function Z(£) for a one-body Hamiltonian: Z = / D « T ) exp[-7rJ" 0 V( T )dT]Z(a Z(S) =

(2a)

Tr{Texp£[-fiH0T + ci(TKndtT-ndiT)]dr},

(2b)

1/2

where c=(.2nPU) , and T is the ordering symbol for the fictitious "imaginary time" T.e To evaluate Eq. (2), it is convenient to Fourier analyze £(T): C(T)=

B' i,e'a«T,

0„ = ari/,

{„ = {_»*.

LETTERS

Kn

:_

i-i vv o, v, n

Gnn+V

(4)

,

where v „ " = -act, „. G„„ • ° is the one-particle Green's function which satisfies the Dyson equation: C M . ° = G i , 0 0 8„.+S»G» 0 , V;»-„G J 1 , a .,

(5a)

where G„°° is the zero-order one-electron Green's function in the presence of the r-averaged potential -ack0, G n 00 = {iu„-fleda+oci;0+ipr

sgnwj"1,

oj„ = (2n+l)v.

(5b)

zV

m~D

G„ ,

K„„ = 0 .

(7)

Clearly K describes the scattering of the virtuallevel electron by the fluctuating field U ( T ) - { „ ] , with the T-averaged effects included in z e r o o r der. The nth t e r m in the power s e r i e s expansion of the Tr In t e r m can be represented by a single closed-loop diagram with nv° lines attached. Z0U0) in Eq. (6) is the value that Z(£) [Eq. (2b)] takes on if £(T) is replaced by its T average £0. Explicitly, for the symmetric case, ed + ?U = 0, and for large 0T,

»[•**']-*«"--'(&)

(3)

1 ^Sf^-<E/. *c«r)«m or > 8X

1969

variables, and is defined by

-?*[••(&;]• <«>

I/ss —»o

By introducing a dimensionless coupling constant, X, multiplying c, it follows that

14 JULY

where A is the partition function for H0 with edo replaced by edo. In Fig. 1 we plot (IT£ 0 a -lnZ 0 )//3, which is the effective free energy for a T-independent field £0, for several values of U/vT.8 When calculating the partition function, it is useful to distinguish four separate regimes of the parameters 0 r and U/vT: (1) U / i r r « l . This corresponds to a nonmagnetic impurity with weak exchange effects. (2) U/ttT» 1 and T»TK, where T K is a Kondo-like temperature of order (U/kt)e~,'U/ar. This corresponds to a strongly localized moment above the Kondo regime. (3) U/vr» 1 and T
2

T is the virtual level width irN(0)jV| . Using the Fredholm solution of Eq. (5a) and the relations TrN = -9D/&\,

D=eTrln<1-/f)

for the Fredholm numerator and denominator, it is straightforward to show that an exact expression for Z is Z = J~„dl0 n 2 M „ e x p { - i r V>Q

£ V'


U-'l 2

- — *>

+ STrln(l-.K:

0

)}2 0 (4o),

(6)

where !d2£u denotes an integral over the complex (,u plane. 7 The Tr In term takes account of the fluctuating parts of the effective field, i.e., the iv's for v*0. K° is a matrix in the frequency

FIG. 1. Effective free energy for a T-independent field i0. The curves a r e labeled by U/rT. 93

382 PHYSICAL

VOLUME 23, NUMBER 2

REVIEW

contribute appreciably to Z, so that the expansion of T r l n ( l - t f ) to order | £ „ | 2 suffices in this c a s e (the linear t e r m s vanish). On performing the diiv integrations one finds Z = fSLdtDe~*to2Zotio)

n

\l-~-
K>OL

"

Oa) J

where the "polarization bubble" i s , for v > 0,
wav{av+2fir) and tpv=
In i +

o„(n„+2flr)

£ 2 r 2 +cV

For | n „ / 8 r | « l ,

1 'Pv—,7r/3r;R Ll-i^H /3r

+

.

,

(9b)

(pv b e c o m e s (10)

where ft = [ l + ( c ^ r 2 ] - 1 . 1 0 In this s m a l l - ( £ / / •nT) limit one can a l s o expand lnZ 0 to order £ 0 2 , a s one can s e e from Fig. 1 since the effective free energy has a s i n g l e minimum at the origin and large positive curvature in this c a s e . This procedure i s exact in this limit and i s equivalent to the random-phase approximation including both bubble and ladder diagrams with correct spin counting. One can s e e from Fig. 1 that in c a s e (2) the dominant contributions to the partition function will c o m e from v a l u e s of £ 0 near the two minima. By inspection of Eq. (9) it i s clear that for | 0 near the minima the v > 0 contribution i s small, thereby justifying the Gaussian approximation for the | „ integrals in this 4 0 neighborhood. For £ 0 far from the minima, a careful treatment of the v * 0 t e r m s shows that the entire £ 0 integrand in Eq. (6) i s negligible. Therefore Eq. (9a) g i v e s an accurate value of Z so long a s the integral i s carried out only near the minima in effective free energy. In c a s e (3) the small energy arising from s c a t tering from the vicinity of one minimum to the other must be carefully included. Fluctuations about a given minimum a r e correctly included in Eq. (9); however, the infrequent hopping from minimum to minimum must be treated s e p a r a t e ly. 1 1 In c a s e (4) a number of low-frequency £„'s give appreciable non-Gaussian contributions to Z. This problem i s p r e s e n t l y under study. To obtain the static magnetic susceptibility of this s y s t e m , w e u s e the relation x = ( l / 9 ) [ 8 2 InZ/ 8h2]j,,0 where h i s a magnetic field applied in the z direction. Since the Zeeman energy enters ad94

LETTERS

14 JULY

1969

ditively in H with ci0/&, one can shift the o r i g i n of i0 by 0/i B A/c s o that, aside from A, the Z e e man energy appears only in the Gaussian factor in Eq. (6). It follows that 2 M B ' , MB'

i " Trr ++ JV| - [ a r < * o V l ] + Xb«d,

(11)

w h e r e (4 0 2 ) i s g i v e n by inserting £ 0 2 into the i n tegrand of Eq. (6) and dividing by Z. The numerical r e s u l t s for the susceptibility a s a function of temperature a r e shown in Fig. 2. It i s interesting to note that for U/nT'»l, the susceptibility i s C u r i e - l i k e over a wide t e m p e r a ture range. For l a r g e U/vV, the s u s c e p t i b i l i t y approaches the Curie law appropriate for a f r e e s p i n - \ moment. F o r £ / / 7 r r « l , \ i s e s s e n t i a l l y temperature independent, corresponding to a weakly-enhanced Pauli susceptibility. In r e g i m e s (1) and (2) the c u r v e s in Fig. 2 w e r e calculated from Eq. (9) as d i s c u s s e d above. In r e g i m e (4), U/•nV~\, the c u r v e s in the figure w e r e calculated using the exact e x p r e s s i o n [Eq. (8) ] for Zn but neglecting the Tr In t e r m of Eq. (6). This approximation c o r r e s p o n d s to n e g l e c t ing finite frequency fluctuations of the effective field. Work i s currently proceeding to include the contribution of t e r m s for finite v in r e g i m e (4). The coupling between moments, which i s a t w o center problem in first approximation, i s c u r rently under study. Extension to degenerate o r bitals will be undertaken in the near future.

FIG. 2. Plots of the dimensionless quantities (x -Xband)r/MB2 vs flT. The full lines are calculated directly from Eqs. (9a) and (11); the dashed lines are calculated by neglecting Tr l n ( l - i 0 in Eq. (6) as e x plained in the text. The asymptote for large U/*T is the correct Curie law for a free spin i. For small U/ irT, the correct exchange-enhanced Pauli susceptibility is obtained.

383 VOLUME 23, NUMBER 2

PHYSICAL

REVIEW

We thank Dr. B. Mtfhlschlegel for showing u s his unpublished work. We also thank Dr. D. R. Hamann for stimulating discussions. *Work supported in part by the National Science Foundation and the Advanced Research Projects Agency. tNational Science Foundation Postdoctoral Fellow. 'C. Herring, in Magnetism, edited by G. T. Rado and H. Suhl (Academic P r e s s , Inc., New York, 1966), Vol. IV, Chap. VI. 2 R. L. Stratonovich, Dokl. Akad. Nauk SSSR 115, 1097 (1957) [translation: Soviet Phys.—Doklady 2_, 416 (1958)]. 3 J. Hubbard, Phys. Rev. Letters 3_, 77 (1959). 4 P. W. Anderson, Phys. Rev. 124, 41 (1961). 5 B. Miihlschlegel, University of Pennsylvania, 1965, unpublished lecture notes. ^ h e r e a r e other ways of writing Z in functional a v e r age form as Miihlschlegel (Ref. 5) has shown. Independently, D. R. Hamann, this issue [Phys. Rev, Letters 23, 95 (1969)1 has employed two random fields to study the Kondo problem.

LETTERS

14JULY1969

T

In polar coordinates, d2^v = ^dRvzdBv. An alternative scheme for the evaluation of Eq. (2) is to perform a coupling-constant integral over the strength of Vk as Miihlschlegel (Ref. 5) has done. Unfortunately, in the full ferromagnetism problem a s o p posed to the one-center approximation, this procedure has the serious drawback of requiring one to integrate through the insulator-metal transition. 9 Large-amplitude localized spin fluctuations have been discussed from other points of view by A. D. Caplin and C. Rizzuto [Phys. Rev. Letters 21., 746 (1968)]; P. Lederer and D. L. Mills I Phys. Rev. Letters 2p_, 1036 (1968)1; N. Rivier and M. J. Zuckerman [Phys. Rev. Letters 21., 904 (1968)]; and M. Levine and H. Suhl [Phys. Rev. 171, 567 (1968)]. 10 We note that the t e r m s 1 and - \nv\R/pT in Eq. (10) correspond to the adiabatic and transient t e r m s , r e spectively, in the Nozidres—de Dominicis ( P . Nozieres and C. de Dominicis, Phys. Rev. 178, 1097 (1969)] solution of the x-ray intensity problem as employed in the magnetic impurity problem by P . W. Anderson and G. Yuval (Phys. Rev. Letters 23, 89 (1969) (this issue)] and by Hamann (Ref. 6). "Hamann, Ref. 6. 8

95

384 P H Y S I C A L REVIEW B

VOLUME 2, NUMBER 7

1 OCTOBER

197

Theory of Itinerant Ferromagnets with Localized-Moment Characteristics: Two-Center Coupling in the Functional-Integral Scheme* W. E. Evenson, S. Q. Wang, and J . R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania (Received 18 May 1970)

19104

The coupling between two magnetic centers in a band is discussed within the framework of the functional-integral scheme. The model used is the single-orbital Hubbard model with Coulomb repulsion only on the two magnetic sites. In lowest approximation, antiferromagnetic Ising coupling is obtained when the moments are nearest neighbors. When the moments are far apart, Ruderman-Kittel-Kasuya-Yosida coupling is the most important of several terms.

A functional-integral technique has recently been applied 1 , 2 to the theory of ferromagnetism in s y s t e m s like iron which exhibit localized-moment b e havior above the Curie point yet show itinerancy. This recent work considered the general formulation of the problem and the details of the one-center problem (i. e . , the ferromagnet at t e m p e r a t u r e s sufficiently high to eliminate s h o r t - r a n g e o r d e r ) . In this paper we study the two-center problem in detail to gain insight into the coupling of moments in this system.

The Tli contain any one-center potential, including Zeeman energy, a s well as the hopping m a t r i x e l e ments or F o u r i e r components of e(k) expanded in the complete set of Wannier functions for one band. Following the p r o c e d u r e discussed in detail in Ref. 2, we obtain the two-center "static approximation" by keeping only the zero-frequency components of the random fields £,a(r) and £ 6 (T). Then we obtain

Considerable previous work has been done on the two-center problem in a two-impurity Anderson model. 3 All of that work was done within the framework of H a r t r e e - F o c k approximation. Alexander and Anderson, Moriya, and Liu studied the case of two close impurities, while Kim and Nagaoka, and Caroli studied mainly the case where the two impurities a r e widely separated. These authors were primarily interested in the coupling energy in the localized-moment r e g i m e , where the Coulomb r e pulsion U is the dominant energy in the problem. Unfortunately, it i s in this case that H a r t r e e - F o c k is least reliable. The differences in approach and in choice of model have made it difficult to compare the p r e s e n t r e s u l t s in detail with the previous work; however, many qualitative features a r e the same.

where

The model we have considered is the nondegenerate orbital Hubbard model 4 with Coulomb repulsion on the two magnetic c e n t e r s of the system, i. e . , a two-impurity Wolff model. (In practice, Hund's rule coupling due to orbital degeneracy is an i m portant effect. However, the introduction of degenerate orbitals involves considerable mathematical complication, so we have r e s t r i c t e d our attention to the simpler nondegenerate c a s e for the time being.) The Hamiltonian i s , then, H = H0 + Hlt where if
(la)

ill

and Hl = U(nainal + nb,nbi).

(lb) 2

*.utt.= LI « - J". #»*•«£>'**> Z.tte«,

U), (2a)

ZJU,

tJ=Z0es,l"a-':°>

•

(2b) 1 n

Z 0 is the partition function for c- (2n^U) before, and the spur (trace) i s Spln(l -K0) = Z l n { [ l + a c l , , 0 G : » ] [ l +

— 0 as

actKGl°{n)]

no

- c*UltoG°JMGl%)}

,

(3a)

where for i, j = a o r 6 G°>) =- I —

~ -

.

(3b)

We can proceed beyond the static approximation by the R P A ' 2 or by the m o r e sophisticated methods that have recently been applied to the one-center p r o b l e m . 5 In both the one- and two-center probl e m s , we find that the static approximation is exact for 7\.j =0 or for U= 0. The RPA' gives the right leading corrections from the latter limit, while the methods of Ref. 5 also fix up the leading c o r r e c t i o n s in the former limit. One obtains a smooth i n t e r polation through the U/A ~ 1 region (A being the bandwidth) by either the static approximation or the approximations of Ref. 5. We have considered two c a s e s in the two-center problem: (i) when atoms a and b a r e n e a r e s t neighb o r s , and (ii) when atoms a and b a r e far apart so that R„,= R„ - R, i s much l a r g e r than a lattice spac2604

385 2

THEORY O FITINERANT

FERROMAGNETS

ing. In both these cases the In t e r m in (3a) can be expanded into a noninteracting t e r m plus coupling t e r m s with the first coupling t e r m dominating the interaction energy when one looks at either V/A or A/V small. This is true because in the expansion for coupling energy, the ratio of successive t e r m s is the small p a r a m e t e r U/A or A/U. This fact i s obvious when a and b are far apart so Cjin) must be small compared to G„°(«). The noninteracting term i s , of course, simply the one-center t e r m which was discussed in Refs. 1 and 2. The coupling part in general, without expansion of the In in (3a), leads to extremely complicated integrals, so we will r e p o r t h e r e on only the leading coupling t e r m . G°J?(n) is given by (3b), but to proceed we must evaluate the k* sum. In Ref. 1 this i s done using a Lorentzian state density. However, now that we a r e looking at coupling t e r m s , the long tails on the Lorentzian lead to unphysical effects in c a s e (i) {a, b near neighbors) when the hopping i s very s m a l l ; in p a r t i c u l a r , G "at r e m a i n s finite a s 7 ^ — 0 for that c a s e . For this reason, we have used a Lorentzian-squared state density [ i . e . , M e ) = 2A 3 / ?(A2+ e 2 ) 2 ] to evaluate G'J in c a s e (i). When a and b a r e nearest neighbors, Gj° i s found from the equation of motion for G°° to be G°°(n)=[MnG:»-l]/z|3r, where a tight-binding band i s assumed, so that Tu = T if t, j a r e nearest neighbors, and T ( J = 0 otherwise (except for the one-center potentials on a and 6), and z is the number of nearest neighbors. If the width A of the normalized Lorentzian-squared state density used in case (i) i s chosen to be z1 nT, then the leading t e r m s in G°° and G°„°as , all agree with those obtained from i n s e r t ing the tight-binding assumption into (3b). In c a s e (ii), we have used a simple Lorentzian state density to evaluate G'J; G^° is left in the form (3b) when a and b a r e widely separated. Since we have assumed a tight-binding form for Tu, the width of the Lorentzian, A, in case (ii) must also be p r o portional to T. For additional simplification we have taken T „ = Tbb= - \V. We have called the dominant coupling t e r m in the cases studied - /3C(£0, 4») so that C i s the coupling free energy for fields £ s , t,t. We will discuss its form when U/A » or « 1. We will first consider C(£„, £,,) when a and 6 a r e nearest neighbors and {//A » 1 . F r o m the noninteracting t e r m in Z((,a, (,b) we s e e that the effective free energy has sharp minima when (,&> KM=±C/2TI, i. e . , c4 o0 , c£ M = ±pU. Then we can expand the general result for pCUao, £«) to obtain in the static approximation

WITH'

2605

If c^a0 = c£,b0 = 0U, i . e . , the two moments a r e t + , C„s>-A/irz+2T*/U,

(5a)

and if c£ o0 = - c£ M = /3£/, i. e . , the two m o m e n t s a r e H,

C,,£r-A/iT£+r2/t/.

(5b)

We s e e that the antiparallel a r r a n g e m e n t i s favored, being lower in energy than the parallel a r r a n g e m e n t by T*/U. In perturbation theory, however, one finds the energy difference to be ZTZ/U. Our p a r t i c u l a r form of the functional-integral formulation of t h i s problem puts 2 f for U in the static approximation, and the techniques of Ref. 5 a r e needed to r e n o r m a l ize 2U back to U. So this i s the expected s t a t i c approximation result, and higher approximations should make a quantitative, but not qualitative, difference. Doing RPA', for example, one o b t a i n s a result which favors antiparallel alignment m o r e strongly than static approximation, as expected from the perturbation-theory r e s u l t . This s y s t e m i s more like an Ising model than a Heisenberg model because the energy i s a function of total SM r a t h e r than total S. To go over to a Heisenberg m o d e l in this formalism one needs to include high-frequency p a r t s of £„(T) and 4„(T) with £„(T) and £ 6 (T) p h a s e coupled to give the p r o p e r mixing of ++ and • • into triplet and singlet S,= 0 s t a t e s . An a l t e r n a t i v e p o s sibility i s to use a vector random field as mentioned in Ref. 2. Other difficulties then a r i s e , however. Again in c a s e (i) if U/A« 1, the effective f r e e energy has a minimum for £.M, £ w near z e r o . Then the static approximation gives C(U,

(6)

«M)2!4E^M/3«0A.

This again favors antiparallel alignment of t h e extremely fuzzy moments distinguishable in t h i s limit. The RPA' calculation gives a factor favoring t h e parallel configuration which i s of comparable o r d e r but smaller in magnitude. In c a s e (ii) with V/A » 1, we obtain for the coupling energy in the static approximation n e a r cl^, „i(t-S')-a,

C ( U 4, ,

2 /,

6A2 \

-2^e-k(i_2^

j

+

!

2^)ln|0ej|], (7a)

with C«-«.

(7b)

386 2606

EVENSON,

WANG, AND S C H R I E F F E R

a n d / ; i s the Fermi distribution function for €5. If ci)lB=ciK=fiU, t h e n ? ) 2 = ^ , and if d^= - ci,^ = PU, then 77*= f/2. The first term in the brackets of (7a) is a part independent of relative spin orientation plus the Ruderman-Kittel-Kasuya-Yosida 6 (RKKY) coupling term with exchange integral, J, given by 2&Z/U ~ 1*/U. In the Schrieffer-Wolff transformation 7 J is given by 87^/1/ when et= - \U; however, the proportionality constant between J and 7*/U here depends on the relation between the bandwidth of our effective Lorentzian band A and the hopping integral T. One should do a calculation which i s consistent in the sense of using the tight-binding band throughout the calculation in order to get the correct J to compare to the Schrieffer-Wolff transformation. The next two terms in brackets in (7a) give contributions to CU„o> £») which fall off much faster with increasing separation distance than does the contribution from the first term if one considers energy bands proportional to kz. In that case, the RKKY term goes as 1/i?3 while the other terms go as \/FC or faster. One would, therefore, not expect these terms to be important except for very unusual energy bands. The last term in brackets in (7a) contains In Ipej | and so looks reminiscent of the Kondo effect, but appears in order J* rather than J 3 as one obtains in Kondo effect. As was mentioned previously,'•* the static approximation does not properly include the Kondo effect in the one-center problem. For essentially the same reasons as in the one-center problem, we do not expect the static approximation to treat

*Work supported in part by the National Science Foundation and the Advanced Research Projects Agency. *S. Q. Wang, W. E. Evenson, and J. R. Schrieffer, Phys. Rev. Letters 23, 92 (1969); J. R. Schrieffer (unpublished). 2 W. E. Evenson, J. R. Schrieffer, and S. Q. Wang, J. Appl. Phys. « , 1199 (1970). 3 D. J. Kim and Y. Nagaoka, Progr. Theoret. Phys. (Kyoto) 30, 743 (1963); S. Alexander and P. W. Anderson, Phys. Rev. 133, A1594 (1964); T. Moriya, Progr. Theoret. Phys. (Kyoto) 33, 157 (1965); B. Caroli, J.

2

InT terms correctly, so this last term should be canceled off in a better solution of the problem. When (//A « 1 in case (ii), the coupling energy for £ao> £M near zero in the static approximation is given by

CU

£ )-

8 » * y . . ; r «"'•*'•*•*

f

(o)

We expect the improved approximations for case (ii) to give quantitative corrections to the static approximation similar to those observed in case (i); however, an explicit calculation i s possible only if we specify a consistent band structure since the E sums will appear in denominators of the RPA' or higher approximation terms. At this stage specialization to a detailed energy band structure does not seem to be sufficiently enlightening to be worthwhile. The regime U/A ~ 1 demands more accurate treatment of | , ( T ) than has been done heretofore. We expect the static approximation to be qualitatively correct in this region (except for the Kondolike term mentioned above), but further work i s necessary to improve the calculations there. We also expect the introduction of orbital degeneracy to produce a ferromagnetic ground state under proper conditions. Work is proceeding on these effects. We wish to acknowledge helpful discussions with Dr. Hellmut Keiter, Dr. Alan J. Heeger, and Dr. John A. Gardner. We also wish to thank Dr. Keiter for a critical reading of the manuscript.

Phys. Chem. Solids 28, 1427 (1967); S. H. Liu, Phys. Rev. 163, 472 (1967). 4 J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963). 5 H. Keiter, Phys. Rev. B (to be published). e M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954); T. Kasuya, Progr. Theoret. Phys. (Kyoto) 1£, 45 (1956); K. Yosida, Phys. Rev. 106, 893 (1957). ! J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).

387 iKeprinted from JOUR.VAL OF APPLIED PHYSICS, VO!. 41, No. .3, 1199-1204, I March 1970 Copyright 1970 by the American Institute of Physics Printed to U. S. A.

New Approach to the Theory of Itinerant Electron Ferromagnets with Local-Moment Characteristics* W.

E . EVENSON, J . R . SCHRTETPEE, AND S. Q. WANG

Department of Physics, University of Pennsylvania, PItiladelphia, Pennsylvania 19104

The understanding of ferromagnets, like iron, which exhibit localized moment behavior above the Curie point yet show itinerancy has long stood as a major theoretical problem. An account will be given of recent progress on this problem which was achieved through functional integral methods. This technique transforms the interacting electron system into an average over a system of noninteracting electrons moving in a Gaussian-weighted external "magnecic" field which acts only on the electronic spins. For a single magnetic impurity in a free electron metal, a single approximation allows one to go from Pauli paramagnetism to localized moment behavior in a smooth manner as the atomic exchange interaction is increased. The two impurity problem leads to an effective exchange coupling as in the Heisenberg model, which is antiferromagnetic for the nondegenerate orbital case studied here. Application of the technique to homogeneous systems leads to damped spin waves in the ferromagnet in lowest approximation. I. INTRODUCTION

can be treated in zero order by a band model with effective exchange interaction parameters, Fe should be viewed as an intermediate coupling case for which neither the band nor localized limits form good zeroorder approximations. This is not to say that many properties of Fe cannot be well accounted for by a band model. Rather most observed properties of Fe are relatively insensitive to its intermediate coupling nature. A viable theory should be able to account for these features plus the apparent "localized moments" as seen in the alloy experiments and in the paramagnetic susceptibility. Recently, the beginnings of a theory to handle this intermediate coupling (or schizophrenic electron) regime have been worked out by Wang, Evenson, and Schrieffer.5 The theory is still in its formative stages yet the general features of the approach look quite promising. In essence, by using an identity of Stratonovich,s one can exactly transform the partition function for the interacting assembly of d electrons to a partition function for noninteracting electrons moving in a spaceand time-varying externally applied "magnetic field," the field being averaged over with Gaussian weight. The method, when applied to the problem of a single impurity atom with intra-atomic exchange (the Anderson model) 7 gives in lowest approximation a continuous variation of the susceptibility from enhanced Pauli paramagnetism (for weak exchange) to an ideal Curie law (for strong exchange), all within an itinerant scheme. Thus, one has hope that a careful application of the technique to a system like pure Fe will adequately treat both its itinerant and localized aspects.

There has been a long-standing debate whether the "d electrons" in Fe, Co, and Ni are better described in zero order by a localized (Heisenberg) model or by an itinerant (band) model. Some of the experimental evidence has been summarized by Herring.1 While much of the experimental evidence originally quoted in support of one or the other limiting model is in fact well accounted for by either limit, the bulk of the relevant data, with a few notable exceptions, supports the itinerant theory. Evidence supporting the band limit includes the large electronic specific heat, nonintegral magneton numbers observed in saturation moments, the magnetoresistance, and Hall effect, etc. for the three elements. This evidence leads to the inescapable conclusion that the d electrons must exhibit at least some itinerant character. To our knowledge only for Fe is there any evidence which unambiguously favors the localized model. Small additions of Cu, Zn, Al, etc. to Ni lower the saturation moment per atom by an amount proportional to the number of valence electrons per solute atom, while addition of these elements to Fe reduces the moment by roughly 2.2 /is (the saturation moment per atom in pure Fe) irrespective of valence.5-3 This behavior suggests that a reasonably well-developed localized moment resides on each Fe atom, while the moment on Ni is strongly influenced by its environment, as one expects in a band scheme. Co appears to be closer to Ni in this respect. In addition, it appears that the ratio of the Curie constant [C=(T—B)x] and the saturation moment observed for Fe is too small to be accounted for by any reasonable choice of parameters H. THE FUNCTIONAL INTEGRAL FORMULATION in a purely band theoretic model. On the other hand, the OF THE HUBBARD MODEL band model gives a rather better account of this ratio The model we choose to consider for the ferromag4 for Ni. Thus, it appears that while Ni (and possibly Co) netic system is the multiorbital Hubbard mode!,* but 1199

388 EVENSON,

1200

SCHR.IEFFER,

for simplicity to begin with we will neglect the degeneracy of the d band and look at the nondegenerateorbital .Hubbard model. (In practice Hund's rule coupling due to orbital degeneracy is an important effect, needed to produce ferromagnetic rather than antiferromagnetic coupling between atoms. However, the essential localized-moment features of our problem are already contained in the simple case from which we start.) The Hamiltonian is

E=Ea+Eu with

AND

Z, for the Hamiltonian in the presence of an external field. Then.we get the susceptibility, x, for example, by differentiation of logZ with field h, where Z = Trexp[-j8(ff-Aitf)]. Stratonvich,' then Hubbard' and Miihlschlegel,"1 developed a functional integral method for calculating Z which begins with the identity (valid for any bounded operator a) exp(iro ; ) :

(2.1a) (2.1b)

and

(2.1c) i

where £/,- is the Coulomb interaction, assumed to act only when electrons are on the same atom (i.e., Wannier state) .TJ i=U for the pure metal; the impurity problem corresponds to L',- varying over the lattice. The c,,+, c> are creation and annihilation operators for electrons in Wannier state * with z component of spin a/2=±.\. The Ti,' include the usual transfer interaction, i.e., hopping integrals, and Zeeman energy for an externally applied field. We measure all energies relative to the chemical potential, /J. If all the Vi were zero, E would be a one-body operator which could be exactly diagonalized in terms of the correct scattering states (or Bloch functions) which properly mix the Wannier states for the many sites. This would give an energy band picture for the resulting metallic system with the usual nearly temperatureindependent Pauli susceptibility. On the other hand, if the hopping were zero, and there were one electron per site, one would have an insulating system. Then the susceptibility would be a spin-J Curie law, x~/3- These are the two important limiting cases: no localized moments or perfectly localized moments, depending on the size of the ratio T/U, where T is the nearestneighbor hopping. As discussed above, for a system like iron one is dealing basically with an intermediate situation in which the magnetic effects appear to be close to the localized (insulator) limit, while transport properties are those of a metal. Unfortunately, it is just in this interesting case that it becomes very difficult to solve this problem. Perturbation theory in the appropriate small parameter, T/U or U/T, will not treat the regime where T/U~l, and moreover, perturbation theory will not take us through the Mott transition where the system goes metallic from the small T/U insulator. We have hopes that the functional integral formulation of the problem will allow in a natural way those approximations which are useful in the regime

T/U~l. To study the magnetic behavior of this system, it is convenient to calculate the grand partition function,

WANG

/:


(2.2)

as seen by completing the square in the exponent. To use this identity we can rewrite Bx in any of several ways, e.g., # > „ « , • , = -iUittm-mi)2+Rri(H,-t

+»,-,),

(2.3)

!

since «,-, = «,-, for fermion occupation number operators. Alternatively, we could write, as Hamann" did recently, Urnum i= iUi(nti+m

i ) 2 - H ' . ( " . t - '». i) 2 ,

(2.4)

or by writing the fermion operators in terms of spin operators V .•«.!",• i = \U ,(«,-, +•«,• t ) - f t',-S,- Si = i*7.(n,-, +*,- , ) * - iU,Sr S,-.

(2.5a) (2.5b)

We will not discuss the relative merits of these ways of writing Ex, but in our present discussion we will use the first expression. We take the second term in (2.3) into Bo so that #0= £ f./*/*>,

fif^Tif+Wi&u,

(2.6a)

•/»

#i=-EKri(K,t-«,,)'.

(2.6b)

i

Now if H0 and Hi commuted, one could write exp(— &E) as exp(-£r? 0 ) exp(— pHi). One could then use the Stratonovich identity to write exp^C/.Crt.t-n,,)2] = f° #.- expC-rfcif QirfiUt) 1/2 («„ - » , i)£,].

(2.7)

J—at

Unfortunately Ho and Hi do not commute, so before we can linearize the exponential we need to use the Feynman time-ordering trick:

^B= T exp (J*dr(Ar+Br)} ,

(2.8)

where r is a fictitious "time;" T is the chronological ordering operator, which orders products like A rBr- A r» chronologically with larger "times" to the left. The A, and Br- can be treated as commuting operators so long as they are acted on by the T operator in the end. Using

389 ITINERANT

LOCALIZED

ELECTRON

this time-orderiog trick then, the /iff,- becomes a functional integral over all possible functions £,(T) , and

txp(-pH)-t

fRSXiir)

FERROMAGNETS

Differentiation of logZk with respect to X yields 3 logZx

ax

« p ( - I dr'lPHtr.

— £ / drrtitiir) (w„r}x, i*

+ E [rfW

-cUr')

{nur-n,„)!))

(2.9a)

with C . = (2T/3 £/',-)I/2.

1201

(2.9b)

JK

(3.1)

'

where the identity on the right-hand side defines ( )x. Since Z(?i, •••,£#) = Zx_i, we would like to integrate (3.1) using the fact that Z\_o is easy to find since it corresponds to C/,-=0. At this point one introduces Green functions to determine {ni,T\. We define

The significant point here is that one has reduced the density matrix exp(— #27) for a system with the twobody Coulomb interaction to the average over the density matrix for a system with only one-body terms but in a random '''magnetic" field f ,(r) in the z direction. [Notice £,-(r) enters #(£,•) with Sz to contribute a Zeeman-Iike energy, where {,•(•»•) looks like a magnetic field.] Thus we replace the exchange field one electron exerts on another by a fictitious external field which varies in time and from site to site, but which is so arranged as to reproduce exactly the effect of the actual interaction. The TrT in Z is applied only to the quantummechanical operators, H0 and «,„ and is not affected by the variable £,(r). Therefore we can interchange the order of doing the functional integral and the TrT to obtain Z as a Gaussian functional average

where Z0 is the partition function for all c, = 0. fJZo just leads to the Pauli susceptibity for the metal with the t/,/2 shift in level position, so exprjSp Iog(l —A.")] contains the interesting physics of local moments if they occur.] K is a known matrix defined by

2 = / I T » € . - ( r ) exp ( - f

where

(3.2)

<«,vr}x=G.-.-x'(r,r+).

(3.3)

so that x

To find G we write its equation of motion. By Fourier transforming all quantities, it is possible to formally solve for G\ The ,\ integral can then be done exactly so that we finally obtain a simple formal expression for *(&. fc,-"-$.v) =Z0 exp[Sp l o g ( l - A - ) ] , (3.4)

K:,

iW £ ^ ( r ' ) ) XZfe.fe.-.-.tv)

GUX'{T, r') E - T r r ^ w ' + p x V T r r p , ,

(2.10)

* ' = - « : , • £ , • , , _ » / £ ; / ( « ' ) &»-i

&M= £

(3.5)

S.»exp(-2ir/V),

of a partition functional 2(&,&,-",fr)=Trr X exp I- J Jr'lfiRtr.-

£
(2.11)

and ga'(n) is the zero-order one-electron Green function Sp means trace in a, i, and n variables. We will find later in considering which approximations are necessary to evaluate Z that it is useful to treat the zero-frequency part of the effective potential, —
This is an exact expression for Z. There are now two KmVg-(Vt+r)g,vhen major pieces in the problem of evaluating Z from this (3.6) JW.<'-"=-«reA,a l V 4.'&..'; expression: (1) evaluate Zfo, f2,- ••, fcv) for arbitrary fields £,(r), 0 < r < l , and (2) carry out the functional thus average of Z(&, &,• • •,fcv).Notice that the functional Splog(l-A-) = Splog(l-F<#) integral is normalized so that if Z(&, £»,• ••>£jvr) is independent of the {,-, say it = Z 0 , then Z=ZQ. This is - f S p l o g C l - F W - F o s ) - ' ] . (3.7) the case, for example, if all Z7, = 0. Now define a new zero-order Green function by the HI. FORMAL EXPRESSION FOR Z(ft, f2, ••• , £y) Dyson-like equation To evaluate Z ( £ v , f.v) we introduce a coupling constant X multiplying the e,-. Then the density matrix and let K0=Vag, AT's V'g; then Px(&,&,•••, £v) is just Sp l o g ( l + ^ ) =Sp l o g ( l - £ 0 ) + S p l o g ( l - A " ) . exp

-(£'**')•

(3.9)

When we put this form back into the functional integral (2.10) to get Z, it is convenient to express the

390 1202

EVENSON,

SCHRI

FFER,

AND

WANG

functional integral in the £,-, variables as we have done forZ(fa,6,--iey).Then

given by

Z= [" II dU [( n Wi.) «p(~ t * Ifr,I1)

Here the interesting number which takes us from the local moment to the completely delocalized state is U/TTT. (Note: One is not obliged to go over to the Anderson model language, but we could stick with the Hubbard form while taking only one of the f/,^0. The advantage of the Anderson model in this case is its familiarity and the ready physical interpretation of the terms. The results are identical if one makes a careful correspondence of the various terms in the two representations of the model.)

X % - , W ,

(3.10)

T^rN(u)\Vuhr.

(4.2)

where {,-,* = £,-._, and V0 which is automatically A. The Anderson Model diagonal in the IT and n variables; then We begin our study of ferromagnets with local- Sp log(l-.ff)-»Sp log(l-/To) = Z l o g ( l + « & f r ' ) . moment behavior above T, by considering temperatures Te«T«T,. (For iron T,~ 1043°K, and 7>~20 000°K.) (4.3) At these high temperatures there is little short-range If we take A=0, (IT large, and assume the so-called order in the spin system, so we can to lowest order consider the problem of the ferromagnet to be a collection symmetric case where u= —U/2, we obtain of one-center problems, i.e., our chosen center thinks it = tan iog I + (4 4) is all alone in some "effective energy band" made up of i g all the spin-orientation averaged electronic orbitals of the system. We will see that the results are relatively fjFor general values of /3T, u, and h, Z(&) can be insensitive to the precise nature of this effective band. expressed in terms of digamma functions.] Then our model has £/\ = 0 except at the particular site In the static approximation, then, we focus on where £/",•= U and in fact reduces to the usual Anderson model with Z = r
° -^ T

^- 7 L

y j• -

J—oO

(4.1) «."*.+V/2,

= f

dbexpL-pF.,($,)],

(4.5)

J—to

and the integral must be done numerically. pF.t(h) = - logZ(&) -f-irfir is plotted in Fig. 1 for U/rT=0.2, 0.5, 1.0, and 5.0. For U/rT«l, only small fluctuations of & where Vu is the hopping potential from d to k, reflecting about the origin enter with appreciable weight, corthe fact that an electron initially on the d state in a responding to weakly exchange-enhanced Pauli paralocalized orbital will decay into the band. The tt. is the magnetism. For U/TTV >0.5, minima develop symmeteffective band mentioned above. The 7 \ / of the Hub- rically 'about the origin and a wide range of £o values bard model can be thought of as Fourier components of enters with appreciable weight, corresponding to large the et. expanded in the complete set of Wannier func- amplitude localized spin fluctuations. r / / V r » l is the tions for a single band. strongly localized-moment regime, the two minima For a band density of states Ar(<), the width of the corresponding to the up and down spin states.12 Un"virtual level" on the d state, i.e., the inverse lifetime of fortunately, the static approximation, while having the state with a single electron on the "impurity," is many features which are correct (e.g., smooth transition

391 ITINERANT

LOCALIZED

ELECTRON

from Pauli to Curie law susceptibility), does not give proper leading order corrections in the two extreme cases.

1 100

-

80

-

1

1

/

(4.6)

where <£,(&) is the "polarization bubble" (4.7)

The first term in (4.6) is just the free energy due to the mean value of the fluctuating field, while the second term represents the free energy of the fluctuations about this mean. Clearly, this approximation is reasonable if

-

m

4-60

/

40

20

t' &

20

1

To evaluate the RPA' we retain the Sp log(l-^T') term in (3.9) to second order in K', while keeping the Splog(l— Ko) exactly. The f^o integrals, being Gaussian, can be performed, and one finds

•

1

U/TT»L

M

yi.O

1

We can improve on the static approximation by enlarging the class of functions £ (r) which we consider. In particular, we can include for each value of fo all Gaussian fluctuations of £( T ) about that average value. We then get proper leading corrections for U/TT<£\ and the leading corrections in the other limit are brought closer to the exact ones. By analogy to the random phase approximation (RPA), we call this scheme the "RPA'." The RPA corresponds to retaining only the K1 terms in the Sp log(l—K) occurring in (3.4). This gives a free energy arising entirely from Gaussian fluctuations in the "time-varying magnetic field." While this approximation is valid for small U, as U/rT approaches unity, the free energy functional becomes strongly anharmonic as shown in Fig. 1 for | 0 . It is in fact this anharmonic nature of F which ultimately leads to the Curie law for

* , ( & ) • » - I E *.'**•••'•

1

\

C. The"KPA Prime"

0/W(6i)-iW.«ttb)+ ZlogCl-(e»/x)*.(6i)],

1203

FERROMAGNETS

^

s

40

0?

——

80

60

100

0r Fio. 2. Plots of the dimensionless quantities (*—xi«oa)r/p s ! versus (3I\ The full lines are calculated directly from (4.9a) and (4.9b) within RPA'; the dashed lines are calculated by the static approximation. The asymptote for large U/rV is the correct Curie law for a free spin J. For small U/TV, the correct exchangeenhanced Pauli susceptibilitv is obtained.

the fluctuations are sufficiently small, as is the case when U/TT«1. As V/TT increases toward unit}-, the fluctuations become large, and hence non-Gaussian. Then the Gaussian terms included in RPA' are not even sufficient to make the fo-integral converge. However, when U/TV becomes very large, the free energy functional develops sharp minima. For reasonable temperatures, the only important contributions to the & integral must come from the neighborhood of these minima, so the integral can be restricted to that neighborhood. Then RPA' again gives well-behaved results. D. The Magnetic Susceptibility

The applied field, /;, will clearly enter Z/,(t)/Z0/, additively with the fictitious field &• Hence, Zk{b, g^o) _ Zn-odo+fttaftA, g ^ ) ZoA

ZoA

Therefore, we put fo'=£o+0MflVc>

an

d

X e x p [ - T ( s V - / W A ) J ] exp[Sp l o g ( l - AT)],

(4.8)

the only h dependence being displayed explicitly in the Gaussian weight factor. Then FIG. 1. Effective free energy F„ for a r—independent field ft, as given by (4.4). The curves are labeled by V/rV. c is (2x;J£/)»».

X= W/U){2*{tf)-\)+x^

(4.9a)

392 1204

EVENSON,

SCHRIEFFER,

AND

WANG

VI. THE tf-CENTER PROBLEM

where

) # e X p(-^) Xexp[Splog(l-K)],

(4.9b)

and xb«nd is the Pauh susceptibility for the band electrons. Numerical results for x are shown in Fig. 2 for several values of U/irT. To our knowledge, this approach achieves, for the first time, a Curie-like susceptibility at high temperatures in the interesting region without starting from a zero-order approximation which already contains a moment. Rather, this theory deduces a Curie law which does not rest on such assumptions. This was one of our original important objectives for this first stage of the problem, and it became clear that in order to obtain a Curie-like susceptibility for temperatures Tt terms will serve mainly to produce an increase in the effective level width. We are pursuing these questions as well as problems including orbital degeneracy in the model.

The treatment of cooperative effects is very natural within the functional integral representation.15 In the ferromagnetic state, the presence of long-range magnetic order enters through the special importance of the time- and space-averaged £,(r), f,_o,,_o, which plays the role of a molecular field. Collective excitations (damped spin waves) enter if one treats the remaining parts of the f,(r) field within a Gaussian approximation analogous to the RPA' discussed in Sec. JVC. Since the spin waves involve transverse fluctuations of the ordered moments, and since approximations do not necessarily preserve the rotational invariance of the system, it is important that one treat the problem in a manifestly rotationally invariant fashion to insure that the spinwave energies properly go to zero for long wavelengths. This can be accomplished by writing the two-body interaction as in (2.5). The interaction is now linearized by a vector field, £,(r). Treating |^_o,,_o in zero order, the RPA' gives the free energy of the ferromagnetic state in terms of the |oo average of the usual RPA partition function in a molecular field, | OT . This approximation does not include the Iocalized-moment behavior, and one must combine the collective effects with those treated in the two preceding sections to obtain a more faithful picture of the intermediatecoupling ferromagnet. While there are yet many difficulties to overcome in finally working out the detailed, quantitative theory of ferromagnetism in iron, we believe that this functional integral technique holds good promise in this and in other very difficult many-body problems where large fluctuations play a crucial role.

• Work supported in part by tie National Science Foundation and the Advanced Research Projects Agency. 1 C. Herring in Magnetism G. T. Rado and H. SuhJ, Eds. (Academic Press Inc., New York, 1966), Vol. IV, Chap. VI. «W. Hume-Rothery and B. R. Coles, Ad van. Phys. 3, 14" (1954). V. THE TWO-CENTER PROBLEM • N. F. Mott and K. W. H. Stevens, Phil. Mag. 2, 1366 (1957). 4 P. Rhodes and E. P. Wohlfarth, Proc. Roy. Soc. (London) To investigate a Heisenberg-like description of the A273, 256 (1963). • S. Q. Wang, W. E. Evenson, and J. R. Schrieffer, Phys. Rev. system in which quasi-localized moments are coupled 23, 92 (1969). via a pairwise interaction, one can study the problem of Lett. «R. L. Stratonovitch, Dokl. AJcad. Nauk SSSR 115, 1097 two magnetic atoms imbedded in a nonmagnetic host; (1957); [English transl.: Soviet Phys.—Doklady 2, 416 (1958).] 7 P. W. Anderson, Phys. Rev. 125, 41 (1961). i.e., Ua= Ub= U, L",=0 if i^a or J in (2.6). Within the •J. Hubbard, Proc. Roy. Soc. (London) A276, 283 (1963); static approximation, one again retains only $l0 (i= a or A277, 237 (1964); A281, 401 (1964). b). Proceeding as in the one-center problem, when a •J. Hubbard, Phys. Rev. Lett. 3, 77 (1959). 10 B. MQhlschlegal, University of Pennsylvania, 1965 (unand b are nearest neighbors, one finds an antiferropublished lecture notes). a magnetic coupling. Presumably by including orbital D. R. Hamann, Phys. Rev. Lett. 23, 95 (1969). degeneracy one would, under suitable circumstances, "Notice that we are considering only the high-temperature obtain ferromagnetic coupling.15 Due to the fact that paramagnetic state, and our discussion may not apply in the lowtemperature Kondo regime. the scalar £ field within the static approximation does " See, for example, J. C. Slater, H. Statz, and G. F. Koster, not account for transverse fluctuations of the moments, Phys. Rev. 91, 1323 (1953); J. R. Schrieffer and D. C. Mattis, the coupling is of the Ising form. For two widely spaced Phys. Rev. 140, A1412 (1965); D. C. Mattis, The Theory of Magnetism (Harper and Row, Inc., New York, 1965). impurities the interaction is of the RKELY form, while " A more detailed account of this phase of the problem is in for intermediate separations several terms of compar- preparation by the present authors. u J. R. Schrieffer, unpublished lecture notes, Canadian Associaable order complicate the interaction." tion of Physics, Summer School, Banff, 1969.

393 RAPID COMMI MCAI'KIVS

PHYSICAL REVIEW B

VOLUME 46, NUMBER 9

1 SEPTEMBER 1992-1

Generalized Ruderman-Kittel-Kasuya-Yosida theory of oscillatory exchange coupling in magnetic multilayers Frank Herman IBM Almaden Research Center, San Jose, California 95120-6099 Robert Schrieffer IBM Almaden Research Center, San Jose, California 95120-6099 and Department of Physics, Florida State University, Tallahassee, Florida 12306 (Received 26 May 1992) In Ruderman-Kittel-Kasuya-Yosida (RKKY) perturbation theories of magnetic multilayers, realistic energy bands are used to obtain the nesting required for oscillatory behavior, but the Bloch functions are generally approximated by plane waves. To account for the observed long-period oscillations of the exchange coupling, we present a generalized RKKY theory, which includes both the Bloch character of the wave functions and the boundary scattering at the film edges. In contrast to existing theories, these two effects lead to long-period oscillations, that are robust with regard to roughness of the spin distribution and to oscillation amplitudes that sharply increase with the localized nature of the Bloch functions, in agreement with experiment.

Long-range oscillatory exchange coupling JU) has been observed between ferromagnetic layers that are separated by paramagnetic spacer layers of thickness t for a large number of transition- and noble-metal multilayers, such as F e / C r and Co/Cu. 1 , 2 / ( f ) exhibits damped oscillations as f is increased, with a period varying between 9 and 18 A, depending on the spacer metal. 2 In addition, short-period oscillations are observed in certain systems, e.g., Fe/Cr, Fe/Al, and Fe/Au multilayers, which have atomically smooth interfaces. 3 Moreover, such multilayers also exhibit giant magnetoresistance, 4,5 with potential in magnetic recording technology. 6 The origin of such long-wavelength oscillatory exchange interactions is generally believed to be associated with the Ruderman-Kittel-Kasuya-Yosida 7 ' 8 (RKKY) interaction J0(r)~cos2kFr/ri between two localized spins separated by distance r in a bulk metal, where kF is the Fermi wave vector. When summed over spins on the interfaces, 9,10 the coupling becomes JRKKy~cos2kFt /t1. Presumably, this type of coupling accounts for the observed rapid oscillations. The current explanation of the long-wavelength oscillations is based on the "aliasing" effect. 1 1 - 1 3 If one samples a rapidly oscillating wave precisely at each maximum, the apparent wavelength is clearly infinite. However, if the wave is discretely sampled at a slightly longer period, a slow spatial oscillation appears. Since for smooth interfaces t is incremented by multiples of the spacer lattice constant a, if the crystal lattice is nearly commensurate with the 2kF wave, a long apparent period will be observed. Recent first-principles, self-consistent energy-band calculations 14 indicate that the exchange coupling between adjacent Co layers in idealized fee Co/Cu multilayers exhibits both long- and short-period oscillations. The coupling also depends on the crystallographic orientation of the interface. One can account for these results in terms 46

the interface. One can account for these results in terms of the prominent nesting vectors of bulk Cu, as shown in Fig. 1, where interzonal transitions q' = 2k|r —g are required to account qualitatively for the numerical results. The aliasing effect provides a natural explanation for the q' = 2 k f — g oscillations. In this paper we point out that in addition to the aliasing effect there exists another source of long-period oscillations, which we term the Bloch modulation effect. This arises when one goes beyond the plane-wave approximation for the basis states in calculating the R K K Y potential and uses the proper one-electron states in the presence of the lattice potential. The R K K Y exchange coupling between isolated spins or atomic moments embedded in a paramagnetic crystal-

FIG. 1. Model Fermi surface for fee Co/Cu multilayers, (a) Fermi surface and reduced zone of bulk fee Cu. (b) Crosssectional view of Fermi surface and reduced zone, (c) Fermi surface of Cu in extended k space showing formation of the dogbone orbits. The longest period for [001] oscillatory coupling arises from the transition vector q' = q —g, where q = k ' —k is a nesting vector and g a reciprocal lattice vector. 5806

©1992 The American Physical Society

394

HAI'II) ( t l . M M I N K A I I O V S

GENERALIZED RUDERMAN-KITTEL-KASUYA-YOSIDA THEORY .

46

line solid at positions R and R' (not necessarily lattice sites) arises from the spin polarization they induce in the solid. Initially, we neglect structural and chemical perturbations arising from the presence of the spin-bearing atoms and consider only the magnetic perturbations induced by the atomic moments m ( r ) . Accordingly, we represent the unperturbed electronic structure of the solid by Bloch functions ^ a ( k , r ) = u a (k,r)exp[i'k-r] and energy eigenvalues £„(k) in the reduced zone scheme of the spacer material, where a is a band index. According to second-order perturbation theory, the R R K Y interaction is proportional to Mao.(k,k',R)-Ma.a(k',k,R') n(R,R')= 2 — r ^ — 7 ^ • JZv E_(k)-E„.(k')

n(R,R')=

2

Mgg.(k,k',6R)-Ma.n(k',k,6R') E„(k)-E„.(k')

acr'kk'

(1)

5807

where the matrix elements are given by Mao.(k,k',R)=//^(k,r)m(r-R)Va.(k',rWr .

(2)

The sums on k and k' are limited to the reduced zone and are taken over occupied and unoccupied states, respectively; m(r) is the net spin density on a magnetic atom. From the Bloch periodicity condition ^ a ( k , r + d) = e x p [ i k - d ] ^ a ( k , r ) , where d is a direct lattice vector, it follows that M can be written in the form Maa.(k,k',R)=exp[-/(k-k')-R0]Moa.(k,k',SR)

(3)

where R 0 is the position of the lattice site in t h e cell containing R, and R = R 0 + 5R. Thus, the exchange coupling becomes

exp[-i(k-k')-(Ro-Ri)]

(4)

This formalism is readily extended to multilayers by summing the pair interactions between all magnetic atoms in successive magnetic slabs. 9 1 0 If the one-electron basis functions are represented by plane waves, the numerator in Eq. (4) reduces to | M ( k - k ' ) | 2 e x p [ - i ' ( k - k ' ) - ( 6 R - 8 R ' ) ] , where M ( q ) = / m ( r ) e x p [ - i q - r j d r . If the positions 6R, 6R' in t h e unit cell are held fixed, and if we allow |R 0 — R^| to change by considering a series of multilayers having progressively thicker spacers, the period of the oscillation appears to be determined by k —k', the intrazonal nesting vector. However, if g is a reciprocal lattice vector, we are free to include a factor of exp[ — / g - I R , ) — R Q ) ] = 1 inside the sum in Eq. (4) without changing fl(R,R'). One is also free to choose g to minimize |k — k' — g| and maximize the period. Thus, a shortwavelength oscillation, when sampled periodically at nearly the same wavelength, appears to have a long wavelength. This is the aliasing effect proposed by several a u t h o r s " ~ 1 3 to account for the observed long-period oscillations. In addition to the aliasing effect, there exists another source of long-period oscillations, the Bloch modulation effect, which arises when one goes beyond the plane-wave approximation for the basis states and uses the proper one-electron Bloch states in the presence of the lattice potential. To see this, we expand the periodic part of the Bloch function in reciprocal space: ua(k,T) = '£(g)Aga(k)exp[ig-T]. The matrix elements M then take the form

M ao .(k,k',5R)= 2

+ 2

A^k)Aga.{k')M(k-r)exp[i(k-k')-6R] ^ ; a ( k M g V ( k ' ) M ( k - k' + g - g ' ) e x p [ - i ( k - k ' + g - g ' ) - 6 R ]

From the vantage point of Eqs. (4) and (5), the aliasing effect arises from the product of the diagonal (g = g') terms in M a a . ( k , k ' , 6 R ) and M B . a ( k ' , k , 8 R ' ) . As already noted, 6R and 6R', must be held fixed, and the factor of unity, exp[ —ig-(R 0 —R^)] = 1, must be introduced in the numerator of Eq. (4). The Bloch modulation effect arises from all remaining terms, since all of these automatically include reciprocal lattice vector through the nondiagonal terms in g. In essence, the "magnetic" k —k' oscillation can be spatially modulated by the periodic lattice potential via g to produce a beat oscillation of small wave vector, k —k' —g, much as the heterodyne effect acts in the time domain. 1 5 Bearing in mind that the aliasing and Bloch modulation effects can give rise to the same set of long-period oscillations with small wave vectors k — k' — g, can we distinguish these two effects from one another apart from the fact that they arise from different terms in the complete expression for the exchange coupling? If one restricts one's attention to periodically sampling the com-

(5)

plete expression based on Bloch functions [cf. Eqs. (4) and (5)] at discrete lattice points, the A's only influence the amplitude of the oscillation. , 6 The distinction between these two effects manifests itself when one considers spins that do not lie on lattice sites, and when we average over random arrangements of such spins. In this context, let us consider multilayers whose interfaces are not perfectly lattice matched and atomically abrupt but are characterized by structural and chemical disorder. Such interfacial roughness is ill defined experimentally and may take many forms, depending on such factors as lattice mismatch, interfacial orientation, spacer polycrystallinity, and method of preparation. For purposes of discussion, we distinguish two types of interfacial disorder. In type (i) disorder, the atoms occupy random sites on one or more well-defined lattice planes in the interfacial region. In magnetic rare-earth multilayers, 17 for example, where the atomic radii of the magnetic and nonmagnetic atoms, M and A*, are nearly identical, the interface

395

HAI'IIHOMMl \1( VIIOVS

5808

FRANK HERMAN AND ROBERT SCHRIEFFER

46

sitions. If one now averages 8R over a region smaller than the long period, the amplitude remains large. Thus, the Bloch modulation effect is robust with regard to this type of randomness for both intraband and interband processes. The corresponding transitions between planewave states do not exhibit this robustness. The relative strengths of the leading g = g' and g ^ g ' terms in M lor transition or noble-metal spacer materials can be estimated by using parametrized band structure schemes." As will be shown in a subsequent paper, 20 the coupling strengths increase as one moves along a transition series toward increasing Z. This increase is related to the contraction of the outermost d orbitals, and is consistent with empirical observations. 2 While we have treated the spacer as a bulk crystalline material, boundary scattering at the spacer-magnetic film interfaces as well as defect scattering in the bulk will alter the detailed form of ft(R,R'). For example, if the spacer is treated as a free-standing film with rigid wall boundary conditions at r = 0 and f,20 the interaction Clmm is described by a m r o (R,R') = n ( R , R ' ) - n ( R , - R ' ) , where ft is given by Eq. (1) with k and k' restricted to values k =nir/t, n = 0 , 1 , 2 , . . . . When R and R' are near opposite interfaces, ft(R —R') is small compared to ft(R,R'). If disorder is present in the spacer, ft is reduced by a factor of exp[ — |R — R ' | / / ] , where / is the mean free path. In summary, we have shown that long-period oscillations in magnetic multilayers can be produced by the aliasing mechanism and also by the Bloch modulation mechanism. The fact that long-period oscillations are observed in multilayer systems having widely different degrees of lattice mismatch 1,2 —and by inference widely different types of interfacial disorder—can be explained by the existence of these two complementary RKKYtype mechanisms. We believe that earlier studies involving the interaction of magnetic impurities in nonmagnetic solids 21 and spinglasses 22 should be reexamined in the light of the generalized RICKY theory described here.

might consist of a single disordered MN plane, with nearly perfect M and N planes on either side. Broader interfaces could be produced by interdiffusion or by growth conditions that favor the formation of steps, terraces, and islands. In type (ii) disorder, the interfacial atoms are no longer preferentially located on well-defined lattice planes. Instead, these atoms are displaced from lattice sites by misfit dislocations, lattice mismatch, or other sources of severe local strain, e.g., interdiffusion of different size atoms such as Co and Ru. A possible analogy is a semicoherent grain boundary separating two differently oriented crystallites. If the out-of-plane displacements are very small, we revert to type (i) disorder and if sufficiently large, to type (ii) disorder. In the latter case, we assume continuous atomic site distributions in three dimensions within the interfacial region, which is of order of a few interatomic distances wide. According to the aliasing effect, interfacial roughness of type (i) has little influence on the coupling strength of long-period oscillations, as can be demonstrated by discrete spacer thickness averaging. I8 The essential point is that in type (i) disorder all spins on successive magnetic slabs are separated from one another by direct lattice vectors, even if they are distributed at random on the same or adjacent lattice planes. The aliasing effect occurs because the factor exp[ —ig-fRo—RJj)] is equal to unity for all pairs of spins. On the other hand, interfacial roughness of type (ii) suppresses the aliasing effect, wiping out the long-period oscillations arising from this mechanism. This can be seen by explicit calculation by averaging the positions of the spins over distances of order the size of the short (nesting) wavelength: Since |k —k'| is generally large for nesting conditions, averaging the g = g' terms in Eq. (S) over a distribution of SR values larger than the short period 2ir/|k —k'| leads to a small result. Clearly, the aliasing mechanism is no longer effective if the spin positions are not located on well-defined lattice planes or, more generally, at lattice sites. The exponential factor is no longer unity, since it includes a distribution of 6R and 6R' values. Turning to the nondiagonal ( g ^ g ' ) terms in Eq. (5), we note that |k —k' + g —g'| can be small for appropriate values of g and g'. This is true even if a = a', i.e., even if one is dealing with intraband rather than interband tran-

The authors wish to acknowledge stimulating discussions with many of their colleagues, particularly D. D. Chambliss, R. F. C. Farrow, B. A. Jones, and S. S. P. Parkin.

'S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 2304(1990). S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991). 3 Fe/Cr: J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. 67, 140 (1991); S. Demokritov, J. A. Wolf, and P. Grunberg, Europhys. Lett. 15, 881 (1991); S. T. Purcell et at, Phys. Rev. Lett. 67, 903 (1991). Fe/Al and Fe/Au: A. Fuss, S. Demokritov, P. Grunberg, and W. Zinn, J. Magn. Magn. Mater. 103, L221 (1992). 4 P. Grunberg, R. Schrieber, Y. Pan, M. B. Brodsky, and H. Owers, Phys. Rev. Lett. 57, 2442 (1986). 5 P. Etienne, J. Chazelas, G. Cruezet, A. Frederick, and J. Mas-

sies, J. Cryst. Growth 95, 410 (1989); M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988); G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 6 For extensive references, see Magnetic Thin Films, Multilayers, and Surfaces, edited by S. S. P. Parkin, H. Hopster, J.-P. Renard, T. Shinjo, and W. Zinn, Vol. 231 of the MRS Symposium Series (Materials Research Society, Pittsburgh, 1992). 7 M. A. Ruderman and C. Kittel, Phys. Rev. 87, 440 (19561; T. Kasuya, Prog. Theor. Phys. (Kyoto) 16, 45 (1956); K. Yosida, Phys. Rev. 106, 893 (1957).

2

396

46

GENERALIZED RUDERMAN-KITTEL-KASUYA-YOSIDA THEORY . . .

8

L. M. Roth, H. J. Zeiger, and T. A. Kaplan, Phys. Rev. 149, 519(1966). 9 C. Kittel, Solid State Phys. 22, 1 (1968). 10 Y. Yafet, J. Appl. Phys. 61, 4058 (1987); W. Baltensperger and J. S. Helman, Appl. Phys. Lett. 57, 2954 (1990); W. M. Fairbairn and S. Y. Yip, J. Phys. Condens. Matter 2, 4197 (1990). " D . M. Deaven, D. Rokhsar, and M. Johnson, Phys. Rev. B 44, 5977(1991). I2 C. Chappert and J. P. Renard, Europhys. Lett. 15, 553 (1991); P. Bruno and C. Chappert, Phys. Rev. Lett. 67, 1602 (1991). 13 R. Coehoorn, Phys. Rev. B 44,9331 (1991). ,4 F. Herman, J. Sticht, and M. Van Schilfgaarde, in Magnetic Thin Films, Multilayers, and Surfaces, Ref. 6, p. 195. See also J. Appl. Phys. 69, 4783 (1991). 15 The asymptotic form of the RKKY interaction comes from states near k f and — kF, where 2k f is the nesting vector. Using |i/_ k (r)| 2 =ju k (r)| 2 , the exchange interaction for the bulk is given by Xbuik
5809

film with sharp edges, we obtain Xtam^,t')=Xo^~r') —jfolr + r') if we use plane waves. If we use Bloch states instead, we obtain ^.. m (r,r , ) = [ ^ o ( r - r ' ) - ; 0 ( r + r ' ) ] | u k f ( r ) | 2 | u k j r ( r ' ) l 2 . For further details, see Ref. 20. If the Bloch functions are represented by orthogonalized plane waves, the A's are given in perturbation theory by v4=2K(g)/[£(k)-E(k' + g)]. Thus, the ^'s are large if k is near a Brillouin-zone boundary. But this is the condition that one has a long-wave oscillation, i.e., q=2k F — g. This shows that the A's are large for long-wavelength oscillations. "For recent reviews, see J. J. Rhyne, Phys. Scr. T 29, 31 (1989); also C. F. Majkrzak et al., Adv. Phys. 40, 99 (1991). 18 Y. Wang, P. M. Levy, and J. L. Fry, Phys. Rev. Lett. 65, 2732 (1990). " D . A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids (Plenum, New York, 1986). 20 R. Schrieffer and F. Herman (unpublished). 21 J. B. Boyce and C. P. Slichter, Phys. Rev. B 13, 39 (1976); Q. Zhang and P. M. Levy, Phys. Rev. 34, 1884 (1986). 22 K. Moorjani and J. M. D. Coey, Magnetic Glasses (Elsevier, Amsterdam, 1984). I6

VI Electrons and Phonons

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399

SCHRIEFFER'S W O R K ON T H E THEORY OF ELECTRONS A N D PHONONS

Hewlett-Packard 2

Department

Alexandre S. Alexandrov 1 ' 2 Laboratories, 1501 Page Mill Road, 1L-12, Palo Alto, CA 94304, USA of Physics, Loughborough University, Loughborough LE11 3TU, UK

The BCS theory of superconductivity [l] was originally derived on the basis of an early demonstration by Frohlich [2] that the conduction electrons could attract each other due to their interaction with vibrating ions of the crystal lattice. In a more complete analysis by Bardeen and Pines [3] in which Coulomb effects and collective plasma excitations were included, interaction between electrons and the phonon field was shown to dominate over the matrix element of the Coulomb interaction near the Fermi surface. The BCS theory predicted a non-Fermi-liquid ground state with gaped fermionic single-particle excitations. No boson excitations were present in the theory, other than phonons. The BCS quasiparticle approximation appeared to be surprisingly successful in explaining thermodynamic and electromagnetic properties of superconductors including the venerable Meissner-Ochsenfeld effect. The BCS derivation of the latter was criticized because it was not strictly gauge invariant. We know that almost every revolutionary theory breaks some old rules, but the lack of the gauge invariance was taken by Bob Schrieffer [4] and others (Anderson [5], Bogoliubov, Tolmachev, and Shirkov [6], Rickayzen [7]) quite seriously. Since Gor'kov [8] introduced the gauge-invariant Green's function formulation of the BCS theory, the problem was viewed as a technical one. Notwithstanding, efforts to resolve the 'gauge' problem resulted in a remarkable series of excellent papers on plasmons and phonons in superconductors and, more generally, on the collective excitations in solid-state plasmas. In their pioneering study ' Collective Behavior in Solid-State Plasmas' [9] David Pines and Bob Schrieffer presented a comprehensive analysis of the so-called two-stream instability (TSI) in semiconductors and semimetals, associated with the acoustic plasmon (AP). TSI were known in hot plasma physics. When electrons and ions flow in opposite directions, ionic sound waves can develop as was predicted by Tonks and Langmuir in 1929 [10]. The AP corresponds to low-frequency oscillations in which positive and negative charges move in phase with one another, in contrast with the usual optical high-frequency plasmon in which the electrons and holes oscillate out of phase. TSI and AP were observed about forty years ago in the nondegenerate high-temperature electron-ion plasma where Landau damping was suppressed due to a large difference in the electron and ion temperatures (Boyd, Field, and Gould [ll], Alexeff and Neidigh [12]). However, relatively little was known about AP and TSI in the electron-hole plasma. On the one hand, the electron-hole plasma possesses an obvious advantage compared with the hot electron-ion plasma because the concentrations and temperatures of two components can be varied over a substantial range. But on the other hand, the AP mode is Landau damped because it lies in the single-particle continuum of the faster moving electrons and holes, which are scattered by phonons and impurities.

400

To study the electron-hole TSI Pines and Schrieffer used an elegant approach, which was different but equivalent to the familiar random phase approximation for the dielectric response function. They made use of the two Boltzmann equations for electrons and holes coupled by a self-consistent electric field, including the scattering in r-approximation, and the drift velocities (replacing the temperatures of each component by complex temperatures). Increments of AP instabilities were calculated in a straightforward manner for any unequal temperatures and densities of the components. TSI in semiconductors was found to be marginal due to relatively short relaxation times. A much better possibility for producing the instability was identified for semimetals such as bismuth. The paper ignited a long-lasting experimental and theoretical interest in acoustic modes and instabilities of the solid-state plasmas. In particular, AP was proposed as a possible source of incoherent light scattering in semiconductors (Platzman [13]) and as a mediator of the attractive interaction in some superconductors. Certain low-frequency features in the light scattering spectra of photoexcited electron-hole plasmas in undoped GaAs were quoted as evidence of AP (Pinczuk, Shah, and Wolff [14]). Recent developments in fabrication techniques have made it possible to create two and one-dimensional electron-hole plasmas. In contrast to higher dimensional systems where the AP mode is often Landau damped, the low-dimensional AP mode is invariably undamped at long wavelength (Das Sarma and Hwang [15] and references therein). The possibility of creating TSI in low-dimensional structures by driving two components in the opposite directions has been studied by several authors, and a few promising directions in the search for TSI have been proposed recently. The second paper of this chapter ' Coupled Electron-Phonon System'' by Stanley Engelsberg and Bob Schrieffer [16] applied field theoretical methods to a many-body problem of interacting electrons and optical phonons. It extended the Migdal theory [17] of electrons coupled to Debye (acoustic) phonons towards coupling to Einstein (optical) phonons. At that time many experiments on many-body systems were interpreted within the framework of the Landau quasiparticle picture of the Fermi liquid. Landau pointed out that the low-lying excited states of a fermionic system had a one-to-one correspondence with the low-lying states of the ideal Fermi gas. Coupled electron-phonon systems in metals and doped semiconductors provided an excellent test of the phenomenological Landau theory. Migdal showed that the contribution of the diagrams with 'crossing' phonon lines (the so called 'vertex' corrections) was small in case of interaction with acoustic phonons. Because the frequency of acoustic phonons sq remains much lower than the 'frequency' of electronhole excitations vq, the first correction to the electron-phonon vertex turns out to be of the order of the sound velocity s divided by the Fermi velocity v, for any wave vector q. The Migdal 'theorem', which states that the vertex corrections are small, is crucial in deriving the closed set of equations for the electron and phonon self-energies. The damping of renormalized (quasi)particles was found to be cubic in energy and hence negligible, in agreement with the Landau theory. Engelsberg and Schrieffer noticed that the Migdal theorem did not hold for the optical phonons in the long-wavelength limit, where the phonon phase velocity is of the order or

401 greater than the Fermi velocity. To deal with the 'vertex corrections', they generalized the Ward identity and calculated for the first time the spectral function, the renormalized energy spectra, and the irreducible polarizability (including the vertex corrections) of electrons coupled with optical phonons. One of the findings was a pronounced incoherent continuum in the spectral function representing a 'dressed' electron and excited phonons. They also carefully analyzed the electron spectrum of the Debye model, confirming the cubic damping rate of low-energy fermionic excitations. The technique of this frequently cited paper has since been applied not only to coupled electrons and optical phonons but also to coupled single particle and collective excitations in 3He (Engelsberg and Platzman [18]) and to other systems. The problem of vertex corrections appeared to be important in superconducting semiconductors (Koonce and Cohen [19]) and in excitonic superconductors (Davis, Gutfreund, and Little [20], Rietschel and Sham [21]). In recent years it has become one of the most challenging problems in various theories of novel high-temperature superconductors, where the carrier kinetic energy is reduced and optical vibrations are hard. In the strong-coupling regime the problem was solved using the canonical Lang-Firsov transformation predicting the bipolaronic superconductivity (Alexandrov and Ranninger [22]) and a softening of phonons (Alexandrov [23]). More recently the vertex corrections in the intermediate region of parameters have been attracting a lot of attention. The first corrections (Kostur and Mitrovic' [24], Pietronero, Strassler and Grimaldi [25]) were calculated and different self-consistent methods involving the Ward identity (for example, Cai, Lei and Xie [26], Takada [27], Bang [28], Cosenza, De Cesare, and Girard [29]) were developed with somewhat conflicting conclusions about a role of the corrections in the enhancement of the superconducting critical temperature. The vertex corrections have been found even more important in some non-phononic mechanisms of high-temperature superconductivity (Schrieffer [30]). The concluding paper of the chapter by Bob Schrieffer and myself [3l] corrects an old mistake in the calculation of the irreducible polarizability of electrons coupled with Einstein phonons. Expanding the electron-phonon vertex function in powers of the phonon momentum and then integrating it over frequency and momentum of the polarization loop leads to a pole in the polarizability as a function of frequency. As a result, one would expect two branches of optical vibrations splitting with the increasing momentum transferred to the lattice. In our paper we show that, in fact, there are no poles in the polarizability, if the integration is performed first and the momentum expansion second. Within the model there appears only a hardening of the renormalized optical phonons but no splitting. The hardening contrasts with the Migdal softening of acoustic phonons. This behavior was explained as the result of the familiar repulsion of energy levels under a perturbation. The acoustic phonon frequency sq is well below the characteristic frequency vq of the electronhole continuum, so that if these two excitations are coupled the lowest 'acoustic level' shifts downwards in the absence of any other collective modes. On the contrary, the upper optical phonon level shifts upwards because its energy is above the electron-hole continuum in the long-wavelength limit. Inclusion of the Coulomb repulsion between the electrons introduces

402

another collective excitation, the electron plasmon. When its frequency is higher than the optical phonon frequency, the latter is 'repelled' downwards, and softens due to dynamic screening (Reizer [32]). Otherwise, it is hardened due to the coupling with electrons. The theory of electrons coupled with optical phonons has been applied to the interpretation of different experiments, in particular, of the phonon spectra in Os metal (Ponosov, Bolotin, Thomsen, and Cardona [33]) and photoemission spectra in high-temperature superconductors (Varelogiannis [34]), in Ga and Be semimetals (Hofmann, Cai, Grutter, and Bilgram [35], Hengsberger, Purdie, Segovia, Garnier, and Baer [36], LaShell, Jensen and Balasubramanian [37]) and in the Chevrel-phase compounds (Kobayashi, Fujimori, Ohtani, Dasgupta, Jepsen, and Andersen [38]).

References J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).* H. Frohlich, Phys. Rev. 79, 845 (1950). J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955). A. Bardasis and J. R. Schrieffer, Phys. Rev. 121, 1050 (1961).* P. W. Anderson, Phys. Rev. 112, 1900 (1958). N. N. Bogoliubov, V. V. Tolmachev, and D. V. Shirkov, A New Method in the Theory of Superconductivity (Consultants Bureau, Inc., New York, 1959). G. Rickayzen, Phys. Rev. 115, 795 (1959). L. P. Gor'kov, Zh. Eksp. Teor. Fiz. 34, 735 (1958) [Sov. Phys. JETP 7, 505 (1958]; Zh. Eksp. Teor. Fiz. 36, 1918 (1959) [Sov. Phys. JETP 9, 1364 (1959)]. D. Pines and J. R. Schrieffer, Phys. Rev. 124, 1387 (1961).* L. Tonks and I. Langmuir, Phys. Rev. 33, 195 (1929). G. D. Boyd, L. M. Field, and R. Gould, Phys. Rev. 109, 1393 (1958). I. Alexeff and R. V. Neidigh, Phys. Rev. 129, 516 (1963). P. M. Platzman, Phys. Rev. 139, A379 (1965). A. Pinczuk, J. Shah, and P. A. Wolff, Phys. Rev. Lett. 47, 1487 (1981). S. Das Sarma and E. H. Hwang, Phys. Rev. B 59, 10730 (1999). S. Engelsberg and J. R. Schrieffer, Phys. Rev. 131, 993 (1963).* A. B. Migdal, Zh. Eksp. Teor. Fiz. 34, 1438 (1958) [Sov. Phys. JETP 7, 996 (1958)]. S. Engelsberg and P. M. Platzman, Phys. Rev. 148, 103 (1966). C. S. Koonce and M. L. Cohen, Phys. Rev. 177, 707 (1969). D. Davis, H. Gutfreund, and W. A. Little, Phys. Rev. B 13, 4766 (1976). H. Rietschel and L. J. Sham, Phys. Rev. B 28, 5100 (1983). A. S. Alexandrov and J. Ranninger, Phys. Rev. B 23, 1796 (1981). A. S. Alexandrov, Phys. Rev. B 46, 2838 (1992). V. N. Kostur and B. Mitrovic, Phys. Rev. B 48, 16388 (1993). L. Pietronero, S. Strassler, and C. Grimaldi, Phys. Rev. B 52, 10516 (1995). J. Cai, X. L. Lei, and L. M. Xie, Phys. Rev. B 39, 11618 (1989).

403

[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

Y. Takada, J. Phys. Chem. Solids 54, 1779 (1993); Phys. Rev. B 52, 12708 (1995). Y. Bang, Phys. Rev. B 60, 7458 (1999). F. Cosenza, L. De Cesare, and M. F. Girard, Phys. Rev. B 59, 3349 (1999). J. R. Schrieffer, J. Low Temp. Phys. 99, 397 (1995).* A. S. Alexandrov and J. R. Schrieffer, Phys. Rev. B 56, 13731 (1997).* M. Reizer, Phys. Rev. B 61, 40 (2000). Y. S. Ponosov, G. A. Bolotin, C. Thomsen, M. Cardona, Phys. Status Solidi B 208, 257 (1998). G. Varelogiannis, Phys. Rev. B 51, 1381 (1995). Ph. Hofmann, Y. Q. Cai, Ch. Grutter, and J. H. Bilgram, Phys. Rev. Lett. 8 1 , 1670 (1998). M. Hengsberger, D. Purdie, P. Segovia, M. Gamier, and Y. Baer, Phys. Rev. Lett. 83, 592 (1999). S. LaShell, E. Jensen, and T. Balasubramanian, Phys. Rev. B 61, 2371 (2000). K. Kobayashi, A. Fujimori, T. Ohtani, I. Dasgupta, O. Jepsen, and O. K. Andersen, Phys. Rev. B 63, 195109 (2001). (The symbol * indicates a paper reprinted in this volume.)

404 PHYSICAL

REVIEW

VOLUME

124, N U M B E R 5

DECEMBER

1, 1961

Collective Behavior in Solid-State Plasmas* DAVID PiNEsf AND J. ROBERT ScHRiEFFERf

John Jay Hopkins Laboratory for Pure and Applied Science, General Atomic, Division of General Dynamics Corporation, San Diego, California (Received July 27, 1961) The conditions for the existence of plasma wave instabilities in the plasma formed by the electrons and holes in semiconductors are discussed. The dispersion relations for both the high-frequency optical mode in which electrons and holes move out of phase, and the low-frequency acoustic mode in which electrons and holes move in phase are calculated. Growing acoustic waves are shown to occur for a sufficiently large relative drift velocity of the electrons and holes, and the boundary between growing and damped waves is determined for various electron-hole temperature ratios. Growth rates are calculated for several cases of interest; when the influence of impurity and phonon scattering on the electron-hole behavior is taken into account it is concluded that InSb is perhaps

T

H I S paper is devoted mainly to the theoretical investigation of certain aspects of collective behavior in the "classical" plasma formed by electrons and holes in semiconductors at n o t too high carrier densities and not too low temperatures. The extension of the ideas developed herein to the quantum electronhole plasma found in semimetals or certain semiconductors is discussed in the Appendix. In general, there may exist two modes of collective oscillation for a two-component plasma, such as the classical plasma of electrons and ions or the electronhole plasma in a semiconductor. One mode consists of a high-frequency oscillation in which the electrons and holes oscillate out of phase with one another; the frequency &)i of a long-wavelength oscillation is Wl2 = W + 2 +&)_ 2 ,

where w + and u>_ are the electron and hole plasma fre" quencies, respectively. The other mode corresponds t o a low-frequency oscillation in which the holes (assumed to be heavy) and electrons (assumed to be light) move in phase with one another. I t is thus a plasma oscillation appropriate to the holes plus their associated screening cloud of electrons; for equal densities of holes and electrons the frequency of a long-wavelength oscillation of wave number k is W2^(r_/2r+)%+=

(m-/2m+ykv-,

provided the ratio of electron temperature to hole temperature, 7 1 / T + , is large enough, and the electronhole mass ratio, m-/m+, is small enough. These are just the requirements that the low-frequency mode possess a frequency which is distinct from typical individual hole and electron excitation frequencies, kv+ and &i>_. * This work was supported by a joint General Atomic-Texas Atomic Energy Research Foundation program on controlled thermonuclear reactions. t Present address: Department of Physics, University of Illinois, Urbana, Illinois.

the most promising semiconductor in which to observe such instabilities. An investigation of the hole and electron temperatures and the relative electron-hole drift velocity as a function of field strength is carried out for InSb. It is shown that moderate field strengths (~100 v/cm) suffice to produce electron-hole drifts of the required order of magnitude for the observation of plasma wave instability; however, the scattering mechanisms present are sufficiently effective that it appears marginal whether the other condition (long hole relaxation times) necessary for the observation of the plasma wave instability is achievable in practice. In an Appendix the conditions for the occurrence of similar plasma wave instabilities in semimetals are analyzed briefly. Here zi+ and »_ are denned by (m-vJ/2)=KT; («t + u + 2 /2) = KT+, and the foregoing requirements are necessary in order that the collective mode not be too strongly damped by the individual particle excitations. B y analogy to t h e vibration spectrum of polar crystals, we may call the high-frequency mode an optical mode of plasma oscillation, the low-frequency mode an acoustic plasma mode. We shall be particularly concerned with the possible existence of a two-stream instability in the semiconductor associated with the electron drift under t h e application of an electric field. Our motivation for this study is the obvious need for an understanding, both theoretical and experimental, of the high-frequency instabilities in fully ionized plasmas. From a theoretical point of view, the conditions for the existence of certain classes of such instabilities (and in particular for the two-stream instability in a homogeneous plasma) are well understood; the calculations of the growth rate of the instability for short times (where the linear approximation is valid) are also reliable. However, once the amplitude of the growing plasma oscillation becomes sufficiently large that nonlinear effects (such as the coupling between plasma modes of different wavelengths) begin to play a role, comparatively little is known of the resulting behavior of the plasma. T h e experimental situation is even less satisfactory. T h e unambiguous observation of the two-stream instability has proved an extremely challenging problem for t h e plasma experimentalist, and only recently has a measure of success been achieved in its detection. 1 I t seems not unlikely that quantitative experiments which bear on the above nonlinear aspects will prove equally difficult to carry out. T h e electron-hole plasma in a semiconductor would seem to offer a promising tool for the investigation of such instabilities. I t possesses the obvious advantage t h a t the relative concentrations and temperatures of

1 G. D. Boyd, L. M. Field, and R, Gould, Phys. Rev. 109, 1393 (1958). 1387

405 1388

D.

PINES

AND

J

the electrons and holes can be measured and varied over quite a substantial range of interest. The principal disadvantage is that the electrons and holes are scattered by phonons, impurities, and, in some cases, one another; hence one is hampered by the need to find temperatures and concentrations such that O J T ± » 1 , where 0 is the characteristic frequency one wishes to study and T + and r_ represent the hole and electron lifetimes against the extraneous scattering mechanisms. As we shall see, therefore, the possibility of achieving conditions for observing the two-stream instability appears somewhat touch and go for semiconductors, where one has to contend with lifetimes =10~ 1 2 sec. The prospect may be brighter in semimetals, where the lifetimes are S 1 0 - 1 0 sec or perhaps an order of magnitude longer for low field strengths; however, interband transitions may act to reduce these lifetimes for a large applied field. In a two-component plasma one finds that for a sufficiently large drift velocity of electrons vs ions (or holes) the plasma becomes unstable against a growing wave of plasma oscillation, corresponding to a coherent excitation of the oscillations by the electron beam. The conditions for the existence of this instability and its growth rate have been studied by Rosenbluth, 2 Buneman, 3 and Jackson 4 for a plasma of electrons and ions at equal temperatures. They find that the instability comes into play for an electron drift velocity Vi% 1.32D_. When the electron-ion temperature ratio, T L / r + , is sufficiently large the critical drift velocity required to produce an instability is reduced, being of order (w_/w + ) ! ti_ or smaller. 5 Drift velocities of this latter order seem definitely achievable in high-mobility semiconductors; the first problem, then, in producing a two-stream instability in a semiconductor is that of achieving a sufficiently large electron-hole temperature ratio. InSb appears a likely material, because the electronhole mass ratio is large, (~ 14) and the electron mobility is quite high (~10 6 cm 2 /v). The high electron mobility means that the application of modest electric field strengths will act to produce substantial deviations from Ohm's law, with appreciable heating of the electrons. On the other hand, the large value of m+/mmeans that the hole mobility is an order of magnitude smaller than the electron mobility; as a result one gets much less hole heating than electron heating for a given field strength, so that a large value of m+/m- tends to favor a large value of T-/T+. This aspect of semiconductor behavior favors the existence of acoustic plasma oscillations and the existence of a lower threshold for the observation of the two-stream instability. 2

M. Rosenbluth (private communication). ' 0. Buneman, Phys. Rev. Letters 1, 8 (1958); Phys. Rev. US, 503 (1959). * J. D. Jackson, J. Nuclear Energy Pt. C: 1, 171 (1960). *M. Rosenbluth (private communication); I. Berstein, E. A. Frieman, R. M. Kulsrud, and M. N. Rosenbluth, Phys. Fluids 3, 136 (1960).

R.

SCHRIEFFER

As we have mentioned, the principle obstacle to carrying out such observations is the incoherent scattering of the electrons and holes. T h e instability will occur only if the product of its growth rate (neglecting particle relaxation times), Cle, and the relevant electronor hole-scattering lifetime, r ± , is greater t h a n unity. The growth rates we calculate for InSb under favorable circumstances are of the order of 01+/10; the reciprocal of the hole lifetime (which is the relevant one here) is of this same order, so that the production and observation of the growth of acoustic plasma oscillations may or may not be feasible for this material. In Sec. II we discuss the dispersion relation for both high- and low-frequency modes in a system of two coupled plasmas a t rest having different masses, densities, and temperatures. The treatment is extended to include the effect of directed particle drift velocities and the scattering of the carriers of the crystal lattice. The effect of an external electric field on the dispersion relation is also included. The conditions under which the electron drift may be just sufficient to excite a growing acoustic wave instability are treated in Sec. I I I . The growth rate of the oscillations is discussed in Sec. IV. In Sec. V we discuss the semiconductor aspects of the problem with particular reference to the magnitude of the particle drift velocities and temperatures one can expect in moderate electric fields. In Sec. VI a brief discussion of the possibility of observing the two-stream instability by pulsed conductivity measurements is given. II We consider an idealized situation in which there are n+ holes and «_ electrons per unit volume distributed uniformly over a large sample. The holes are assumed to be free particles of effective mass m + ; the electrons likewise possess an effective mass m— For the low frequencies of interest to us here the interaction between the charged particles is well described b y #/eo\ r , ~ r,-|, where e0 is the static dielectric constant of the semiconductor. The coupled electron-hole plasma oscillations of this system have been studied using the collective variables approach and the random phase approximation by Nozieres and Pines. 6 In this report we shall use an alternative approach; we work with the Boltzmann equation for the one-particle distribution functions, /±(r,v,2), and take into account the influence of charged particle interaction by means of a selfconsistent field.7 The equivalence of the collectivevariables random phase approximation approach with use of the collisionless Boltzmann equation plus the self-consistent field is by now well understood. 9 6 D. Pines, Can. J. Phys. 34, 1379 (1956); P. Nozieres and D. Pines, Phys. Rev. 109,1062 (1958); P. Nozieres, Ann. phys. 4, 8657 (1959). A. Vlasov, J. Phys. U.S.S.R. 9, 25, 130 (1945); L. D. Landau, ibid. 10, 25 (1946). 8 J. Goldstone and K. Gottfried, Nuovo cimento 13, 849 (1959); M. Cohen and H. Ehrenreich, Phys. Rev. 115, 786 (1959); D. Pines, J. Nuclear Energy Pt. C: 2, 5 (1960).

406 COLLECTIVE

BEHAVIOR

IN

SOLID-STATE

The random phase approximation is valid when the average potential energy per electron is small compared to the average kinetic energy. Thus our treatment will be valid when e2k± -<1

or

1<)KT±

/4irn±e2\' f — ] \ (OKT+

<1,

(2.2) 16

For electron and hole densities of the order of 10 , the weak coupling condition (2.1) is satisfied for temperatures in excess of 20°K for InSb (eo=16). Therefore, we shall see the weak coupling condition is satisfied for the experimental condition of interest to us in what follows. The time evolution of the distribution function of a two-component plasma in the presence of an external electric field and an external scattering mechanism is extremely complicated. No general solution of this problem has thus far been obtained. Our treatment is based on the assumption that there are two reasonably distinct time scales which characterize the change in the distribution function. The first time scale is associated with the macroscopic drift of the particles induced by the external field. The second time scale, which we assume to be short compared to the first, is that which characterizes the coherent behavior of the system; that is, the collective oscillations and their growth for sufficiently large relative hole-electron drift velocity. Thus we regard the drift velocity of the particles as changing adiabatically with regard to the times characteristic of plasma effects. Consider, for the moment, the collisionless Boltzmann equation for the distribution functions / ± ( r , v , / ) in the absence of an external electric field: a/±/d<+v-V/±±(«/»»±)E.V,/±=0.

-*"/

<*v(/+-/-).

(2.4)

Fourier transform in space and time of (2.3) and (2.4); one then arrives at the dispersion relation for plasma oscillations: Awe1 1+-

k v »/o+

/ *

o>— k-v+*5

+-

4«2

r

\d\

£„*%. J

k -i k-V./o-

-=0. c o - k v+id

(2.6)

The small imaginary part ib arises from the choice of a boundary condition corresponding to a retarded solution of (2.3) and (2.4). Such a solution is valid only for ImoiJjO. For lmw<0, it is necessary to analytically continue the functions of a appearing here; this may be accomplished by the choice of the contour of integration indicated in Fig. 1. This prescription is equivalent to the result obtained by Landau using Laplace transforms and treating the problem as an initial value one; it leads to damping of the plasma oscillations by the individual electron excitations in the absence of directed motion of the electrons. It is convenient to write the dispersion relation (2.6) in the following form: \+—W{ k>

— )+—W[—1=0, \kv+J

1

k

(2.7)

\kvJ

where the response function, W, is given b y 1

W(Z) = lim MO

/•/ dq (2»)V-^

""©'

qexp(-
Zexp(-Z2/2) + l

The dispersion relation for the collective modes of the system is determined by solving (2.3) and (2.4) simultaneously; this may be done easily if one assumes the departures of / ± from their respective Maxwellian values, /o±, are small. Thus one writes /±-/o±+/i±

R»v,

FIG. 1. Contour of integration for evaluating integrals appearing in (2.6); »z is k- v/k.

(2.3)

In (2.3), the electric field E is that arising from the averaged field of the charged particles, and is determined by Poisson's equation: «oV-E

-&r*

(2.1)

4JT«±

4m±e?/t<jKT±.

1389

IfflV,

1

where k± are the hole and electron Debye screening wave vectors denned by k±2 —

PLASMAS

(2.5)

and linearizes the resulting equations by neglecting the cross-terms E-V»/i ± as being of higher order. The solution is perhaps most easily obtained by taking the

Z e x p ( - Z 2 / 2» ) // d 9 exp(g»/2).

(2.8)

Jo

The result (2.8) follows after substitution of the Maxwellian values, /o ± = a± exp( — mrf/luT^ in (2.6) and a certain amount of straightforward algebra. As we shall see, the W function plays a central role in the discussion of dynamics of plasma oscillations. It has the following expansions for large and small

407 1390

D.

PINES

A N D J.

argument: Z»l: 1

W(Z) = »7r»Zexp(--Z 2 )

3 2

15 4

2Z

4Z

8Z6 (2n+l)l 2"Z2"

Z«l:

(2.9)

W(Z)=iw'>Z exp(--Z 2 )+l 2 4 (-2)"Z 2 »- 2 Z 2 1-—Z 2 -)-—Z'+ • • (2«+l)!U 3 15

(2.10)

where ( 2 « - f l ) ! ! = l X 3 X 5 - • - X ( 2 n + 1 ) . T h e highfrequency plasma oscillation solution of (2.7), coi, is obtained by replacing the W functions by their highfrequency expansions; one obtains 1=

o)+2 « _ ' 1a>2 oj2

rkJ as tir ! expi L k* kv.. \ A+2 +

JfcVV

u>

/ exp W kv+ \

«2 \ - | , kWJA

(2.11)

where the plasma frequency for each type of carrier is defined by a)±2 = 4ir« :t e 2 /m :t ee. (2-12)

R.

SCHRIEFFER

T h e physical origin of the damping is simply understood by considering particles whose velocity is approximately equal to the phase velocity of the wave, vp=a/k. If the wave has sufficiently large amplitude, particles with velocities slightly greater than vv will be trapped in a potential trough and decrease their average velocity to vT transferring their extra kinetic energy to the wave. Particles with velocities slightly less than vp will be accelerated by this trapping mechanism and absorb energy from the wave. Therefore if the distribution function decreases with increasing velocity, damping takes place since there will be more slow particles to absorb energy than fast particles to transfer energy to the wave. If, on the other hand, the distribution function increases with velocity near vP it is possible that the wave will grow in amplitude. The low-frequency mode of the electron-hole system takes on a simple form if the phase velocity of the wave is large compared to the hole thermal velocity and small compared to the electron thermal velocity. I n this case the small- and large-Z approximations can be made in the dispersion relation for the electrons and holes, respectively. T h e dispersion relation then becomes co + 7 3**11+.* \ 1=—(H +•••) 2 2 a> \ 2 a. /

kJ *2

a kJ kJ w V / —-H expf kv-

oP- \ )

For long wavelengths (kv-
(2.15)

^ + -01 (

since in this limit the damping of the wave due to individual electron and hole excitation is slight. The real part of the frequency is, as we have indicated,

[

4re2/n+ «—(—-1— to \m+

nt-

)]'

(2.14)

Thus if »+. = »_, the plasma frequency of the coupled system is given by the plasma frequency appropriate to the reduced mass, 1

1

() which is to be expected since the holes and electrons are moving completely out of phase with respect to each other.

C02

=(

+—)*

(2.16)

(2.17)

We see directly from (2.16) that the damping of the long-wavelength acoustic wave will be small only if the following conditions are satisfied:

w+.

«1, ra_

»1. T+ n_

(2.18)

These are just the conditions, ui22»£2u+2, which are necessary in order that we be justified in our use of the low-frequency and high-frequency expansions, respectively, of the electron and hole response functions, W, in the long-wavelength limit. Thus where the frequency of the acoustic wave satisfies the condition *2»+.2««22«*2t)_2,

(2.19)

408 COLLECTIVE

BEHAVIOR

IN

the damping of the wave will be small, and one finds a simple analytic form (2.17) for w2; where (2.19) is not satisfied, one must use the exact values of the W functions, and the damping is appreciable. We now consider the behavior of the system if the holes and electrons possess net drift velocities IM+ and Vd~, respectively. If collisions between the particles are more efficient in relaxing the momentum and energy distributions of the holes and electrons than their interactions with the lattice, then the distribution functions in the presence of an external electric field, £o, will take the form of displaced Maxwellians : / ± (A'o) = a ± exp[—m±(v — ti
(2.20)

The temperatures of the holes and electrons may differ from the lattice temperature as a result of the energy absorbed from the external field. With the aid of (2.20) it is easy to show that the dispersion relation for plasma oscillations is given by a modified form of (2.7), in which Z± = (<x> — k • Vd±)/kv± replaces the arguments w/kv± appearing there. This result follows simply by noting that the integration over velocity in (2.6) can be reduced to that for Vd = 0 by the change of variables v—> v + v ^ , the only change being that w is replaced by the Doppler shifted value a—k-v,j. As we discussed above, the Landau damping term depends upon the sign of o>—k- \d- If co— k- Vd<0 the Landau prescription for treating the pole in the integrand tends to make the wave grow rather than decay. This result follows simply from the fact that for vp=a/k while for u/k>Vd the maximum of the Maxwellian has been passed and damping results, in agreement with the physical description of the effect given earlier. The growth of oscillations due to the relative drift of the particles is known as the two-stream instability; a study of this instability in semiconductors will form the major part of this paper. The particle-lattice interactions can be included in the Boltzmann equation by a relaxation time approximation in the case of impurity scattering while, in general, the phonon scattering requires a more detailed treatment. If one uses a collision time approximation,

(a//aO-u=-(/-/o)/rW,

(2.21)

then the only effect of the collisions on the dispersion relation is to replace u/kv± by [u-\-i/T±{v)~\/kv±. It is clear that the collision term leads to damping of the oscillation since if w is satisfied the dispersion relation with T + = T _ = « > , then u—i/r satisfies the dispersion relation when T + = T _ = T . The external field Eo which produces the drift velocities v
SOLID-STATE

PLASMAS

— )}x=±e dvz'

1391

v./o±,

(2.22)

tn±

where E = E o + E , . I t is straightforward to solve (2.22) for / ± and combine the result with Poisson's equation, i k E s = 47re /

(/1+-/i-)<^'>

(2.23)

to obtain the modified dispersion relation. T h e result is equivalent to replacing the temperatures T± by complex temperatures,

zv=r.

1=F-

j e iEo io

k (2.24)

k\T±

in both W and &±2. The external electric field can therefore have a strong effect on the dispersion relation if a particle gains an energy from the external field which is large compared to KT while moving a distance of one wavelength. The sign of the effect depends upon both the charge of the particle and the angle between E 0 and k. This correction to the dispersion relation vanishes for Z5>\ since in this limit thermal effects are negligible compared to potential energy effects. Ill In order to obtain a feeling for the conditions under which an instability corresponding to growing waves of frequency u will occur it is convenient to calculate the wave number k for which lmoj = 0, as a function of the drift velocity of the electrons. This yields the boundary between the growing waves and the damped waves for a given drift velocity. We choose a coordinate system in which the holes have zero drift velocity a n d neglect the damping due to particle-lattice interactions and the external field Eo. The basic equation we wish to investigate is, therefore, from (2.7) and (2.20) 1+—W(Z+)-

-W(ZJ) = 0,

(3.1)

P

where Z+ = u/kv+;

Z_ = (o) — k • \d)/kv—

Since there are a large number of parameters to be specified, i.e., m±, T±, n±, and Vd, we choose some typical cases of interest to s t u d y :

1. n+ = ti-, r + =r_ ; 2. r+=o, T_>0; 3. n+=w_, r_=ior+ 4. » + = » - , r_=4r +; 5. re+ = 50n_, r_=4r+.

409 1392

PINES

D

2.0

/

AND J.

R.

SCHRIEFFER

The relation Z+=—Z- gives

\

—

V

r,_;VT-

— Vn.;V0

\

—

V

\

(3.6)

n+.n_;T..|0T+

(m+/mJ)*+l

n • n_ ;T_« 4T.

Thus for long wavelengths and low velocities, the waves have a dispersion law corresponding to phonons, that is, a) is proportional to h. For high velocities, (3.5) and (3.6) lead to u=v2u+. (3.7) Thus, we see that if m+2>m^ the unstable oscillations behave much like the low-frequency acoustic oscillations in which the holes and electrons move essentially in phase with each other. In case 2, because r + = 0 the hole response function may be approximated by its high-frequency limit while the electrons may be treated in the low-frequency approximation at long wavelengths. The real part of the dispersion relation (3.1) becomes

i=(o,+y
•u+s=s{k)k;

(3.8) (3.9)

V„/V. FIG. 2. The boundary between growing waves and damped waves for an electron-hole plasma with «+ = «_, m + —14w_, and varying values of T-/T+.

the imaginary part gives Z_ = 0 or

In all of these cases we keep the mass ratio, m+/pt-, fixed at 14. The mass ratio of 14 is appropriate to the heavy holes and low-energy electrons in InSb, which is likely to be one of the most favorable materials for observing this instability. Case 1 has been investigated by Buneman3 and Jackson.4 Under the condition Imu=0, Z+, and Z_ are real and therefore the imaginary part of (3.1) reduces to

On combining (3.9) and (3.10) we see that the relation between k and »d_ for which oscillations are just stable is given by

2

2

Z+exp(-Z+ )=-Z_exp(-Z_ ), which may be satisfied by Z+= — ZsZ. of (3.1) becomes

(3.2)

The real part

1 = 2 — / - 1 + 22 e x p ( - Z 2 ) ( exp(/2)rfA

(3.i)

The relation between k and v* is plotted in Fig. 2. The critical drift velocity for which infinitely long wavelengths just begin to grow is given by Brf=0.926»_[l+ (»L_/m+)*].

(3.4)

As the drift velocity increases shorter wavelength oscillations will grow. This condition persists until one attains the velocity l.S2v-£l + (m-/m+)i2; at this value of Vd the maximum range of wave numbers is unstable; the maximum wave number of a growing wave is given by 0.722£_. For very large velocities, the maximum wave number for which plasma oscillations can grow is given by k=V2(a,_/tv)[l+ ( » > + ) i ] . (3.5)

(3.10)

w=k-Vrf.

vd=

<"+

.

(3.11)

Thus, for sufficiently large k, unstable acoustic plasma oscillations can be excited by electrons with a vanishing small drift velocity. For vd>u+/k^ the spectrum of growing waves extends down to k = 0. The boundary given by (3.11) is likewise plotted in Fig. 2. The distinctly different behavior between cases 1 and 2 should be noted. The essential reason for the great difference in the boundaries is the lack of Landau damping for the holes in case 2. This has the effect of allowing growth of oscillations whenever the drift velocity of the electrons is greater than the sound velocity s(k) for these oscillations. Since s(k) decreases as l/k for large k, the condition for growing waves can always be satisfied for sufficiently large k. We conclude that if the temperature of the holes can be made very small, the critical velocity for excitation of plasma oscillations can be appreciably lowered. Thus, it is desirable to obtain a material with a high rate of energy loss for one type of carrier, e.g., the holes, so that their temperature can be maintained at a low value. Also these carriers should have a large effective mass since from (3.11) the minimum drift velocity for a given wave to grow is proportional to l/\/m+. However, as we shall

410 COLLECTIVE

BEHAVIOR

I N SOLID-STATE

PLASMAS

1393

see in the next section, the growth rate w+ is proportional to l/\/m+ so that there will be an optimum value of m+ for a given drift velocity to maximize the growth rate. The boundary of the region of growing waves for case 3 is also shown in Fig. 2. For the temperature ratio T-/T+= 10 the critical drift velocity is »
r_ (2+2.r 2 )H\7V

\

2r+(l-*2)/

+

0'1' (3-i2)

where k/k^=x. In order for the curve to fold in toward the k axis, it follows from (3.12) that

For a mass ratio of 14 the criterion is satisfied for r _ / 7 + ^ 1 7 . Thus, the temperature ratio must be quite large in order for the high wave numbers to become unstable before the low wave numbers do so. We remark that T_ 1 T_ Zi^ - ; Z£* ZiexpC-Z, 2 ), 2 27V1+* T+ so that for T~/T+> 10, the parameters satisfy Z i » l and | Z 2 | « 1 . Therefore the expansions used in deriving (3.12) and (3.13) are valid. Case 4 is, as one would expect, intermediate between case 3 and case 1. The influence of having unequal hole and electron densities (always a possibility since the lack of charge balance may be supplied by ionized impurity atoms) is considered in case S, and the boundary between growing waves and damped waves is shown in Fig. 3. The increase of hole density by a factor of 50 has the effect of increasing the threshold for growth of long-wavelength oscillations relative to the other cases considered; however, very short-wavelength oscillations continue to grow at lower drift velocities. For a given drift velocity, a considerably larger number of wave numbers can grow than for case 3 which has the same temperature ratio but equal densities. It is clear from the foregoing considerations that an appreciable lowering of the threshold drift velocity for creating unstable waves can be attained by maintain-

FIG. 3. The boundary between growing waves and damped waves for an electron-hole plasma such that m+—14m_, «+=50w_, and r_=4T+. ing one of the plasmas at a low temperature. Of course it would be best if both plasmas could be maintained at a low temperature with a large relative drift velocity; this is made difficult due to the scattering of the particles with the lattice and with each other. For the best situations in semiconductors we will see that the ratio of electron to hole temperatures is between 5 and 10; thus, neglecting other relaxation processes, the electrons would be required to have a drift velocity between 0.22 and 0.40 times the electron thermal velocity in order that a growing wave instability be excited. In general, somewhat higher drift velocities are required due to particle-lattice collisions. Clearly any drift that the holes attain from the external field works in favor of instabilities since the holes have a drift velocity which is opposite to that of the electrons and only the relative drift velocity enters the criterion for growth. However, if the holes have a high mobility their temperature will increase and therefore a balance is required between the hole drift velocity, temperature increase, and relaxation time. IV Growth rates for the instabilities we have been considering maybe calculated in a straightforward fashion as long as we stay within the linear region for which the dispersion relations of the preceding sections apply. In the calculations which follow we take into account the influence of hole lattice collisions by introduction of the relaxation time T + ; the electron-lattice collision time r_ will be an order of magnitude larger than r+ for the cases which interest us, and its influence on the electronic polarizability may be safely be neglected. The introduction of T + has the effect of simply reducing the growth rates by an amount 1/V+. Approximate expressions can be derived for the growth rates for cases 2 and 5 in the region of interest, i.e., Vd
411 1394

D.

PINES

AND

J.

R.

expansions of the polarizabilities may be used for holes and electrons respectively. The real part of the dispersion relation is given by 1=

<*+'

« r — ( W,H

SCHRIEFFER

- 7>4T«.V/n,-4/3

3

j

[»,+ (»,•+I/T-OJ

2(0>r-kvdY\k.

(kv-y

ft

(4.1)

here w = a)r+iio„ and we have assumed that fei_T_»l. Typical growth rates for T + = a> are of the order of W+/20 to a>+/10 so that it is necessary for W+T+ to be of FIG. 4. Growth rate curves for several cases of interest. the order of 20 to obtain growth. Since the waves with maximum growth rate occur for &~£_, it follows that w~cop and it is legitimate to neglect 1/T+2 in the real Vd/v- and a large relaxation time T+. A plot of the part of the dispersion relation compared to w2. Thus, growth rate based on (4.5) for r + = 0, w+/m_=14, w + =»_, tJ<j = 5i)_, and T + = oo is given in Fig. 4. The (4.2) u&s(k)k, maximum growth rate is approximately 0.107OJ + and where therefore it is necessary to attain a relaxation time T + , 01+' such that &)+7-+> 10 for unstable oscillations to occur. (4.3) Thus far the direct influence of the external field on kt+{l-2[s(k)-viJ/vJ}kJ k>+kJ the plasma oscillations has been neglected. This apThe imaginary part of the dispersion relation may be proximation is valid if the high frequency expansion of expressed as the hole polarizability is appropriate; however the correction to the imaginary part of the low-frequency 7u,—k-v electron polarizability leads to an extra contribution o>i= 2 1 to the growth rate of the form T+ 2 OJ+ k 3

r

(4.4)

a-iZ, exp(-Z, 2 ) -* T'Zt exp(-Z 1 2 )+V(l+'? 2 ),

where 7j = eEo-k/fc2Kr_. For k in the direction of the drift velocity, TJ is negative and hence the external field on inserting the expression for to we obtain the growth diminishes the instability. For typical values of parate rameters, e.g., £o=100 v/cm, k = ko, »-=3X10 1 6 , and 1 ir ! w+x 2 o>>^ , _ Z 1 _exp(-Z 1 _ ), (4.5) r_=150°K, 7i2= 0.05 which is to be compared with fl^Zi exp(—Zi ) = 0.55 at the fastest growing k. Thus, T + 2 (14-*2)' where the influence of the external field is relatively weak in this case. £i-=Growth curves for cases 3 and 4 were calculated by [2(1+**)]» to_ expanding the W functions about their values for real There are a number of conclusions one can draw from arguments and keeping first order terms in the imagithis expression. For growth of oscillations k must be nary parts. The results are also plotted in Fig. 4. Case 3 essentially in the direction of the drift velocity. In the was calculated for a drift velocity Vd=i>-/2 and exabsence of l/r + ) there always exists a value of k above hibits a maximum growth rate which is approximately which growth can occur for arbitrarily small drift ve- half of that given by case 2, that is, T+ = 0. Since case 4 locity; however, the growth rate decreases as l/x3 for would give a very small growth rate for nrf/n_ = 5, we x2>l. If we had chosen a low but finite temperature for have chosen the value of J in drawing our plot. The the holes, Landau damping of the holes would become maximum growth rate is approximately 0.067w,^- for important for large k and eventually lead to damped this drift velocity. Case 5 was treated by using the exoscillations in accordance with the boundary curves pression (4.5) and including the Landau damping of discussed above. We note that the growth rate in- the holes. The results are shown in Fig. 5 for Vd/v-~ j . creases linearly with o>+ for fixed w+/w_ so that it is The small apparent value of the maximum growth rate desirable to have a high density of carriers. Also, the is somewhat misleading, since the quantity plotted is growth rate is increased if &J+/O>_ is decreased; this can W2/W+- Due to the fact that n+ is increased by a factor be attained either by increasing the number of elec- of 50 over the value in case 4, the maximum growth trons or by choosing a material with small electron rate is actually larger by a factor of 4.5 than that in mass. It is of course desirable to have a large ratio of case 4.

412 COLLECTIVE

BEHAVIOR

We conclude from this brief analysis that in order to obtain growth of unstable oscillations it is desirable to work with a material with high densities of carriers and long relaxation times for particle-lattice scattering. For very high carrier densities hole-electron scattering via the screened Coulomb potential must be taken into account. The plasmas should have a relatively large temperature ratio and the relative drift velocity should be of the order of the thermal velocity of the hot plasma. The possibility of attaining these conditions is discussed in the next section.

IN

SOLID-STATE

PLASMAS

when we assumed that the influence of the external field could be characterized by a drift velocity, and that the particle-lattice interaction could be characterized by a relaxation time. We postpone until the latter part of this section a discussion of the validity of this assumption for the case under consideration. We assume, therefore, that the hole and electron distribution functions in the presence of external field, E0, / ± (Eo), may be written as m±(v-vd±y\

/ ± (£ 0 ) = a±exp'

("

In order to decide upon the feasibility of producing and observing the two-stream instability in solid-state plasmas, it is necessary that we investigate in some detail the extent to which an applied electric field E0 may give rise to an appreciable relative electron-hole drift velocity in the solid. Such electric fields not only shift the average electron and hole velocities but also alter the effective electron and hole temperatures. An added complication is the presence of a variety of temperaturedependent, lattice-scattering mechanisms for both the holes and electrons: we consider acoustic phonon scattering, optical phonon scattering, and ionized impurity scattering here. We carry out our calculations under the assumption that inter-electron or inter-hole collisions dominate in determining the form of the distribution function; the particle distribution function in the presence of the applied field may then be taken as Maxwellian with respect to an average drift velocity vd±, and an effective temperature (greater than that of the lattice) T±. This assumption, which was introduced by Frohlich8 for the hot-electron problem, obviously requires that the exchange of energy and momentum among the electrons (or holes) take place at a rate which is fast compared to the rate r ± , which characterizes the particle-lattice interaction. We have, in fact, made a similar assumption in our treatment of the two-stream instability, 004

'

JL

1

1

-

-

003

oxa

-

0.01

-

0

3

4

5

L»/t»

6

?

FIG. 5. Growth rate curve for » + = 50»_; T- = iT+. »H. Frolich, Proc. Roy. Soc. (London) A188, 521 (1947).

1395

(5.1)

2KT±

For definiteness, we consider the electron distribution function which satisfies the Boltzmann equation sE„ •V,/-

-F) •

(5.2)

\ dt /coll

To determine the two parameters va- and T-, it is convenient to take the momentum and energy moments of (3.2). In this manner one obtains the relations eE0 f

d

m J

dvt

— I p . -fdh = — neEo = fp>(-)

d°v=F(vd.,T),

(5.3)


(5.4)

eEoi [mv* Cmv1 d •/ f
= /—(-) J 2

\di/mn

The electron-electron collisions conserve energy and momentum; therefore their influence may be neglected in computing the right-hand sides of (5.3) and (5.4). The simultaneous solution of this pair of equations determines Vd- and 7\_ as a function of £oBefore proceeding to a detailed consideration of these equations, we make the following qualitative remarks concerning the role of the different scattering mechanisms. First, a given scattering mechanism may play a quite different role for energy transfer than for momentum transfer, that is, the dominant contributions to F and G may be from different mechanisms. Thus the scattering by ionized impurities is the dominant scattering mechanism at low temperatures, but is ineffective as an energy transfer mechanism because of the large impurity atom mass. On the other hand, the scattering by optical phonons (which possess an approximately constant energy hw^icd, where 8 is the Debye temperature) provides the most effective energy transfer mechanism over a wide range of temperatures for which the acoustic scattering provides the dominant momentum transfer mechanism. Indium antimonide seems in many respects one of the

413 1396

D. P I N E S

AND

J . R.

most promising semiconductors for the possible observation of the two-stream instability. As we have mentioned, the electrons and holes possess quite different masses, so that their mobilities are correspondingly different. The low-field mobilities of both holes and electrons in InSb have been measured as a function of temperature by Putley.10 He finds that for a concentration of 2X1014, electrons/cc the electron mobility at 20CK is ^ 1 0 6 cm2/volt sec. As the temperature increases, the impurity scattering decreases in effectiveness, so that the electron mobility increases until it reaches a maximum of S7X10 6 cm2/volt sec at 60°K after which it falls off. For holes, the dominant scattering mechanism for the specimen used by Putley from 30CK upwards appears to be acoustic scattering; Putley finds a fit for his measured values of mobility of the form

where T is in °K. Thus at 20°K, for a sufficiently low impurity atom concentration, the hole mobility would be 7X104 cm2/volt sec. If one uses the Brooks-Herring formula for the scattering of the holes by ionized impurities, 2"2 t
1 b)-b/(l+b)

(5.5)

tQtnixT)2

b= ;—, r n+h'e2

(5.6a)

one finds that for a density of 3X1016 holes/cc, the mobility due to impurities is~10 4 cm2/volt sec, assuming w + = 0.18?n. In general, as one goes to larger field strengths, the mobility will become field-dependent, since the particle temperature increases with increasing electric field. Indeed, one may pass from a region in which impurity scattering is dominant in determining the relaxation time through a region in which acoustic scattering is dominant up to a region in which optical scattering determines the relaxation time, by simply increasing the field strength. The detailed calculations of the variation on the drift velocity and particle temperature with field strength may be carried out using (5.3) and (5.4). The calculation of the contributions to F and G due to acoustic phonon and optical phonon scattering has been carried out by Stratton.11 He evaluates F and G by expanding f(Et>) to first order in vj- so that 2

( —T-\'<) nevd-, T /

KT«(KT-nt.Mi*)i

(5.7)

Mao

/ T _ \ 2 5 6 V 2 / W _ « . E M nevd-

\ 7 V 2 7 T » \ KT I M.c(r)' 2 2 K r»(Kr_w_«i. ) ,

1 F„p=

• N„F0

(5.8)

3\Ar

1je-r-n

(2mjc0)i XLle^T—

l}A'o(r-/2)

+ {fr->-+1 }Ki

,

where n± is the impurity density, and 2

an approximation which is valid if mvd-2/2icT_d-2/2KT-^\ and therefore we might expect to obtain a drift velocity which is accurate to within 25% by following this procedure. Stratton gives the following expressions for F due to both acoustic and polar optical mode scattering:

where UL is the longitudinal sound velocity and Mac is the zero-field mobility due to acoustic phonon scattering;

M = 5.4Xl0 6 r- 1 ' 16 cm2/volt sec,

M=

SCHRIEFFER

mv \f 1+w_v- v<;_"| ( -nA -^r}

(56)

10 E. Putley, Proc. Phys. Soc. (London) 73, 128 (1959); 73, 280 (1959). " R. Stratton, Proc. Roy. Soc. (London) A246, 406 (1958).

{yj2)nemvd-~},

where J V ^ I V - 1 } - 1 , y=0/T, Y _ = 0 / r _ , and Ka and K\ are modified Bessel functions of the second kind. Also, K8 is the energy of optical phonons, taken to be a constant and the parameter Fa, having the dimensions of an electric field strength, is given by F0±=

(t^-to-^emiKB/h2,

where to and e„ are the static and optical dielectric constants of the material. The Brooks-Herring formula for ionized impurity scattering leads to the expression F, =

87r2 eiNim^n-Vd- 0(6)

3

£o2(2m_Kr_)*

=

nevd-f T \* 1 — 1,

(5.10)

Mv(r)\r_/

where <j>(b) — \n(l-\-b) — b/(l+b) and b is given by (5.6a). m is the zero-field mobility due to impurity scattering and Ni is the ionized impurity concentration. The requirement of a long relaxation time demands that the experiments be carried out at a low lattice temperature so that the phonon scattering is small. As pointed out above, the effect of impurity scattering will decrease due to heating of the carriers by the electric field and for this reason it is desirable to carry out experiments at temperatures below the zero-field mobility maximum. The G functions are given by 3'H€U i ^

G»C=

**,(r)

(i-r/r_)avr)i,

(5.ii)

414 COLLECTIVE

BEHAVIOR

IN S O L I D - S T A T E

/2K0\l Gop^neNaFol )

PLASMAS 1

1.0 0.9

X[> G, =

-l>_»«WJP,( T -/2),

\HneKT{ T \» (

M^(T)\T.

)(-£>

(5.12)

"

(5.13)

where M is the mass of an impurity atom. Calculations of the hole and electron temperatures and the ratio of the electron drift velocity to the electron thermal velocity for p-type InSb at T= 20°K have been carried out for A7,= 1016/cm3 on the basis of the above formulas. The results of these calculations are plotted in Fig. 5. In carrying through the analysis, we have attempted to take into account the variation of the electron effective mass in InSb with temperature (the variation arising from the nonparabolic form of the bands due to spin orbit coupling) by using an effective mass appropriate to hot electrons, w_ = 0.03m. The calculations for electrons show that the optical phonons dominate both the momentum loss and energy loss for electron temperatures greater than 100°K. For T_ = 60°K, the impurity and optical phonon scattering contribute essentially equally to the momentum loss of the electrons while the acoustic phonon contribution is smaller by a factor of 5; the optical phonons however, still dominate the energy loss by a factor of 20 over the acoustic phonons at this temperature. The situation is somewhat more complicated for the holes where the momentum loss is primarily due to acoustic phonons up to T' + =80 o K, at which point the optical phonons take over and become the dominant scattering mechanism; the impurity scattering plays essentially no roll for T+>40CK., due to the decrease in the Coulomb crosssection for high velocities. The energy loss for the holes is primarily due to optical phonons down to the hole temperature r + =30°K, where the acoustic phonons contribute approximately 10% of the energy loss. The impurity scattering is ineffective in the energy loss mechanism over the entire range of temperatures for both holes and electrons. It should be emphasized that these calculations have been based on a value of e0= 17.5 taken from the infrared measurements of Spitzer and Fan.12 Ehrenreich13 finds that eo= 18.9 leads to better agreement between his calculation of the mobility associated with mode scattering and the experimental value. If this larger value of e<, is chosen, Fo should be increased by a factor 1.7 and a somewhat smaller temperature ratio will obtain for a given value of Ee. Since the optical modes give by far the largest contribution to both the F and G functions for fields greater than 100 v/cm, the increase in F 0 will simply scale the En axis by a factor of 1.7. The curves for the holes may be approximately 11

R. Spitzer and H. Y. Fan, Phys. Rev. 106, 882 (1958). " H . Ehrenreich, J. Phys. Chem. Solids 2, 131 (1957).

0.7 0.6

/

0.5

i

0.4

V-.., _

O.I

-VI

•

/ /

/

" ^y~—*^

***~

\

s

/

0.3 O.i

.,,,..,

v*.

ve „

FIG. 6. Hole and electron temperatures and electron drift velocity as a function of field strength in InSb, assuming the following initial conditions: 7"iMtiee = 20°K; wi_=0.03m; m+=0.18m; iV< = 10>Vcc.

1

v«

OB

and

1397

•S. "

"

•

•

•

•

•

.

.

.

.

_

y >

100

200

300

E. (V6ll/cm)

adjusted by scaling the Eo axis by y/1.7 for r + < 8 0 ° K and 1.7 for T + >80°K. The foregoing analysis of the hot-electron problem shows that substantial temperature ratios T_/T+ and drift velocities Vd/v- may be achieved at moderate electric fields. Although we have not carried out detailed calculations of the growth rate for the conditions obtaining for the curves of Fig. 6, we can draw some tentative conclusions as to whether the conditions for the existence of the two-stream instability can be met in InSb. We first remark that on comparing Fig. 5 with Figs. 3 and 4 we may conclude that growth rates of the order of u+/l5 appear definitely achievable, provided hole-lattice scattering effects are negligible. Thus one requires, at the very least, a hole-lattice relaxation time T + , which is sufficiently long that C0+T+

3:15.

u + is dependent on the hole density, and T + is dependent on both the impurity atom density and on the temperature. The scattering due to acoustic and optical phonons can presumably be eliminated by going to a sufficiently low initial lattice temperature (and correspondingly lower particle temperatures) so that it is the holeimpurity scattering mechanism which will cause the greatest trouble. For example, for n + = 3X10 16 , one finds o>+=1.8X1012 sec -1 ; on the other hand, for this concentration of ionized impurity atoms at T'+ = 20oK one finds T + = 2 X10 -12 sec, so that W+T+^3.6 in this case. Matters improve slightly as one reduces the ionized impurity concentrations; at r=20°K, the best one can do is to reduce it so much that acoustic scattering is dominant; in this case, one would have T + = 7 . 2 5 X 1 0 ~ 1 2 sec. What is wanted then is a higher density of holes than of impurity centers, a condition which can be achieved through ionization across the gap by the strong electric fields, according to Glicksman and Steele.14 Thus one might start with a p-type sample, containing ~10 14 impurity centers and holes/cc; on applying a strong JEO, one could produce perhaps several times 1016 electrons and holes/cc; an u+r + of the order " M. Glicksman and M. C. Steele, J. Phys. Chem. Solids 8, 242 (1959).

415 1398

D.

PINES

AND

J.

of 15 would then seem not completely out of the question provided one further worked at values of initial lattice temperature such that the hole temperature T+, for the value of Eo in question, was somewhat lower than 20°K. I t should be added that under these circumstances it is probably necessary to consider the relaxation time associated with the impact ionization. Detailed considerations are required to decide what would be the best experimental setup, though it does seem likely that unless the ionized impurity concentration is substantially below 3X10 1 6 /cc, and the initial lattice temperature below 20°K, the conditions for the existence of the two-stream instability cannot be met in InSb. We conclude this section with a brief discussion of the validity of our assumption of a displaced Maxwellian distribution function for the holes and electrons. Following Frohlich and Paranjape 15 we may regard the electron-electron collisions as being more effective provided the energy loss per unit time by an electron (moving at a velocity greater than vJ) in electronelectron collisions is_ greater than that in electronphonon collisions* Frohlich and Paranjape find that this will be true for acoustic phonon collisions if 1 E*mJu?t»„3=4rr KT<*TM(E)

(5.14) '

where E is the average electron energy, and T.,, is the relaxation time for collisions between electrons of energy E and acoustic phonons of lattice temperature T. This condition may be expressed in the form

R.

SCHRIEFFER

and therefore the interparticle scattering dominates the particle-acoustic phonon interaction in determining the distribution function for all concentrations of practical interest at r = 2 0 ° K in InSb. The corresponding criterion for interparticle collisions to dominate the optical mode scattering in determining the distribution function is n>n =

tJF„K0( T. (•

1

(5-15)

where /i 60 is the zero-field mobility due to acoustic phonon scattering a t the temperature T. For electrons in InSb with r = 2 0 ° K and choosing £ = 0 . 0 1 ev, m-/m0=0.0S, M;=3.6X10 6 cm/sec, eo=17.6, and C M,c (20 K) = 4.3X10' c m ' / v sec corresponding to a deformation potential constant E\=— 7.2 ev, the criterion becomes re_>2.4X106cm-3, which is of course always met. As pointed out earlier, Putley finds the hole mobility in InSb can be well represented by a combination of impurity scattering and a mechanism giving rise to a mobility which varies as 5.4X10 e r~ 1 - 46 . We have assumed in our discussion that this mechanism is in fact acoustic mode scattering. The criterion (5.15) for holes then becomes n + > 7.8X10" cm"8, 15

H. Frohlich and B. Paranjape, Proc Phys. Soc. (London) B69, 21 (1956).

(5.16)

r_=145°K, and m_/»«o

f2m±K.Td±\

3*A

\mj

T-<6.

I t is impossible to satisfy both of these criteria for steady state conditions in weak external fields a t r = 2 0 ° K . However, due to impact ionization of electrons across the energy gap b y hot carriers, it may be possible to increase the minority carrier concentration sufficiently to satisfy these conditions for b o t h holes and electrons. I t would appear from the preceding discussion t h a t the growth rate could be increased by increasing the concentration of carriers. This could be accomplished by using a pulsed beam of high-energy electrons. Several difficulties arise however, if one goes to high carrier concentrations. The use of classical statistics is not longer valid if T±
\293/ \KTJ \ 1 0 /

\10 / \»J

for

\

For InSb, taking r = 2 0 ° K , 0=29O°K, r + = 5 0 ° K , f 0 =17.5, e«,= 16, m+/mo=0.18, = 0 . 0 3 , we have »+>2.8X1014cm-3 and «^>3.7X10i6cm-3.

«_>10«(—V—)(-)

X( —6 ) ( — ) ( — ) .

e-"IT-

T)'<

27TC3

V

J

--n±.

The concentrations appropriate to InSb for degeneracy temperatures r d ± = 1 0 0 o K are » + = 5.7X10 16 cm" 3 and » _ = 1.6X10" cm" 1 . There is also the problem of a decrease in the relaxation time due to electron-hole scattering as the concentration is increased. T h e relaxation time for electrons may be estimated from an extension of the Brooks-Herring formula to include screening due to both holes and electrons. B y treating the holes as being fixed scattering centers, we have the relation 1

(4ire2)!

r(k)

8jrA»A»e,o

-m_«Jln(l+&) 1, I 1+bJ

where b=W/{kDJ+kDJ). For H ± = 1 0 " cm" 3 , T+ = 50°K, r _ = 2 0 0 ° K , and * 2 A 2 /2m_=0.01 ev, t h e screening parameter b is equal to 7.65 and the electron relaxation time is 1.16X10 -13 sec. The relaxation time for the holes is more difficult to estimate due to the large mass ratio of holes and electrons. The electron relaxation time in any event would appear to be sufficiently short

416 COLLECTIVE

BEHAVIOR

to make the approach of dubious merit in generating plasma instabilities.

IIN

SOLID-STATE

PLASMAS

1399

tive drift velocity will tend to saturate near the threshold for creation of unstable oscillations. Thus the effectiveness of the mechanism in producing a saturation drift velocity (or current) depends upon the VI effectiveness of the energy transfer from the growing The behavior of the system once the threshold for oscillations to fine grained thermal motions. Due to the growth of instabilities is reached is quite complicated; lack of theoretical and experimental understanding of we can here only make some qualitative remarks, and this nonlinear decay process it is impossible to make hope to draw attention to some of the interesting fea- predictions in this regard and indeed it is for this tures of the problem. In thermal equilibrium, each reason that experiments on the two-stream instability relatively undamped plasma mode will possess a small would be most interesting to carry out. amplitude of vibration due to thermal excitation. As In our analysis we have assumed that the major the particle drift velocity increases just beyond the portion of the distribution function in the quasithreshold for growth of oscillations, a small number of steady state can be described by a displaced Maxmodes will have their thermal level of oscillation ampli- wellian. The validity of this approximation rests upon fied by the two-stream mechanism. These growing waves the level of plasma oscillations which are excited and will absorb energy from the directed particle drift in turn depends upon the nonlinear effects about which motion and therefore decrease the drift velocity slightly. little can be said at this time. It would appear, however, The decreased drift velocity will then support fewer that the qualitative arguments in favor of a saturation growing modes. This process will continue until the current are likely to be correct. It should be noted that energy supplied by the external field is appropriately a saturation drift velocity may result directly from the distributed amongst the lattice waves and the growing optical phonon scattering, a fact which should be taken acoustic plasma modes into account in the analysis of experimental data reIn order to calculate this distribution, it is necessary lating to the generation of the instability in polar to know the level of plasma oscillation which exists, crystals. and a knowledge of this in turn requires an underWe mention that another type of electrostatic instanding of the nonlinear coupling between the plasma stability may occur under special circumstances. If the modes of different wavelength. This coupling causes a distribution of one of the carriers exhibits a hump other long wave-length mode to decay into higher wave- than the main hump of the Maxwellian, plasma oscillanumber modes. The higher wave-number modes decay tions may be exicted by particles whose velocities are in turn into still higher wave-number oscillations until in the region where the distribution function is increaspresumably a fine grained random or "thermal" motion ing with velocity. Such a situation might exist when one will be set up. Since the particles are coupled to the lattice is dealing with a low-density plasma in a polar crystal. by the various interactions discussed in the last section In this case the optical phonons are more effective in (i.e., impurity and phonon scattering, etc.) a large part the energy loss of high-energy particles than the of this thermal energy will be given to the lattice. It particle-particle collisions which tend to restore the would appear that a quasi-steady state would eventu- distribution to a Maxwellian form. Thus one might ally be set up in which the growth of the oscillation due expect a hump to develop in the particle distribution to the two-stream mechanism is just balanced by a just above the threshold for optical phonon emission. decay due to nonlinear interactions. The situation would The hump would then lead to coherent excitation of not be a true steady state since the temperatures of the plasma oscillations analogous to that expected for particles and lattice will increase slowly with time and runaway electrons in a hot plasma. as a result the threshold for excitation of unstable In conclusion, the considerations presented above oscillations and the various scattering mechanisms will indicate that the production of a two-stream instability be time dependent. For weak nonlinear effects the in the coupled hole-electron plasmas of a semiconductor modes will attain a large amplitude before this quasi- is marginal due to the relatively short relaxation times steady state sets in and the rate at which each mode which are attainable in practice. There is a good possiabsorbs directed drift energy will be correspondingly bility that the situation would be brighter for producing large. Now the external field delivers energy to the the instability in semimetals such as bismuth where resystem at the rate of j • Eo, part of which goes directly laxation times for both holes and electrons of the order into thermal motion of the lattice by the usual Joule of 10-10 second or longer are observed. This case is disheating process and the remainder goes into excitation cussed briefly in the appendix. of plasma oscillations. It follows that for a given value of JSo the weaker are the nonlinear effects, the larger is ACKNOWLEDGMENT the portion of the available drift energy going into each plasma oscillation and therefore the fewer the modes It is a pleasure to thank Dr. Marshall Rosenbluth for that will be excited. many stimulating discussions on these and related These arguments lead to the conclusion that the rela- topics.

417 1400

D.

PINES

AND

J

APPENDIX

R.

SCHRI EFFER

that ?V-A»-»SV-

Two-Stream Instability in Quantum Plasmas

(A.5)

The discussion of the two-stream instability given above is easily extended to the case in which one must take account of quantum effects. The random-phase approximation for the polarizability, which holds for both classical and quantum plasmas for long wavelengths, is

the solution of Eq. (A.3) for Uq = Qiq+jUlg is

-4xe2 4wa(q,a) = —-~1 £ { / ( * ) - / ( * + ? ) } q *

where

3irm+rs(q)l3 fi2,=

4x«e 2 i (qf =

, hQ-Ek+,+

(A.l)

Ek+iS

where for Fermi-Dirac statistics f(k) is

/ W = l/[c <Et -" ) " ,T +l]. Here sider mass spect fi9 of

l + 4 T O + ( g , n 5 ) + 4 7 r a - ( 9 ) f i , - q - v d ) = 0.

(A.3)

If the effective temperature T+ of the particles of mass m+ is sufficiently small such that 2

z

? Kr+/m+«n, ,

(A.4)

the polarizability 4ira+ may be replaced by its highfrequency limit — w ^ / O , 2 where Uj H . 2 =47r«e 2 /m + ; this result is identical to that obtained for high frequencies in the classical plasma. If in addition the particles of mass m_ have a sufficiently large Fermi energy / j _ so

v FJ 3 m+

for 92i>F_2«47rwe2/ra_. Here VF~ is the Fermi velocity of the m_ particles. In order to have growing wave solutions ( l i 2 t > 0 ) it is required that v,>s(q)^(w-/3m+)hF^.

(A.2)

y. is the Fermi energy and Ek=h2k!/2m. We cona two-component plasma formed by particles of «t_ drifting with an average velocity v«j with reto a set of particles of mass tn+. The frequency the acoustic plasmon is given by

(A.6)

1 w_ ^

f» + [l+47ra_(9,fi„)]

1 X

[>(?)?-v
(A.7)

One might hope to observe this instability in bismuth where the hole mass m+ is roughly thirty times that of an electron w_. Also, one has conductivity relaxation times for both holes and electrons longer than lO - 1 0 second at liquid helium temperatures in this case. If we use iij?_= 10r cm/sec, «„ = 0.017 ev and T + = r_=10~ 1 0 sec, the growth rate for wave numbers q less than the screening wave number k,^107 c m - 1 is ik^W10{ 6

10-&q(vd/s-1)

-1}.

where J = 1 0 cm/sec. Thus with drift velocities of the order of 2X10 6 cm/sec ( = I > F _ / 1 0 ) , a two-stream instability might be observed. In practice, this high drift velocity is likely to be difficult to attain due to excitation of electrons from the heavy hole band into the conduction band by high-energy conduction electrons.

418 PHYSICAL

VOLUME

REVIEW

131,

NUMBER

3

1

AUGUST

1963

Coupled Electron-Phonon System* S. ENGELSBERG

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey

J. R. SCHKIEFFER

University of Pennsylvania, Philadelphia, Pennsylvania (Received 26 March 1963) The coupled electron-phonon system is considered for phonon spectra of Einstein and Debye forms. The single-particle electron Green's function G is calculated in a nonperturbative manner in both models, and its spectral weight function is examined to determine the validity of a quasiparticle picture. The weight function and the poles of G both lead to several branches of excitations rather than a single "dressed" electron. The asymptotic time dependence of the G is found, and the effect of multiphonon processes on the electron decay rate is discussed. The electronic polarizability, P of the interacting system is calculated with the aid of a generalized Ward's identity for the electron-phonon vertex. This identity, which is a consequence of electronic charge conservation, is derived in an Appendix. The calculation of P is carried out in the limit that the Fermi velocity is small compared with the phase velocity of the polarization field. An Appendix on the formal development of the Green's function equations is included. INTRODUCTION

far from the Fermi surface. In Sec. II, t h e result for E investigate the coupled electron-phonon system G(p,/>o) is used to calculate the asymptotic behavior of using a field-theoretic scheme 1 2 that goes beyond C(p,<) as t—>». In Sec. I, it is found that the electronic decay r a t e lowest order processes. Two model phonon spectra of shows no anomalies near the thresholds for multiplethe coupled system are studied; these are the Einstein and Debye models. 8 We view these models as two phonon emission. The constant density of states limiting cases, neither of which is realized in practice. approximation is relaxed in Sec. I l l , and such anomalies their relative magnitude is again of The Einstein model is a reasonable approximation for appear. However, lli and they are not to be trusted if vertex short-wavelength phonons while the Debye model is order (m/M) preferable for processes emphasizing long-wavelength corrections are neglected. The electronic polarizability P(n,q<s) is discussed in phonons. Sec. IV with the aid of a generalized W a r d ' s identity We begin with the Einstein model in Sec. I. By approximating the electron-phonon vertex function T for the electron-phonon vertex which follows from by unity and the free electron density of states in energy charge conservation. This identity is derived in by a constant, the integral equation for the electronic Appendix B. With the approximation of Sec. I for 2 , self-energy 2 can be trivially solved. This approximation the Ward identity is satisfied by the ladder approxiof r for the Debye model has been discussed by Migdal 4 mation for the vertex function which enters the expreswho argues that the error thereby introduced is of the sion for 2 P. The polarizability is calculated to order order of the square root of the electron to ion mass ratio (qflF/90) , where vr is the Fermi velocity. This calcu(m/M)w for normal metals and is therefore negligible. lation allows one to investigate the long-wavelength His argument also applies to short-wavelength modes phonon spectrum. The Debye model is considered in Sec. V. Within the in the Einstein model but fails when the phonon's phase velocity is of the order or greater than the Fermi same approximations used for the Einstein model, the spectral function and poles of the electron Green's velocity. The poles of the electron Green's function are given function are derived and plotted for the D e b y e case. in several figures. Several branches of excitations appear. Several branches are again found in the single-particle The quasiparticle picture is examined for this model spectrum. The definition of the Green's functions and the and found to be inappropriate except very near to or formal development of the equations determining these functions are given in Appendix A. * This work has been supported in part by the U. S. Air Force Office of Scientific Research, the Advanced Research Projects Agency and the National Science Foundation. I, ELECTRON SPECTRUM IN THE •J. Schwinger, Proc. Natl. Acad. Sci. U.S. 37, 452 (1951); EINSTEIN MODEL N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience Publishers, Inc., New York, 1959). The model we first consider is one in which the * V. M. Galitskii and A. B. Migdal, Zh. Eksperim. i Teor. Fiz. lattice is composed of independent oscillators, each 34, 96 (1958) [translation:Soviet Phys.—JETP7,139 (1958)]. * C. Kittel, Introduction to Solid State Physics (John Wiley & having a single characteristic frequency. T h u s , a Sons, Inc., New York, 1955). 4 A. B. Migdal, Zh. Eksperim. i Teor. Fiz. 34, 1438 (1958) phonon can carry only a fixed energy, «, independent [translation: Soviet Phys—JETP 7, 9% (1958)]. of its momentum. Since the phonon damping is expected 993

W

419 994

S.

ENGELSBERG

AND

D(p-k) FIG. 1. Diagram corresponding to the approximation for the self-energy. Gin)

to be small, our initial choice is that of undamped oscillators. We start with the approximation to the self-energy (A19), illustrated in Fig. 1, which neglects vertex corrections r dAk X(p) = i£

D(p-k)G(k)

J

(2TTY

dk d
f

i

{iTrYiiPo-hY-rf+ib] 1

x-r>-*(k)-2(*)]

where

-0+

(11)

J.

R.

SCHRlEFFER

2(/>o) may now be completely determined if we use the relation lmkoZ(ko)>0 for ko>Q < 0 for k0<0 which follows from the Fourier transform of the definition (A4). Closing the e contour in the upper half-plane we obtain the expression

fN

[ dk0- f

2 (/>») =

J„

Joo

dkA)

, (1.4) (po-koy-u?+iS

valid for po on the real axis. I t is interesting to investigate whether a simple quasiparticle picture of our model leads to a reasonable approximation for G{p). A quasiparticle picture assumes that the weight function .<4(ep,/>o) = (l/ir)|ImG(p,/>o)|, which appears in the spectral representation G

/•-

A(tf,p0')

d

(P>A>)=/

f

P»'

Jo

H po—po+w

A A(e (6 Pv,p ,K) 0')

d

P»

J-*>

. C1-5)

pe—Po'—rt

can be well represented by a single Lorentzian function

e(k) = £ 2 /27»-M.

af|rp|/»

Here p. is the chemical potential and g, the coupling constant, is chosen to be independent of wave vector. The propagator for the electron field is written in its most general form, (A9). The phonon propagator has the characteristics mentioned above and includes an infinitesimal imaginary part to insure temporal outgoing wave boundary conditions which is required by the definition given in (A4). We see that for this model 2 (p) is independent of p ; we therefore define

The quantities E„ and 2\TP\ are interpreted as the energy and decay rate of the quasiparticle. Were this picture exact, the quasiparticle peak would exhaust the sum rule

ko—S(^o) = koZ{ko) .

and the parameter ar would be unity. In general, the weight function is given by

(1-2)

To perform the d3k integration, we approximate the free-electron density of states N(e) = mk/2ir2 by its value at the Fermi surface mkF/2r2. Furthermore, we maintain particle-hole symmetry by carrying out the e integration symmetrically about the Fermi surface, « = 0, and for convenience extend the limits of integration to infinity. The justification for these approximations is that the major contribution to the self-energy integral comes from electron states with energies «<w, the Einstein energy, which is typically one hundredth of the Fermi energy. 3 Thus, we expect the characteristic features of our results to be insensitive to the changes of the integration region far from the Fermi surface. Within this approximation the electron self-energy may be written as /•" dk0

2(Po) = ig>N

—

J-„

2ir

f° X

1 de

koZ(k0)

•/-» (po-koY-ot+i&ikozy-?

.

(1.3)

A(t„po') =

(po'-Epy+Tp*

A(et,Po')dpo=l,

/

1 A((„Po)=-

\Zi(P)\

(1.6)

(1.7)

wfa-*,-Zi{p)J+XS(p) where 2 « and 2 / are the real and imaginary parts of 2 , respectively. It follows from (1.4) that for real pa 2rW=

/ 4w Jo

dk0{S(Po-ko-u)+S(po-ko+u)

-&(po—ko—w) — S(—po—ko—u)} •Sgnpo

for

\po\>a

2u

=0

for

\p0\
(1.8)

Thus, within our approximation 2 j is a constant for energies above the threshold for phonon emission. Were we to include damping of the phonons, due to their interaction with electrons, 2 / would be nonzero for \po\
420 COUPLED ELECTRON-PHONON emission thresnold. The real part of X for real po is 2

gN

po+<*

2u>

po—oi

(19)

Due to the assumed particle-hole symmetry we find 2u(0) = 0, so that the chemical potential is unaffected by the interactions. On combining (1.7), (1.8), and (1.9), one finds the weight function is

\lmG(p,Po)\=&(po-H+(?N/2u)ln\(p<,+o>)/(Po-w)\)

for

|/>„| (1.10)

g2N/2u ~ LP«- «f + («W/2«) In| (*•+«)/(*>-«) I ] 2 +1 rfN/2a This function is plotted for several values of £p in Figs. 2 through 5. For electrons injected in momentum states just above the Fermi surface (co»«p>0) a "quasiparticle peak," here a delta function, occurs at the energy (i.ii) £p=«p/(i+gWM with a strength (1.12) at=i/(i+«w/«»). For large g'N/ai1 the major contribution to the sum rule (1.6) comes from the continua above thephonon emission threshold \po\ =oi. The maximum value of the weight function in the continuum occurs at po=u{i+giN/o>2)U2. The continuum for p0<0 enters only when an electron is extracted from state p (i.e., hole injection) and describes a dressed hole and a phonon being excited. As ep increases, the delta-function peak moves toward u, approaching u asymptotically as tr—>•». If we continue to interpret this peak as a quasiparticle even for € p »u, this branch of the excitation spectrum exponentially approaches a constant energy, a>, at large momentum. The strength of the peak decreases however as exp[—2«pw/gW3, as «p—» «>. There is an important change in the continuum as ep increases beyond oi in that the simple peak occurring for ep oo, while the upper peak approaches the freeparticle energy as ep —> oo. Since the upper peak almost completely exhausts the sum rule for large ep, it appears natural to call this peak the quasiparticle peak for large e„. Therefore, the quasiparticle representation of G(j>) is reasonable in the regions | ep | »w, however for | ev | ~w the picture is clearly incorrect. Another procedure often used to define quasiparticles2 is to seek poles of the analytic continuation of G into the lower (upper) half-plane of the complex po plane for po>0 (p„<0). If a simple pole is the only singularity of this function near the real axis, the asymptotic time dependence of G(p,t) in the limit I—* °o will contain the term op«^'jr»'«^|I,»|,l where the pole is located at po~Et-t-irt, with residue a p . In our problem,

995

SYSTEM

for

\p0\>a.

|«

the analytic structure of G is complicated by the branch cut due to the logarithm in 2 B . While poles do occur on the second Riemann sheet, the branch point at po=i» leads to contributions to G(p,<) which cannot be neglected as t —» °°. The analytic continuation of G across the cut from u to » and from — » to -co is accomplished by subtracting from Sr(p,^>o) the jump in its value as po crosses the real axis from above to below for />o>0 and vice versa for po<0. Defining pa=E+iT, we have the continued expressions 2*(£+*T) =

-g2N

2/(£+*T)=

4u

In

(£+w)2+r=

(1.13a)

CE-o>)H-r2 2uT

tan- 1

sgn£ . (1.13b)

To define the energy spectrum of the quasiparticles by this approach we look for zeros of G~l{p) = po— «p—2(/>o). The solutions of this equation are presented in Figs. 6 and 7 and were obtained by numerically solving the transcendental equation for T as a function of E, then plotting £ as a function of £—2ii(£+jT) = e. Typical values of gW/w2 range from \ to 5 in metals. Figures 6 and 7 have been drawn for gW/u>2=5. The branch labeled I arises from the delta-

- 6 - 5 - 1 - 3 - 2 - 1

0 X

I

2

3

4

5

6

FIG. 2. Spectral weight function A (tt,xai) appropriate to the electron Green's function for «p = 0, gW/w* = i. There is a deltafiraction contribution at x—0.

421 S.

996

ENGELSBERG

AND

J.

S C H R I E F F E R

has poles at the characteristic frequencies a>„ and fi„ of the coupled system. X„2+An2 fi* = +i[(X» 2 -A„ 2 ) 2 +g„ 2 ] 1/2 , 2 (1.16) X„2+An2 2 "„ = 1[(X,»-A,«)»+^»]W. 2

FIG. 3. The spectral weight function A(ev,xu) for ep = 0.75a> and g1N/al = i. The heavy vertical line represents a delta-function contribution.

function term in the weight function. Branch II arises from a pole on the second sheet. It should be emphasized that a quasiparticle picture based on these two branches alone does not give an accurate representation of G(p,/>o), as is clear from the plots of the weight function. This physical statement is true since G(p0) is not meromorphic but has branch-point singularities at energies pn=±u which cannot be neglected in determining the time dependence of G. The origin of the two branches or alternatively of the peaks in the spectral weight function might be partially clarified by considering the soluble problem of N pairs of coupled harmonic oscillators. Consider the Hamiltonian

(/>»2+x„v+p„2+A„2e„2)

H E

+ E«n? n Qn.

(1-14)

n-l

The Green's function for the operator
J

CuA(5ttl,Mu) 2.01.81.61.41.21.0.8.6.4.2-

'

1

1

1

1

1

FIG. S. The spectral weight function A (er,xu) for ep = 5a> and !?N/a* = i. The heavy vertical line represents a delta-function contribution.

n-l

*e iE '<0|{n 9 „(7) ?n (0)}|0>

Gt%m = il

If the starting Hamiltonian is positive-definite the system is stable, and both u„2 and fl„2 are positive. In a similar manner, our approximation for the electron Green's function has manifestations of two types of single-particle excitations. Roughly speaking, the peak at £ p =fp/(l+g 2 iV/uj 2 ) corresponds to an electron "clothed" by a cloud of phonons analogous to a polaron in insulators, while the positive energy continuum for fixed momentum represents a clothed electron and a phonon being excited, the sum of their momenta being

(1.15)

BO

p. We note that for large tf the poles of G(pd) on the first and second sheets have the bare phonon-like behavior on branch I and the bare electron-like behavior on branch II. Within the approximations made we have a result which is only of order the coupling constant squared. Our approximation for the self-energy contains implicitly the possibility for multiple phonon processes. Why is their effect not seen? The reason for these processes not contributing may be understood most simply by examining the relevant diagrams in the perturbation expansion. For example, consider doublephonon emission shown in Fig. 8:

2««

tfkid'ktD^Di^Goip-ktfGoip-h-kt).

This expression contains as a factor

since FIG. 4. The spectral weight function A(ep,xw) for t9 — 2ta and g'N/t^ = i. The heavy vertical line represents a delta-function contribution.

f d3k' r / G
/

dz

1 de = 0, (e-ko'T =0

422 COUPLED ELECTRON-PHONON by Cauchy's theorem. Thus, we see we may even include some "phonon line crossing" diagrams which would arise from inclusion of the full vertex function; as long as there are two symmetrically placed electron lines, the contribution will be zero. In Sec. I l l we will consider the effects of multiple-phonon emission.

The integral is to be closed in the lower half-plane. For «>w we choose the contour of integration as in Fig. 9. The result of the integration may be written as exp(—iEit) G(e,/) = - i l+?W/V-£i2)

II. EVALUATION OF G(t,l) FOR LARGE t

exp(-i£n(—TuO

We now consider the evaluation of G(t,t) for large positive /. The Fourier transform oiG(e,pti) gives us the amplitude for a particle which is placed in the system at time zero with kinetic energy, «, remaining in the same state after a time, I.

2a-

2T

-e+— In 4OJ

\u-zJ -

e

dr

/ = JL

2oi

exp(-izt)

2TT

z-e+-

fN

~

1

ln(Vr')]2+

exp[-*'£i(e)0

l+gW/[ W 2 -£,He)] exp[-iEii(e)+TuQ 1+gWuVCu2- (£n+tTn)]2 2a>e-<" g iV/(ln()

ing as it does only on the behavior at the point z = a>, remains unchanged.

In Sec. I we saw t h a t our approximation for 2(/>o) did not reflect multiple-phonon processes in that 2 was a smooth function of p0 near the multiphonon emission thresholds 2w, 3a, • • •. We show below t h a t this result, while approximately correct, is strictly valid only if the 6 integral is extended symmetrically to infinity. We break particle-hole symmetry to investigate the approximations of Sec. I . Results identical with this section,

Thus, the resulting expression is

2

(irfN/^)2

Ill. MULTIPLE-PHONON PRODUCTION

dytr".

2a./ [(gW/2w) In/] 2 J o

"[Q]

If we change the variable in the line integral to T' = i(z—o>), then the dominant contribution comes from the branch point T' = 0. Therefore, placing T' = 0 in those terms of the denominator which give finite contributions, we obtain

\jji—ig' N*/k*+(gN/2o})

Jo

If we place y=T't and neglect all but the singular behavior as l —-> «>, we obtain

G(€,0=-^

L

+

dz

4o>

(2.1)

/ -

\+fN/t<s-(Eu+irun

"lGU

r dz J

997

SYSTEM

2

(2.2)

Of course, for / increasingly large the damped term will be negligible not only compared with the undamped term but also with respect to the l//(ln/) 2 term. In fact, the technique used in the evaluation of the line integral supposed / much greater than all the energies which entered. Thus, when this technique applies, the damped term may be mathematically neglected. However, we include it since it represents a physically important state of the system. When i is less than w we use the same integration technique but change the contour so that both poles are included. T h e result of the line integration, depend-

b

1

1

1

1

X '

i

4

/it

3

-

2

i

s^

1 I

i

i

i

3

4

i

i

C/Ql

FIG. 6. Poles of the electron Green's function on the first and second branches. For a given kinetic energyj«p, the poles are labeled by E.

423 S.

998

ENGELSBERG

A N D J.

except for numerical factors, are obtained if a large symmetric cutoff is chosen. Thus, we will again use the expression (1.3) for 2(p) b u t replace the lower limit

d*k

/

1 4

[

z

R.

S C H R I E F F E R

— oo by —fi in the t integration. When the imaginary part is taken and the spectral representation for G inserted we find

f

- /

2

(2ir) (p0— *o) —oi +iSl

Jn

tr A0—Ao'+tt;

.+ / } S,v-*0+. / _ „ ir k0— ko'—it]

(3.1)

We perform the ko integration by closing the contour so that only the poles of the phonon propagator contribute

i r
imG(k,*0')

ImG- 1 (/'o) = I m ~ / / dk0'a J (2ir) 4 Uo po—u—h'+ir) = Since ImG^ipo) result

r°

ImG(k,4o) / dko PO-J-LI—ko'—it) J-„

g> C d3k I /•" / / dko'ImG(k,k0,)6(po-<*-ko')+ 2a, J ( 2 j r ) s l ; 0

f0 / dk0'ImG(k,h')S(p0+oi-k0') J-„

} . (3.2) I

is an odd function of po we need consider only positive values of pt>. Equation (3.2) leads to the g2

1

ImG- (* 0 )= 2uJ

f d3k / ImG(k,/> 0 -co) ;

for po>i>

(3.3)

(2TT) 3 (2TT)

= 0

for

0
Thus, continuing the constant density of states approximation, we have for po>ai 2 g

lmG

ImG

N r

2oi ./_,•

,

(po—w)

[p0-u-t-XR(p0-w)J+tImG-1(p0-u)y

>+#o—<•>—Re2(£ g*Nir 0—<«>)" S(#o-u)-]j l-H-tan- 1 lmG~'(po—u) 2« 12

As required, this result agrees with (1.8) in the limit y.—»oo. We m a y make use of t h e fact that n » [ ImG-1(/>o—w) | and consider the energy region where M»|^o— w— R e 2 | by expanding (3.4) t o form a linear difference equation ImG" 1 (/>„) =

g>N

ImG_1(/>0—w)

(3.4)

Jl '

-u)

denote 7*=ImG - 1 (/>o), where k gives the integer p a r t of Pa/a, the difference equation (3.5) may be written (3.6)

(3.5) 2vfi

2w

2o>

with the boundary condition I m G - I ( ^ o = 0 ) = 0. If we The solution to (3.6) is 7o = 0 g2Nr

i

^ (-*)•• = ! — ( E """

7~ =

2a> \

2u

£-5

' ) ,

l+ X

(3.7)

J

where

x=

«1. 2w/i

3 E/<0

Thus, as the energy goes through multiples of the Einstein frequency, I m G - 1 goes through oscillations of FIG. 7. Imaginary part of the energy, r , as a function of E for the order of magnitude 1/800 times ImG _1 (aj). We the poles of the electron Green's function corresponding to the have considered only the no lines crossing diagrams in curves drawn in Fig. 6.

424 COUPLED ELECTRON-PHONON our approximation for the self-energy. We cannot trust the accuracy of these small fluctuations since the contribution of the crossed line diagrams which arise by including the full vertex may be of the same order of magnitude. To summarize, although b y taking a finite cutoff, the effects of multiple-phonon processes may be seen, their effect is too small to be considered reliable in our approximation. IV. ELECTRONIC POLARIZABILITY To investigate the effect the modified electron spectrum has on the response of the system to external fields, we consider the irreducible polarizability P(q) defined by 5

3 plana FIG. 9. Integration contour used in evaluating G(e,<) for large /.

~w

We introduce a vector vertex, r , as the solution to the equation 2p+q

r(M)=

*f

+#

d*k ~G(k+q)

2m

r #p P(q)=-2i

999

SYSTEM

G(p+q)G(p)T{p,q),

XG(k)D(P-k)r(k,q).

(4.1)

J {2KY and illustrated in Fig. 10. At first sight one might be tempted to replace T by unity as in the expression for the electron self-energy part 2 (p). That such an approxi-

FIG. 8. A multiple-phonon process (4th order in g) included in the approximation for the electron selfenergy.

(4.5)

The fact that solutions of these equations satisfy q0T{p,q)-q-r(p,q)

= G-\p+q)-G->(p)

(4.6)

follows by taking a linear combination of E q s . (4.4) and (4.5) to form an equation for (qoT — q - r ) . By direct substitution of the assumed relation (4.6) into this equation one finds q°r(p,q)-n-r(j>,q) = ?o-«p+,+ e,-2(H-?)+2(/.) = G-'{p+q)-G-\p),

(4.6)

as required, since in our approximation mation is inconsistent is most readily seen from the Ward identity 6 (4.2) dpo

dpo

This relation and a "generalized" Ward identity of the form q0r(p,q) = G-\p+q)-G-i(p) (4.3)

f d'k 2 ( # ) = - t f / —-G(k)D{p-k).

(4.7)

J (2*Y The Ward identity (4.2) follows if we first take the limit q—> 0 and then let q0—* 0 : r(/>,0)=lim 50-*0

G- 1 (p+q0) -G~*(p) ?o

dG-Kp) Spa

Therefore, in the limit q0—> 0 and qvr/qo—• 0, we have is valid in the limit q —> 0. These relations are a consed2 g*N quence of charge and current conservation. The proof (4.8) of these relations is given in Appendix B. If 2 is apr(#,o)=i =i+dpo -pt proximated by (1.1) it is straightforward to show that (4.2) and (4.3) are satisfied if V is evaluated within the Since fN/u' is typically ~ J, the corrections to the free ladder approximation, 7 illustrated in Fig. 11. T h a t is, vertex are at least of order unity in this limit rather the scalar vertex, r is approximated by the solution of than ~ u / £ j « l which Migdal has found for the integral equation \q\»{o>/EF)pF. T(p,q) = l+ig*

d'k

/

G(p+q)

•G(k+q)G(k) (2T>

XD(p-k)T(k,q).

(4.4)

6

See D. Pines, The Many-Body Problem (W. A. Benjamin, Inc., New York, 1962). • J. C. Ward, Phys. Rev. 78, 182 (1950). ' Y. Namhu, Phys. Rev. 117, 648 (1960).

G(p)

FIG. 10. Diagram corresponding to the definition of the. irreducible polarizability, P(q).

425 1000

S. P4"

F. N G E L S B E R G p+q

AN D

J.

R.

SCHRIEFFER

and the angular average of 8 is defined to be

k+H

— [<3(P,q)dVp= £ 4lT .>

FIG. 11. Ladder approximation for the vertex function.

We now turn to the problem of determining the polarizability in the long-wavelength limit subject to the condition that the phase velocity ?o/|q| of the wave being considered is large compared to the Fermi velocity vFIt is convenient to rewrite P(g) as f d*p 2 P(q>= - */7rTS
J (2*y

a?)

where g is defined as 2(t,q)-G(p+q/2)G(p-q/2).

(4.10)

j9o(P,q«)Xr(P,qa)

f d*k X,(p,qo) = 8,.*+i? / LSo(k,qo)X,(k,qB) J {2ir)* + ---^{k,qo)Xll{k,q,)-]D{p-k).

r d4k X(p,q) = l+ig> / G(k+q/2)

I

2 T —

(4.11)

Since the right-hand side of (4.11) is independent of p, we assume X can be expanded for small | q as a power series in q2: X(p,q)=

£

|q|**.(M>).

(4.12)

If P and G are also expanded as a power series in |q|, it is clear that only terms even in | q | enter and that only the spherical average of g need be considered; thus

O

I

2

3

4

(4.16)

Fio. 13. Intersection of ReII(y) and q^—w* giving the phonon frequencies for fixed q in the limit jolql^i*.

4

XG(k-q/2)D(p-k)X(k,q).

(4.15)

While all the 8's are known explicitly, we know X, explicitly only for y = 0. Nevertheless, an exact solution can be given for Po(?o) and Pi(qH). To see this consider the q2 expansion of the vertex equation (4.11):

2

(2ir)

d4p

-2ij,„

+ &(P,qo)X,-2(P,q
The vertex function X(p,q) = r(p—q/2,q) satisfies the equation

J

(4.14)

Thus, Eq. (4.9) becomes

°,(q0) =

Long-Wavelength Polarizability

|q|"S^,!Zo).

*—0 (even)

5

For v^0 multiply by <3<s(p,qo)Xa(p,q<s) and integrate over p: dtp

/

<3Q(p,qo)XD(p,qo)X,(p,qa) d'p

P(q)^

L

|q|'P.(?o)

'I (M

(4.13)

l<3o(P,qo)Xr(P,qo)+ • •

-Q^p^XoipM

X£X0(p,g0)-lJ.

(4.17)

In Eq. (4.17) we have used Eq. (4.16) for v=0 to simplify the right-hand side. By rearranging Eq. (4.17) we find the integral required to obtain P,(qo) can be expressed as FIG. 12. RePfa) plotted as a function of jo in the limit ?o/|q|»HF.

d"p

/

[So{p,qo)Xr{p,qo)+

d4p

"V

2

3

CM

••

•SAp,9«)Xo(p,qo)]

i
(4-18)

426 COUPLED ELECTRON-PHONON

1001

SYSTEM

Thus for i/= 2 we find Pi(«o) = - 2 *

f #P

— - 92(/>,?o)AV(/»,go), J (2x)4

(4.19)

which can be evaluated directly since 92 a n d Xo a r e known functions. We note that as in the exact expression for the polarizability, in our approximation, -Po(
\ J

-G(p+q/2)G(p-q/2)X0(p,q0) (2TT) 4

r d'p = 2*/ — • LG(p+q/2)-G(p-q/2n

= 0. (4.20)

J (2*y

FIG. 15. Spectral weight function in the Debye model for ep = 0.75w and a~\.

2

Thus, to order q the real part of P(q) for qo/ | q [ »flf is 11(9), since U(q) = fP(q). But Dyson's equation gives D(q) = Do(q)+D0(q)n(q)D(q) or D0-l(q) =

FIG. 14. Spectral weight function A (cf,xu>) for the electron Green's function in the Debye model for e9 = 0 and the coupling strength a = i- There is a delta-function contribution at x=0.

given by ReP(q)-

3\

qj (1+gW/u,2)2 qo/ X{2+

3gW or

g*N

(4.21)

ur—qo

qo2-rf-n(,q).

Since we are willing to accept a spatially nonlocal interaction, II may be a function of q. However, our result (4.21), is also a function of qa not linear in qo2. Thus, we could not start with a Hamiltonian formulation and arrive at an interacting phonon spectra with a single frequency for small wavelengths. Another question we may put forward is the following. Suppose the initial Hamiltonian had an Einstein spectrum; using the perturbation approach of the previous sections, what is the interacting phonon spectrum? The condition go/'\q\^>vF used in the calculation for the polarizability is appropriate for the long-wavelength Einstein phonons. The dressed phonon frequencies are given by the poles of the dressed phonon propagator. D-i(g) = qo'-ui-n{q,qo)

= 0,

A plot of ReP(j) as a function of 90 is shown in Fig. 12for?0/|qi»F. We initially chose a phonon propagator corresponding to an undamped Einstein spectrum. Experimentally this is a valid assumption for large momentum transfers. One question we may answer at this stage is whether it is possible to arrive at this spectrum for the interacting phonon field starting from a Hamiltonian formulation. That is, our interacting phonon propagator is chosen to be IM(?) = ?o ! -u ! . We have implicitly calculated the phonon self-energy

FIG. 16. Spectral weight function in the Debye model for t9 = 2& and a — \.

(4.22)

427 1002

S. E N G E L S B E R G A N D J. R. S C H R I E F F E R becomes g<

mkr

mkp

where

13-

/

\

3"

/

\

a .2 -

/

c2 = mvF!/3M. The variables of integration are changed by introducing

\

q2=(p-k)2,

2jq|
Equation (5.1) may then be written

X(p)=-

\P\Pr

FIG. 17. Spectral weight function in the Debye model for ep = 5u and a = J.

1

iT^mc' f dkodt Z9d9~,

(2TT)3

(Po-koy-cY+i& koZ-\~€

x-

(5.3) 2

• (koZ) where TJ(q) = g2P(q). Thus, the dressed frequencies are given by the intersection of the curves 11(g) and qa2—c^ The only dependence of 2 on p is in the factor 1/1 p |. in Fig. 13. It is interesting to note that two roots exist If we put | p | =&*• then 2 and Z are independent of e, for each value of q, one above and one below the bare and the techniques used in the treatment of the Einstein phonon frequency. As q —> 0, the roots approach a>. spectrum are applicable. Since the dominant contriSo t h a t if an Einstem spectrum is a reasonable choice bution to 2 comes from states of energy about the for the noninteracting phonons, we would expect to Fermi surface, this replacement is justified. If particleobserve experimentally the two branches splitting as hole symmetry is maintained and the cutoff extended to we increase the momentum transferred to the lattice. infinity the self-energy is

V. ELECTRON SPECTRUM IN THE DEBYE MODEL

2 (/>„) =

We now consider a model of the lattice in which the phonons are described by a Debye spectrum, 4 that is, there exists a linear relation between the energy and momentum carried by a phonon, u , = c | q [ , where c denotes the velocity of sound and q ranges from zero to qo, the Debye wave number. The corresponding maximum phonon energy is denoted by 01. Zero phonon damping is again assumed. We begin immediately with the approximation for the self-energy used previously in which r(p,q) = l,

2(p) =

d*k

•'I

&'

lp?

If

dk0-f

dko

" dqq3 Jo

(2ir)2

(pa-k<s)2-c2q!S+i

(5.4)

As in the Einstein case, the boundary value of I m Z from above for po>0 is obtained from the delta-

1

(2TY (> 0 -*o) J -Wp_ k 2 +tS k<>Z(k)- t

(5.1)

To obtain an estimate of the coupling constant we use the deformation potential model without corrections for umklapp processes 8 :

|4irZeyAr\"2 "p-k

I ]q| \M)

q2

q2+*.'

(5.2)

where we choose the static approximation to the dielectric function K and k, is the Fermi-Thomas screening length k,2 = ie2mkr/ir, where Z is the atomic number, N the ion density, M the ionic mass. In the long-wavelength limit, (2) »J. Bardeen and D. Pines, Phys. Rev. 99,1140 (1955).

Fio. 18. Poles of the electron Green's function on the first and second branches in the Debye model. The real part of the energy, E, is plotted as a function of the kinetic energy tp measured relative to the Fermi surface. The parameter a was chosen to be 1/24 for this calculation.

428 COUPLED ELECTRO N-PHONON

1003

SYSTEM

function contribution and is given by

for

pn>o>.

— irpi for

pa
The results of the integration are

ipW

Re2(£+*T) =

-1

( [ C E + < o ) » + r ' I ( £ - « ) ! + F 3 co» (£+o>) 2 +r 2 2 3 2 f£u> +K-E --Er ) In —2 2 2 - + - In 2 2 2

16C^F 1

(£ +T )

3

(£-u>) +r

+ (fr 3 -2I\E 2 )tanImZ(£+*T) = —

u2r

8c 2 /»/

/r» £ T \

1 \«,6

2£IV 2

2 2

(£ +r ) +(r 2 -£ 2 )a) 2 )

[(£+«) 2 +r 2 ][(£-to) 2 +r 2 ] (£ s +r 2 ) 2

2 /

+(

£ r 2 ) tan-

2£IV 2

2 2

2

2I\ 2

(£ +r ) +(r -£ )«

~ 1 — tan-

2

3

2

£ +r 2 +« 2

(5.5)

The spectral weight function A (ct,po) which is denned by (1.7) can be calculated from (5.5) with r —* 0. One finds uA (tp,xu) =

• {x-e„+a[x+x*In11-1/^|

,

(5.6)

+ln| ( * + ! ) / ( * - 1 ) | ] } » + ( < W ) V ( * )

expect in general multiple peaks in the electron spectral weight function which describe various decay modes 1 a majority of the total weight in limiting cases (i.e., and e—>0 and «—>»), a quasiparticle picture is expected CO to be insufficient for calculating system properties (5.8) involving finite wavelength and frequency excitations. 24ppV This does not preclude the use of Green's function Plots of A for a=i are shown for ep = 0, 0.75a>, 2u, and methods since the approximations used above may 5w in Figs. 14, 15, 16, and 17, respectively. This value well give an adequate description of these properties of a was chosen to make the damping rate for the even though the quasiparticle scheme is inappropriate. Debye model agree with the calculation presented above for the Einstein model when * > 1 . It should be noted that in Fig. 14 the delta function at a = 0 (for ep = 0) is the dominant part of the weight function. It is clear from the plots that as in the Einstein model, the quasiparticle description breaks down unless e<3Cu or £t>o>. The poles of G{p) on the first and second sheets are given for the Debye model in Fig. 18 fora= 1/24. There is a strong similarity between this plot and the corresponding curves for the Einstein model shown in Fig. 6. The imaginary part of the electron self-energy 2/(£) is plotted in Fig. 19 for the two branches of Fig. 18, showing the well-known cubic damping rate for small E on branch I. where

\x\l;

0<|*|
2

CONCLUSION While the above calculations are for highly idealized models of the coupled electron-phonon system, we believe the qualitative features of the results are characteristic of physical systems. Specifically, we

FIG. 19. The imaginary part of the energy I", plotted as a function of E corresponding to the curves given in Fig. 18.

429 1004

S. E N G E L S B E R G

A N D J. R. S C H R I E F F E R

ACKNOWLEDGMENTS

use of the commutation relations

We would like to thank Dr. T . T . Wu for indicating the integration technique used in Sec. I I . We are also grateful to Dr. Kenneth Johnson and Dr. E. P . Wigner for several helpful discussions.

[p(M,nt(k',0J=«'«*.k' [*(k>o,vt(k'><)]=[n(k,0,nt(k')o>o. Since we choose the phonon field Hermitian in coordinate space, we have

APPENDIX A. FORMAL DEVELOPMENT OF EQUATIONS

*t(k) = * ( - k ) ,

The Hamiltonian we start with is1-8 (h= 1)

n+(k)=n(-k).

The equation of motion obtained is

ff=£e(k)c,,t(k)c„(k)

d2

k.«

+h Z [nt(k)n(k)+«»(kW(k)*(k)] k

+ £

-V>(k,0=-o>(k)V(k,0

dt2

-2g(k) £ et(k',0c(k'+k, /)-/(k,0 , (A3) j(k')e.t(k)e,(k-k')*(k')

tt.k'.ff

+£J(-k)*(k),

(Al)

k

where e(k) = *(|J6|) is the electron's kinetic energy as a function of momentum, minus the chemical potential ii; g{k) = g(\k\) is the electron-phonon coupling function and u (| k |) is the phonon frequency. J (k) is an external source of phonons which we will use in generating the

where the spin sum has been accounted for by the factor of 2. The electron and phonon time-ordered Green's functions are defined by the equations 1 G(k,/;kY)=-J-

($, oo j $, — « )

= -* D(k,t; k'/) =

Dl-q)

($,+»|r(c(k,0ct(k'/) ) ! * , - • >

6<*(k,0> *<*(-k',0> = «/(k',/') SJ(-k,t)

= -»{(r(v(k,0t»t(k'>o)> -(^(k,0)(^(k',r))}. (A4)

G(p)

FIG. 20. Diagram representing Dyson's equation for the electron Green's function. Green's function equations and then set equal to zero. We work with a box of unit volume and use periodic boundary conditions. The equation of motion for the electron field is obtained by considering the commutator dc(k,0 [e(k,0,ff] = * -€(k)e(k,/) at +£«(k'Mk-k',0*(kV),

The Heisenberg state vector !f>, — » ) represents the ground state of the electron-phonon system, specified by a complete set of observables whose eigenvalues are given at the time minus infinity. T h e definition of the electron Green's function and the field equation (A2) are sufficient to obtain

* e(k)|G(k,/;kr) . dt J = 5 k , k .o(/-0-ȣg(k")

(A2)

k'

where we have used the anticommutation relations {c„(k,/)A. t (k',<)} = 0,.
{c,(k,0,c.'(k')<)) = {c/(k,0,<:,'t(kV)} = 0. Since the interaction is spin-independent, the spin variables,
X(T((k",t)c(k-k",

/)ct(k',0)> • (AS)

The phonon Green's function equation is generated by taking the functional derivative of the ground-state expectation of the phonon field equation with respect to / ( £ ' / )

a2

-D(k,l,k',t')+<S(k)D(k,t; —1 dt2

k't')

8 =ok,k-o(/-/')-2g(k)£ k"

. (A6) o/(k'/)

430 COUPLED

ELECTRON-PHONON

SYSTEM

We use the following identity to rewrite the equation for the electron Green's function

Then (A5) becomes ,_ „

-*<7Xk"/'MMc+(k'/)}>

\i «(k)]c(k,<;kr) L dt J +* E «(k")[

- « / ( - k " , t")

G(k,t; k'l')-

—7

-+i(v(k",t))}

X G ( k - k " , /; k',/') = 8k,k-a(/-0 which ma

-i((p(k'V")>{r{c(k,/)(;t(k',/')}>-

K-*»]

1005

E g ( k - k ' " ) ( v ( k - k ' " , t))G(k,l;

y

be

rewritten as

k'/)

k"'

/• SG-UWijkj^) +i / <*M/« £ g(k-k'")G(k'",<; k,,/0 G(k2,(2; k ' / ) = «k.k<«('-/'), y k'"*,*, «y(k"'-k, o where k'" = k - k " . Thus, \i—«(k)lc(k,/; L d/

k ' / ) - E * ( k - k " ) M k - k ' " , 0>G(k"7; k',/')+« /"*iAt*i

J

k'"

E

7

g(k-k'")G(k"V; k./O

k'",ki, k2 , kj

aG-'Ck^r.kj,^) x

By denning the vertex function as

G(k 2 ,/ 2 ;kV')C(k 8 ,/ 3 ;k'"-k, t) = KvS('-t')-

8<*>(kS)/.))

r(ki,
1

SG-'Ck^uWs)

g(k.)

«{*.(k,,/,))

,

(A7)

(A8)

Dyson's equation for the electron Green's function may be written [dt" E [Go-1 {k,l; k " / ' ) - 2 ( k , i ; k",/")]G(k"/'; k ' / ) = S k . k -8(/-«'). y

(A9)

k"

In (A9) we define the inverse of the free-electron Green's function by

G0-1(kJ/;k"/')=j[i—€(*)\lk..-g(k-k")<#>(k-k",0>)«a-<") and the self-energy, 2(k,i; k"t") = i [dhdh 7

E

«(k-k'")g(kj)G(k'",<'; k lF
k ' " - k , /).

(AlO)

k"',ki,ki

We may take advantage of the translational invariance of the system for the source term set equal to zero, and define G(k"',/;k 1 ,/ 1 ) = 8k-.k1G(k1;/,/1)

rdH

= 8k»-.k,/

e-w-^GikJ

(All)

and similarly for the other two-point functions. The three-point function, due to translational invariance depends only on two 4-momenta r(k,,/ i ; k " / ' ; ka,/3) = S kl _ k «, k ,r(k",k 3 ; *,,/",/») r dko" dk3a = 8k 1 -k". l ,/ _e*»"<"-'")e- i t ' 0 ( "-«r(k",* 3 ). J 2r 2w

(A12)

431 1006

S. E N G E L S B E R G

AND J .

R.

SCHRIEFFER

These properties when used in (A10), lead to the standard form (Fig. 18) 2 W = +* / —f(
,

(A13)

when we take the limit of an infinite volume continuum. The phonon Green's function equation, (A6), may be written in a similar form: /d1 \ ( —+o>2(k) )D(k,l; k ' / ) = -«k.k-«(/-0+2*g(k) £ W / k"

6G(k"+k,t;k"t+) • 67(k'/)

= -«k,k'«(«-<')+2»«(k)f (k') / dhdhdl, /

£

G(k"+k, /; ki,/,)r(k 1; /,; k2,/2; k,,*,)

k",ki,k!,ki

XG(k 2 ,k; k",t*-)D0t,,tt; k ' , 0 -

(A14)

When translational invariance, the infinite volume limit and time Fourier transforms are used, [AT 1 (k)-Tl(k)jD(k)=

1,

(A15)

if the inverse of the free phonon Green's function and the phonon self-energy are defined by DQ~i(k) = k02-<J(k) r d*p

n(*)=-2#(*) /

(A16)

—-C(p+k)G(p)r(p,k).

J f»*

To complete the formal development we consider the equation obeyed by the vertex function. Using (A8), (A9), and (A10), we obtain

r(k,<; k'V"i k'/) = 8k-k".k'«(<-<")«(<-'')+* Uhdl^kdh y

£

g(k-k'")g(k2)G(k"7; k,,<,)r(k,,/,; k4,*<; k ' / )

k",ki,k 2

XG(k4)/4; W ^ k , , / , ; k " / ' ; k2,/2)£>(k2,/2; k'"-*, l)+i I dhdh E y

XG(k'"f<;k^i)C

——-Vr(k 1 ) < 1 ;k'/';k s ,/ 2 )Z)(k 2 ,/ 2 ;k'-k,0]. (A17)

The last term of (A17) represents the introduction of a new function since the functional derivative of TD with respect to (
^§P^

f\/\/VA =-AAJH/IA + V W t / Dtl D l » o<" W^^-JHMr

The electron self-energy (A13) then becomes V ,. N ) = t

^

.[ j

d

1 ,/„...-,,.. \ n / \ ^ W ^ f - f W ? ) .

M,M (A19)

The vertex equation, derived using (A8), (A9), and (A18) corresponds to the ladder approximation' shown in Fig. 11, when the variation of D with (ip) is neglected,

_ . . . , . f **** , . .._., , . T(p,q)=l+i g 2 (p-k)G(*+ 9 ) (A 18) J (2ir)4 V ' XT(k,q)G(k)D(p~k). (A20) This approximation for r is used in Sec. IV to discuss the electronic polarizability.

|T(p.q)|v/W\r °<«>

FIG. 21. Diagram representing Dyson's equation for

the phonon Green's function,

g(k-k'")g(k 2 )

k'",k l ,kj

APPENDIX B. WARD IDENTITIES T o derive the W a r d

identity* for T Used in Sec. IV

we consider the related vertex function

Tf{x,y,z)

432 COUPLED

ELECTRON-PHONON

(p.=0, 1, 2, 3), defined by

SYSTEM

1007

the relations

,

D'o(2).^(*)3=-V'(«)«,(*-x), (zo=*o)

A,(*,y,z)= = - / Pf(x',y',z)G(x,x')G(y',y)dx'dy'.

(Bl)

_, , . ., , , The current density 4-vector j? (for electrons of one spin orientation) is defined by

D'o( 2 ),^(y)]=^W« 3 (i-y), ( B6 ) c a n b e re duced to

(

(2o=yo)

3 dkAx,y,z) v - = — iG(z—y)h(z—x) «-o dz„ +iG{x-z)b{z-y).

J

(B8)

j,.(z) = *t(*¥00, M = 0, i. */ \ „ ., \ = 2 ^ W ^ - W W > ( , ) } l , = 1,2,3. (B2) In the absence of spin-dependent forces, j„ satisfies the continuity equation , . E — i , W = 0, *„=(/,*). (B3) "~° dzf • j j i f forces, /n,\ u ifT T ., In the presence ofr spin-dependent (B3) i.holds j? includes electrons of both spin orientation. If we assume the system to be translationally invariant it follows that the Fourier transforms of G and V can be represented by d*p

/

G(fi)e^*~''>—-

By inserting the Fourier representations of G and T into (Bl), it is easily seen that the 4-divergence of the right-hand side of (Bl) is 3 dK{x%£)

£

f

^

'J ^

^'^(P^~9^o(.P,q)lG(p)G(_p+q) dipdiq X e x p p # ( * - y ) + j j ( * - * ) > - — - . (B9) V

' Since the right-hand sides of (B8) and (B9) are equal, so are their Fourier transforms: i[G(p+q) — G{p)^\ = *'{q-f (p,q)-q(p,q))G(p)G(p+q).

(BIO)

This relation is equivalent to the Ward identity (B5) stated above. To connect T and T„ we note that r satisfies the equation

<0|r{ V (q,/")c( P +q,/'y(p,0}|0> ,

r„(*'y,») = / f,(p,q) exP[t> (*' - y) +iq(xf-i)y?-l.

(B4)

(2"")s

XG(p+q,/-li)G(p,(!-*')AiW!.

By taking the 4-divergence of (Bl) and using the continuity equation (B3) one readily obtains the generalized Ward identity qoT«(p,q)-1-r(P,q) = G-i(p+q)-G-i(p)

rj S e e

(B5)

for the vertex VK. To see this in detail, consider the 4-divergence of A,, with respect to z„

(A5)

satisfies t h e

(Bll)

a n d ( A 8)-(A10).] The phonon field, „, equation of motion

-(—+«/L(q,<") = .D<")*l(q><") = ?«p(q/'),

(B12)

where

» dK(x,y,z)

p(q/') = j V(z)4>(*)
/Tfr'n(Mi(t/M\./ N = (r{Do(«),*(*)>t(y)})«(zo-»o) +
(5^T^ ( ^ , °' ) ) > ' ^^

The two commutator terms arise from differentiating the time ordering operator with respect to z0, while the last term vanishes according to (B3). With the aid of

Therefore, by applying the operator A r K q . O (Bll) we obtain

to

<0| r{p (q/')c(p+q, /V(p,fl> |0> =_

/r(P+<1' 'i;P.*'A'»)D«-,(«.'")D(q, «,-<") XG(p+q, t'-h)G{p, tt-t)cU1dttitt.

(B13)

We note, however, that aside from spatial coordinate

433 1008

S. E N G E L S B E R G

AN ' J .

Fourier transforms, (Bl) and (B13) differ for j» = 0 only by f o being replaced by TDtrlD. Hence, the relation Ta(p,q) = Do-1(.q)D(q)T(p,q) (B14) is valid. If we assume that Tu(p,q) is analytic in q so that q-T(p,q) vanishes for q—* 0, we need only show D~l(q)Do(q) = l in the limit q—> 0, in order to prove (4.3). The relation q0T(p,q) = G~1(p+q)-G-i(p) then follows from (B5), (B14) in the limit q—> 0. To prove

R.

SCHRIEFFER

D-1(qo)D0(ga) = l or equivalents Do~1(qo)D(q0)'= 1 we useEq. (B12): D0-K
434 PHYSICAL REVIEW B

VOLUME 56, NUMBER 21

1 DECEMBER 1997-]

Radiative corrections to the long-wavelength optical-mode spectrum of the electron-phonon model: Absence of mode-splitting effects and hardening of the mode A. S. Alexandrov Department of Physics, Loughborough University, Loughborough LEI 1 3TU, United Kingdom J. R. Schrieffer National High Magnetic Field Laboratory and Department of Physics, Florida State University, 1800 E. Paul Dirac Drive, Tallahassee, Florida 32306 (Received 21 July 1997) Renormalization of the phonon spectrum of the Einstein form due to a weak coupling with electrons is calculated. No splitting of the phonon mode is found. A ^-dependent hardening of the long-wave optical phonons contrasts with the softening of acoustic phonons in a weakly coupled electron-phonon system and with the softening of all phonons in the strong-coupling polaronic regime. [S0163-1829(97)02646-5]

The problem of coupled electrons and phonons was formulated by Migdal1 and by Engelsberg and Schrieffer2 for the Debye (acoustic) and Einstein (molecular) phonons, respectively. As Migdal showed, one can omit the vertex corrections for Debye phonons arriving at the renormalized sound velocity F— s(l —2\)m, where K is the coupling constant. However, in many doped semiconductors, there are optical phonons which are practically dispersionless in the long-wave limit, td(q) = a>. In this case the vertex corrections appear to be important when the phonon's phase velocity is of the order or greater than the Fermi velocity, i.e., m ^\q\vF,2 so that Migdal's theorem does not hold even for weak coupling, \ < 1. Adopting the ladder approximation for the vertex function and expanding it for small |q| as a power series in |q| 2 , it was found that the irreducible polarizability P(q) has a pole at q0 = ai:2 N lvF\q\\22

p{q)=

3g2N/w2-g2N/(w2-q2)

+

^[-^-j

a + ,W)'

' (1)

with N=mpF/2tr2 the density of states at the Fermi level, pF=mvF, and g the electron-phonon coupling constant. Here and below qs(q,qo) and h = c=l. This expression can be simplified near the pole q0 = oo and \=g2N/b)2<^l. Keeping first order terms in \ , one obtains

finite |q|.2 Later on, several authors3"7 discussed the vertex corrections in connection with the superconducting Tc. In particular, Grimaldi et al? obtained a strong enhancement of Tc from the first-order vertex correction for a long-range electron-phonon interaction. In this paper we recalculate the irreducible polarizability and show that the pole of P(q) in Eq. (1) is an artifact of the expansion in powers of |q| of the vertex function prior to the integration over the frequency and momentum of the polarization loop. As a result, there is only a hardening of the Einstein phonon mode without any splitting in the weakcoupling regime contrasting with the softening in the strongcoupling polaronic regime, \ > 1 . 8 Thus the vertex corrections play no qualitative role in the renormalization of the Einstein phonons. To illustrate the point, it is sufficient to calculate the zeroth- and first-order diagrams in powers of X as shown in Figs. 1(a) and 1(b), respectively, in a weak-coupling regime, \<1. The zeroth-order electronic propagator is determined as G-\p)=p0-t+iSsgn(£)^p0-{+iSsgn(Po), where £=vF(\p\—pF) andp=(p,p0). larizability, Fig. 1(a), is

(5)

The zeroth-order po-

which yields two poles of the dressed phonon propagator D-\q)

= q2-u>2-g2P(q)

= 0.

(3)

The two branch splittings as one increases the momentum transfer to the lattice are \ 6>==& ±

' 2vfVF^

A(q)

(4)

AM

B<<0

b

for \q\<€a>/vF. Therefore, if an Einstein spectrum is a reasonable choice for the noninteracting phonons, from Eq. (2) one would expect to observe the two branches splitting at

FIG. 1. Zeroth (a) andfirstorder (b) in \ irreducible polarizabilities.

0163-1829/97/56(21)/13731(3)/$lO.O0

13 731

56

© 1997 The American Physical Society

435 BRIEF REPORTS

13 732

p<0){

^-2'i^G^+^G^-i^Ldxi

i

56 dtjdpo

[po + qo~i-vF\q\x+iS

sgn(p0 + q0)][p0-^

+ iS

sgn(p0)]' (6)

with 8= + 0. The result depends on the order of integrations if the limits for £ are extended up to ± ° ° . Therefore the frequency (p 0 ) integral should be calculated first. For [q|
Pm(q)

=N

9o

tfo+fj-lql

vF\q\

«o-i>Flq|

In 9o,


\<0+qo\ . k o | - « > . . In | j — iv W(\q0\ — (x>) (13)

(7)

As a result, the first-order polarizability, Eq. (8), is given by

which is the same as Eqs. (1) and (2) with \ = 0 in the long-wave limit, \q\
PaHq) = 2g2N:

XhV

«5-Mql) 2 1o + vF\q\

In

•(2vF\q\)' go

°> 2

2|

to - g 0 l

°>

\qo\ x l n r — — r — i - n - — - — © ( ? o -<>>) P(u(q)

= A(q) + A(-q)

+ B(q),

(8) (14)

where

Near ^ 0

A ( ? ) = - 2 i | ^ T C 2 ( P ) G ( P +4)2(P)

(9)

=

<"*

we

obtain in the long-wave limit

P ( ? ) = / ,<0 >(«) + /

, < , )

t e ) - ^ ( ^ )

(l-2\ln2), (15)

and

B(q) = 2g

d^pd^p' TrG(p (2 IT)

II

XG(p'

+

+

q)G{p)G(p')

q)D(p-p')

(10)

with the bare phonon propagator £> \q) = ql~w2 + iS. The electron self-energy 2(/>) was calculated in Ref. 2 as

* W / \po + u\ 2-(/') = - -y^-1 In rr—ZT^jr + ' ^ sgn(/? 0 )©(l/'o| —&>)) (11) where ®(x) = 1 forjc>0 and zero otherwise. The remaining integrals in Eq. (9) lead to g2N2 A(q)=^

ql-^FHY

In i—— \(o--q-0\ « ( ?0 - t o )

ITT

<7o , \<»+cio\ •Ina> \<0-qo\ (12)

to

By integrating in Eq. (10) first over momenta and then over frequencies, we obtain

which is drastically different from Eq. (2). There is no pole in the polarizability, and consequently there is no splitting of the Einstein phonon. There are no poles in higher orders either. The absence of any extra branches of collective excitations is a consequence of the conservation of the number of degrees of freedom in a coupled electron-phonon system. Because the Coulomb repulsion is absent from our consideration, there is no other collective excitation except the dressed Einstein mode. The vertex and self-energy corrections are negligible as one can see from Eq. (15) if \<\. There is only a hardening of the renormalized optical phonons depending on the momentum as |q| 2 :

=

to+3

.

(16)

to

The hardening, Eq. (16), contrasts with Migdal's softening of acoustic phonons and with the softening of all phonons in the strong-coupling polaronic regime. 8 This contrasting behavior is the result of the familiar repulsion of energy levels under any perturbation. The acoustic phonon energy sq is well below the characteristic energy vFq of the electron-hole continuum, so that if these two excitations are coupled, the lowest acoustic phonon level shifts downwards. On the contrary, the energy of the long-wave optical phonon is well above the electron-hole continuum, w>vFq, and so the upper optical phonon level shifts upwards. In the polaronic

436 56

BRIEF REPORTS

regime the spectral width of the polaron spectral function is about g2/o>2. It plays the role of an electron-hole continuum being much wider than the width of the phonon spectral function. This is why in the strong-coupling polaronic regime the lowest phonon level is shifted down by the phononpolaron coupling.8 In conclusion, we have calculated the electronic polarizability of a coupled electron-phonon system including the first-order corrections in powers of X. These corrections have

13 733

no role to play while K<1. We have found a q-dependent hardening of the Einstein phonons rather than their splitting. The hardening in the weak-coupling regime and the softening in the strong-coupling polaronic regime of optical phonons would allow one to distinguish these two regimes experimentally in doped semiconductors. J.R.S. acknowledges support of this work by DOE Grant No. DE-FG05-94ER45518.

5 'A. B. Migdal, Zh. Eksp. Teor. Fiz. 34, 1438 (1958). M. Grabowsky and L. J. Sham, Phys. Rev. B 29, 6132 (1984). 6 S. Engelsberg and J. R. Schrieffer, Phys. Rev. 131, 993 (1963). Y. Takada, J. Phys. Chem. Solids 54, 1779 (1993). 3 P. B. Allen and B. Mitrovic, in Solid State Physics, edited by H. 7C. Grimaldi, L. Pietronero, and S. Srassler, Phys. Rev. Lett. 75, Ehrenreich, F. Seitz, and D. Tumbull (Academic, New York, 1158 (1995); L. Pietronero, S. Strassler, and C. Grimaldi, Phys. 1982), Vol. 37, p. 1. Rev. B 52, 10 516 (1995). A High Temperature Superconductivity, edited by V. L. Ginzburg8A. S. Alexandrov, Phys. Rev. B 46, 2838 (1992); A. S. Alexanand D. A. Kirzhnits (Consultant Bureau, New York, 1982). dra, H. Capellmann, and U. Gobel, ibid. 46, 4374 (1992). 2

VII High-Tc Superconductivity

This page is intentionally left blank

439 SCHRIEFFER'S P A P E R S ON H I G H - r c S U P E R C O N D U C T I V I T Y

Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305-4045,

USA

Almost immediately after the discovery of high-Tc superconductivity (HTSC), Bob Schrieffer became actively involved in this field, and made influential contributions in a wide range of subjects. In 1987, I joined the Institute for Theoretical Physics at UC Santa Barbara as a postdoc, where he was the director. I still vividly remember Bob's tremendous excitement about the discovery, and its profound theoretical implications. "It feels like 1956 again!", he said to me at our first meeting. More than ten years have passed since then and, as we look back at Bob's contributions to the high-Tc problem, we see that his far reaching insights have always guided the community in the right directions. ^From the very beginning, Bob identified antiferromagnetic (AF) correlation as a central piece of physics in HTSC. As Doug Scalapino mentioned in his commentary, Berk and Schrieffer discussed the possibility of superconductivity (SC) mediated by spin fluctuation as early as 1966. This idea was applied to the superfluid state of3 He, with some success. Even before the discovery of HTSC, Scalapino and collaborators [l] had considered d-wave pairing mediated by antiferromagnetic spin fluctuations. They showed that a purely repulsive potential centered around the (7r,7r) point in momentum space can lead to an oscillating potential in real space, which is repulsive at the origin, but attractive on the near neighbor sites. A d pair wave function best takes advantage of this form of the potential. Bob was very interested in this approach, but he also felt that this formalism would be a more appropriate starting point when one approaches the SC phase from the overdoped Fermi liquid side. At that time, this was a physical intuition, but later Bob put this idea behind more rigorous theoretical reasoning [2]. Therefore, Bob was strongly motivated to develop a theory starting from the AF insulating state at half-filling. Borrowing from his early work on polyacetylene with Su and Heeger, Bob developed the idea of a spin bag. The idea is that a hole doped into a AF insulator would locally suppress the AF order, forming a "spin bag." The distortion of the AF background follows the coherent motion of the hole, in a similar fashion to the polaron, where the electron motion is accompanied by the coherent lattice distortion. The spin bag like distortion of the AF background costs AF correlation energy, therefore, it is more favorable for two electrons to share the distortion. This effect creates a pairing correlation among the doped holes and could lead to a SC state at relatively high temperature, governed by the AF energy scale. This idea was the starting point of a series of papers Bob worked on with myself, Xiaogang Wen [3; 4] and Arno Kampf [5]. A qualitatively important finding of the spin bag approach is the SC pairing in the presence of long ranged static AF background [3; 4]. Therefore, the resulting state is a coexistence state of AF and SC. At the time of this theoretical work, there was not much experimental evidence for such a coexistence state, and indeed Bob's later work with

440

Arno Kampf [5] modified the formalism such that the pairing also works in the presence of dynamically fluctuating AF order. However, more recent experiments seem to indicate that the AF long range order could persist over a much wider doping range, and sometimes overlap with the SC phase [6]. Therefore, in the coexistence regime, the spin bag picture could provide a viable pairing mechanism. This work also gives the indication that the interplay and the competition between the AF and SC phases is a crucial ingredient of the HTSC problem, and this idea inspired subsequent theoretical developments [7]. The phase diagram of the t — J model is still an unsolved problem. But numerical calculations based on variational MC calculations [8] suggest that the AF phase at half-filling and the pure d-wave SC phase for doping large than x = 10% are indeed separated by a regime in 0 < x < 10%, where both orders coexist. The spin bag theory also produces a side result which has important implications on the Mott transition. The basic starting point of the spin bag theory is the spin-densitywave (SDW) approximation of the half-filled insulator. There was a common belief that the SDW approximation is based on weak coupling, while the effective spin-1/2 Heisenberg Hamiltonian would be the appropriate starting point at strong coupling. Since the HTSC problem is believed to be in the strong coupling limit, there were some initial criticism because the spin bag theory started from the SDW approach. This issue was resolved definitely in the long version of the spin bag paper [4]. It is shown that at half-filling, the AF phase of the Hubbard model on a square lattice is continuous as the interaction parameter U/t is varied. The ground state is always AF ordered in both the strong and weak coupling limits. Moreover the SDW approach works well quantitatively in both limits. In the large U/t limit, both the size of the ordered AF moment and the spin wave dispersion relation obtained from the SDW approximation agree quantitatively with the results obtained from the Heisenberg model. In this sense, there is no "Mott transition" at half-filling. Throughout his theoretical work on HTSC, Bob paid close attention to experiments. From the classic work of Schrieffer, Scalapino and Wilkins [9], it is known that the phonon mediated pairing interaction manifests itself directly in the single electron spectra as measured in the tunneling experiments. In his work with Z. X. Shen [10], Bob focuses his attention on a striking feature in the angle-resolved-photo-emission spectra, namely the peak-dip-hump feature. This feature reminds one of the phonon resonance observed in the tunneling spectra, and this leads Shen and Schrieffer to conjecture that the bosonic mode responsible for the peak-dip-hump feature is also directly responsible for the pairing interaction in the HTSC cuprates [10]. There are two natural candidates for this bosonic mode. One is the TV resonance mode observed in the neutron scattering experiment below Tc, while the other is a phonon mode. At this moment, the nature of this bosonic mode is still being actively debated [ll; 12], but it is clear that the Shen-Schriefer analysis had a major impact on the subsequent developments. As mentioned earlier, Bob held the point of view that the conventional spin fluctuation mediated pairing works only when one approaches the SC phase from the overdoped, Fermi liquid side of the phase diagram. As we know, this theory successfully predicted the d-wave

441

nature of the SC pairing state, in qualitative agreement with experiments. However, at a quantitative level, this picture encounter difficulties in the underdoped regime, where the AF order competes with the SC order. In a deeply insightful paper [2], Bob constructed a general argument based on symmetry principle and Ward's identity, showing that the pairing strength is significantly weakened by vertex corrections. In conventional SC, this type of vertex corrections can be safely neglected, since Migdal's theorem states that they are of the order of m/M, the ratio of the electron and ion masses, which is very small. On the other hand, in the HTSC cuprates, the boson that mediates the pairing can itself become soft, and lead to an ordering instability which competes with SC. This is perhaps the most interesting aspect of HTSC. The relationship between AF and SC has both a cooperative aspect, since the SC condensation energy does arise from the AF correlation energy, but also a competing aspect, in the sense that one form of order suppresses the other kind. Bob's argument based on Ward's identity does not exclude the possibility of SC pairing mediated by AF spin fluctuations, but strongly suggests a more unified treatment of both types of order. HTSC is one of the most important problems in condensed matter physics. Exciting and new theoretical frameworks still remain to be discovered. On the other hand, it is also remarkable that the basic BCS theory, coupled with the d-wave pairing symmetry, gives a qualitatively correct description of the SC properties. The most important outstanding problem is to understand the transition from the half-filled AF state to the of-wave SC state at finite doping. In this respect, Bob's work on the spin bag theory, on the coexistence of the AF and the SC orders and on the importance of the vertex correction will guide us for many years to come. References

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986). J. R. Schrieffer, J. Low Temp. Phys. 99, 397 (1995).* J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. Lett. 60, 944 (1988). J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. B 39, 11663 (1989).* A. Kampf and J. R. Schrieffer, Phys. Rev. B 41, 6399 (1990).* Y. Sidis et al., Phys. Rev. Lett. 86, 4100 (2001). S. C. Zhang, Science 275, 1089 (1997). A. Himeda and M. Ogata, Phys. Rev. B 60, (1990). J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Lett. 10, 336 (1963). Z. X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997).* H. Y. Kee, S. A. Kivelson, and G. Aeppli, Phys. Rev. Lett. 88, 257002 (2002). A. Abanov et al., cond-mat/0112126.

(The symbol * indicates a paper reprinted in this volume.)

442 VOLUME61, NUMBER23

PHYSICAL REVIEW LETTERS

5 DECEMBER 1988

Pairing-Bag Excitations in Small-Coherence-Length Superconductors A. R. Bishop, P. S. Lomdahl, J. R. Schrieffer,(a) and S. A. Trugman Theoretical Division and Advanced Studies Program in High Temperature Superconductivity Theory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received 6 June 1988) Localized baglike solutions in the pairing theory of superconductivity are studied. Starting from the Bogoliubov-de Gennes equations on a two-dimensional square lattice for a half-filled negative-t/ Hubbard model, cigar- and star-shaped bags are numerically obtained, inside of which the order parameter is reduced, self-consistently trapping an added quasiparticle. These nonlinear excitations are important when the coherence length is small as for the new high-temperature superconductors. Several experimental consequences are discussed. PACS numbers: 74.65.+n, 74.20.-z, 74.70.Vy

In the pairing theory of superconductivity, the ground-state energy gap A is uniform in space for a translationally invariant system.1 When a single quasiparticle is added to the system, the gap is assumed to remain uniform so that the excitation is in a plane-wave state k. As in Koopmans's theorem2 for the HartreeFock approximation to extended systems, A is unaltered by the presence of the excitation as the volume of the system tends to infinity. For nonzero temperature, the gap decreases because of the finite density of quasiparticles, with A vanishing at the transition temperature Tc, where the quasiparticle density is of order (%/a)2 per coherence volume £3, with a the mean electron spacing and S, the zero-temperature coherence length. Since £/a « 103 for conventional superconductors, the depression of the gap by the addition of one quasiparticle in a coherence volume is extremely small. However, for the new layered high-temperature superconductors the corresponding £/a is of order 1 —10.3 Therefore, when a quasiparticle is excited in a coherence volume the gap is locally substantially reduced even at zero temperature. This reduction forms a baglike potential which, if sufficiently strong, self-consistently traps the quasiparticle, as in a self-trapped polaron.4 Here the pairing field plays the role of the phonon field of the polaron. Preliminary studies of this problem have been carried out by Weinstein5 and by Coffey, Sham, and Lin-Liu.6 In this Letter we study the structure of such localization pairing-bag solutions to the Bogoliubov-de Gennes equations7 on a two-dimensional square lattice in the context of the "negative-t/" Hubbard model. As in the case of spin bags,8 for a half-filled band we find selfconsistent solutions for A which have "cigar" or "star" shapes depending on the symmetry of the orbital in which the quasiparticle is initially placed, as well as the initial spatial form of A. While these solutions break translational symmetry, this can be restored by forming linear combinations of such localized configurations suit-

ably phased to create a momentum eigenstate. Our numerical calculations as well as analytical estimates show that the mean bag size is of the order of the coherence length £,. For a/§ ~ A/EF =0.075, the pairing field amplitude in a star bag is typically reduced by =s 38% within the bag and a localized quasiparticle level appears at « 0 . 9 2 of the homogeneous gap. For A/E> =0.145, the corresponding numbers are 88% and 0.75, respectively. There are a number of physical consequences of these baglike quasiparticles. Since the conventional spatially uniform mean-field approximation is invalid for systems with such bag excitations, the statistical mechanics takes on a different form— a gas of nonlinear excitations which locally reduce the gap gives a very different free energy than the conventional mean-field Fermi gas of the weakcoupling theory. Also the infrared absorption and tunneling spectra should exhibit a substantial broadening of the gap due to rapid decay of injected quasiparticles into the lower-energy states. Again the near-normal core regions of deep bags can strongly modify other BCS consequences. We consider the negative-t/ Hubbard model on a 2D square lattice to model the CuC>2 planes in the oxide superconductors. The Hamiltonian is **

2* tnmCmsCns nms

2u ' nm

nmCm\Cn\Cn[Cm\

.

\i)

In the mean-field pairing approximation for the superconducting phase one has "

Zu InmCmsCns nms

JLi\J^nmCn\Cm\ nm

+ H.c.-|A„J7K„m], (2) with the self-consistency condition A*m = V„m(\cm1cn\\(p),

(3)

where Vnm is positive for an attractive potential and | 0>

© 1988 The American Physical Society

2709

443

VOLUME 61, NUMBER 23

PHYSICAL REVIEW

LETTERS

5 DECEMBER

1988

is a quasiparticle occupation number state. H can be diagonalized by making the Bogoliubov-Valatin transformation9: Cm 2*i*-UniYis ' VnisYi,-sl » (4)

Cns sss2*i lUni Yis ' vnis Yi, —s ' •

-4r _|_ _ i i

Here YL are the quasiparticle creation operators, [H,yl]=EisrL

[H,risl = -Eisyis,

Eis>0.

(5)

Without loss of generality we have chosen phases so that u„iS is spin independent.10 The u and v amplitudes can be determined by our taking matrix elements of the equation of motion of the bare operators c between an initial state | 0> and the state having one more quasiparticle in orbital i. As usual, we work within the grand canonical ensemble. Using vnis = —svni one finds EiU„i=2jm[ — tnmumi + A„mvmj] ,

._

j ]

! !"

! !

I

!\

! [ ! ! j 1 ! ] I 1 ! !

t ~i

L

44-

J '' ^ L i ! 1 ! 1 j ! T [ i 1I | » _ M i ~i! !j jI i1 i] i I •_ TBIj l j i l j r i ! TriTiT" _ ^

i

14

(6)

0.45 0.25 0.65 FIG. 1. One quasiparticle has nucleated a cigar-shaped bag EiV„j =Xm \-tnm Vmi+A*mUmi] , (7) on a 14x14 lattice with periodic boundary conditions and Ko = 2.5t. The top part of the figure shows A vs the site index n and the self-consistency condition becomes in a linear array (row by row). The lower part shows A on the A„m =Vnm'£lilumv*,(l -Nn) -um v*iNn] , (8) square lattice as a gray-scale plot. The bag was seeded at the sites where it nucleated. The energy level of the quasiparticle where Nis is the quasiparticle occupation number. is pulled into the gap. Its value is 0.430? and the band of Evaluating the expectation value of H in the state | 0>, higher levels starts at 0.565r. we find the total energy given by

£,=-Z£,(i-2>,,)+2;%^. i

(9)

I s ) nm V nm For an NxN lattice, we note that there are IN1 amplitudes u„ and v„. While this leads to IN2 eigenvalues Ej, only the positive values of £, correspond to the energy required to create a quasiparticle, whether it be a quasielectron above the Fermi surface, or a quasihole below; all physical excitation energies are necessarily positive. The negative eigenvalues correspond to the energy released when a quasiparticle (either electronlike or holelike) is destroyed. Therefore, we are only interested in the positive-energy solutions, and sum only over Ei > 0 in (9). Like most mean-field theories, the Bogoliubov-de Gennes equations are only qualitatively correct for very strong coupling ( K n / r » l ) . In this limit, they yield a symmetric self-consistent energy gap for the addition or removal of a particle of Ko/4. This is half the exact value, which may be obtained by direct calculation. For Vo/t—*0, the mean-field equations obtain some properties exactly. The small corrections for weak but finite coupling can be obtained by perturbatively including the difference between the mean-field and the exact Hamiltonians. One can calculate the dressing of the quasiparticle in a state in which the quasiparticle is localized at the origin, measuring the pairing field a distance r away. There is a zero-order effect from the Pauli principle that suppresses A on the scale of §, and a similar effect in first 2710

order from dynamic correlations. Details will be given elsewhere. This paper considers the intermediate coupling regime, with VQ less than half the bandwidth, where the equations are expected to be quantitatively accurate. We have solved the equations numerically on periodic NxN square lattices with TV = 4 to 16. Extremal solutions to the coupled equations (6) and (7) were sought by iteration in {«m,vm} until self-consistency (8) was obtained" for a given quasiparticle occupation, with various initial A„m profiles as "seeds." The eigenvalue distribution E, and symmetries of the associated eigenvectors {«„,-, v„,} were studied. To illustrate bag states we report here results for a local negative U, Vnm =Ko5 nm , and pure near-neighbor, isotropic hopping of strength t. In Fig. 1 we show the result of placing one additional quasiparticle in an orbital near the flat region of the square Fermi surface. Localization occurs, mirroring the symmetry of this orbital and producing a cigar shape. If orbitals near corners of the square Fermi surface are occupied, the localized state assumes their symmetry—the crossed cigar or star shown in Fig. 2. The local deformations of A„„ are accompanied (see figure captions) by an eigenlevel being drawn into the uniform gap present in the absence of added quasiparticles: the greater the localization, the deeper the gap state. The remaining continuum states suffer energy-level shifts which are limited to near-gap states as the bag becomes more delocalized.

444 PHYSICAL REVIEW

VOLUME 61, NUMBER 23

LETTERS

5 DECEMBER

1988

0.40

A 0.25 0.10 113

•

_i_j.--X~y-

--;-1"

._j..i. ft

15

- U - L - j - { -4, x _ _ | „

r

„ ;

-

r

- f -

- 1 - -

"f"r _

zt'tltk

it i

„

!

;' r

..:..;.

! j

! ! ! ]

! ] i

1 1 - ) +-T- 1 T

c

T't

J._

!

0.35

; ] :

+ 1 - I " (—!—

-|-| -|--

! ! ! i [ S ] 1 ] ! ] < : t T l T !

15 0.20

n 225 ' !'

! iTW

-:-]-{-!- f - -i - f "i <

-;-i

1 -I

i

i"! -ih i~

J.„;JPTJI

- - f r - -l-j-i-ff

•Lj.

113

2'

0.50

FIG. 2. Two quasiparticles occupying the two lowest energy levels have formed two star-shaped bags on a 15x15 lattice for Vo = 2.25t. The energy levels are again pulled into the gap and are almost degenerate, with values 0.363; and 0.366J. The band of higher energy levels starts at 0.464r. Note that the orientation of the stars was seeded to be slightly off diagonal in this case.

Finite-size effects enter sensitively as a function of the chemical potential and even or odd N for small systems12; it is important to choose the orbitals to be occupied by quasiparticles from appropriate positions in k space and to respect the symmetry of orbitals with seeding profiles. In addition a large V0 leads to highly localized bags which experience a Peierls-Nabarro-type lattice pinning and metastability. The ground-state energy gap decreases and £ increases as Vo/t decreases. Correspondingly the bag becomes shallower and increases its mean radius.13 A typical extended bag is shown in Fig. 3. We have not found a minimum value of V0 for localization at this mean-field level, regardless of the bag shape. Simple analytical estimates8 suggest that the localization radius = §. However, the effective £ [ « ( 5 f c ) - 1 , with Sk the range of states strongly affected by Vol is now strongly direction dependent because of the square Fermi surface. With increasing Vo/t the bags are increasingly isotropically localized on a single site. However, for weak coupling they are nearly isotropically extended except in very focused directions (two or four, for edge and corner states, respectively) for Fermi-surface orbitals. The binding energy Eb of a bag also increases with Vo/t, where the binding energy is defined relative to an added quasiparticle with uniform A. For example, on a 10x10 lattice and for star bags in the same initial Fermi-surface orbital, we find (V0/t,A/t,Eb/t) with values (2.5,0.615,0.088) and

I :1

15 0,20

0.30

0.40

FIG. 3. One quasiparticle has formed an extended bag on a 15x15 lattice for a smaller K0 = 2.0(. The subgap energy level is at 0.308f and the band of higher energy levels starts at 0.356(.

(3.0,0.854,0.147). In general the star bags seem to have a few percent greater binding energy than the corresponding cigar bags. We have also made preliminary studies of the interaction between two bags by placing one quasiparticle in each of two orbitals of appropriate symmetry. We can expect some bag attraction since (as for the bipolaron) the energy cost of reducing A locally is shared by the two quasiparticles when they are within the bi-bag area. However, the kinetic energy of localization competes with this effect, and it appears that the bag-bag interaction depends sensitively on the relative orbitals occupied and the bag sizes, symmetries, and orientations. Two quasiparticles (spin up and down) in the same orbital produce the most strongly attractive bags, resulting in a deeper single bi-bag. Two star bags arising from different Fermi-surface corner-point orbitals lower energy by orienting along a common diagonal (see Fig. 2). They appear to bind at a smaller separation as Ko increases. However, for K 0 ^ 3 , a hard-core potential13 prevents binding. The interaction is independent of relative spin—we have not included explicit spin-dependent interactions. It appears that two cigar bags may repel, attract as parallel cigars, or attract in a crossed-cigar (star) configuration, depending on the initial k states occupied. In all cases two bags are accompanied by two subgap energy levels. These levels are more strongly tunnel split as the bags approach or for stronger interactions. In summary, we have found baglike localized solutions to the discrete Bogoliubov-de Gennes equations on a 2711

445 VOLUME 61, NUMBER23

PHYSICAL

REVIEW

square lattice within a local negative-l/ model. The bag shape is determined by the symmetry of the bound hole (or electron). These excitations have the same spin and charge as the bare hole (electron), namely j and +e ( — e), respectively. As in the strong-coupling polaron, the true quantum mechanical states must be constructed out of the present broken-symmetry states by a phased, translationally invariant sum, to yield a band of bag states. Note that the hole and order-parameter field must move together at a large velocity (vy). This motion is expected to deform the bag, determine limiting velocities, and even prevent the condensate from following the hole in some cases. These issues will be discussed elsewhere, as will continuum limits and k-dependent scaling4 for this self-trapping problem. The influence of a chemical potential will also be reported elsewhere—the large density of states at half-filling is most favorable for superconductivity, but varying electron density changes the Fermi-surface structure and therefore bag shapes. The physical consequences of such pairing-bag states are under investigation, including electromagnetic absorption, quasiparticle and Josephson tunneling, thermodynamics, fluctuation effects, quasiparticle recombination, etc. We have also developed a time-dependent extension of the present theory, which will allow us to study dynamics of bag-formation and their interactions, localized bag oscillations, and decay channels—for instance of quasiparticle excited states to the uniform ground state by phonon and microwave emission. This work was supported in part by the U.S. DOE. One of us (J.R.S.) would like to acknowledge the hospitality of the Los Alamos Advanced Studies Program in High Temperature Superconductivity Theory, supported in part by the Public Service Company of New Mexico.

2712

LETTERS

5 DECEMBER 1988

'"'Permanent address: Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106. 'J. Bardeen, C. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). 2 T. Koopmans, Physica (Utrecht) 1, 104 (1933). 3 For example, Y. lye et al., Physica (Amsterdam) 148B, 224 (1987); L. P. Gor'kov and N. B. Kopnin, to be published. 4 For example, D. Emin and T. Holstein, Phys. Rev. Lett. 36, 323 (1976). 5 M. Weinstein, Mod. Phys. Lett. B 1, 327 (1987). 6 D. Coffey, L. J. Sham, and Y. R. Lin-Liu, Bull. Am. Phys. Soc. 33, 294 (1988). 7 P. G. de Gennes, Superconductivity in Metals and Alloys, (Benjamin, New York, 1966). 8 J. R. Schrieffer, X.-G. Wen, and S.-C. Zhang, Phys. Rev. Lett. 60, 944 (1988). Note that spin bags occur in an antiferromagnetic background as the system tries to maintain nesting of the Fermi surface. Superconductivity, in contrast, is insensitive to nesting but sensitive to the density of states. 9 N. N. Bogoliubov, Nuovo Cimento 1, 794 (1958); J. Valatin, Nuovo Cimento 1, 843 (1958). 10 The complex order parameter is essential for time- and magnetic field-dependent effects, as will be described elsewhere. "We have also investigated an equivalent minimization of the total energy (9). 12 For zero chemical potential, BCS pairing in a small system occurs in a set of degenerate states for TV even and in a small insulating gap for TV odd. Thus as Vo—• 0, A is proportional to Vo for TV even and zero for TV odd, in contrast to the exponential dependence in an infinite system. When Vo increases sufficiently that | is less than the system size L, a crossover occurs and A approximates the infinite-system value. 13 There are two important length scales: the mean £, (determined by Volt) and the range of interaction in V„m, which determines a "core" size within |.

446 PHYSICAL REVIEW B

VOLUME 39, NUMBER 16

1 JUNE 1989

Dynamic spin fluctuations and the bag mechanism of high- Tc superconductivity J. R. Schrieffer, X. G. Wen, and S. C. Zhang Institute for Theoretical Physics, University of California. Santa Barbara, California 93106 (Received 16 November 1988) The spin-bag approach to the high-temperature superconductivity is presented in detail. The genera) argument that the local supression of the electronic pseudogap leads to an attractive interaction of the quasiparticles is substantiated by the detailed calculation of the pairing potential mediated by the collective modes of the spin-density-wave background. In particular, the spinwave spectrum, the sublattice magnetization, and the spectral distribution of the collective modes are studied within the random-phase approximation. In the low-doping limit, different shapes of the Fermi surface give rise to a superconducting gap which formally has rf-wave- or p-wave-like symmetry, however the gap has no nodes on the Fermi surface. Therefore, the superconducting properties of our model are analogous to those of a conventional j-wave (i.e., nodeless) BCS superconductor. We also discuss possible bag effects in the large-C/ Hubbard model and in chargedensity-wave systems. Finally, the relation of our work with other approaches and with experiment are discussed briefly.

I. INTRODUCTION There is considerable experimental evidence that the properties of the newly discovered high-temperature oxide superconductors, 1 such as conventional superconductors, can be accounted for in terms of the pairing theory. 2 This evidence includes the factor of 2e occurring in the flux quantum and in the Josephson effect, 3 as well as the electrodynamic and thermodynamic properties 4 of these materials. While there remain experimental uncertainties, largely due to materials preparation problems, the central features of the data support the pairing approach. However, because of the short coherence length £ s c ~ 12-20 A in the planes of the oxide superconductors, fluctuation effects are expected to be considerably larger than in conventional superconductors and a complete theory must include such effects. The ingredients of the pairing theory are (1) the existence of spin- j fermion quasiparticles in the normal phase of the material and (2) an attractive effective potential V between these quasiparticles. This attraction causes a cooperative phase transition to the superconducting state at a temperature Tc which is large compared to the temperature at which a single pair binds. For conventional superconductors, V arises from the exchange of phonons between quasiparticles and is opposed by the screened Coulomb repulsion. Because of the retarded nature of the phonon interaction, the quasiparticles largely avoid the short-range Coulomb interaction yet benefit from the time-delayed phonon attraction. The phonon mechanism leads to condensation into a state having both orbital and spin angular momentum zero. In view of the reduced or negligible isotope effect in the layered oxide superconductors and the fact that the phonon mechanism is likely unable to account for transition temperatures in excess of 4 0 - 5 0 K, a new mechanism must be operating. Other pairing interactions are observed, e.g., in superfluidity 3 He, where the attraction is due to the ex39

change of spin fluctuations. Because of the smallness of the Fermi energy and the weakness of the pairing interaction, Tc is extremely small ( ~ 2 x 10 ~ 3 K) for 3 He. The condensation in this case consists of pairs having orbital and spin angular momentum one. It appears that spin fluctuations also lead to superconducting in the actinides such as U B e n and UPt3, etc. Pairing in atomic nuclei arises from the attractive nucleon potential, with Tc —106 K. Thus, the pairing theory has provided a framework which has successfully explained superconductivity in systems with widely different interactions involving energy scales spanning 13 orders of magnitude. In oxide superconductors, the central theoretical questions are (1) what is the nature of the quasiparticle excitations in the normal phase and (2) what is the origin of the pairing interaction between these excitations? In a recent paper we have proposed that the normal-phase excitations are spin-y fermions, corresponding to a hole surrounded by a region of reduced spin or charge-densitywave order. These "bag" excitations attract each other, as in the case of bipolarons. In the presence of the Fermi sea, the cooperative pairing condensation occurs at a temperature higher than that at which bipolaron formation occurs, leading to high-temperature superconductivity. An important clue as to the nature of the normal and superconducting states of these oxides was recently provided by neutron 6 and Raman 7 scattering experiments. These data show that strong finite-range antiferromagnetic correlations exist in the superconducting phase and that long-range antiferromagnetic order occurs in the phase diagram near the superconducting phase. Thus it appears that antiferromagnetism and superconductivity are intimately related in these materials, in contrast to magnetism opposing superconductivity in conventional superconductors. In LajCuCi, antiferromagnetic order occurs with a commensurate wave vector Q — (.fc/a,n/b,0), where Q is observed to remain commensurate for a finite level of doping. The sublattice magnetization is found to be large, of 11 663

©1989 The American Physical Society

447 11 664

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

order y of a spin- y per Cu site. This spin density appears to be largely on the copper atoms. These observations can be interpreted8 in terms of a Mott-Hubbard localized electron picture based on the electron-electron interaction U being assumed to be larger than the valence bandwidth W. Alternatively, a commensurate spindensity-wave interpretation based on an itinerant electron approach is also possible. Which picture is closer to reality is yet to be decided by experiment, although U~ fV/2 may be realized in nature. Returning to the bag excitations, the internal structure of the quasiparticles as well as the pairing attraction arise from a common source—the local suppression of antiferromagnetic order in the vicinity of the quasiparticle. As discussed below for U < W, a hole injected into a spindensity-wave (SDW) system depresses the staggered magnetization S surrounding the hole, in a region whose size L and shape depend on the nature of the Fermi surface as well as the mean SDW amplitude. (A numerical study of the spin bag is performed in Ref. 9.) This region of depressed S provides a bag inside of which the hole is self-consistently trapped. The bag containing the hole moves as an entity and acts as a quasiparticle of spin- f and charge e. It was shown5 that by temporarily sharing a common bag, two such quasiparticles interact via an attractive pairing potential, when proper account is taken of short-range correlations induced by the screened Coulomb potential. In the opposite limit U > W,10 it has been recently shown by several authors that a similar result occurs if finite-range antiferromagnetic order exists. In essence, an added hole leads to a reduction of the local staggered order parameter by disordering the spins in the vicinity of the hole. Therefore, frustration disorders the spins inside the bag for V > W while for U < W the hole reduces the local Fermi-surface nesting and reduces the amplitude of the SDW inside the bag. The bag is formed by longitudinal (Sz) spin waves for U<£.W and transverse spin waves for W»W. Presumably for U—W, both spin-disorder and spin-amplitude reduction play a role in dressing the quasiparticle and in producing the pairing attraction. If local antiferromagnetic order were not to exist, it has been proposed that a local coupling of pairs of spins to total spin zero might occur, leading to a resonatingvalence-bond-type (RVB) picture.'' In this approach, the excitations are presumed to be spinless charge e bosons and chargeless spin- y fermions, as in the quasi-onedimensional conductor polyacetylene.12 This scheme is distinct from the pairing theory since spinless charged particles rather than spin-} charged particles are the building blocks of superconductivity, a crucial difference. Finally, we note that the spin-bag approach discussed above may be extended to systems like Ba(Pb,Bi)C>3 (Ref. 13) which exhibit charge-density-wave-type (CDW) ordering. As in the SDW case, an added hole depresses the CDW order parameter in a region surrounding the hole whose size and shape depend on the Fermi surface and the CDW amplitude. Two such quasiparticles can interact via an attractive pairing potential as stated above. These charge-bag effects may also account for superconductivity in (Ba,K)Bi0 3 . 13

39

In this paper we present a formalism which enables us to calculate the dynamical pairing potential between spin bags. The bag idea suggests that holes sharing a local depression of the SDW amplitude experience an attractive potential. It is therefore natural to consider the pairing interaction of the quasiparticles in the presence of the SDW background through the exchange of the collective amplitude fluctuation of the SDW condensate. Strictly speaking, such a calculation is only valid if there is infinitely long-ranged antiferromagnetic order present. While this does not happen at high doping concentrations in the superconducting regime, neutron scattering experiments give evidence for an antiferromagnetic correlation length of order 10-20 A and fluctuations of energy do ^ 0.02 eV. This local slowly fluctuating antiferromagnetic order still causes a SDW pseudogap to persist since 010*^2^—1 —2 eV. In general, if the frequency scale associated with the fluctuation of the antiferromagnetically ordered domain is small compared with the pseudogap and if the length scale of the local antiferromagnetic order is large compared with the SDW coherence length, £ SDW = ftuf/A, where vF is the Fermi velocity in the absence of the SDW, the above starting point is qualitatively correct. We note that the spin-bag approach is to be contrasted with conventional paramagnon theory.14 There one considers the pairing interaction between the electrons arising from the exchange of spin fluctuation above the normal state in the absence of the SDW. The interaction is purely repulsive in momentum space, peaked at the nesting wave vector Q and leads to a weak (/-wave pairing with four nodes on Fermi surface. What is neglected in this approach is the effect of the local antiferromagnetic order on the self-energy of the quasiparticles and, consequently, on their interaction. In our approach, however, the effect of the antiferromagnetic background is built into the hole wave function so that the attractive interaction mediated by the collective amplitude fluctuation of the SDW leads to a nodeless pairing gap over the Fermi surface, although the formal symmetry can be of the p-wave or rf-wave type, depending on the shape of the hole Fermi surface. In the case of the normal-state perturbation theory, nodes of the pairing gap are unavoidable since the Fermi surface is a continuous loop around k~0 where in the present case, with the SDW background present, the hole Fermi surface forms pockets at the magnetic zone boundary so that the nodes of the pairing gap can be avoided at the hole Fermi surface. The nodes of the pairing gap appear at the region of momentum space where there is a SDW gap, and has no effect on the superconducting properties. In building a theory for high-7V superconductivity based on magnetic interaction, the main challenge is to find a consistent solution of the gap equation so that the pairing gap is nodeless over the Fermi surface, as most experiments indicate.4 The fact that our magnetic mechanism can indeed give rise to a nodeless pairing gap is quite remarkable. The pairing interaction in the presence of the SDW has been studied extensively by Fenton in Ref. 15, in a somewhat different context. The relation between his approach and ours will be discussed at the end of Sec. III.

448 DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM .

39

11 665

This paper is organized as follows: In Sec. II we review the basic formalism of the spin-density-wave approach to the Hubbard model and calculate the collective charge and spin-density fluctuation modes above the S D W condensate within the random-phase approximation. The sublattice magnetization is also obtained within the mean-field theory and with the first-order fluctuation effects included. In Sec. Ill, we present the detailed calculation of the pairing potential. The solution of the superconducting gap equation and the effect of fluctuations on the quasiparticle self-energy are presented in Sec. IV. Finally, we discuss our results and comment on charge bags in charge-density-wave systems in Sec. V.

II. COLLECTIVE MODES Let us start by considering the two-dimensional Hubbard model 16 on a square lattice H--1

X (c,Vy a + H.c.) + U^nnnn

U.j) a

,

(2.1)

where (ij) denotes nearest neighbors. By transforming to momentum space, H can be expressed as k,a

FIG. 1. The large square represents the first Brillouin zone; the small square represents the reduced Brillouin zone Tor the Bloch state in the SDW background, which is also the Fermi surface for free electrons at half-filling. Q is the nesting vector.

£. Pi k.k'q a,a

where £ > is the Fermi energy. At half-filling, the Fermi surface is a perfect square [the zero-energy contour of (2.3); see Fig. ll. In this case, the ground state is unstable against spin-density fluctuations. To see this, let us look at the charge- and spin-density correlation functions defined by

t.fi XC-k + q.fiCk.a,

(2.2)

(2.5)

where

XiJ(q,t)

ek — —2t(coskxa+coskya)

,

(2.3)

a,p are the indices for spin quantization along the i direction, a is the lattice constant, and N is the total number of sites. All the momentum summations extend over the first Brillouin zone (Fig. 1). The zeroth-order (U— 0 ) ground state of H is defined by c*„|0>-0,

ek>EF,

(2.4) cL\0)-0,

<*<£>,

**•<«..>-£$

« * 0 —flk+q) (O + Ek— Ek+g~iS

*8°(g,«) l+t/*S%7,«u)

- + -£-<0 | TS!,U)SL9iO) 2N

10>,

where (2.6) k.a

is the charge-density operator and

(2.7) k,a,fi

is the spin-density operator, with
nk+A\-nk)

(2.8)

10 + Ek —Bk+q+iS

with x(/(q,o)) ~S'Jx8°(q,a>) where /i* is the electron occupation number. In the presence of the interaction, these correlation functions can be calculated by the randomphase approximation (RPA), which, in graphical representation, sums over an infinite series of bubble graphs and ladder graphs (Fig. 2). This summation can be easily performed by using Dyson's equation. In the charge channel, one finds

*8M9,«)'

z°°(\ r P ,G)p-,(o) |o>,

(2.9)

'0000 s s^"

>

FIG. 2. The Feynman diagrams used in RPA to calculate £ $ A and *#PA. The single line represents the electron propagator in the normal state (no SDW). The dotted line represents the interaction V.

449 11 666

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

and in the spin channel, the result is 00

xdpA(q,a>)-

*o (?,a>) \-UXS°(q,
S".

(2.10)

At half-filling, the Fermi surface has the nesting property. Namely, there exists a vector Q which connects opposite sides of the Fermi surface. This property leads to vanishing energy denominator e* — e*+g in ; c o ° ( 9 ~ C &>=»0), and, consequently, one obtains a purely imaginary pole in ^RPACG'*") a s a function of co. In this case, XHPA(Q,I)~exp(flg/) for large and positive / where (IQ is real and positive. This means that since the normal Fermi surface |0> defined by (4) is unstable and the true ground state | fl) is the one with a static spin-density wave present (for a review of the S D W formulism see Ref. 17). Without loss of generality, let us assume a S D W mean field to be polarized in the z direction (2.11)


where the variational parameter 5 will be determined later by a self-consistency condition. In the presence of this mean field, the Hartree-Fock factorized form of (2.2) is „_V t US (2.12) ci+Q,a°la'Ck,J" 2*ekCk,aCk,a — ——-N £ *,« 2 This one-body Hamiltonian can be diagonalized by the transformation (
(2.13a)

Yk,a~VkCk,a - Uk 2 (o-3)„0C* +Q,p .

(2.13b)

ri,a

—UkCk,a+

Vk £

To avoid double counting, k is restricted to the magnetic zone, i.e., one-half of the first Brillouin zone (see Fig. 1). The superscripts c and u refer to the conduction and the valence bands split by the exchange Bragg scattering from the SDW. The transformation amplitudes are

39

±Ek. The S D W ground state | n> for a half-filled band is defined by rti:.\n)—rl,a\ti)-Q.

(2.16)

Using this definition of the ground state | n ) in Eq. (2.11), one obtains a self-consistency condition determining the gap parameter A,

-.-4E'-f k

±2

I

N r (el + A2)

^-N

2h.k

(2.17)

U

(2.18)

U

Due to the singularity in the density of states at the Fermi surface, the solution of this gap equation for small U is given by •te

-irJtTU

(2.19)

while for t « £ / , 2A—U, the Mott-Hubbard gap. Having discussed the single-particle spectrum in the SDW background, let us now turn our attention to the collective density fluctuation. To do that, we study the charge- and spin-density-correlation function as defined in Eq. (2.5) but the expectation value is taken with respect to the SDW ground state | ft) as defined in Eq. (2.16). Since there is a mean field which breaks the original crystal symmetry and doubles the unit-cell area, the correlation functions have off-diagonal terms in momentum space representation. These off-diagonal terms arise from the umklapp processes with respect to Q. W e therefore define the correlation functions in the presence of the S D W background by

* 0< W,<)-

2N


**W.«> —=^, Bk

«*'

1 +E

Vk '

1-

2N

1 1/2

k

1/2 Ek

Ek (2.14)

£*-(ej2 + A2),/2, A

„ _

where pq and Sq are given in Eqs. (2.6) and (2.7) and | n ) is defined by (2.16). To calculate these correlation functions in this mean-field approximation is straightforward. One transforms pq and Sq into the y representation by inverting Eq. (2.13). The time dependence of the y operators is given simply by yt.aU)-e-"Etyi,a(.0)

US and

where A is the S D W energy gap parameter. The diagonalized Hamiltonian is given by

H-ZEkhkrL-rhrL),

(2.15)

7k.aU)> •e"E,yk\a(0).

(2.21)

In evaluating the average, the only nonvanishing terms arise from the combinations such as

k,a

where 2 * means that the sum extends over the magnetic zone. The single-particle energy spectrum is given by

(2.22) and one obtains

_J 0

X0 (q,q';a>)-8(q-q')xl?(q,co),

xo^~(q,q'^'>-S(q-q')xo~(.q,(o)

+ S(q-q,+

Q)xQ''(.q,to),

(2.23)

450 DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM . .

39

11 667

where EkSk+q+A EkEk+q

*8°<9.»)-*S'(9.«>)--TkZ' 27V k

*r(9,*>)--^r

1 -

1 oi — Ek+Q ~~Ek+i5

1 a> — Ek+q~ Ek+i8

SkEk+g— A

EkEk+q

-co —

1 —w—Ek+ti~Ek 1 Ek+q—Ek+iS

+ i8

(2.24)

(2.25)

The umklapp transverse susceptibility is given by XQ

( g , < » ) - — 2 , -=2N k Ek

co — Ek+q — Ek+iS

Xo"(q ,a>) —%o' + ( q ,co), XQ + ~

(2.26)

— co — Ek+q~ Ek + i6

-XQ+(q,co).

Equations (2.23)-(2.26) give the response functions in absence of the interact-ion. The RPA response functions can be obtained by solving the Dyson's equation (Fig. 3) and one finds *R°PA(g,9';
%RPA(q ,q';a>) •

(2.27) \+Uxo(q,co)

'

Xo(q,q'\co)

(2.28)

\-Ux"iq,io)

'

and *RPA()~]C;t:o+ [1 — Uxo ~ (qi,q,o>)]

_1

iq,q\,co)[\

— Uxo

{q\,q';co)]

(2.29)

is a matrix inverse in momentum space and is given by

(l-t/^o+_(?i,9';
l\-UXo-(ql

+ Q,a>)]8(ql-q,)

[l-Uz£~(qua))][l-Uxo~(qi

From Eqs. (2.27)-(2.29) we observe a general feature of the spontaneous symmetry breaking: If the ground state is spin rotationally invariant, symmetry guarantees x""2x+~. In our case, however, the ground state | n> breaks the continuous spin rotational invariance of H and this relation no longer holds. As a result, x+~ contains a gapless pole, as predicted by the Goldstone theorem. In fact, from (2.29) and (2.30) one can study the pole near q ~Q, and find that the spin-wave spectrum is indeed gap-

#RPA()a* +

'.

U

2

(l/t/-A2x)g(g-q') <,coAx) -ll/U-A2x)U2ySq2-(co2/4)x] 2

where

+ Ux$~(.q,,co)8(q]-q,

+ Q)

Q,
+

(2.30)

less if 1 y 1 _ 1 (2.31) NT Ek"' V But (2.31) is identical to the gap Eq. (2.18) and this provides a nontrivial check of the self-consistency of the R P A calculation. In order to find the spin-wave velocity v, we expand (2.29) and (2.30) for small q - Q and small co to locate the pole. Using the gap Eq. (2.18) we find

'

(2.32)

conclude that the spin-wave velocity in the large U limit is 2

—» sin «c^a) """ T

E?'

N *

(2.33)

El

and £ q = ( q — Q)a. Equation (2.32) determines the spin-wave velocity for all values of t/Ugiven by t2y(\/U-A2x) 2 2 A x +(x/4)(l/l/-A2x)

1/2

(2.34)

In the large limit, x — 1/2A 3 , > — 1/4A 3 , 2A—• U, and \/U — A 2 x —* 2f 2 /A 3 as one can see directly by taking the large (/solution of the gap Eq. (2.18). From (2.34) we

(2.35) V2 where J~4t2/U. This agrees exactly with Anderson's 1 8 calculation of the spin-wave velocity of the s — y Heisenberg model. In fact, with little more effort, one can work out the entire spin-wave spectrum in the large U limit for general q by expanding only to small
451 11668

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

39

> FIG. 3. The Feynman diagrams used to calculate £RPA and ,£RV* in the SDW background. The double line represents the electron propagator in presence of the SDW background.

By numerically solving the gap Eq. (2.18), we have determined the spin-wave velocity for all ranges of t/U from Eq. (2.32). The results are plotted in the Fig. 4. Besides the spin-wave spectrum, the sublattice magnetization in the antiferromagnetic state is an important and experimentally accessible quantity. At large U, the effective Hamiltonian for the half-filled Hubbard model is known to be the Heisenberg antiferromagnetic Hamiltonian. The quantum-spin-wave fluctuations are known to reduce the mean-field sublattice magnetization of one /i« per lattice site to about 0.6MB- 1 8 _ 2 0 It is very important to study the spin-wave-reduced sublattice magnetization in the present itinerant spin-density-wave approach as well. The mean-field approximation to the sublattice magnetization can be easily obtained by solving the gap Eq. (2.18) and using the definition (2.14), \S\ -2A/U. The sublattice magnetization given by the numerical solution of the gap equation is plotted in Fig. 5. For small U the solution follows the behavior of Eq. (2.19), while for large l / , A » £ * , so that the left-hand side of Eq. (16) can be approximated by 1/2A, therefore, A—-U/2 and | S | —» 1 just as in the localized limit. To study the effect of fluctuations on the sublattice magnetization, we first derive a trivial identity

10

21

.10

40

SD

80

?O

80

90

U/t FIG. 5. The sublattice magnetization S — 2A/1/ obtained in mean-field theory.

where G„p(x\t\\x2t2)" ~i(Tca(x\t\)cjl(.x2(2))fore the sublattice magnetization is given by S-TT<5e>--^2/-^TrCT3G(*.C-*;<*>).

There-

(2.37)

For the mean-field S D W vacuum | O) as defined by Eq. (2.14), the one-particle Green's function is given by

^,0, Oa/)(p,p

, ,

ep)SafS(p-p,)+^pS{p-p'+Q)

(w +

;a>) —

—2j — „ 2_i_ _• e


+ iS

•

(2.38) In this case, -E£ + i8

<S"(x)> - Z < c J ( x , 0 ) < T i ^ ( x ( 0 ) > -=-/TrCT'G(x,0";x,0),

0.0

_ Ay J_-_i4(2.36)

N T Ek "

U '

(2.39)

just as given by Eq. (2.14). In order to calculate the fluctuation effects on the sublattice magnetization, one simply has to calculate the self-energy correction to the one-particle Green's function. The self-energy matrix ~La,p(p,p',a>) is approximated by the one-loop Feynman diagram (Fig. 6), where ZRPA, *RPA, and £R>A are given by Eqs. ( 2 . 2 7 ) , (2.28), and

(2.29), respectively. The full one-particle Green's function Gafi(p,p',a>) is then obtained by the Dyson's equation

GV W , o > ) -G2fi ' (p,p',a))-ZaJ1(.p,p\a>) .

(2.40)

We find that in the large U limit the charge fluctuations £RPA and the amplitude spin fluctuations ZM>A contribute

U/t FIG. 4. The solid line represents the spin-wave velocity calculated using RPA. The dashed line plots v,/a—(.\y/l)J the spinwave velocity expected for Heisenberg model.

FIG. 6. The one-loop paramagnon correction to the selfenergy of electron in the SDW state.

452 39

DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM . . .

11 669

states," so that we calculate these interaction matrix elements first and then perform the transformation (2.13) to obtain the interaction matrix elements between the y states. Within the RPA, the interaction results from the exchange of the charge and spin collective modes given by (2.27)-(2.29) (Fig. 8). It is convenient to approximate the frequency dependence of each term in the interaction (0,z, ± ) by the static limit and a cutoff frequency, casting the results into an effective Hamiltonian. We find in the charge-fluctuation channel

Hc"4r, 2 Z[2C/-M*-*')]S„'«5/r/> 4/V k.k'.q o,o'

W

x

U/t

FIG. 7. The solid line represents the sublattice magnetization including the fluctuation effects. The dashed line is the meanfield result.

Q * o ' C -k'+q./fC

-k+q.fiCk.a

(3.1)

.

in the amplitude spin-fluctuation channel

Hz---^-

2

£K z tt-*')
4 / V k,k\q a.o'

little to the reduction of the sublattice magnetization. In fact, their contribution to i8Sb_)/N is of order t2/U2 whereas the contribution from transverse spin fluctuations is of order 1. The numerical results for the transverse spin waves are summarized in Fig. 7. In the small U limit, the charge and the spin amplitude fluctuations are as important as the transverse one. However, in this case the effects of all fluctuations are small and (SQ)/N is close to the mean-field value 2A/U. Therefore we expect the numerical result in Fig. 7 to be close to the true sublattice magnetization in both small and large U limit. It should also be good approximation for intermediate range of U. Actually, we can compare our RPA result with the recent computer simulations of Hirsch and Tang20 and find fairly good agreements. Neutron experiments seem to exclude the possibility of 5 5 0 . 4 . This suggests that A//S0.8 or Uh'Z.2.9 from our calculation. Within the range 0.8 5 A / / 5 2.5 (or 2.9 £ I//f £ 5 . 8 ) the width of conduction band is larger than A and the itinerant electron picture describes the system more accurately than the localized electron picture. In conclusion we find that the RPA calculations above the SDW background not only gives correct results in the small V limit, but also extrapolates sensibly to the large U limit, as reflected in the calculation of the spin-wave spectrum and the sublattice magnetization. In particular, these calculations clearly demonstrate the consistency of the itinerant picture with the neutron experiments.

XCk'JC -k'+q.fC-k+q.pCk.a

(3.2)

,

and in the orientational spin-fluctuation channel (k — k')o£aopp

# H

47V

k.k'.q'.a XCk'a'C-k'

+ q.pC-k+q.pCk.a

,

(3.3)

where

l+UZ§°(q) Vz(q)-

V2X*'W

r+-(q) — c/V

l-l/#>

(3.4) (?)

(
The total effective Hamiltonian is given by

Htn-Hc+Hz+H+-+Hl-

.

(3.5)

Notice that we have explicitly included the original Hubbard interaction U in the charge channel. In Eqs. (3.1)-(3.3) %8°(q), Zo'lq), and xo'iq) are the a>-0 components of the dynamical susceptibilities defined in Eqs. (2.24) and (2.25). Since XQ "(?•<») «s odd in to and

III. PAIRING POTENTIAL Having discussed both the single particle and the collective modes in the presence of the SDW background, we are now in the position to study the interactions of two holes doped into the half-filled band. The unperturbed states available to these holes are the eigenstates meanfield SDW Hamiltonian, the "y states" as given by Eq. (2.13). However, the interaction is easier to calculate as matrix elements between the original Bloch states, the "c

V

s^ FIG. 8. The Feynman diagrams used in RPA to calculate the pairing potential in the SDW background.

453 11 670

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

vanishes for co— 0, the momentum nonconserving term does not appear in H+ -. We are now in the position to determine the matrix elements of the effective Hamiltonian in the /-state basis and use them as the pairing potential. Transforming Heg to the /-state basis, one obtains the matrix elements for both intraband and interband transitions. At half-filling, the valence band is full and the conduction band is empty.

39

Weak doping removes electrons from the top of the valence band and it is these mobile holes which are responsible for the metallic behavior and superconductivity. Since the SDW gap 2A is large compared to the cutoff frequencies for the attractive parts of the pairing potential, one needs only retain the intra-valence-band matrix elements between hole pairs of opposite momentum, and we find

Hc — ^ E ' £ U 2 l / - K c a - r ) ] / 2 ( j M ' ) < S a ' 0 f y , > + [ 2 t f - K c t t ^ 4/V

klk'

a.a fi.fi"

(3.6) Hz

^E'lllVz(k-kllHk,kl^aa^ 4W

ktk'

+ yAk-k'

+ Q)mHk,k,)6aaS^]rkVar±klfr-k,prLa,

(3.7)

+ Q)pHk,k,)\a^^y^ar±hfir''-k.0rl.a,

(3.8)

a.a

H+-^-^Tll\V+-(k-klnHkX)-V+-{k-k, 4/V k,k' a.a' 0,0'

where m(k,k') — ukvk'+vkuk-,

(3.9)

l(k,k')-ukuk-+vkvk',

(3.10)

p(k,k') — ukvk' — vkuk',

(3.11)

n(k,k') ~ukuk'~vkvk',

(3.12)

are the so-called coherence factors. Equations (3.6)-(3.8) summarize the effective interactions in all channels between holes in the valence band, as calculated in the self-consistent random-phase approximation in the presence of the SDW background. We see Hes has a fairly complicated structure. However, if we restrict ourselves to the weak doping limit, qualitative conclusions can be easily drawn. In this limit, the holes are concentrated at the top of the valence band, where ul = vl» j . One can therefore approximate the four coherence factors by i » U , * ' ) « / ( * , * ' ) » 1 , p(k,k')««(*,A:')«0. An immediate consequence of this approximation is that these hole states are decoupled to the spin orientation fluctuations and the spin-flip scattering vanishes, i.e., H+- » 0 . Later we will see that only two states near the top of the valence band have a dominant contribution to the superconducting gap equation. The spin-orientation fluctuations are not important to the superconductivity in our model. In the charge channel the pairing interaction between holes of opposite spin is given by l2U-Vc(k-k')]-l2U-Vc(.k-k' --Vc(k-k')

+ Q)) + Vc(.k-k' + Q).

(3.13)

However, Vc(q) is nothing but the original Coulomb interaction dressed by charge fluctuations. In contrast with the spin channel, the charge channel does not exhibit an instability and the renormalization effects are weak, with Vc(q) having a weak momentum dependence. If we neglect the momentum dependence all together, (3.13) vanishes. This important result has a physical interpreta-

tion. The states we are pairing are the eigenstates in the presence of the SDW, i.e., the y states of Eq. (2.13). At the top of the valence band (or at the bottom of the conduction band) for which uk » c ^ « T , these states are superpositions of equal weight with states having momentum k and k+Q. In real space, if the wave function of an up-spin hole vanishes on the even sublattice due to destructive interference, between k and k + Q, the wave function of a down-spin hole vanishes on the odd sublattice. Therefore the matrix element of the on-site Coulomb repulsion vanishes due to the vanishingly small overlap of the up- and down-spin y wave functions on the same site. Now consider the interaction matrix element (3.7) in the amplitude fluctuation channel, Vz(k-k')-V2(k-k'

+ Q).

(3.14)

Unlike Kf(fl),Kz(g), the longitudinal (non-spin-flip) exchange interaction is strongly enhanced by the magnetic fluctuations for q — Q, leading to a sharp momentum dependence. The reason for this is clear, Vz(q) is proportional to the RPA magnetic susceptibility, which in the absence of SDW would diverge logarithmically at q " Q, indicating the antiferromagnetic instability. However, V,(q) is the RPA magnetic susceptibility in the presence of the mean-field SDW, and the system exhibits stable spin-wave excitations. For small A, a sharp peak of Vz(q) is present at q "Q, with a width of the order of the inverse SDW coherence length {sbW~A/f. From this structure of Vz{q), one observes that the pairing interaction in the amplitude fluctuation channel (3.14) is strongly attractive for small momentum transfer and outweights the interactions in ail the other channels for holes near the gap edge. Therefore at relatively low doping the superconductivity is mainly due to the pairing potential in amplitude spinfluctuation channel. Strictly speaking, one can always work within the magnetic zone inside of which the y states are originally defined. However, for most discussions, it is useful to work with the extended zone scheme. One can continu-

454 DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM

39

11 671

ously extend the definition of y states outside of the magnetic zone using expression (2.13), since £*+g —— e* so that by (2.14), Uk+g~Vk and V^+Q— Uk- Using this definition, we find

yi'.i--rt'+e.t

and

(3.15)

ri'i ~ri'+gi •

The first consequence of this definition is the fact that the pairing potential between holes with up and down spins are all antiperiodic functions in the magnetic zone, i.e., Vk.k'"" — Vk+Q,k,m" ~ Vky+Q as one sees directly from Eqs. (3.6)-(3.8). We note that the antiperiodicity of both the pairing potential and the basis states ensures that the total summands in the Hamiltonians are all periodic functions, and all the physical properties are invariant under the shifts of the origin of the magnetic zone. This unusual antiperiodicity arises from the fact that for a given k,yi; to/ t and fk\ describe different orbital wave functions. One can FIG. 10. The momentum dependence of J?RPA(9, the spectral weight \m%fo\{q,w) near a> —2A de&fc-I,fVkkA (?•<») From the numerical calculations we find for t — 1, A —0.5, for ) are plotted as functions of to in Fig. 9. We find a large portion of the spectral A.=0.9a , weight concentrated near •»—2A. Although no physical pole is found, the amplitude fluctuation of the SDW order A =0.58, parameter behaves approximately like a physical mode and for / - 1 , A - 1 , and U-3.29, X=0.7a, 0

0

20

30

4 0

SO

60

'0

BO

9

A =0.45. The dynamic pairing potential, between spin-up and spin-

1

( 1 ll / ll ; l\ 3

1 \ 1 \ 1 x

i

1 1 1

) \

1

—s

-

\

(0/ t

FIG. 9. The frequency dependence of the susceptibility %tir\(Q,o>). The solid line is Irn?RpA. The dashed line is ReJf&A.

FIG. 11. The energy band of electrons in the SDW state. At finite doping er is the Fermi energy of the holes.

455 J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

11672

is the energy spectrum of the electron (Fig. 11), and

down holes, is given by -mHk,k')lU2z&A(k-k' +

,

39

+ Q,a)] ,

7

lHk,k )W x&A(k-k;co)].

e(x)s (3.17)

Therefore, if we use the effective Hamiltonian (3.7) to approximate the dynamic interaction given by (3.16) and (3.17), V2{k-k') in (3.7) may be taken to be K 2 a - * ' ) = t / 2 * f t A ( * - * > - 0 ) e U a - \Ek - £ > | ) xe(aj0-|£V-£>l), (318) where w„ is the cutoff energy, EF is the Fermi energy, Ek Vu-[-mHkX)U2zfoAtt-k,

1,

JC>0,

0, x < 0 .

Unlike £RPA(^,<»). the spectral weight of £RPA(?>G>) is broadly distributed in the interval — 4r < a> < At and has only a weak momentum dependence. Furthermore, according to our previous discussion, the pairing potential in the charge channel VIk' is ineffective for k and k' near the hole Fermi surface, since the holes are concentrated at the top of the valence band in the weak doping limit in which case spin-flip hole and spin-down hole occupy different sublattices. Therefore, in the following discussion, we shall simply take

+ Q)+lHk,k,)U2x&Att-k')]e(,a>a-\Ek-EF\)e(a>l,-\Ek--EF\)

to be our pairing potential. For larger doping, we should include the scattering far away from the hole Fermi surface due to the interactions in the charge channel. Since there is no frequency cutoff for this interaction the charge channel potential leads to an effective pseudopotehtial acting in the region below u>a which is much reduced in strength. Let us conclude this section with a discussion of the relation between our work and that of Fenton.15 His main result is that the interaction entering the superconducting gap equation in the presence of the static SDW is the same as the interaction in the absence of the SDW, in apparent contradiction with our result. However, his conclusion is derived under the assumption that uk~uk, vk — — v * so that uk vk is purely imaginary. It is important to note that the phase of ukvk is determined by the phase of the SDW relative to the underlying lattice. Choosing ukok to be purely imaginary corresponds to a SDW with spin-density peaked on the bonds and vanishing on the lattice sites. Therefore, it is quite conceivable that such a SDW does not modify the site interactions. However, in the presence case, the SDW results from a self-consistent solution of the Hubbard model and the spin-density necessarily has peaked on the sites. In this case ukvk is real. As we showed in this section, the presence of such a SDW does in fact change the interaction in a dramatic fashion, producing an attractive potential. As one can see, we have fully included the interactions in all possible channels. The extra terms found by Fenton vanish identically in our case since uk and vk are both real.

(3.19)

(3.20)

tion, the single-electron energy spectrum +E(k) is very special. For nearest-neighbor hopping the maximum of the valence-band energy_ — E(k) is_ highly degenerate, consisting of two lines, ky ""0 and kx ~ 0 in the shifted magnetic Brillouin zone (Fig. 12). While this property is very unusual, one should realize that the degeneracy of the single-electron energies along the lines kx—0 and ky —0 is totally accidental and not guaranteed by symmetry. Injeality, the single-electron energy E(k) — —E(k) +AEik) for a valence-band electron has corrections AE(k) arising from the higher-order corrections to the Hartree-Fock approximation, next-nearest-neighbor hopping and interactions between electrons on different sites, etc. For small concentrations of_holes, the Fermi surface of Hartree-Fock spectrum — E(k) (Fig. 13) is highly sensitive to these perturbations. For example, if we add the energy correction AE(k) which lifts the degeneracy such that the exact^single-electron energy E(k) has a single maximum at kx~ky— 0, then holes near ic—O have the lowest energy and holes will concentrate in this vicinity. The resulting Fermi surface is a small circle around ic ""0 (Figs. 12 and 14). If we assume, instead, that the corrected energy E(k) has maxima at {kx,ky)—(0,+n/J2a) and (±;r/\/2fl,0) lE(0,±n/^2a)-E(±7t/-J2a,0) due to the x— y symmetry], holes tend to concentrate near

IV. GAP EQUATION Having completed our discussion of the pairing potential, we now proceed to solve the superconducting gap equation. At half-filling, there are no free carriers and the system is insulating due to the presence of the SDW band gap. With finite doping, the holes form a Fermi surface. Its shape in momentum space is most important for the nature of the superconducting gap. One may notice that in the Hartree-Fock approxima-

FIG. 12. The small square outlined by the bold line represents the shifted magnetic Brillouin zone. The circle is the Fermi surface at low doping, assuming the quasiparticle energy spectrum E(k) satisfies E(k — 0) < Elk — (0,>rA/2a)]. The superconducting gap is positive along the Fermi surface.

456 39

DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM .

11 673

\

FIG. 13. The small square with the rounded corners is the Fermi surface for the mean-field energy spectrum Ek. The dotted lines are the Fermi surfaces in the shifted magnetic Brillouin zone.

(0, ± n/^/la) and (±;r/>/5a,0). For a small concentration of holes there are two separate Fermi surfaces as shown in Fig. 15. Therefore, at low doping the shape of the Fermi surface is not determined by — E(k) but by the energy correction AE(.k). In order to determine the shape of the Fermi surface, we must first evaluate AE(k). Since AE(k) is composed of many contributions, it is very difficult to evaluate AE(k) reliably. For example, the one-loop spin-fluctuation self-energy_ correction (Fig. 6) tends to make valence electrons near k "O have higher energy. At k — 0, there is no quadratic term in the momentum expansion of the mean-field energy E(k). However, the self-energy correction contributes to the quadratic term. Using the RPA susceptibility Z$>A (3.16), we find the effective mass of holes at k— 0 is approximately given by m* = l l a _ 2 / _ 1 — 6m, | , - i c V V - 1 2 A 1 f° r D o t n A/<—0.5 and A/f""l. In the following we first discuss the case where the valence electrons at k ""0 have highest energy and the Fermi surface for a small concentration of holes is a small circle around &—0, as shown in Fig. 14. The superconducting properties for the Fermi surface as shown in Fig. 15 will be discussed later. The relevant Fermi energy in the problem is the Fermi

FIG. 15. The Fermi surface of electrons with spectrum E(.k), assuming Elk -(0,*-/<j)] > Elk -(.n/2a,n/2a)].

energy of holes EF which is measured from the top of the valence band to the Fermi_ surface which has energy £> (see Fig. II). Thus eF—E(k—0)-EF. At low doping, e f is less than the frequency cutoff of the pairing potential. Actually, CF=0.1 eV for w* ~ 6 m , and hole density of 0.1 hole per site. Therefore, EF also serves as a frequency cutoff in the superconducting gap equation in addition to the frequency cutoff provided by the pairing potential. This leads to a superconducting gap and transition temperature which is rather sensitive to the doping concentration. Due to the small frequency cutoff the empty conduction band can be ignored and the superconducting gap equation may be written as

Afc

ZVtt-^-, *'

(4.1)

2Ek'

where

Ek-J(Ek-Er)2

+ (AF)2.

(4.2)

Strictly speaking the states very close to the bottom of the conduction band may contribute to the superconducting gap equation, since the frequency cutoff ioa — 2A is close to the gap between the conduction and valence bands. But such a contribution is small compared with that from the states in the valence band. Therefore the contribution from the conduction band can be safely ignored. Because of the antiperiodicity of the pairing potential Vk.k

K*,f +c --n + e . A .,

(4.3)

one can easily check that the solution of the superconducting gap satisfies

Afc--Af$e. c

FIG. 14. The same Fermi surface as in Fig. 12 is represented in the (unshifted) magnetic Brillouin zone. The superconducting gap has alternating signs around the corners. The dotted lines are the node lines of the superconducting gap.

(4.4)

Therefore Af must have lines of zeros in the magnetic zone. However, this does not necessarily imply that the total gap, antiferromagnetic-plus-superconducting, has zeros on the Fermi surface. This depends on the shape of the Fermi surface and the properties of the pairing potential. To see this, let us first look at thejproperties of the gap equation in the shifted zone where k "0 is the center of the Fermi surface (see Fig. 12). In the case when the Fermi surface is a small circle around A: — 0, only small momentum transfers are involved in scattering around the Fermi surface. Since the pairing potential is negative for

457 11 674

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

all small momentum transfers, a uniform superconducting gap near the Fermi surface is the solution. The lines of zeros of Af c arising from (4.4) are on the magnetic zone boundary where a large S D W gap exists. Thus, A^ c is nonzero over the entire free Fermi surface and no powerlaw temperature dependence is expected. One can also solve the gap equation directly in the original magnetic Brillouin zone where the Fermi surface is distributed at the corners (see Fig. 14). In this case, the superconducting gap Afc has alternating signs as shown in Fig. 14. This is because of the strong positive peak of the pairing potential for momentum transfers near Q. The resulting superconducting gaps for these two different choices of the magnetic Brillouin zone center can be mapped into each other by Eq. (4.4) (see Fig. 16). In both cases, the superconducting gap is nodeless at the Fermi surface since the S D W gap A preserves a gap when A s c — • 0. Thus, the zeros of superconducting gap are located in the regions of the Brillouin zone where there is no free Fermi surface. To understand the symmetry of the superconducting order in SDW state, we calculate the superconducting order parameter in real space and find
+ ( - 1 )'ukUvk

- ( - 1 Vuk]

k

x.

(4.5)

where the summation extends over the magnetic zone in Fig. 1. (y+k\y~k])—Afcl2Ek is nonzero only near the Fermi surfaces, i.e., near the four corners of the magnetic zone (Fig. 14) in low-doping limit, (yi'iy 1 -*)) has the same alternating signs around the four corners of the magnetic zone as Af . Since we are only interested in the symmetry properties of superconducting order parameter we may assume the order parameter to be proportioned to S functions at the four corners and write

a

H

-,o

k+

]"

k+

a We find that in real space (4.6) gives

39

FIG. 16. The same Fermi surface and superconducting gap as in Figs. 12 and 14 is now represented in the repeated zone scheme. can be suppressed by nonmagnetic impurities in the Cu-O plane similar to ordinary rf-wave superconductivity. 21 In order to estimate the magnitude of the superconducting gap, it is convenient to work in the shifted magnetic zone as shown in Fig. 12. In this case only the pairing potential for small momentum transfer is important and one can ignore the positive peak near k — k'~Q. Note that in addition to the frequency cutoff, the pairing potential also has a momentum cutoff of order 1/|SDWIf the Fermi surface is small, i.e., the momentum transfer across the Fermi surface is less than 1/£SDW we may rewrite the momentum cutoff as an effective frequency cutoff of order 1/2/M*£SDW where m* is the effective mass of holes. Therefore the effective frequency cutoff ton is the smaller one of the two energies ma and } m ' £SDW. In this case, we replace Vkk' by

a

Vkk'=-Voe(.too~\Ek-EF\)e(a0-\Ek'-EF\)r,

-,o]

(4.8) where K 0 = j iV2(Q) - K 2 ( 0 ) ] = \V2Al
a

J

c c [ l + ( - ! ) ' ] x[l-(-l)-''][(-l)''"A-(-l)'»~'''],

(4.7)

which is illustrated in Fig. 17. From (4.7) we find that pairing is only between holes on different sublattices. This simply reflects the fact that at low doping the spin-up holes and spin-down holes live on different sublattices. Another property of the superconducting order parameter is that it changes sign under 90° rotation. Thus the summetry of the order parameter resembles a rf-wave pairing. However, as we emphasized earlier, there is no node on the Fermi surface in the lowdoping limit and most properties of the superconductors are essentially the same as the conventional i-wave superconductor. But the sign change of the superconducting gap from corner to corner does have experimental significance. We expect that the superconducting state

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.

o

FIG. 17. The superconducting order parameter plotted as a function of j . x marks the position of i, O represents a positive value, • represents a negative value, and + represents zero.

458 39

DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM .

tant in solving the gap equation. Thus we can approximate Ek by — £2+const. 2m* near k ""0. Now Ek becomes 2

Ek<

2m

^ - e

F

1/2

(4.10)

+A|C

In

( W + A j c ) 1 / 2 + w0 , , Asc

et ""2ei>oexp

An

U} + A$c)in

+ ln—

Asc

+ eF

—

An V0m*a2 (4.11)

which reduces to Asc "-2-x/e7a>oexp

In Vom'a2

(4.12)

in the limit ASC"^«F and oo. Similarly, one finds the critical temperature ke Tc — 1.14-v/cfflJoexp

2K

V0m*a2

(4.13)

The ratio 2Asc/7V~3.52 is not changed by the appearance of the second frequency cutoff due to the small Fermi energy. It is clear that the critical temperature Tc is very sensitive_ to _the _ effective mass of the holes. _ Since E(k)~(k2 — k2)2 near £ — 0 the quadratic term k2/2m* in E(k) — — £ ( f c ) + A£(fc) is completely due to the energy correction AE(k). Therefore, the effective mass is determined by A£(k) alone. Since we are not able to evaluate the energy correction AE(k) reliably, we cannot estimate the effective mass m * reliably from the first principle. Experiments suggest that r w * ~ 5 - 1 0 m , . We would like to remark that the above analysis is correct only when Asc % eF. This is because the superconducting gap equation is derived using mean-field theory. The mean-field theory is correct only when there are many Cooper pairs in an area of the coherence length squared. In our case this leads to

(4.15)

Vom'a"

Such a bound state can appear only when the temperature 7";S ej. Comparing e» with the superconducting transition temperature Tc, we find In V0m*a2

"2-Jioo/eFexp

The superconducting gap equation can be easily solved. One finds ,

ing energy is given by

(4.9)

£* =

(4.14)

"tic

1 ttnef A^c

AneF -exp a>o

An V0m*a2

This justifies that Asc £ eF is the correct condition for the mean-field theory to be valid. One may also consider the Cooper problem of two electrons binding above the rigid Fermi surface. The two electrons may form a bound state due to the attractive interaction between them. The bind-

(4.16)

Above discussion we have discussed the superconducting properties for the Fermi surface in Fig. 14. With a little modification the above discussion also applies to the Fermi surface in Fig. 15. In this case the superconducting gaps have the sign indicated in Fig. 18. T h e superconducting gap still has no nodes on the Fermi surface. The 90° rotation is spontaneously broken in the superconducting state. The real-space superconducting order parameter ee U)'-

+,

'~J'~J'll

+(.-

l) y *"'"][l ~ ( ~ 1 ) * " ' ' ] (4.17)

or

—

where £ s c 2nvF/&sc is the superconducting coherence length, vF is the Fermi velocity, and n is the density of the holes. Using vF — (2eF/m*)U2 and n~(2m*ln)eF, we find

Asc eF

Therefore, when Asc ~ eF, hence e/, ;S Tc, the collective pairing of electrons (i.e., the superconducting transition) happens at a temperature higher than that at which two electrons can form a bound state in real space. We would also like to remark that the spin-up and spin-down holes being paired are admixtures of singlet and triplet states. This is due to the fact that the spinrotation symmetry is broken by the S D W mean-field vacuum and electron spin is not a good quantum number. This can be seen more clearly by considering the wave function of two paired electrons ^ ( * i ° i , * 2 " 2 ) , where a, = ± 1 are the spin variables of electrons. The fact that the spin is not a good quantum number is reflected in that the wave function yr(&iai,&2<>2) cannot be rewritten in a factorized form ^ 0 (fci,*2)Vj(oi.«2), since the orbital wave functions of spin-up electrons and spin-down electrons are different, e.g., for the electrons near the top of the valence band, the wave functions vanish on the even lattice site for spin-up electrons while the wave functions for the spin-down electrons vanish on the odd site. Therefore, in our case the pairing wave function is the mixture of singlet and triplet spin states.

x[l+(-l)''][l-(-l)'']

«
11 675

ce U)''

+,

'~J'~J'U

- ( - 1 )J'~'*Hl

+ ( - 1 )*"'']

x[l + ( - l ) ' ] [ i - ( - l H I and is demonstrated in Fig. 19. The pairing wave function is odd under the 180° rotation which resembles a p-v/ave pairing. We notice that in our «/-wave-like superconducting state (for the Fermi surface in Fig. 14) the superconducting order parameter is a single complex scalar since the spacial rotation symmetry is broken by the lattice. But for our pwave-like superconducting state for the Fermi surface in Fig. 15 the situation is very different. One may notice that there are two separated Fermi surfaces in Fig. 15 and

459 11676

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

39

parameters is the Goldstone mode which is pushed to plasma frequency when the long-ranged Coloumb interaction is included, while the difference of the two-phase order parameter 0\ — 02 remains to be a soft mode and is gapless in the limit when the two Fermi liquids decouple completely. The existence of this soft bosonic mode may have some important consequences on specific heat in the superconducting regime and the critical properties of the superconducting phase transition. The superconducting state with rf-wave-like symmetry is also obtained in Ref. 22. The pairing potential obtained there agrees with the one in Ref. 5 and in Sec. Ill of this paper. But as we have emphasized in Ref. S and in this paper, the superconducting state should be a nodeless one at least in the low-doping limit. A p-wave superconducting state in the presence of SDW is obtained in Ref. 23 in a somewhat different approach. Higher angularmomentum pairing with complex nodeless-pairing order parameter has been considered in Ref. 24.

FIG. 18. The two possible sign configurations of the superconducting gaps for the Fermi surface in Fig. IS. The dotted lines are node lines of the superconducting gap.

V. DISCUSSION

In developing a theory of high-temperature oxide superconductivity, several fundamental issues must be addressed which ultimately will be answered by experiment. (1) Is the BCS formalism, involving the condensation of pairs of fermions, qualitatively correct or is a totally the electrons on different Fermi surfaces have little indifferent approach necessary? (2) Are the observed teraction between them, at least in the low-doping limit. finite-range strong spin correlations in the superconductIn this case the electrons near the two Fermi surfaces ing state crucial to the occurrence of high Tc, or is some behave like two independent superfluids coupled by a other mechanism operative in these materials? (3) Is the weak Josephson-type coupling. Therefore, there are two complex superconducting order parameters {e' ' and itinerant electron or the Mott-Hubbard localized approach better suited to treating superconductivity in these je and the Ginzburg-Landau theory must be generalmaterials? (4) Does the problem fundamentally involve ized to this case. The sum 0\ + 02 of the two-phase-order one, two, or more bands or orbitals per unit cell? There have been a number of Monte Carlo studies of the single-band Hubbard model.25 The results of these studies show a suppression of the bare-particle-pairing susceptibility as the Hubbard interaction V is increased. However, when the renormalized one-particle Green's function is included,26 a slight enhancement in the rf-wave pairing susceptibility is found. This
460 39

DYNAMIC SPIN FLUCTUATIONS AND THE BAG MECHANISM . . .

spin bag, is likely to lead to a small overlap between the bare particle states created by the operators used to define the gap parameter in the Monte Carlo studies and the quasiparticle states. If this overlap is indeed small, it is unclear if the Monte Carlo analysis is sufficiently accurate to properly recognize pairing were it to occur. It would be helpful if the Monte Carlo studies were able to study the quasiparticles in the normal state as a prelude to studying superconductivity. In particular, it is important to investigate the correlation functions giving the spin distribution and the spin-spin correlations in the vicinity of a dynamic (versus localized) hole. The exact diagonalization performed by Dagotto et al., clearly demonstrates the continuity of the spin-bag picture as a function of the coupling.27 In our approach, we started from the single-band Hubbard model with the antibonding Cu dxi-yi and O p„ orbitals being the relevant energy band. While this is an extremely simplified model, considering the complexity of these materials, we believe that this model incorporates the basic physics responsible for the high-temperature superconductivity. Within the weak to intermediate coupling regime we are considering, only the states close to the Fermi surface are important, while the details of the localized nature of the corresponding Wannier functions in the position space are less relevant. For a discussion between the relation of the one- and two-band model, see Zhang and Rice.28 A more realistic model for the Cu-O plane of the highTc superconductors should involve three bands, from the bonding, nonbonding, and the antibonding combinations of the Cu dxi-yi and O pa orbitals. In the undoped samples, the bonding and the nonbonding bands are filled, while the antibonding band is half-filled. Due to the strong antiferromagnetic fluctuations, a SDW (pseudo) gap is opened at the Fermi surface. An important issue is whether the top of another energy band (e.g., the bonding or the nonbonding bands) happens to lie in the gap opened by the SDW. If this is the case, the doped hole will be at the top of this band rather than in the antibonding band and a genuine two-band model is needed to study the interaction of the holes and the antiferromagnetic fluctuations.29 However, electron rather than hole-doped high-7"c materials have recently been synthesized, supporting the particle-hole symmetry characteristic of a one-band model. In general, the Fermi surface of the doped holes and the couplings to the spin fluctuations depend on which band is populated by the holes. However, our one-band spin-bag model and many other models involving two bands share the same physical feature that a doped hole destroys the antiferromagnetic order in its vicinity and thereby leads to an attractive interaction when two holes share the same region with the depressed antiferromagnetic order. Whether this attractive interaction can lead to a nodeless superconducting gap certainly depends on the shape of the hole Fermi surface. Within the two-band model, there are possibilities where the holes form pockets at the magnetic zone boundary,30 just as the case of the one-band spin-bag model. In this case, it is likely that the superconducting gap is nonzero everywhere at the hole Fermi surface.

11 677

In this work we have assumed that strong spin correlations are intimately related to the large value of Tc in the cuprates. An important question is whether the spin order in the superconductor corresponds to local antiferromagnetic order or a coupling of pairs of spins to angular momentum zero as in the resonating-valence-bond-type state. Excitations in these two schemes are very different, the former leading to spin charge e fermions while the latter appears to support two types of excitations, spin- 7 neutral fermions and spin-0 charge e bosons. We have taken the point of view that the spin correlations are locally antiferromagnetic in character so that the excitations are spin- y charged fermions, consistent with a BCS pairing approach. While the pairing theory is generally phrased in terms of a Fermi-liquid scheme based on an itinerant electron approach, the theory also applies to the holelike excitations in the Mott Hubbard regime if these excitations carry spin- 7 and charge e, as in the weak-coupling regime. Several authors have discussed this possibility.I0 As in the itinerant approach presented in this paper, the local antiferromagnetic order is reduced by the presence of a hole forming a bag or antiferromagnetic polaron type of excitation. It is possible that the bag simply changes in size as one goes from the weak-coupling (small U/t) to the strong-coupling (large U/t) limit, maintaining its spin and charge quantum numbers. In this case, one would expect that the correct physics of the materials could be treated starting from either limit. Beginning with the itinerant scheme, we have shown that a hole in a commensurate spin-density-wave system preserves its spin and charge but clothes itself with a region of reduced antiferromagnetic order, the size of this region is given by the SDW coherence length 4SDW = hvF/n&- For typical values of the SDW gap A—0.8 eV and the Fermi velocity of the nonmagnetic phase U F ~ 5 X 107 cm/sec, £SDW is of order 2a-3a. The hole and its surrounding bag move as a quasiparticle whose mass we estimate to be of order mB — 6me. When a second quasiparticle is added, it is attracted to the other excitation, as worked out above. While this result is intuitively clear since the two holes can share (at least temporarily) a common bag thereby lowering their energy, this is in contrast with the corresponding situation in the absence of at least local antiferromagnetic order. In the nonmagnetic phase, one finds a repulsive interaction between quasiparticles in momentum space. Therefore, the existence of the SDW gap gives an increase of the hole energy which is reduced by the presence of a second hole, producing an effective attraction. Whether these excitations actually bind in pairs or not in the weakly doped insulator is unclear, since the acceptor ions, e.g., the effective negatively charged Si\ ions in the 2:1:4 material or the O ions in the 1:2:3 06+ x material, may act as trapping centers for the positively charge bags. In Fig. 20 a schematic phase diagram of the cuprate superconductors is shown as a function of the hole concentration x. The antiferromagnetic insulator is separated from the metallic region by an intermediate phase which has been described as spin-glass-like. One possible

461 11 678

J. R. SCHRIEFFER, X. G. WEN, AND S. C. ZHANG

FIG. 20. A schematic phase diagram of the cuprate superconductors. AF, antiferromagnetic; SG, spin glass; SC, superconducting, x is the hole concentration.

39

ample one can transform from rf-like to j-like behavior by shifting the origin of k space, as shown above (see Figs. 12 and 14). The essential point is that the magnitude of &fc is found to be nonzero at all points of the Fermi surface and no power-law temperature dependence is expected as in conventional / > 0 superconductors. The nodeless properties of the superconducting state have been observed in many experiments. 4 Analogous bag effects may be operating in C D W superconductors, like Ba(Pb,Bi)0 3 and possibly ( B a , K ) B i 0 3 . 1 3 It is known that CDW correlations occur in the former and lead to a pseudogap. We have found that charge-bags form in this case as well and such bags are attractive, however the charge fluctuations are coupled to phonons leading to a partial isotope effect. A number of questions remain to be resolved regarding the bag approach. (1) Are the spin and charge of the excitations correctly predicted in the antiferromagnetic insulator and in the metallic phase? (2) What is the actual shape of the Fermi surface in the metal and is the pairing-order parameter free of nodes on the Fermi surface? (3) What is the influence of quantum fluctuations in the superconducting properties? (4) Can the bag approach account for the general experimental facts which are not subject to materials difficulties? Hopefully, these issues will be clarified in the forseeable future.

scenario for this behavior is that the excitations are pinned by the charged acceptors in both the antiferromagnetically ordered phase as well as in the spin-glass phase. When the bag density increases beyond a critical value xc, the exclusion principle increases the kinetic energy of the bags sufficiently to melt the glass, leading to a Fermi liquid of spin bags. Regardless of the details of this intermediate phase, the bags are assumed to form a degenerate Fermi liquid in the metallic phase. While we have explicitly calculated the bag-bag interaction only in the long-range spin-ordered phase, it is plausible that a similar attraction occurs in the metallic phase if the range of spin order is larger than the bag size. It may be that this restriction is not essential for an attractive interaction to occur, a question we are now studying. We have solved the BCS gap equation using the interaction between spin bags derived above. We find, as reported in our earlier publication, that Afc is nodeless over the Fermi surface. However, the traditional characterization of A | C as s, p, or d like is not an invariant concept in the presence of strong-band-structure effects, since for ex-

We would like to acknowledge helpful discussions with E. Abrahams, S. Chakravaty, S. Kivelson, H. Nakanishi, D. J. Scalapino, and K. Schonhammer. This work was supported in part (J.R.S. and S.C.Z.) by the National Science Foundation Grant No. DMR 85-17276 and also (X.G.W.) No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration.

'J. G. Bednorz and K. A. Muller, Z. Phys. B 64 189 (1986); C. W. Chu et al., Phys. Rev. Lett. 58, 405 (1987). 2 J. Bardeen, L. N. Cooper, and J. R. SchriefTer, Phys. Rev. 108, 1175 (1957). 3 P. L. Gammel et al., Phys. Rev. Lett. 59, 2592 (1987); C. E. Gough et al. Nature (London) 326, 855 (1987); T. Witt, Phys. Rev. Lett. 61, 1423 (1988). 4 Y. J. Uemura et al., Phys. Rev. B 38, 909 (1988); D. R. Harshman et al., ibid. 36, 2386 (1987); S. E. Inderhees et al., ibid. 36, 2401 (1987); G. A. Thomas et al., Phys. Rev. Lett. 61, 1313 (1988). 5 J. R. SchriefTer, X. G. Wen, and S. C. Zhang, Phys. Rev. Lett. 60,944(1988). 6 G. Shirane et al., Phys. Rev. Lett. 59, 1613 (1987); J. Transquada et at., ibid. 60, 156 (1988). 7 K. B. Lyons et al., Phys. Rev. B 37, 2353 (1988); K. B. Lyons et al., Phys. Rev. Lett. 60, 732 (1988). 8 S. Chakravaty, B. I. Halperin, and D. R. Nelson, Phys. Rev. Lett. 60, 1057 (1988). 9 W. P. Su, Phys. Rev. B 37, 9904 (1988). ,0 Y. Nagaoka, Phys. Rev. 147, 392 (1966); W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 1324 (1970); B. Shraiman and E. Siggia, Phys. Rev. Lett. 60, 740 (1988); S. Schmitt-

Rink, C. M. Varma, and A. E. Ruckenstein, ibid. 60, 2793 (1988); S. A. Trugman, Phys. Rev. B 37, 1597 (1988); C. L. Kane, P. A. Lee, and N. Reed (unpublished); A. Aharony et al., Phys. Rev. Lett. 60, 1330 (1988). 11 P. W. Anderson, Science 235, 1196 (1987); P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett. 58, 2790 (1987); S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Phys. Rev. B 35, 8865 (1987); R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988). ,2 W. P. Su, J. R. SchriefTer, and A. J. Heeger, Phys. Rev. B 22, 2099 (1980). ,3 A. W. Sleight, J. L. Gillson, and P. E. Bierstedt, Solid State Commun. 17, 27 (1975); R. J. Cara et al., Nature (London) 332,814(1988). I4 K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986); D. J. Scalapino, E. Loh, Jr., and J. E. Hirsch, ibid. 34, 8190 (1986); D. J. Scalapino and E. Loh, Jr., ibid. 35,6694(1987). 15 E. W. Fenton, Solid State Commun. 65, 343 (1988), and references therein. I6 J. Hubbard, Proc. R. Soc. London, Ser. A 276, 283 (1963); see also J. E. Hirsch, Phys. Rev. B 31, 4403 (1985). 17 E. Fawcett, Rev. Mod. Phys. 60, 209 (1988).

ACKNOWLEDGMENTS

462 Physica B 163 (1990) 267-270 North-Holland

PSEUDO GAPS AND SPIN BAGS A. KAMPF 1 and J.R. SCHRJEFFER 2 Advanced Studies Program in high-temperature superconductivity theory, Los Alamos National Laboratory, Los Alamos, NM 87545. USA It is shown that antiferromagnetic spin fluctuations in a two-dimensionai metal, such as heavily doped cuprate superconductors, lead to a pseudo gap in the electronic spectrum. The self-energy of spin bags and their pairing interaction in the paramagnetic metal are calculated. These results are consistent with the corresponding results in the weakly doped ordered antiferromagnet.

I. Introduction The existence of antiferromagnetism and superconductivity in nearby regions of the phase diagram of cuprates, as shown in fig. 1, suggests that local spin order plays an important role in bringing about superconductivity at high temperature. Finite range antiferromagnetic spin correlations are observed for doping concentrations in the superconducting range and larger [1], with these correlations eventually vanishing in the strongly doped metallic phase. It was proposed [2] that bag-tike excitations exist in both the paramagnetic metal and antiferromagnetic phases, in which there is a pseudo gap or actual gap for adding a carrier. These spin bags correspond to a local reduction of spin order in the vicinity of an added hole (or electron for the n type doped materials). The spin bags attract by sharing each other's region of reduced spin order, or equivalently reduced gap energy. This proposal was investigated in the antiferromagnetic phase by Wen, Zhang and one of the authors (SWZ) [3], who showed that the pairing interaction between spin bags is in fact attractive, in contrast to the one fluctuation repulsion between quasi particles occurring in the strongly metallic phase [4]. In this paper, we discuss the problem starting from the large doping limit for which the material is a good paramagnetic metal rather than an antiferromagnetic insulator. We show that while the one particle selfenergy is of the Fermi liquid form for large doping, as 1

On leave of absence from Institut fur Theoretische Physik, Universitat zu Koln, FRO. 2 Permanent address: Department of Physics, University of California, Santa Barbara, CA 93106, USA. 0921-4526/90/$03.50 © Elsevier Science Publishers B.V (North-Holland)

x is reduced, antiferromagnetic spin fluctuations grow in strength with a pseudo gap in the electronic spectrum gradually setting in for the doping level approaching the superconducting regime. In the pseudo gap regime, the one particle spectral weight function has two peaks rather than one as in the Fermi liquid regime. The mechanism which brings about the pseudo gap can be seen by dividing the self-energy in the spin polaron Spo, and vacuum fluctuation Xv, parts. It is shown that the pseudo gap arises from the spin susceptibility peaking at the antiferromagnetic wave vector Q. This causes the energy lowering due to polaron effects in the Fermi liquid to be sharply reduced while the Pauli principle suppression of vacuum fluctuations leads to an increase of energy, forming the pseudo gap. A spin bag corresponds to the decrease of the magnitude of the pseudo gap in a region of order of the spin coherence length surrounding the hole. We show that the pairing interaction between spin bags in the paramagnetic regime is attractive for small r

A

Normal

Doping

Fig. 1. A schematic phase diagram of the cuprate supercon ductors. AF, antiferromagnetic; SG, spin glass; SC, super conducting, x is the hole concentration.

463 268

Kampf and JR. Sckrieffer I Pseudo gaps and spin bags

m o m e n t u m transfer, as S W Z found in the spin ordered phase.

2. Self-energy We consider the 2d one orbital H u b b a r d model on a square lattice. At half filling the nesting property of the Fermi surface is known t o produce antiferromagnetic spin order, i.e., a spin density wave ( S D W ) of wave vector Q = (ir/a, i r / a ) . For doping x larger than a critical value, xe, spin order exists only over a finite range L s s , with the spin susceptibility x(q, <") exhibiting a peak around Q with a half width of order L ~ \ T o study how the onset of spin fluctuations affects the self-energy, we consider X(k, )• In the Fermi liquid regime, x varies smoothly with q on the scale of the Fermi m o m e n t u m . In this case Spol dominates and the spectral function has one peak. As

Fig. 2. "Backward" a) and "forward propagation" timeordered Feynman diagram b). a) represents the contribution from the suppression of vacuum fluctuations and b) represents the spin-polaron contribution, s is up or down corresponding to the spin-flip or non spin-flip susceptibility, respectively .

antiferromagnetic spin fluctuations build u p , x D e " comes p e a k e d about the nesting wave vector Q = (IT/ a, via). T h e essential point is that if an electron is injected in a state k above t h e Fermi m o m e n t u m , t h e intermediate state k + Q is necessarily below t h e Fermi surface. This is possible only if t h e i n t e r m e d i a t e state is a hole, as in fig. 2a. I n this limit Spol is small compared to 2„, and a p s e u d o gap is formed. To demonstrate this effect consider t h e simple model susceptibility [5]

X(q, a>)= - \ do>'g(oj') —

;y— iS

2_, >-(*»/«. £*/„) {q -' Q)~ + r '

(1)

which has the basic feature that it is e n h a n c e d a r o u n d the nesting wave vectors Q = ( ± i r / a , ± n 7 a ) . T h e static L e e - R i c e - A n d e r s o n like model [6] is recovered for g(oj) = lim^ _ 0 8(a> - coa). We consider, for example, a linear frequency distribution . ,

2 oi «) .

(2)

Using this susceptibility, we evaluate the self-energy £ ( * , » ) - - l i t / 2 ^? S / | ^ G(k-q,w-

u)

*x(q,")

(3a)

= £ v , +- 2„.

(3b)

Fig. 3. Density of states as calculated from the model susceptibility for different values of T

464 A. Kampf and J.R Schrieffer I Pseudo gaps and spin bags

T " 1 is a measure for the effective spin-spin correlation length measured in units of the lattice spacing. By varying r we can continuously cross over from the paramagnetic metal regime for f" ~ 1 to the pseudo gap behavior for r
oa o* s.t

Ak(a>)=

- |ImG(fc, (0 + IM)\

269

i-a

(4)

IT

is shown in fig. 3. Similarly, the pseudo gap develops within the 2d single orbital Hubbard model away from half-filling by varying the Coulomb energy U or the hole doping concentration [5],

(a)

3. Pairing interaction Based on the spin bag approach we have proposed an alternative pairing mechanism for high-Tc materials. As demonstrated above strong antiferromagnetic spin correlations lead to a pseudo gap in the quasi particle spectrum. The pairing attraction Vkk. arises in this approach from the lowering of the system energy when two quasi particles share the region of reduced antiferromagnetic correlations surrounding each of these excitations. The range of this attraction is of order the SDW coherence length fSDW- In momentum space, this corresponds to an attraction for momentum transfer q smaller than f SDW Contributions to the effective interaction between a pair of holes (or particles) can be obtained by accounting for the effects on the self-energy due to the presence of the second hole. Starting with the selfenergy contribution infig.4a, the presence of a second particle with the same spin and momentum as the intermediate fermion line (not involving the bubbles) requires one to include the exchange graphfig.4b in order to restore the Pauli principle. Stretching out the lines it is clear that this diagram is just the repulsive one spin fluctuation exchange process [7]. The corresponding diagram for particles of opposite spin involves an even number of bubbles. In addition, the spin-flip interaction corresponding to the particle-hole ladder must be added to preserve spin rotation invariance. However, the existence of the pseudo gap does provide a new mechanism for an attractive spin bag pairing potential. It arises from the exclusion principle violation shown in fig. 5a, where a fermion line inside

(b) Fig. 4 Self-energy diagram in the presence of a second injected particle a): exchange diagram restoring the Pauli principle b).

one of the bubbles representing the susceptibility is equal to the momentum of the injected particle. This diagram is compensated by its exchange counterpart shown in fig. 5b, or the crossed line diagram shown in fig. 5c. The fact that this diagram leads to an attraction is readily seen. As discussed above, the self-energy due to spin flucutations is positive near the SDW instability and leads to a pseudo gap. A second particle added to the system suppresses these fluctuations through the Pauli principle, reducing the phase space for electron-hole excitations and hence reducing \This real space reduction of susceptibility extends over a range of order fSDW around each quasiparticle and reduces the self-energy for another particle in its vicinity, thereby giving an effective attraction. The effective pairing potential Veff (k, k') is the sum of the attractive spin bag interaction VSB(k, k') and the repulsive contribution from the antiparamagnon exchange processes VAPM(A, k'). Since VSB is largest for small q = k- k' and VAPM is largest for large ? = G, the effective pairing potential is attractive for small momentum transfer and repulsive for large momentum transfer [5]. This is the same behavior that

465 A. Kampf and J.R. Schrieffer I Pseudo gaps and spin bags

270

—-

M

rrrO-r

,©"1(I>'-©.

©

©

©

£1

©

41

(a)

-is: i

(b)

Op (c) Fig. 5 Lowest order spin bag contribution to the pairing interaction. SWZ [3] deduced for Vsf! in the antiferromagnetic phase

4. Conclusion We have shown how antiferromagnetic spin fluctuations depress the density of states in the vicinity of the Fermi surface, leading to a pseudo gap. Injected particles reduce the amplitude of fluctuations and thereby reduce locally the size of the pseudo gap. This combination of a hole (electron) moving with its region of reduced antiferromagnetic correlations is termed a spin bag. We have shown that the pairing interaction between two spin bags is attractive for small momentum transfer due to the holes sharing each other's bag and repulsive for large momentum transfer reflecting the repulsive nature of the spin fluctuation exchange process. This repulsive feature of the pairing interaction suggests that the pairing gap parameter Asc formally has d-wave like symmetry. However, if hole doping at the four corners of the magnetic zone leads to hole pockets which do not overlap due to the small density of states in the pseudo gap the system will appear to have a nodeless order parameter on the whole Fermi surface and to be very different from a conventional d-wave superconductor.

Acknowledgement The Public Service Company of New Mexico and the US Department of Energy are acknowledged for their support of the Los Alamos Advanced Studies Program in high-temperature superconductivity theory.

References [1] G. Shirane et al., Phys. Rev. Lett. 59 (1987) 1613. Y. Endoh et al., Phys. Rev. B 37 (1988) 7663. J. Tranquada et al., Phys. Rev. Lett. 60 (1988) 156 [2] J.R. Schrieffer, unpublished preprint, august 1987. [3] J.R. Schrieffer, X.G. Wen and S.C. Zhang, Phys. Rev. Lett. 60 (1988) 944, ibid., Phys. Rev. B 39 (1989) 11663. [4] D.J. Scalapino, E. Loh, Jr. and J.E. Hirsch, Phys Rev. B 34 (1986) 8190. [5] For details the reader is referred to: A. Kampf and J.R. Schrieffer, Phys. Rev. B, submitted (preprint). [6] P.A. Lee, T.M. Rice and P.W. Anderson, Phys. Rev. Lett. 31 (1973) 462. [7] D.J. Scalapino, E. Loh, Jr. and J.E. Hirsch, Phys. Rev B 34 (1986) 8190. D.J. Scalapino and E. Loh, Jr., Phys. Rev. B 35 (1987) 6694. K. Miyake, S. Schmitt-Rink and CM. Varma, Phys. Rev. B 34 (1986) 6554.

466 PHYSICAL REVIEW B

VOLUME 41, NUMBER 10

1 APRIL 1990

Pseudogaps and the spin-bag approach to high-7,(. superconductivity A. Kampf* and J. R. Schrieffer+ Advanced Studies Program in High Temperature Superconductivity Theory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received 2 October 1989) It is shown that anliferromagnetic spin fluctuations in a two-dimensional metal, such as heavily doped cuprate superconductors, lead to a pseudogap in the electronic spectrum. The spectral function evolves from one peak in the Fermi-liquid regime to two peaks, one for particles and one for holes. The self-energy of spin bags and their pairing interaction are calculated. These results are consistent with the corresponding results in the weakly doped ordered antiferromagnet.

I. INTRODUCTION High-temperature oxide superconductors exhibit antiferromagnetism and superconductivity in nearby regions of the phase diagram, as shown schematically in Fig. 1, where x denotes the doping concentration. Recent experiments show that the qualitative features of the phase diagram are the same for either hole or electron doping,' suggesting that particle-hole symmetric models are favored. Furthermore, neutron 2 and Raman 3 scattering experiments have established in the superconducting phase, the existence of antiferromagnetic spin correlations whose range is large compared with the lattice spacing. It is plausible that the existence of high-r c superconductivity is intimately related to these spin correlations. In developing a theoretical understanding of these phenomena, two limiting points of view have been taken. The first, advanced by Anderson, 4 assumes that the onsite effective electron-electron Coulomb repulsion U is large compared to the bandwidth W. In this approach one starts from the Mott Hubbard insulating state in which all sites are singly occupied. While this state is antiferromagnetic, it is presumed that for doping beyond a critical concentration the system enters a phase in which pairs of spins are coupled to total spin zero, with these pair bonds resonating over all possible spin configurations. Excitations in this scheme are assumed to carry charge e and spin 0 (holons) or charge zero and spin { (spinons). This resonating-valence-bond (RVB) approach leads to superconductivity only when interplanar electron hopping is taken into account. In related work, a number of authors have explored the existence of neutral spin-{ excitations in frustrated antiferromagnets. Wen, Wilczek, and Zee 5 have shown that a parity- and time-reversal-violating spin-order parameter of the form Si-(S 2 XS 3 ) can exist if the frustration is of the proper form. Laughlin 6 has proposed that in such frustrated antiferromagnets, quasiparticles exhibiting fractional statistics can lead to superconductivity. In addition there are several informative studies 7 of the dynamics of one or more holes hopping in a Heisenberg antiferromagnet having long range order. Thus, large- V models exhibit a rich set of phenomena. Whether these 41

include a valid theory of high-temperature superconductivity in cuprates is unclear at present. The other limiting point of view is that U is smaller than the bandwidth W. In this case the electrons are taken to zero order as being itinerant, with antiferromagnetism described as a spin-density wave (SDW). If the Fermi surface has appropriate nesting properties, as in the half-filled two-dimensional square lattice with nearestneighbor hopping, a gap A S D W extends over the entire Fermi surface and the system is an antiferromagnetic insulator, as in the large- V Hubbard model. The quasiparticles in the SDW phase carry charge e ( — e) and spin \ as conventional holes and electrons. However, a quasiparticle alters the sublattice magnetization in its vicinity and forms a region of reduced antiferromagnetic order termed a spin bag extending over a distance equal to the SDW coherence length, ^ = ^ u f / A S D W (Ref. 8), which moves with the particle. As Wen, Zhang, and one of the present authors (SWZ) (Ref. 9) have shown, the pairing interaction between two of these spin bags is attractive at small momentum transfer, reflecting the reduced exchange gap inside the SDW coherence volume surrounding each excitation. Within the pairing theory, one finds that the doped material exhibits superconductivity with a gap A s c which is nodeless over the entire hole (electron) Fermi surface.

Normal

SG

SC Doping

FIG. 1. A schematic phase diagram of the cuprate superconductors. AF, antiferromagnetic; SG, spin glass; SC, superconducting, x is the hole concentration. 6399

© 1990 The American Physical Society

467 6400

41

A. KAMPF AND J. R. SCHRIEFFER

Therefore, the temperature dependence of quantities such as the N M R relaxation and acoustic attenuation rates, surface resistance, penetration depth, etc. are expected to be activated at low temperatures, as in conventional swave superconductors. We note that the pairing attraction in this approach basically arises from the dynamics of the holes and spin degrees of freedom in the twodimensional layers, in contrast with the resonatingvalence-bond (RVB) approach which relies on interplanar coupling. In the spin-bag approach, three-dimensional superconductivity in these layered materials results from holes hopping between planes coupling the phases of the finite-range pairing correlations, already well developed in the planes into a coherent three-dimensional state. While the SWZ analysis is carried out in the antiferromagnetic phase, for which spin-up and spin-down holes tend to live on different sublattices, they argue that the resulting pairing interaction is insensitive to the presence of long-range spin order so long as the spin-spin correlation length L s s is large compared to the in-plane superconducting coherence length £sc = fivF/Tr&sc. Experimentally, it appears that these quantities are comparable for many high-temperature superconductors, although there remains considerable uncertainty about both quantities for most materials. In the present paper, we approach the problem from the paramagnetic metal phase (large x) rather than the antiferromagnetic (AF) phase (small x). First, the one particle self-energy is studied for a model form of the dynamic spin susceptibility, ;f(q,w). The familiar mass renormalization of Fermi liquid theory is obtained if X varies smoothly on the scale of the Fermi momentum and the Fermi energy, appropriate to U small compared to W For suitable nesting properties, as U increases x becomes peaked as a function of q in the vicinity of the nesting wave vector Q and its characteristic frequency is reduced from the Fermi energy to the spin-fluctuation energy. In this regime, the one-particle spectral weight begins to develop two new quasiparticle peaks, rather than the one peak of conventional Fermi liquid theory whose weight vanishes as A F order sets in. One finds that the density of states smoothly evolves from a modest renormalization at the Fermi surface for small U to a suppression when strong antiferromagnetic fluctuations build up. As longrange spin order sets in, the pseudogap becomes the SDW gapWhile the model susceptibility calculations illustrate the physics of the problem, to obtain a more quantitative understanding of the pseudogap formation we have carried out detailed numerical calculations of the susceptibility within a random-phase approximation (RPA), as a function of U and doping level x. These results are consistent with the model studies mentioned above and show the smooth evolution of Fermi liquid to pseudogap behavior. The pairing interaction arising from the local suppression of antiferromagnetic fluctuations is then calculated using the model forms of X- While the conventional oneantiparamagnon exchange leads to a repulsive interaction, it is readily seen that the existence of the pseudogap provides a qualitatively new mechanism for the pairing

attraction in the paramagnetic metal (PM) phase 8 just as SWZ (Ref. 9) deduced in the A F phase. In the P M phase, the leading-wrder pairing interaction is represented by a diagram consisting of two crossed spin fluctuations lines. The physical meaning of this diagram is discussed and the results are compared with those obtained in the A F phase. II. MODEL SUSCEPTIBILITY We consider the 2D single-orbital H u b b a r d model on a square lattice H = ~t

2

^ClCla

+

ClClo)+U^nnnn-^nia, (i)

where C]a and Cia are creation and annihilation operators for localized electron states of spin a on site /, and nia = CjaCia. /x is the chemical potential. It is well known that with increasing Coulomb repulsion U, the Hubbard model develops strong antiferromagnetic correlations between spins on nearest-neighbor sites. At precisely half-filling, the nesting property of the Fermi surface of the tight-binding band E t = — 2t {cosk.a

(2)

+ cosk„a)

gives rise to a ground-state instability leading to a spindensity wave. This becomes most evident in the strongcoupling limit U/t»\, where to leading order in t/U the Hubbard model is approximately equivalent to an antiferromagnetic (AF) spin-} Heisenberg model. 1 0 The 2D s p i n - j Heisenberg model itself establishes long-range A F order at zero t e m p e r a t u r e . ' ' Away from half-filling ft < 0 or at finite temperature, antiferromagnetic correlations persist giving rise to a finite spin-spin correlation length, ^•ssIn a R P A treatment of the longitudinal spin-density correlation function, the instability to an S D W is determined by the Stoner criterion (3)

l = £/*0(q,&> = 0) .

Xo(q,co) is the correlation function in the absence of the interaction and is given by

x 0 (q>«>> = N

wk(l— n t+ q) » + Ek-Ek + 'k + q

q

-l8

(1-nJ

W + E k - E k + q + l'S

(4)

with the electron occupation number n k . At half-filling <,fi = 0) ^o(q,
468 41

PSEUDOGAPS A N D THE SPIN-BAG APPROACH TO HIGH-7; . . .

mensurate depending on temperature and band filling.13 In the presence of a spin-density wave with long-range order a gap opens up in the single particle excitation spectrum. In weak coupling U < t the gap A S D W is exponentially small, while in strong coupling A S D W is of order U (Ref. 9i. In the paramagnetic metal regime, finiterange spin correlations cause the density of states to show a depletion of spectral weight near the Fermi level, reflecting the nearby instability to antiferromagnetic ordering. The formation of this pseudogap may be viewed as a precursor effect to the metal-insulator transition. The situation is similar for the BaPbxBil_xOi compounds where the instability arises from the formation of a charge-density wave rather than a spin-density wave 14 (SDW) (see Fig. 2). For a finite-hole concentration we study the effects of spin fluctuations on the hole self-energy within the random-phase approximation (RPA). While the RPA does not provide a fully accurate description of ^(q,a>), the qualitative effects are already apparent in this approach. In Figs. 4(a) and 4(b) the S2 and S± spin fluctuation contributions to the ground-state energy are shown in the presence of an added hole in momentum state k. Two effects occur. The first is that the Pauli principle is violated by intermediate hole lines of the vacuum fluctuations which also have momentum k, as shown in Figs. 3(a) and 3(b) for Sz and S± modes, respectively. This Pauli principle suppression of the vacuum fluctuations is accounted for by exchanging the lines of a vacuum fluctuation hole with that of the added hole, thereby canceling the Pauli principle violating diagram as in Figs. 3(c) and 3(d). This gives a positive contribution to 2 for electrons (co>0) and a negative contribution for holes ( G J < 0 ) . The second effect is the conventional polaronlike shift which opposes the first effect. One finds to this order that the total hole self-energy is given by

*(q,v) =

6401

* 0 (q>v)

(6)

i-tfXo
Representing *(q,
.0. k t — * ^ ~ * ~ Tt

kt (a)

(b)

(c)

W)

2(k.w)= - jw2jf 2 /|JG
BaPbxBi^xQ,

X < 0.65

L N(E)

- 0.7

•0.8

• 0.8

IW

FIG. 2. Schematic representation of the change in the electronic density of states for the compositions near the metalsemiconductor transition (taken from Ref. 14).

FIG. 3. Contributions to the hole self-energy £(k,&)) due to (a) S, vacuum fluctuation violating the Pauli principle in the presence of the injected hole; (b) the exchange version of (a) restoring the Pauli principle; (c) and (d) the 5 ± analogs of (a) and (b), (e) the "backward propagation" time ordered Feynman diagram representing (b) and (d); (f) the "forward propagation" diagram representing the polaron-like contribution to 2. In (e) and (f) s is up or down corresponding to the spin-flip or non-spin-flip susceptibility, respectively.

469 41

A. KAMPF AND J. R. SCHRIEFFER

6402

paramagnetic metal regime. We choose a simple linear (normalized) frequency distribution

the spin susceptibility by X(q,)

(7)

Q-<S.±,) (q-Q)2+r2

g(o>) =

The Lorentzians are centered around the four nesting wave vectors (±ir,±ir) their width T is determined by the inverse of an effective spin-spin correlation length and A is a coupling constant. In essence, ^(q,a)) represents a potential-potential correlation function (V(T)V(T')) for a slowly fluctuating one-body interaction V{i), whose expectation value is zero, < V) = 0 . An analogous susceptibility has been used by Lee, Rice, and Anderson to study fluctuations in a one-dimensional electron phonon system close to a Peierls instability. The model susceptibility (7) can be easily extended to allow for finite frequency spin dynamics. For this purpose we introduce a frequency distribution g{o)) = (\/Na) ~£a8(a) — coa) for a set of Na characteristic frequencies ioa, each contributing as a dispersionless propagating mode to the spin susceptibility. ^(q,
v

f j

. ,

.i

y ( q , < u ) = - J dcogico )-

2(o'X

4X2

Rex(q,co)-

1

(10b)

where we choose


2

(q-Q) 2 + r 2

(11)

III. SELF-ENERGY Using the model susceptibility Eq. (10) we can evaluate the real and imaginary part of the one-particle selfenergy, where for simplicity we begin by replacing the full propagator G by G0 in the RPA formula (5) and neglect vertex corrections. One finds

0)0

(0 — z k - q

N

"k-q

H-q

— In

1+-

a>0 t-k-q COQ

-(l-«k-,) l

co — co0

(8)

The frequency distribution g (ai) is assumed to be approximately independent of q since there are no collective modes with a well-defined dispersion o(q) in the

{UXY

O] + (O0

Imjr(q,a>)= ^ - - ^ - 0 ( < u o - \o>\ ){q)

Q=(±ir,:

The static Lee-Rice-Anderson-like model is recovered in the limit g(
Im2(k,o)) = 3ir

— In

o)0 2

(10a)

*(q)=

(o)')2 + i5

r

co0

with a cutoff frequency co0, determining the characteristic frequency scale for spin fluctuations. However, in the adiabatic limit, when a0 is small compared to the SDW gap, the results of our analysis are not sensitive to the detailed form of g(
2

2 — -Q) 2 +r 2

(TJX)2 \

(9)

6(a) 0 — co) co0 u0

+ ln

1--

"•k-q

L

(I2a)

k-q

L

2*(q)-

k-q [ + « k _ „ 9 ( e k _ „ — iu)0(a) o — e k _ q +
( l - n k _ Q ) 0 ( £ t f - £•^k k ++ qq)0(4)o-GJ + £ k _-)] . For a quantitative discussion of the self-energy (12) at 7" = 0 we first consider the limit where 0(q) is a 5 function centered at the SDW nesting wave vector Q =(ir,ir). 2(q,a>) is as sketched in Fig. 4 for a given momentum k above the Fermi surface. Due to the artificial sharp cutoff of the frequency distribution g () = [-i.

(13) For a), and o>3 the

(12b)

imaginary part of the self-energy vanishes and therefore these solutions correspond to two well defined quasiparticles with infinite lifetime, indicating the presence of two quasiparticle bands. The intermediate solution a>2, however, may be identified with the simultaneous excitation of an electron (hole) plus an incoherent cloud of spin fluctuations with momentum Q, giving rise to a finite lifetime of the order of co0/( UX)1 as determined by the imaginary part of the self-energy Im£(k,tt>2)- This intermediate solution therefore introduces incoherent spectral weight into

470 41

PSEUDOGAPS AND THE SPIN-BAG APPROACH TO HIGH-r c

6403

R e l ( k , to) —

FIG. 4. Hole self-energy calculated from the model susceptibility (8) with a linear frequency distribution for spin fluctuations and in the limit $(q) = (2jr)28(q — Q). The thin lines indicate the changes if #(q) is broadened into a Lorentzian.

The 0)2 solution is responsible for the finite density of states in the gap leading to a pseudogap. However, we find that by using G rather than G0 in Eq. (5) for 2 , the incoherent density of state is substantially reduced in magnitude since the decay of a quasiparticle of energy A leads to another particle of energy > A and a spin fluctuation of energy > 0. This mechanism acts in a feedback sense, further suppressing the density of states in the gap. In the limit (u0—>0 the lifetime of the excitations in the pseudogap, associated with the a>2 solution, vanishes. In this limit a static SDW gap A S D W = 2X.[/ exists with longrange spin order, i.e., #(q) = (2ir) 2 5(q—Q). It is important to note that in the limit Ta « 1 of the Lorentzian momentum distribution of the susceptibility the self-energy is strongly momentum dependent and the real part of 2(k,&>) of quasiparticles (holes) is positive (negative) indicating that their energies are shifted away from the Fermi level compared to the energies ek in the absence of interactions. This shift is precisely the origin for the gap in the excitation spectrum. It also tells us that it is the suppression of vacuum fluctuations, as represented by the "backward propagation" Feynman diagram Fig. 3(e), which dominates the physics close to the metal-insulator transition. The "forward propagation" diagram shown in Fig. 3(f) for a particle injected above the Fermi surface is suppressed because close to the antiferromagnetic instability ^"(q,6>) predominantly transfers momentum close to the nesting wave vector Q to the particle. Thus, the added particles would be scattered from a state above to a state below the Fermi surface, violating the Pauli principle. For finite increasing width T of the Lorentzian momentum distribution of the susceptibility the two quasiparticle peaks from &>, and ) broaden and develop into incoherent background contributions, while the intermediate co2 solution develops into the single quasiparticle of the weakly correlated paramagnetic metal. With increasing T the pseudogap is therefore less and less pronounced as shown by the densities of states in Fig. 5, calculated as the momentum space average of the spectral function

FIG. 5. Density of states as calculated from the model susceptibility for different values of T.

(15) •(V

1,

In the limit where the spin-spin correlation length is reduced to a single lattice spacing, To = 1, the self-energy is qualitatively different as shown in Fig. 6. In this limit antiferromagnetic spin fluctuations are substantially reduced, and the self-energy is essentially momentum independent. Most importantly, the real part of the selfenergy for quasiparticles (holes) does to the Fermi energy is now negative (positive). This indicates that the quasiparticle energies are now shifted towards the Fermi level. Since the real part of the self-energy of the quasiparticles (holes) is negative (positive) it is now the "forward propagation" diagram Fig. 3(f) representing the magnetic polaron that gives the dominant contribution to the selfenergy. The smooth crossover from a Fermi liquid-like

UU)-*4.0 -0.4

-12

0)„-2 r= 1

0 M/t

12

FIG. 6. Real part of the self-energy calculated from the model susceptibility for T ~' = 1 (in units of the lattice spacing). Energies are measured in units of the hopping matrix element t.

471 A. KAMPF AND J. R. SCHRIEFFER

6404

metal to the pseudogap situation is therefore determined by the change in the relative importance of the "forward and backward propagation" Feynman diagrams. Despite its simplicity the model susceptibility provides qualitative insight into how the pseudogap develops when the antiferromagnetic instability is approached. For a commensurate spin-density wave (SDW) the pseudogap will develop around the Fermi level of the undoped system. For zero damping the Fermi level falls into the pseudogap leading to insulating behavior. For finite doping concentrations the chemical potential is shifted into the large density of states regime of the broadened peaks around the edge of the pseudogap and the system will therefore behave as a Fermi liquid-like metal.

dv G0(k — q,co — 2ir

2(k,
IV. NUMERICAL CALCULATION FOR THE HUBBARD MODEL In order to show that the model susceptibility (8) indeed contains the basic physics of the single band Hubbard model in the intermediate coupling regime we calculate the self-energy (5) numerically using the randomphase-approximation (RPA) result for the susceptibility (6). For simplicity we replace the full propagator G by G0. The frequency integral along the real axis is evaluated by deforming the integration contour into the complex frequency plane, 16 splitting up 1 into two contributions: a line integral along the purely imaginary frequency axis plus two residue contributions from the poles of G0

iv)x{q,iv)

-}t/2-jF2[(1-"k-q)©(«-Ek-q)-''k^qe(Ek-q-w)k(q,^-Ek-q)

Since ^0(q,&)) is an even function of frequency the integral along the imaginary axis is purely real. The imaginary part of the self-energy is therefore entirely determined by the residue contributions, imZ(k,6J)=-|-^-2[(1-"k-q)e(6J-Ek_q) -nk_qe(ek_q-
(17)

The real part of the self-energy is then obtained by the Kramers-Kronig transformation: Re2(k, &))==/>/

dv

ImZ(k.v)

(18)

Using these formulas we can straightforwardly calculate

A

U =-0.375 I 1

I ; :

O

States

S" ~

•-,

& ' to c

v^

\0 /

-. "

_ -

J

?m

0>

u-o — U-05 -•-u.i — U -2

f

,ai 0 .4

'A

0.1

Q

/•* /

•—-*"?t"r'i"* J

|'\ ' >

0

1

1

1

(16)

the spectral function (14) and the density of states (15). The Coulomb energy U and the chemical potential (or the particle number) are chosen to ensure that we are in the paramagnetic metal regime, not too close to the antiferromagnetic instability as determined by the Stoner criterion (3). Figure 7 shows the evolution of the zero temperature density of states with increasing U for a fixed chemical potential y. = — 0.375f. For this value of the chemical potential the critical Hubbard U for the Stoner instability to occur is approximately 3f. The density of states for U=0 shows the usual van Hove singularity of the tight binding band in two dimensions. The momentum integrals have been performed by a coarse graining procedure on a finite lattice. The resulting density of states has then been smoothened over the discrete level spacing of the finite lattice leading to the rounding of the singularity and the band edges. As expected, with increasing U the density of states develops the pseudogap around the Fermi level. In addition, due to the increasing amplitude for spin fluctuations long tails of incoherent spectral weight appear extending far beyond the band edges. The pseudogap develops similarly for a fixed Hubbard U by increasing the chemical potential or equivalently by reducing the hole doping concentration. V. PAIRING INTERACTION AND THE GAP EQUATION

\\

fi'i ..•"'

41

A ;-•-,-..»-.

Cfl/t

FIG. 7. Random-phase approximation density of states of the Hubbard model for different values of U.

In conventional superconductors, the exchange of one phonon between quasiparticles leads to a spin independent attractive interaction so long as the quasiparticles are within the Debye energy of the Fermi surface. However, in high-Tc materials the corresponding exchange of one antiferromagnetic spin fluctuation gives rise to a repulsion near the Fermi surface, leading to a low value of Tc and a pairing gap having rf-wave symmetry. 1 7 , 1 8 We have proposed an alternative pairing mechanism

472 PSEUDOGAPS AND THE SPIN-BAG APPROACH TO HIGH- Tc

41

for high-7" c materials based on the spin-bag approach. As demonstrated above, strong antiferromagnetic spin correlations lead to a pseudogap in the quasiparticle spectrum. The pairing attraction Kkk. arises in this approach from the lowering of the system energy when two quasiparticles share the region of reduced antiferromagnetic correlations surrounding each of these excitations. The range of this attraction is of the order of the SDW coherence length ISDW- " n momentum space, this corresponds to an attraction for momentum transfer q smaller than

However, the existence of the pseudogap does provide a new mechanism for an attractive spin-bag pairing potential. It arises from the exclusion principle violation shown in Fig. 9(a), where a fermion line inside one of the bubbles representing the susceptibility is equal to the momentum of the injected particle. This diagram is compensated by its exchange counterpart shown in Fig. 9(b),

to

£SDW-

In the antiferromagnetic phase, SWZ have shown 9 that the attraction between bags at small q is accompanied by a repulsion at large q ~ Q , the latter being the analog of the repulsion arising from the exchange of one spin fluctuation in the paramagnetic phase. It was shown that for moderate levels of hole doping, where the Fermi surface is likely composed of several isolated pockets, the pairing gap is nonzero over the entire hole Fermi surface. However, due to the repulsive nature of V^. for k —k' of order Q, the gap parameter changes sign between different pockets. Here we investigate the pairing interaction in the paramagnetic metal phase, based on the above discussion of the pseudogap. The intent is to explore to what extent the results of the spin-bag approach join continuously as a function of doping. Just as one can deduce contributions to the self-energy from the influence of an added hole on vacuum fluctuations (see Fig. 3) we can obtain the contributions to the effective interaction between a pair of particles by accounting for the effects on the self-energy due to the presence of the second hole. Starting with the self-energy contribution in Fig. 8(a), the presence of a second particle with the same spin and momentum as the intermediate Fermion line (not involving the bubbles) requires one to include the exchange graph Fig. 8(b) in order to restore the Pauli principle. Stretching out the lines as in Fig. 8(c), it is clear that this diagram is just the repulsive one spin fluctuation exchange process. I 7 The corresponding diagram for particles of opposite spin involves an even number of bubbles. In addition, the spin flip interaction corresponding to the particle-hole ladder diagram shown in Fig. 8(d) must be added to preserve spin rotation invariance. The effective pairing potential arising from the one spin fluctuation exchange processes is in the singlet channel given by the well-known antiparamagnon result. " l/^lk'-k) 2

l-t/ ^(k'-k')

6405

o (a)

'o-o-o'

O'1 O kl

kl

W 1

J1

2

U x0(k'-k) l-t/Jodc'-k)

(19)

As mentioned above, this antiparamagnon exchange interaction is repulsive at low frequencies and gives rise to even-parity anisotropic pairing, i.e., the gap order parameter A(k) within this mechanism has d-like symmetry. 1 7 , 1 8 In essence, one hole sets up in its vicinity a cloud of short wavelength spin fluctuations from which the other hole scatters. The d character of the gap ensures that the holes sample the attractive regions of the potential in real space and avoid the repulsive regions.

-kl

(d)

FIG. 8. (a) Self-energy diagram in the presence of a second injected particle; (b) exchange diagram restoring the Pauli principle; (c) same as in (b) with the lines stretched out; (d) particlehole ladder diagram for the one spin-fluctuation exchange process.

473 6406

41

A. KAMPF AND J. R. SCHRIEFFER

or the crossed line diagram of Fig. 9(c). The fact that this diagram leads to an attraction is readily seen. As discussed above, the self-energy due to spin fluctuations is positive near the SDW instability and leads to a pseudogap. A second particle added to the system suppresses these fluctuations through the Pauli principle, reducing the phase space for electron-hole excitations and hence reducing X- This real space reduction of susceptibility extends over a range of order £SDW around each quasiparticle and reduces the self-energy for another particle in its I Vll(co,(ol) = i~

T

N ~~

J

vicinity, thereby giving an effective attraction. The corresponding diagram for parallel spin pairing is given by spin fluctuations lines having an odd number o f bubbles. To investigate the form and strength of the interaction between bags, we evaluate the crossed line diagram of Fig. 9(c) using the model susceptibility given in (8). Using bare propagators for the intermediate fermion lines the spin-bag pairing potential represented by this diagram is given by:

f ^ • G 0 ( f l ) - v > k - q ) G 0 ( - < u , - v , - k ' - q ) l / 2 ^ ( q , v ) C / ^ ( q + k ' - k , v + fi)'-£

(20)

ITT

Insight in the structure of this pairing potential is provided by first evaluating Kkk.(o),0, i.e., g(co) = b(co — a>0), and the Lorentzian momentum dependence is replaced by a 8 function centered at Q, i£(q) = (2n-)28(q — Q). In this limit we find for w — m': (d)0—ek)(3fi>0 — e k ) — a)2

Kg.(
'k-Q'"

(a>0 + E k )(3(D 0 +E k ) — co1

,

+ nk - Q (O0(o)a — (o — zv) (co0 rta — t^) ' 2

j

!

£[) 0 (a) 0 +
£k)

(21) Due to the S-function approximation in the model susceptibility only momentum Q can be exchanged across the interaction lines giving rise to the 8-function 8(k —k') for vanishing net momentum transfer. The pairing potential VSB is attractive for low frequencies co S a0 and repulsive for high sufficiently frequencies o»co0. This

(a) I

I

i

A

> IS 3 -100.0 Q.

]

\

\

sS 200.0 Q-

,©""®-©, © ©

r

w

J

v

( k , k ) » ( i U 3 , n/3) I l

-10.0 -8.0

-6.0

-

i

-4.0 -2.0

-

V

V

XU-2.0 w0 = 20

1

0.0

2.0

4.0

i 6.0

i 8.0

10.0

(0/t (a)

lb) v

©

kx

©

••..

-Q

il

-k'l

\

1 ^ 0

'Q 0

P'°-.

kk'

."J

5,

<>

(c)

FIG. 9. Lowest-order spin-bag contribution to the pairing interaction.

FIG. 10. (a) Frequency dependence of the spin-bag pairing potential evaluated with the model susceptibility (10) for ^(q) = (27r)25(q —Q): (b) Momentum dependence of the effective zero-frequency pairing potential Ke,r(k,k') = K SB (k,k')+r APM
474 4J

PSEUDOGAPS AND THE SPIN -BAG APPROACH TO HIGH-T, . . .

feature of VSB is not altered by using the linear frequency distribution (8) instead of a single fluctuation frequency CJ0 [see Fig. 10(a)]. In addition, the zero-frequency limit of KSB becomes strongly enhanced, diverging logarithmically at the Fermi surface precisely at half-filling. For the case of a Lorentzian momentum distribution in the model susceptibility the spin-bag interaction broadens into a Lorentzian around Ak = k —k' —0 with a width determined by £SDW <->n t n e o t n e r band, the contribution from the antiparamagnon exchange processes is largest for momentum transfer close to Q leading to an effective pairing potential ^ e i r = ^SB - 1 " ^APM which is attractive for small momentum transfer and repulsive for large momentum transfer, see Fig. 10(b). This behavior of Veff is independent of any special choice for the model susceptibility and merely a consequence of the Stoner enhancement for wave vectors close to Q = (±TT,±TT). The contribution from the effective Coulomb repulsion in the charge channel still has to be added to Keff giving rise to an overall upward shift. However, the effect of the Coulomb potential is weakened due to the pseudopotential scatterings far from the Fermi surface leading to a smaller probability for two electrons being within the range of the Coulomb potential. 2 0 VI. SUMMARY AND CONCLUSION We have shown how antiferromagnetic spin fluctuations depress the density of states N(E) in the vicinity of the Fermi surface, leading to a pseudogap. As the system becomes antiferromagnetically ordered, the pseudogap sharpens to become the spin-density-wave (SDW) exchange gap. While it might appear that the formation of a pseudogap would require higher-order perturbation theory, the above analysis shows that the essence of the problem is already treated in the one-loop approximation for the one-particle self-energy 2(k,oi). The origin of the pseudogap can be understood by dividing £ into two parts, a spin-polaron-like part 2 , and a vacuum-fluctuation part 2 v f . If the spin susceptibility ^•(q,v) varies smoothly with q as in a normal metal, 2p 0 i dominates 2. and Fermi liquid behavior holds. However, if ^ ( q . v ) is peaked near a wave vector Q, as occurs when antiferromagnetic (AF) spin fluctuations are present, 2 v f is strongly enhanced and 2 , is reduced. Since 2 v f is due to the Pauli principle suppressing spin correlations which lower the energy in the absence of the quasiparticle, Xvf is positive for 0, raising the energy of particle states near EF and lowering the energy of states below EF which accommodate holes [ 2 v f ( k , G > ) < 0 for w<EF]. Therefore, the pseudogap is due to exchange suppression of vacuum fluctuations rather than a magnetopolaron effect. Accompanying the pseudogap is a qualitative change of the one-particle spectral function Ak{ai). While A in the Fermi liquid regime exhibits one peak at the quasiparticle energy plus incoherent continua for both particles and holes, we find that A in the pseudogap regime exhibits two main peaks, one for particles (oi> EF) and one for holes (co<EF) for each value of k. This behavior is analogous to a superconductor and arises from a strong

6407

FIG. 11. Lowest-order spin-bag contribution to the selfenergy. The wavy lines represent antiparamagnons.

smearing of (nk) about the Fermi surface, so that a given k state can accommodate both holes and electrons. Alternatively, one can view the two peaks as t h e precursor of the valence and conduction bands of t h e antiferromagnet which are folded back into the magnetic zone when long-range order sets in. We note t h a t the two peaks come from a sharpening up of the incoherent continua of the Fermi liquid regime with the conventional quasiparticle peak being broadened and reduced to small weight, eventually vanishing as SDW order sets in. At this level of calculation, the spectrum arises from the scattering of the injected particle by unperturbed (free) spin fluctuations present in the ground state in the absence of the added carrier. However, in next order as shown in Fig. 11, the injected particle reduces the amplitude of the fluctuations thereby reducing the size of the pseudogap over a distance of order the pseudogap coherence length £s~fivF / i r A s , where vF is the noninteracting Fermi velocity and As is t h e pseudogap parameter. This combination of a hole (electron) moving with its region of reduced antiferromagnetic correlations is termed a spin bag, in essence because in strong coupling the hole is self-consistently trapped inside of the bag of reduced spin order. We have shown that the pairing interaction K kk . between two spin bags is attractive for m o m e n t u m transfer q = |k —k'| smaller than g~] due to the holes sharing each other's bags. However, Kkk. is repulsive for q = Q, reflecting the repulsive nature of the one spin fluctuation exchange process for antiparallel spins, in contrast with the usual phonon interaction. These results are consistent with those derived by SWZ for the ordered antiferromagnet. Thus, one has a smooth evolution as the hole doping decreases from metallic to doped pseudogap and doped antiferromagnetic behavior. While we have approximately evaluated the pairing potential in the spin-bag approach, the shape of the hole Fermi surface is also needed before one can reliably predict the detailed shape of the pairing gap parameter A s c ( k). If we consider hole doping at the four corners of the magnetic zone, then the repulsive nature of Kkk. for k — k' = Q suggests that A s c ( k ) has d-wave-like symmetry A s c ( 0 ) = A o cos(20) where 0 is the direction of k in the x-y plane. If the density of states in the pseudogap is small and the hole pockets d o not overlap, the system will appear to be very different from a conventional rf-wave superconductor in that the hole Fermi surface samples A s c ( 0 ) only near 0 „ = n 7 r / 2 , where c o s ( 2 0 n ) = ± l and |A SC (G„ ) | = A 0 , i.e., nodeless on the hole Fermi surface. Alternatively, if the Fermi surface is connected, a more conventional d-wave superconductor would be expected.

475 6408

A. KAMPF AND JJ. R. SCHRIEFFER

4J

The above treatment can be improved in a number of directions. For example, if the self-consistent G is used to calculate 2 [Eq. (5)], a feedback effect occurs which considerably deepens the pseudogap. Vertex corrections will then have to be included in order to maintain a conserving approximation. Based on the above treatment of spin bags in the paramagnetic phase we are presently calculating a number of experimentally observed properties of both the normal and the superconducting state. The results will be presented in a forthcoming publication.

T h e Public Service Company of New Mexico and the U.S. Department of Energy are acknowledged for their support of the Los Alamos Advanced Studies Program in High-Temperature Superconductivity Theory. A.K.. gratefully acknowledges the hospitality of the Institute for Theoretical Physics in Santa Barbara. Both of us would like to acknowledge helpful discussions with E. Abrahams, S. C. Zhang, and X. G. Wen.

"On leave of absence from Institut fur Theoretische Physik, Universitat zu Koln, Federal Republic of Germany. ^Permanent address: Department of Physics, University of California, Santa Barbara, CA 93106. 'Y. Tokura, H. Takagi, and S. Uchida, Nature (London) 337, 345 (1989); A. C. W. James, S. M. Zahurak, and D. W. Murphy, ibid. 338, 240 (1989). 2 G. Shirane et al., Phys. Rev. Lett. 59, 1613 (1987); Y. Endoh et al., Phys. Rev. B 37, 7663 (1988); J. Tranquada et al., Phys. Rev. Lett. 60, 156(1988). 3 K. B. Lyons et al., Phys. Rev. B 37, 2353 (1988); K. B. Lyons et al., Phys. Rev. Lett. 60, 732 (1988). 4 P. W. Anderson, Science 235, 1196 (1987); P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett. 58, 2790 (1987); S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Phys. Rev. B 35, 8865(1987). 5 X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989). 6 R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988); Science 242, 525(1988). 7 S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988); C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39, 6880 (1989); B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett. 60, 740 (1988); ibid. 61, 467 (1988); R. Shankar, ibid. 63, 203 (1989); S. A. Trugman (unpublished); Y. Hasegawa and D. Poilblanc (unpublished); E. Dagotto, A. Moreo, R. Joynt, S. Bacci, and E. Gagliano (unpublished). 8 The existence of spin bags and the physical mechanism by

which they produce an attractive pairing interaction in the paramagnetic metal phase was discussed by J. R. Schrieffer (unpublished). 9 J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. Lett. 60, 944 (1988); Phys. Rev. B 39, 11663 (1989). I0 A. B. Harris and R. V. Lange, Phys. Rev. 157, 295 (1967). "J. D. Reger and A. P. Young, Phys. Rev. B 37, 5978 (1988). 12 J. E. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 56, 2732 (1986). n Y . Hasegawa and H. Fukuyama, Jpn. J. Appl. Phys. 26, L322 (1987); H. J. Schulz (unpublished). I4 K. Kitazawa, S. Uchida, and S. Tanaka, Physica (Utrecht) 135B, 505 (1985); S. Uchida, K. Kitazawa, and S. Tanaka, Phase Transitions 8, 95 (1987). 15 P. A. Lee, T. M. Rice, and P. W. Anderson, Phys. Rev. Lett. 31,462(1973). "J. P. Blaizot and B. L. Friman, Nucl. Phys. A372, 69 (1981). 17 D. J. Scalapino, E. Loh, Jr., and J. E. Hirsch, Phys. Rev. B 34, 8190 (1966); D. J. Scalapino and E. Loh, Jr., ibid. 35, 6694 (1987); K. Miyake, S. Schmitt-Rink, and C. M. Varma, ibid. 34,6554(1986). 18 J. E. Hirsch, Phys. Rev. Lett. 54, 1317 (1985). " S . Nakajima, Prog. Theor. Phys. Jpn. 50, 1101 (1973); P. W. Anderson and W. F. Brinkman, Phys. Rev. Lett. 30, 1108 (1979); D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980). 20 N. N. Bogoliubov, Nuovo Cimento 7, 6 (1958); P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962); J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).

ACKNOWLEDGMENTS

476

RAPID C O M M l M C A I IONS

PHYSICAL REVIEW B

VOLUME 43, NUMBER 10

1 APRIL 1991

Constructing quasiparticle operators in strongly correlated systems E. Dagotto and J. R. Schrieffer Institute for Theoretical Physics and Department of Physics, University of California at Santa Barbara, Santa Barbara, California 93106 (Received 25 January 1991) In models of strongly correlated electrons the wave-function renormalization Z* of a hole injected in a half-filled band is small compared to unity. This result reduces the accuracy of numerical and analytical predictions of superconductivity for the Hubbard and related models. A simple method is described to deduce approximate quasiparticle operators whose Z» is closer to 1 than for the standard bare operators. The method is tested for the t -J model. We find that pairing operators constructed out of these quasiparticles are also considerably improved. For example, at J/t—0.2 an improvement by a factor of 10 or more is expected in the signal of superconductivity in a Monte Carlo simulation if these quasiparticle operators are used rather than the standard bare operators. High-7V superconductor materials exhibit many anomalous properties in the normal phase. Understanding these properties is important in seeking an explanation of their superconducting phase at low temperatures and small doping of holes. An important guide for theoretical models of these materials is provided by angle-resolvedphotoemission experiments 1 which determine the spectral function of holes in the superconductors. There are two limiting interpretations of these results. Anderson 2 conjectured that these photoemission spectra can be interpreted as showing a cusp singularity which may be explained within a Luttinger-liquid description of the normal state of the cuprate superconductors. In analogy with the one-dimensional ( I D ) case, it is suggested that the wave-function-renormalization factor Z/, of one hole vanishes at the Fermi surface, and the Fermi-liquid fixed point does not exist in either ID or 2D. The theory of a marginal Fermi liquid, 3 proposed by Varma etal. also interprets the photoemission results in terms of Z/,—*0 at However, it has been recently shown 4 that the experimental data can also be explained within the context of a pseudogap approach. Using self-consistent perturbative techniques, it was found that Z* in the 2D Hubbard model is small and there is an appreciable shift of spectral weight from the quasiparticle peak to peaks which evolve from the incoherent background. At large U/t and near half filling these background peaks approach the MottHubbard bands. Thus strong antiferromagnetic fluctuations lead to important changes in the spectral function of a metallic system, as seen in photoemission and bremsstrahlung-isochromat-spectroscopy (BIS) spectra. The fact that Z* is small in the 2D Hubbard model also affects numerical studies of strongly correlated electrons. Currently, there are no convincing numerical calculations showing the existence of superconductivity in the one- or three-band Hubbard models 5 or the t-J model 6 in 2D. This result is puzzling considering that some properties of the normal state, like the existence of antiferromagnetic correlations, optical conductivity at low doping, and others seem to be qualitatively described by these models. However, if Z*, is a small number, then the absence of 41

clear numerical indications of superconductivity may be due to the fact that the pairing-pairing correlation function is suppressed by Z ^ if bare, rather than dressed, fermions are used in the definition of the pairing operators in the simulation. The strong suppression caused by the small value of Z/, suggests that the starting point of both analytic and numerical calculations should be revised. Since the overlap •|Vnkk,alvn-i)|/(|l)

1/2

is small [with y„ being the normalized ground state of the model with n holes and k(o-) is the momentum (spin) of the fermion], the state Ck,0|i//„) is not a good approximation to the state of one less particle on a given background, at least at low hole densities and large couplings. It would be highly desirable to improve on this situation. To obtain robust results with bare-pairing operators, larger lattices and much better statistics would be required in a Monte Carlo simulation. In this paper we present a simple method to construct quasiparticle (dressed) operators whose renormalization factor Z/, approaches unity, compared to that of the bare fermionic operators c^a which is small compared to 1. The method is similar in spirit to the Lanczos approach and it can be implemented for any model of strongly correlated electrons. 7 We will illustrate the technique using the / -J model which is defined by the Hamiltonian

H-jJ^iSiSj-imnj)-

' 2

WiVia+H.C.),

(1)

(i.j).
where J is the exchange constant of the spin-spin interaction and / corresponds to the hopping parameter of the kinetic energy of holes. d\,a are hole operators at site i of a 2D square lattice with periodic boundary conditions (W sites) and a is its spin \d\,<,~c\,a(\ — M|,-„)]. There is no double occupancy in this model. (i,j> denotes nearestneighbor sites and n-, is the number operator. We will concentrate on the case of half-filling with one hole, where many numerical results are available, but the technique can be used for any doping fraction. First of all it is important to verify if Z* is indeed a small number in the physically interesting regime 8705

'1991 The American Physical Society

477

E. DAGOTTO AND J. R. SCHRIEFFER

8706

y/r —0.1-0.5 of the / - / model. The calculation has been carried out numerically on a 4x4 lattice using Lanczos techniques.8 The result is shown in Fig. 1 for the case of the bare operators. We observe that Z» —0.28 at Jit ~0.2, implying that for a pairing operator involving two fermions, the suppression factor Zh will be approximately 0.08, i.e., — 13 times smaller than it would be if Z* were — 1. For smaller values of Jit the situation is worse. For example, at Jit —0.1 the suppression factor Zl is -— if. Thus, numerical studies indeed suggest that Z/, is small in the realistic regime of the t -J model.9 To construct the quasiparticle operators we note that the ground-state energy scales approximately as ~J7'3 U *"1). I0 This result is in agreement with calculations in the Ising model" and is caused by string states. The hole, injected originally on a single site, will spread to a size d given approximately by the extent of the ground-state wave function of the Schrodinger equation in a linear potential, 10 " i.e., d~\Uuy ( / / f « l ) . Of course, this localization describes the internal structure of a quasiparticle whose center of mass must then be displaced phase coherently to form a plane-wave excitation overall. Thus, (or small Jit, creating a hole excitation using a local (one site) operator is a poor approximation. An operator generating a more extended internal structure will have a

41

larger factor Z*. The spatial extent of this operator should grow like ~ 1 / / 1 / 3 to account for this effect. This situation occurs in many contexts, such as near critical points where the physical correlation lengths in units of the lattice spacing are diverging, and local operators are not appropriate. Also in particle physics in the context of lattice-gauge theories (where the continuum limit is obtained near second order phase transitions), it has been shown that to obtain better results for glueball masses in QCD it is necessary to use extended operators. I2 To improve on the bare-hole operators, consider a Lanczos-type calculation. The Hamiltonian is applied to the initial ansatz (which we take to be the bare-hole state lyi > ~ rfk.alyo). where |vo> 1S the ground state at half filling). W|vi> is decomposed into a vector proportional to | Vi) and another perpendicular to it. A better ansatz corresponds to a linear combination of these two vectors. Thus a renormalized fermionic operator d is obtained as

/

•EMt,

]|vo>,

where a is a constant, E\ =, and / labels sites of the lattice. For the / - / model with one hole, this result can be rewritten as (3)

,rf/!J)]|vo>,

i

where H, (HHc^) is the hopping (exchange) interaction in the t-J model and p is another constant. The commutator guarantees that to this order the spinlike excitations relevant for the dressing of the hole are created in its neighborhood. The hopping term moves the hole at a dis-

I ' ' ' I ' ' ' I ' ' ' I 1.0

zh 0.8

t-J,

1 hole

J/t FIG. 1. Zh of one hole as a function of J/t for the / -J model on a 4x4 lattice. Results are shown using the bare fermionic operator (•), strings of length 1 (A), and strings of length 2 excitations (•) in the "dressed" fermionic operator.

(2)

i

tance of one lattice spacing from its original position /, and we call these excitations "strings" of length one. I0 For J<£.t, the operator d is approximated by

31.-1,e'kl

(*% + /»« X


.

(4)

where fi — ± x, ± y; x,y are unit vectors in the crystal directions. The constant p can be fixed by either minimizing the energy of the state | \jf\) or by maximizing the overlap of this state with the exact ground state (if available) in a numerical study. Below we followed the latter procedure. ' 3 This method can be applied many times replacing d by d on the right-hand side of Eqs. (2) and (3). Although the algebra is involved, it is not difficult to show that further corrections will arise from application of higher powers of H in the neighborhood of the hole. These operators contain many different types of excitations including spin flips, strings of length m (where the hole is moved m steps away from its original position by the hopping term), and other combinations. In Fig. 1 we show the effect on ZH of adding string excitations of length 1 and 2 which, at small Jit, we expect will be the dominant excitations.10,14 For the case of strings of length 1 we have only one free parameter /J, while for strings of length 2 there are three others. They correspond to moving the hole from site / to the position l±2x, l±2y with a parameter P\, to I ± (x + y) with p2 and to / ± ( x — y ) with Pi [for the hole we have considered a momentum k""(n/2,»/2)l. The renormalization factor Zj increases by adding these excitations. At J/t —0.2, Zh changes from 0.28 to 0.86. We also find that it is not necessary to tune carefully the set of variational

478

R A I'll) ( ' O M M I N H

CONSTRUCTING QUASIPARTICLE OPERATORS IN STRONGLY . . .

il

parameters to obtain a larger Z*. For example, at J/t —0.2, Z* —0.86 when Z* is maximized in the space of the parameters 0,0i,02,03, but Zh is still large (—0.77) for the special case 0—0i — 0 2 " 0 3 " 1 , i.e., when all excitations enter with equal weight and no maximization is performed. Thus, it is important that the dressed operator for the hole includes the correct relevant excitations even if they do not enter in the operator with their optimal weight. This is important for applications in Monte Carlo simulations where calculating Z* accurately might be difficult. In addition, to string excitations there are spinlike excitations present in Eq. (3) and in further iterations of the method. However, we find their influence is less important. When we applied the spin-flip operator Sj + Sj~ to the links surrounding the hole with no string excitations included, Zh changed from 0.28 to 0.39 at Jit - 0 . 2 which is considerably smaller than the increase we obtained with string excitations. Increasing Jit, the improvement from Sj + Sj~ is even smaller. We also tried with many other types of excitations. For example, it has been proposed that operators involving a hopping term W) within the same sublattice may be important for the coherent motion of the hole. We find that such processes do not contribute appreciably to increasing Z* compared with the string excitations. Of course, including spin flip and other excitations marginally improve further the results; e.g., for Jit —0.2, Zh is increased to —0.96 by combining string and spin-flip excitations. In Fig. 2 we show AE ~(E^t — E„aa)/Ecxaa which is the relative difference between the energy of the state |
- i — I — l — i -

I

'

'

'

I

'

'

'

AIIONS

8707

holes. For that purpose we have obtained the exact ground state of 2 holes in the t -J model using a Lanczos technique on a 4 x 4 lattice. Then, we constructed the state A + |vo>' • £ r f i V i W ( « ) l v o > , i.S

where S — ± x, ± y and f(S) (which defines the rotational symmetry of the operator) takes the v a l u e s / ( ± y ) — — 1 and / ( ± x ) — 1 for a rf-wave ground state. I 5 Note that in the operator A* the holes are created at a distance of one lattice spacing which is reasonable if the bound state of two holes has a small size (for a generalization see below). This form of the pairing operator has been extensively used in Monte Carlo simulations 5 of the Hubbard model. The overlap in the 2-holes subspace

JzTk-\WiWWoW\Wo\A^Wo)\)m for the case where bare operators are used is shown in Fig. 3. Note that Z2» is very close to Z%, i.e., the renormalization factor that we would have obtained by simply squaring our results for one hole. Note also that for J/t —0.2 the suppression factor Z2V is indeed large as expected. In Fig. 3 we also present results including strings of length 1 and 2 for each fermionic operator i r in A together with their corresponding zl for comparison. The increase in Z21, is clear. For J/t —0.2 and including strings of length 2, Z2A is —0.6, i.e., —10 times larger than when bare operators are used; for J/t — \, it is 3 times better. Z2/1 can be easily made closer to unity by further tuning the

I

AE 0.40

t - J , 1 hole

_ string 2

FIG. 2. A£ as defined in the text as a function of J/t for fermionic operators using bare, string 1, and string 2 excitations on a 4 x 4 lattice.

FIG. 3. Zu of two holes as a function of J/t for the t -J model on a 4 x 4 lattice. The continuous lines denote results for Z2* using bare fermionic operators (•), operators with strings of length 1 (A), and strings of length 2 (•). The dashed lines correspond to Z£ using the same string excitations. The dot-dashed lines represent Zv, for these excitations when the holes are created at distances larger than one lattice spacing (as explained in the text).

479

8708

E. DAGOTTO AND [. R. SCHRIEFFER

variational parameters or adding other excitations. Thus, we believe that the method works correctly for two holes. 16 Another simple way to increase Zih >s to create the pair of holes at distances larger than one lattice spacing. In other words, while the quasiparticle operators are individually improved by increasing their size by adding string excitations, we also have the possibility of optimizing the distance between the "centers" of each operator. It is clear that the average distance between the 2 holes will increase when Jit is decreased. Then, in the evaluation of Zii, we also have used the operator

where f(5) is / ( ± y ) •= — 1 and / ( ± x ) •" 1 allowing the holes to be created at distances of both 1 and 2 lattice spacings. The results are also shown in Fig. 3. Z21, is appreciably larger than before. Additional improvements can be obtained by allowing the holes to be created at other distances and by tuning variationally the relative weight corresponding to each distance.

'C. G. Olson etal., Science 245, 731 (1989); Phys. Rev. B 42, 381 (1990); see also G. Sawatzky, Nature (London) 342, 480 (1989). 2 P. W. Anderson, Phys. Rev. Lett. 64, 1839 (1990). 3 C. Varma, P. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. Ruckenstein, Phys. Rev. Lett. 63, 1996 (1989). 4 A. Kampf and J. R. Schrieffer, Phys. Rev. B 42, 7967 (1990); see also Hayoung Kim and Peter S. Riseborough, Phys. Rev. B 42, 7975 (1990). 5 A. Moreo and D. Scalapino, this issue, Phys. Rev. B 43, 8211 (1991), and references therein. Recent studies in the threeband Hubbard model are G. Dopf etal., Phys. Rev. B 41, 9264 (1990); M. Frick etal., ibid. 42, 2665 (1990). 6 E. Dagotto, D. Poilblanc, and J. Riera, University of California, Santa Barbara, Report No. NSF-ITP-90-159. 7 A procedure similar in spirit has been recently applied to the Hubbard model by D. C. Mattis, M. Dzierzawa, and X. Zotos, Phys. Rev. B 42, 6787 (1990). Upon completion of this work we received papers [C. Y. Lo, E Manousakis, and Y. L. Wang (unpublished); Assa Auerbach and Brond Larson (unpublished)] where related ideas are discussed for one hole. 8 For a recent review, see E. Dagotto, Int. J. Mod. Phys. B (to be published), and references therein. 'Although numerically we found that Z/, is nonzero (although small) this is not enough to show the validity of a Fermi-liquid picture. A proper finite-size scaling should complement our

U

Using the dressed quasiparticle operators will help in numerical simulations and analytical treatments of the Hubbard model by removing the noise due to normal-state dressing processes of a pairing analysis. If, in the pairing operator, the bare fermions are replaced by the dressed ones, the operator becomes an extended object and a better "antenna" to capture the possible weak signals of superconductivity in the 2D Hubbard model or other models of strongly correlated systems. Summarizing, we have shown that quasiparticle operators having a Z* close to 1 can be easily constructed in models of strongly correlated electrons. These operators have potential applications in Monte Carlo simulations, variational studies, and also in analytic approaches. Work is in progress along these lines. We thank Adriana Moreo and Robert Sugar for useful discussions. This project was supported by N S F Grants No. PHY89-04035 and No. DMR89-18307. The computer simulations were done on the Cray-2 at the National Center for Supercomputing Applications ( N C S A ) , Urbana, IL. We thank N C S A for their support.

study of Zh. It may occur that long-range distortions of the spin background conspire to produce Z* —0 in the bulk limit. l0 Elbio Dagotto, Robert Joynt, Adriana Moreo, Silvia Bacci, and Eduardo Gagliano, Phys. Rev. B 41, 9049 (1990), and references therein. The scaling of the energy as J2'3 was also noticed by Y. Hasegawa and D. Poilblanc, Phys. Rev. B 40, 9035 (1989). See also K. J. von Szczepanski el ai, ibid. 41, 2017 (1990); S. Trugman, ibid. 41, 892 (1990). "B. Shraiman and E. Siggia, Phys. Rev. Lett. 60, 740 (1988). I2 M. Teper, Phys. Lett. B 183, 345 (1986). l3 Thc maximization was done numerically but the optimal P can also be derived analytically as a function of matrix elements of appropriate operators. Details will be given elsewhere. In practice for Jit— 0.2 ( / / r - 1 . 0 ) , the optimal /5 was 0.50 (0.35) with an error of 0.05. l4 This detail was also noticed by J. Inoue and S. Maekawa, J. Phys. Soc. Jpn. 59, 2110 (1990). l5 It has been shown numerically by various groups that the ground state of two holes in the t-J model has rf-wave symmetry. See, for example, Hasegawa and Poilblanc (Ref. 10), and references listed in Ref. 8. l6 The improved pairing operator does not simply correspond to injecting the pair of holes at distances larger than one lattice spacing. Note, for example, that the new operators include terms with the product
480 PHYSICAL REVIEW B

VOLUME 52, NUMBER 2

1 JULY 1995-11

Properties of odd-gap superconductors Elihu Abrahams Serin Physics Laboratory, Rutgers University, P.O. Box 849, Piscataway, New Jersey 08855 Alexander Balatsky Theory Division, T-ll, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Landau Institute for Theoretical Physics, Moscow, Russia D.J. Scalapino Department of Physics, University of California, Santa Barbara, California 93106 J.R. Schrieffer Department of Physics, Florida State University, Tallahassee, Florida 32310 (Received 13 March 1995) We discuss the class of superconductors that have pairing correlations which are odd in frequency, as introduced originally by Berezinskii and more recently by Balatsky and Abrahams. As follows from the equations of motion, a natural definition of the thermodynamic order parameter of the odd-pairing state is the expectation value of a composite operator which couples a Cooper pair to a spin or charge fluctuation. We use a model pairing Hamiltonian to describe properties of the oddpairing composite-operator condensate. We show that the superfluid stiffness is positive, we discuss superconductive tunneling with an ordinary superconductor, and we derive other thermodynamic and transport properties.

I. I N T R O D U C T I O N

Novel symmetry types of the energy gap function A ( k , w ) in spin-singlet superconductors were discussed recently 1 in the context of high-T c superconductivity, 2 , 3 in order to investigate ways of constructing a superconducting state which is not inhibited by the presence of strong short-range repulsion. An anisotropic superconducting gap parameter ("p wave" or "d wave") which changes sign as k moves around the Fermi surface is the simplest way of achieving such a state. An alternative posssibility was proposed some time ago by Berezinskii, 4 who pointed out in the context of 3 He that it is permissible to have triplet (S = 1) pairing in which the gap function A ( k , u ) is even in momentum k (for example, "s wave"), provided it is odd in frequency u>. Recently, this suggestion was extended to the singlet case, 1 where it was argued that a new class of singlet superconductors can exist for which the gap function is an odd function of both k and w. Such a superconducting state is thus odd under both parity and time reversal and, as in the case discussed in Refs. 2, 3, the effect of short-range repulsion is suppressed, here because the equal-time pair amplitude is zero.

in a recent paper 5 where a specific example based on spin-fluctuation exchange in the two-dimensional Hubbard model was discussed. If an interaction a d m i t s an odd solution, it still must be checked whether it only describes a metastable phase with free energy higher t h a n t h a t of an even-frequency solution. In the present paper, we explore a n o t h e r issue which arises because there is an instability which was not mentioned 6 in all t h e known translationally invariant odd-gap solutions considered in Refs. 1—4. Namely, because of the symmetry properties of the odd-gap pairing amplitudes, there is a change in sign of t h e anomalous parts of BCS coherence factors. A consequence is t h a t a calculation of the Meissner effect in lowest order 7 gives an opposite sign to t h a t of t h e usual BCS case so t h a t the superfluid density appears to be negative. T h u s , although the free energy of the uniform odd gap state is less t h a n t h a t of the normal state, 1 the system prefers a nonuniform distribution of supercurrents, i.e., some sort of vortex state which could in principle be determined in a calculation which goes beyond the linearized Eliashberg equation which is used to find Tc. A possible realization of triplet odd-frequency pairing which is induced by t h e presence of disorder has been discussed recently by Kirkpatrick and Belitz. 8 Although translational invariance is not present, their state presumably is subject to t h e same problem with the Meissner effect.

An odd-in-frequency solution to the Eliashberg equations can only exist for certain forms of the pairing kernel which scatters a pair from k, —k; u;, — u> to k',—k'; w',—ui'. In the singlet case, to obtain an oddui gap parameter, the kernel must have a contribution which is odd in b o t h k and w. The problem of how an appropriate interaction might arise has been discussed

In this paper, we shall show how a stable Meissner state may be achieved by the introduction of a composite condensate which is characterized by t h e joint condensation of a Cooper pair and a density fluctuation. Our point of view is motivated by the following ques-

0163-1829/95/52(2)/1271(8)/$O6.00

1271

52

©1995 The American Physical Society

481 1272

ABRAHAMS, BALATSKY, SCALAPINO, AND SCHRIEFFER

tion: If the anomalous Green's function vanishes at equal times, as it does for an odd gap, what is the appropriate macroscopic thermodynamic order parameter? For example, in the usual BCS case it is the expectation value F ( r , t ; r ' , t ' -> t) = (i>(r,t)ip(r',t)) for | r - r ' | < £, where £ is the BCS coherence length. Since this vanishes in the odd case (it is odd in t — t'), we are led naturally to consider the quantity [<£F(r, t; r', t')/dt] for t —> t' as the equal-time correlation function to be used as the order parameter. Given a pairing Hamiltonian, this quantity can be obtained from the equations of motion and it consists of the expectation value of the product of a pair operator and, depending on the nature of the pairing interaction, a spin or charge density fluctuation. This quantity is of the form (r,t)V>(r,t' -•*)>, (1.1) with an appropriate spin structure which will be discussed below. The operator M contains a spin or charge density fluctuation: M ( r , t ) = I drdr'V>1(r'\t

+ T)II>(T',t

+ r) Z>(r-r',-r),

(1.2) where P is a generalized nonlocal retarded interaction. In Sec. II, we present a symmetry analysis of the Green's functions which leads to this new type of superconductor. We show that for the spin-singlet (-triplet) case, the only symmetry requirement is t h a t the gap function A ( k , w) be even (odd) under simultaneous change of sign of momentum and frequency. Consequently, apart from the conventional even-frequency gaps which are even (odd) in momentum for singlet (triplet) pairing, there can be solutions of the gap equation which are odd (even) in momentum and odd in frequency in the singlet (triplet) case. In Sec. Ill, we give an illustration of how a composite operator arises naturally as the order parameter for an example system. For purposes of calculation, we introduce a model Hamiltonian which can lead to oddfrequency pairing and we derive the gap equation near the critical temperature in terms of the parameters of the model. We find that the strength of the interaction must exceed a critical value in order that the odd solution exist. This is a rather general property of odd-gap solutions 4 ' 3 and is in contrast to the BCS case where the logarithmic behavior of the gap equation ensures a nontrivial solution no matter how weak the coupling. In our case, the kernel of the gap equation, being odd in w, must vanish at zero frequency if the kernel varies smoothly in this vicinity; this removes the logarithmic divergence. In Sec. IV, we calculate the Green's functions in the composite-operator condensate and show explicitly that the superfluid stiffness is positive. Section V contains a discussion of the Josephson effect between odd-gap and even-gap superconductors. In Sec. VI we calculate the N M R relaxation rate and in Sec. VII the effect of impurity scattering for the odd-gap composite superconductor. Subsequent to the singlet odd-gap discussion of Balatsky and Abrahams, 1 several authors, in particular Emery and Kivelson, 9 Balatsky and Bonca, 1 0 and Cole-

52

man, Miranda, and Tsvelik, 1 1 have proposed other scenarios for odd pairing involving composite operators. These examples indicate that the odd-frequency-gap superconducting correlations are relevant for some physical models. Although a robust realization of odd-gap superconductivity remains to be found, in w h a t follows we consider a simple model in which some properties of the odd-gap superconductors, such as Meissner, N M R , and Josephson effects, can be analyzed. II. S Y M M E T R Y OF T H E G A P We introduce the anomalous Green's function JF(k,u,„) = Y,

[

dr e " i " » r
J

o,0 °

xc_k,/3(-T/2))-^(k),

(2.1)

where u „ = (2n 4- l)7rX and 4>ap{k) determines the spin symmetry of the pairing amplitude and hence the gap function. For example, for singlet pairing, ag = *( < T ")a^/2, where o~y is the Pauli spin matrix. For triplet pairing (as in 3 He, for example) ap(ln) = (i„) follows from the anticommutativity of the i/> operators in T'. Using this a n d the antisymmetry of the Pauli matrices we immediately get, for the singlet case, 1 2 ^(k,u-n) = . F ( - k , - u \ , ) ,

A(k,a,n) =

A(-k,-u>„)

and, for the triplet, 4

' '

JF(k,a, n ) = - ^ ( - k , - u ; n ) , A(k,wn) = - A ( - k , - u v ) .

(2.3)

From these relations, it can be seen t h a t the conventional even-spatial-parity singlet (conventional metals) and odd-parity triplet ( 3 He) states have gap functions which are even in M a t s u b a r a frequency. Then the equal-time anomalous Green's function T is nonzero and this leads to the usual off-diagonal long-range order (ODLRO). However, it has been shown 1 ' 4 ' 5 that gap functions which are odd in M a t s u b a r a frequency are admissible. In the odd case, Eq. (2.2) is satisfied with singlet superconducting pairing with a pair amplitude which is odd in both k and ui„: jF(k,ov,) =

-jr(-k,u„)

=

-T(*,-un),

A(k,u;„) = - A ( - k , w n )

=

-A(k,-wn).

(2.4)

Similarly, for the triplet case, Eq. (2.3) is satisfied for triplet superconducting pairing with a gap which is even in k and odd in wn: JF(k,w„) = ^ ( - k . u , , , )

=

-JF(k,-w„),

A(k,w„) = A ( - k , u , „ ) =

-A(k,-w„).

(2.5)

482 52

PROPERTIES OF ODD-GAP SUPERCONDUCTORS

Equations (2.4), (2.5) show t h a t the spin-singlet (-triplet) gap is described in terms of an odd (even) orbital function, while, a t the same time, the spin function is odd (even). There is no violation of the Pauli principle because t h e equal-time g a p function, or pair amplitude, for singlet or triplet vanishes since the gap is odd in w^. 1

1273

interaction between t h e local spins can b e neglected in t h e present context. As in Eq. (3.3), t h e pairing is studied by means of t h e anomalous amplitude ^ ( T ) which is determined by

Ci^lHc

+ Hx,*]:

* i ( T ) = - l > , < 7 V ^ T ( T ) c , . t ( 0 ) ) - J3W

•L,„(T),

(3.6)

i III. EQUATIONS FOR COMPOSITE ODD SUPERCONDUCTIVITY A . Model for t h e c o m p o s i t e o p e r a t o r For simplicity, we consider the m = 1 triplet Berezinskii 4 state. T h e conclusions are similar for t h e singlet case, but the notation is somewhat more cumbersome. From Eq. (2.1), we write the anomalous Green's function as JP(k,r) = £ < r T C k , 0 ( T ) c_K0(O)){ica")a0

- d ) , (3.1)

a/3

which, for d = —(dx + idy)/2

(for example), reduces to

^(k,T) = .

(3.2)

We are considering the odd-gap case in which F(T) is odd in T and even in k, an even-parity triplet odd gap. Since F[T) is odd, it vanishes a t r = 0. However, its r derivative is even and is generally nonzero a t T = 0. 1 4 Therefore, we study the anomalous amplitude

'

-

^

-

(

^

<-<•>)•

<«>

The behavior of f is deduced from the equations of motion. T h e Hamiltonian is

where tij is the lattice kinetic energy m a t r i x element a n d Lipir) is the composite anomalous a m p l i t u d e which characterizes t h e condensate. L is a vector in spin space, and for the m — I odd condensate, it has t h e form Up(r) = (7Vc i / 3 (T)c i t (0)S i (T)). We shall be interested in a spin-1 condensate in which a spin-0 Cooper pair is coupled t o a spin-1 excitation (magnon) S * . Therefore, we shall study L(l,2,3) = L„(9(l,2,3)Ea0/V^ = (T T c l Q (T 1 )c 2 / 3 (T 2 )S3(r 3 ))e c ,^/ % /2,

where 1,2,3 label space and imaginary t i m e points a n d ea/3 is i<Jva0. B . R e d u c e d H a m i l t o n i a n a n d mean-field e q u a t i o n s Our aim is to calculate properties of t h e odd-pairing state which depend on the structure a n d s y m m e t r y of t h e composite order parameter. For this p u r p o s e , we s t a r t with the simplest possible weak-coupling model which gives rise to a condensate characterized b y a nonzero value of L, the expectation value of t h e composite operator discussed in t h e previous subsection, a n d which we can solve analytically. In direct analogy t o B C S theory, 1 6 we write a "reduced" Hamiltonian Hred = H0 + ^T ct^cJgSi • Vij c^CjiS)

H = HC + Hs + Hx,

(3.4)

where Hc is the kinetic energy of the conduction electrons, Hs is the Hamiltonian for some spin or charge excitation, and Hx couples the electrons to the excitations. If one integrates out the excitations to produce an effective retarded (perhaps spin-dependent) electronelectron interaction as in t h e Eliashberg 1 5 treatment of superconductivity, then T can be evaluated by commuting c^f(r) with the effective Hamiltonian. In t h a t case, !F has a term of the form of Eq. (1.1). In order to motivate the structure of the odd composite condensate, we consider a spin fermion model in which the conduction electrons are coupled via Hx to localized spin modes which arise from a second band of electrons. Thus, Hx has the form

HX = J^24aca0ci0-Si,

(3.5)

i

where Si is a spin-1 excitation at site i of the lattice. In what follows, we shall assume t h a t both the possibility of Kondo screening (for J > 0) and the induced ("RKKY")

(3.7)

(ea/3ES-,)-

(3.8)

•3

Here V^ is an attractive (i.e., V < 0) short-range, instantaneous interaction which mediates t h e condensation a n d H0 contains the kinetic energy of quasiparticles a n d t h e noninteracting Hamiltonian for the spins. We should stress t h a t t h e above Hamiltonian is only a n instantaneous approximation to a dynamical six-point interaction. A dynamical theory which takes into account the retardation in L ( l , 2 , 3 ) can be constructed, for example, starting with the interaction Hx of t h e previous subsection 1 7 , 1 8 [Eq. (3.5)] and perhaps o t h e r couplings as well. For the reasons stated above, we work w i t h HTe^. Such a reduced Hamiltonian might be obtained a s an approximation to the repeated interparticle scattering of two quasiparticles with a spin, i.e., the Faddeev scattering of two quasiparticles with a spin fluctuation. Alternatively, if we take the simple interaction of t h e previous subsection and add a n electron-electron interaction V, t h e n the irreducible six-point coupling of Hrcd could arise as shown in the diagram of Fig. 1. T h e six-point coupling admits a mean-field treatment of the L condensate com-

483 ABRAHAMS, BALATSKY, SCALAPINO, AND SCHRIEFFER

1274

52

FIG. 3. Linearized gap equation for composite condensate.

The linearized gap equation, valid near t h e transition, is obtained from Fig. 2 and Eq. (3.10). It is shown in Fig. 3 and its analytic expression is FIG. 1. Six point coupling arising from electron-electron interaction (U) and electron-localized spin coupling (J).

pletely analogous t o B C S . The mean-field decomposition of the reduced Hamiltonian is HIAF = Ho — E [

A

i •

(c^CjsS^es-y

+*j • ( s V ^ S , ) ^ ] ,

(3.9)

where A

< = - E V^(T = °) = - E

^(wiss])^.

A = V J drg(r)g(-T)V{T)\,

(3.11)

where V = —Vij. T h e gap equation differs from the BCS case by the presence of the spin propagator in the integral. It is essential t o recognize t h a t t h e boson described by "D is not what is responsible for forming t h e Cooper pair part of t h e condensate. T h e interaction for the condensate here is t h e instantaneous V . We shall show in the next subsection t h a t for a simple choice for t h e propagator Z>, provided V exceeds a minimum coupling, there is an instability which determines T c .

C. Behavior near t h e transition t e m p e r a t u r e

We have now defined t h e anomalous vertex L ( r ) [Eq. (3.7)] and the "gap" function A [Eq. (3.10)] in complete analogy with t h e Green's function treatment of the BCS theory. We can therefore construct a diagrammatic analysis containing the usual quasiparticle propagators G, spin (magnon) propagators t o be specified D and in which the anomalous vertex L Q £ enters. This vertex, in which two quasiparticle lines (spins a, 0) and one magnon (wavy) line enter or leave t h e condensate represented by a circle, is shown in Fig. 2.

In contrast t o t h e BCS case, t h e mean-field Hamiltonian, Eq. (3.9), is not quadratic in t h e fields so that a n immediate diagonalization is not possible. Instead, we shall assume t h a t t h e transition is second order everywhere so t h a t we may construct a perturbation theory in A, which should be valid both near T c a n d near the minimum coupling, where A is small. This approach is similar t o t h a t used by Abrikosov et al. in a microscopic derivation 7 of Ginzburg-Landau equations. In this way, we can obtain t h e transition temperature a n d various response functions close t o the transition. T h e elementary vertices A, A are shown in Fig. 4. We begin with the anomalous condensate amplitude I J ; ( T ) . TO third order, it is given by t h e diagrams in Fig. 5 and t h e analytic expression is

FIG. 2. Composite condensate vertex L. Solid lines are conduction electrons; wavy line is a magnon.

FIG. 4. Elementary anomalous vertices for perturbative treatment.

A,- = - E v<Mr = O) = - E v-i (44*s<>£°0i

t

(3.10)

484 PROPERTIES OF ODD-GAP SUPERCONDUCTORS

52

1275

7

1(1,2 ,3) = / .d4 0(1,4)A(4)£?(2,4)27(3,4) + / d4d5d6 5 ( 1 , 4)A(4) • 0 ( 5 , 4 ) A ( 5 ) 0 ( 5 , 6 ) A ( 6 ) 0 ( 2 , 6 ) 2 7 ( 4 , 6)27(3, 5). (3.12)

The propagators Q and 27 are for quasiparticles and spins in the normal state. T h e transition temperature is found from Eq. (3.10) a n d t h e linearized equation for L [compare Eq. (3.11)]: A ( l ) = V L ( 1 , 1 , 1 ) = V Jd2

0(1,2)A(2)0(1,2)27(1,2). (3.13)

Here we have taken the interaction V to be of zero range. The Tc equation is therefore given by

1 = VT? Y, 5 3 G&> ")S(-k + «fc -<" + •')»(* «0. (3.14) where a) and v are fermion (odd) and boson (even) Matsubara frequencies, respectively. The precise result for Tc depends on the form of t h e magnon propagator. It is obvious t h a t 27(1,2) = const leads to an ordinary BCS expression for T c . To introduce the new physics of the composite condensate we choose 27(q, v) = T/[(ii/)2 — T2], which could represent the dynamics of noninteracting localized Kondo-like spin resonances. Since 27 is independent of momentum, we carry out the momentum sums in Eq. (3.14) and find 1 = -2TT2NIVT2

Y

[D(W +

W')-T>(U>-U>')],

u>>0, w ' > 0

where g = 2NQVF and uic is t h e ultraviolet cutoff of the interaction. In Fig. 6, we show a plot of Tc against the effective coupling g. As expected, there is a minimum value of g to achieve a transition a n d t h e curve suggests t h a t for a range of g above t h e minimum value, there is reentrant behavior. This may be checked by evaluating the right-hand side (RHS) of Eq. (3.14) near Tc. T h e condensed state occurs for those t e m p e r a t u r e s larger or smaller t h a n T c for which the t e m p e r a t u r e derivative of the RHS is negative. In this way, we find t h a t in the (smaller coupling) region of Fig. 6 where Tc has two values Tc+, Tc- the condensate exists between t h e m and the system is normal for T < Tc- or T > T c + . In the larger coupling region, only T c + exists a n d t h e system is in a condensed phase down t o T = 0. Another possibility is a spatially uniform propagator of the form 27(q, v) = -<Sq

1 =

2N0VT?

(w

4r-

w2 + r 2 w'2 + r 2 '

(3.18)

1

+ r 2 w + u>'

. (3.19)

T h e result of the frequency sums gives t h e transition temperature as Tc =

227,odd

- T2'

In this case, the momentum S function removes one integration and an infrared logarithmic integral (as in BCS) is recovered. The T c equation becomes

(3.15) where JVo is the spin-1 quasiparticle density of states at the Fermi surface. We shall see below that the sums are dominated by their low-frequency behavior. Therefore, for simplicity, we approximate the square bracket of Eq. (3.15), i.e., the odd p a r t of 27, by the separable form

(iv)2

magnon

1.56u>ce-"/N°v.

(3.20)

In order to find the quasiparticle s p e c t r u m in the condensed state, we examine the single-particle G r e e n ' s function. For this, we calculate t h e irreducible self-energy to second order in A. T h e diagram is given in Fig. 7 and

(3.16)

Equation (3.15) can now be evaluated explicitly for T c . In terms of digamma functions, the result is ffrve

[W\2

2irTc J

1

W

\2 2

2nTcJ

FIG. 5. Perturbation expansion for L.

(3.17)

FIG. 6. Phase diagram for composite odd-gap state, showing reentrant behavior.

485 ABRAHAMS, BALATSKY, SCALAPINO, AND SCHRIEFFER

1276

From Eq. (3.23), we find that the density of states in the superconducting state is unchanged from t h a t of the normal state, a situation which always obtains when the self-energy is momentum independent (as in the case, e.g., of electron-phonon interactions). However, the interaction produces mass renormalization. Here, there is no gap, but the quasiparticle spectrum is modified to the form

FIG. 7. Diagram for self-energy.

the expression is E(1,2) = ( 4 / 3 ) A ( 1 ) A ( 2 ) S ( 2 , 1 ) Z > ( 1 , 2 ) .

(3.21)

After Fourier transform, Eq. (3.18) takes the form 2

£(k,uO = (8rrt/3)Ar 0 |A| T J^

VoAd{u> + J).

(3.22)

u>'>0

With the use of the separable propagator of Eq. (3.16), we find 0 = 0°[1+E£] =

iu>Z(iu>) — e'

(3.23)

where Q° is the normal state Green's function, c is the norma] state quasiparticle kinetic energy measured from the Fermi energy, and 2

2

2

z(iw) = i - w /[(iu>) - r ], 2

W

2

=

(4n/3)\\\ ,/f/2V.

(3.24)

' k i f

52

E(k) a e(k)

1+

W2
(3.25)

which is valid near the transition where | A | 2 and hence W2 is small. For small excitation energy, there is a dynamical mass enhancement which does not appear in the density of states, but which is important in response functions. IV. M E I S S N E R E F F E C T The Meissner effect occurs when the paramagnetic electrodynamic response is less than the diamagnetic part. The dc response is given by ji(q) = - Q y ( q ) i M q ) , Q i ; i (q) = ^ J V e 2 / m + Q?.(q). The paramagnetic kernel is

(4.1)

1

dT
(4.2)

a0 kk

where k i = k ± q / 2 . As in the previous section, we evaluate Qp near Tc by perturbation in the order parameter A. The relevant diagrams are shown in Fig. 8. The analytic expression for q —¥ 0 is

e2r:

Q & - « 5 = -—5-|A| a

£

kik' [ g 2 ( k , a ; ) e 2 ( k ' , a , ' ) - 2 e 3 ( k , l i , ) e ( k ' , W ' ) P ( k + k ' , a ; + a , ' ) .

T h e result Qp — Q" > 0 indicates a Meissner effect and a positive superfluid density. If the magnon propagator is momentum independent, there is no contribution to the Meissner kernel since the momentum summands are odd. Therefore, we generalize the magnon propagator to include momentum dependence. An extreme possibility is the case of a static, spatially uniform magnon of t h e factorized form

^

>

*

^

-

^

>

FIG. 8. Current-current correlation function for the Meissner effect.

Z>(q, u) = -<5, * „ / T .

(4.3)

(4.4)

When this is used in Eq. (4.3), the result is Q" - Qn > 0 and is numerically precisely t h a t which is found in ordinary BCS theory near Tc. This is no surprise; the same uniform V gives the BCS expression when used in the Tc equation, Eq. (3.14). If we spread out the 6 functions, as would occur for a more realistic magnon propagator, the sign of Qp — Qn will not change, and the superfluid density will be positive with a value intermediate between zero and the BCS number. As an illustration, we consider a magnon propagator describing a resonance, as before, but one which is spatially uniform: P(q.«0 = -*q

(iu)

r2'

(4.5)

486

W h e n this is used t o evaluate Eq. (4.3), we find a result which may b e smaller t h a n the BCS value, and, we emphasize, of the same sign:

QBCS

|tk.kp|2^dd(k,a,)Fev„(kp,uO+

Yl k,kp,w

for the second case. T h e calculation of other response functions is similar to t h a t for the Meissner effect a n d we give an example in Sec. VI.

(4.6)

r

This result has been obtained as t h e leading term in an expansion in Tc/V which has been assumed to be small. Therefore, there is a Meissner effect for the odd-frequency composite condensate with a reduction of the superfluid density from the B C S value by a factor of order of the ratio of the transition t e m p e r a t u r e to the resonance width. We have also checked the superfluid density for the case t h a t the magnon propagator T> sharply peaked in momentum space a t either q = 0 or q ~ (7r/a,7r/a,7r/a). In each case it is again positive and it is somewhat smaller

J~

1277

PROPERTIES OF ODD-GAP SUPERCONDUCTORS

52

V. JOSEPHSON E F F E C T B E T W E E N ODD AND BCS S U P E R C O N D U C T O R S Here we briefly consider the Josephson effect between odd a n d even (BCS) superconductors. To b e specific, consider a S-I-S junction between an odd-frequency triplet superconductor on t h e left and an even-frequency triplet superconductor on t h e right. By applying the s t a n d a r d tunneling Hamiltonian to this system, we find t h a t the tunneling current has the form

|tw,k'ritk».k'"| 2 i : 'odd(k,a,)F o d d (k",< i -)F e « n (kp,a,)F e v e n (k'", W ),

J2 k,kp,k",k'",u>

(5.1) where tk.kp is the hopping matrix element and we retain only the relevant fourth-order t e r m in the tunneling matrix element expansion. 1 9 For tunneling between either odd-odd or even-even gap superconductors the first t e r m in the above equation is nonzero and dominates the Josephson current. T h e situation is drastically different in the present case. As long as there is no inelastic scattering in the junction, the leading term in the Josephson current between odd and even superconductors vanishes since the frequency integral in the first term is exactly zero. However, the next term is nonzero and produces a current which corresponds to the tunneling of "pairs of pairs" with charge 4e. The very important implication of this result is for flux quantization. If we consider a combined ring, made out of even and odd superconductors, then the flux quantization inside this ring will be in units of

*o/2

he 4e'

VI. N U C L E A R M A G N E T I C R E L A X A T I O N R A T E As another example of a response function, we calculate here the imaginary p a r t of t h e spin susceptibility X which enters the expression for the nuclear magnetic relaxation rate 1/T\: \/TxT

• x"(q,")

(6.1)

V

In the familiar BCS case, t h e behavior of response functions is governed by the combined effect of t h e singular density of states in the superconducting state and the coherence factors which modify t h e matrix elements. However, if one calculates close to Tc by expansion in powers of the gap function, it is not possible to separate these two effects. Near Tc, the correction to the sum on t h e RHS of Eq. (6.1) in the BCS case is

(5.2)

£

We stress t h a t the arguments about the vanishing leading term do not rely on any orthogonality in the spin or orbital channels, such as in the tunneling between s-wave and p- or ti-wave superconductors, where the orthogonality can be easily spoiled by impurities or spin-orbit interaction inside the junction. In our case the vanishing of the pair tunneling occurs as the result of the zero overlap in frequency between the even- and odd-gap superconductors .

oc

X"(q,")

X"(q,")

7C(3)JV 0 2 A 2 (T)

(6.2) 2

Due to the temperature dependence of A ( X ) , this gives an increase in 1 / 7 \ T as the t e m p e r a t u r e is lowered below Tc, presaging the familiar Hebel-Slichter peak. T h e correction to the susceptibility which is quadratic in the order parameter in the composite phase we are considering is given by diagrams like those in Fig. 8. T h e analytic expression is

I

I?

x"(q,")

x"(q^)

= -|A|2Im -Q(Vltw

£

[2g2{k1,u,

+

+ i / ) 0 ( k 2 , w ) 0 ( k 3 , w ' - u)g{k!

u)g(k2,w)g(k3,w'-iy) + k 2 - V.3,io'))V(w + a / ) .

(6.3)

487 A B R A H A M S , B A L A T S K Y , SCALAPINO, A N D SCHRIEFFER

1278

52

T h e result of the i n t e g r a t i o n s is

X"(q,")

£

X"(q,")

7r3JV03|Al2(r)(l-8/->r2) AEF

(6.4)

T h e sign of t h e correction a n d i t s t e m p e r a t u r e d e p e n d e n c e is t h e s a m e a s t h a t of t h e B C S c a s e , b u t t h e m a g n i t u d e is vastly reduced, by a factor of at least (Tc/Ep)2. T h i s is d u e t o t h e fact t h a t t h e d e n s i t y of s t a t e s i n the c o m p o s i t e case near Tc is u n c h a n g e d from t h a t in the normal s t a t e , while it is precisely t h e singular d e n s i t y of s t a t e s in t h e B C S case w h i c h is r e s p o n s i b l e for t h e Hebel-Slichter peak.

VII. I M P U R I T Y

SCATTERING

W e n o w address t h e q u e s t i o n of the effect of s t a t i c impurity scattering o n Tc. Since t h e order p a r a m e n t e r A is m o m e n t u m i n d e p e n d e n t , we m a y e x p e c t t h a t the transit i o n t e m p e r a t u r e is unaffected by a low c o n c e n t r a t i o n of i m p u r i t i e s . T h i s proves t o b e t h e case, as we n o w d i s c u s s . T h e d i a g r a m for the linearized g a p e q u a t i o n is t h a t of F i g . 3 dressed w i t h i m p u r i t y lines. It is s h o w n in F i g . 9, w h e r e the solid lines are i m p u r i t y - a v e r a g e d elect r o n Green's f u n c t i o n s . 7 T h e Tc e q u a t i o n w h i c h replaces E q . (3.14) is 1 = VTl

£

£

<7( k , u)G(Vl,

u / ) D ( k + k', w + « ' )

k,k' u),u>'

x P ( k + k',w + u/),

(7.1)

w h e r e P ( k + k',w-l-w') is t h e i m p u r i t y vertex ladder s e e n in F i g . 9. It is given by

' A . Balatsky and E. Abrahams, Phys. Rev. B 4 5 , 13 125 (1992). 2 P.W. Anderson, J. Phys. Chem. Solids 5 2 , 1313 (1991). 3 F . Mila and E. Abrahams, Phys. Rev. Lett. 6 7 , 2379 (1991). 4 V.L. Berezinskii, Pis'ma Zh. Eksp. Teor. Fiz. 2 0 , 628 (1974) [JETP Lett. 2 0 , 287 (1974)]. 5 E. Abrahams, A. Balatsky, J.R. Schrieffer, and P.B. Allen, Phys. Rev. B 4 7 , 513 (1993). 6 S. Yip and A. Leggett (unpublished). 7 A. Abrikosov, L.P. Gor'kov, and I.E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1975). 8 T.R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. 6 6 , 1533 (1991); Phys. Rev. B 4 6 , 8393 (1992). 9 V.J. Emery and S. Kivelson, Phys. Rev. B 2 6 , 10 812 (1992). 10 A. Balatsky and J. Bonca, Phys. Rev. B 4 8 , 7445 (1993). 11 P. Coleman, E. Miranda, and A. Tsvelik, Phys. Rev. B 4 9 , 8955 (1994); Phys. Rev. Lett. 7 4 , 1653 (1995). 12 Recall that W and A are related by A = W/Z, where Z

FIG. 9. Linearized gap equation in the presence of impurities. Solid lines are impurity-averaged Green's functions. Dotted lines are electron-impurity interactions.

P ( k + k ' , w + u / ) =T[D

(k + k ' ) 2 + ( " -uj')a].

(7.2)

Here, 1 / r is t h e i m p u r i t y s c a t t e r i n g rate, D i s t h e diffusion c o n s t a n t V2FT/3 and s — [sgn(w) - s g n ( o / ) ] / 2 . T h i s result for t h e v e r t e x is a p p r o p r i a t e i n t h e h y d r o d y n a m i c r e g i m e |k + k'| < \/vpT a n d ui — w' < 1/2-r. O t h e r w i s e , P = 1. W h e n Eq. (7.1) is e v a l u a t e d w i t h either o f t h e m a g n o n p r o p a g a t o r s m e n t i o n e d in Sec. I l l C, i.e., t h e m o m e n t u m i n d e p e n d e n t o n e or t h e s p a t i a l l y uniform o n e , o n e finds a T c w h i c h is u n c h a n g e d from the c l e a n c a s e u p to corrections of order (1/EFT)2. ACKNOWLEDGMENTS

P a r t of the work r e p o r t e d here w a s p e r f o r m e d u n d e r the auspices of the P r o g r a m o n Strongly C o r r e l a t e d E l e c t r o n S y s t e m s a t t h e Los A l a m o s N a t i o n a l L a b o r a t o r y . T h e h o s p i t a l i t y of t h e A s p e n C e n t e r for P h y s i c s , w h e r e p a r t of t h e work w a s carried o u t , is a c k n o w l e d g e d b y E . A . a n d A . V . B . T h e s u p p o r t of N S F Grant N o s . D M R 9 2 21907 (E.A.), DMR92-25027 (D.J.S.), and D M R 9 2 - 2 2 6 8 2 ( J . R . S . ) is also a c k n o w l e d g e d . We are g r a t e f u l to t h e following for helpful d i s c u s s i o n s a n d useful c r i t i c i s m : P. Coleman, A. Dugaev, G.M. Eliashberg, L.P. Gor'kov, a n d

P. Wolfle.

13

14

16 16

17

18 19

is the self-energy renormalization in the normal channel, which we take to be even in k and u;,,. Note also that the odd gap cannot be obtained within the weak-coupling BCS approximation which assumes no frequency dependence of the electron-electron interaction and therefore none in the gap A ( k , w„). Should the first derivative vanish at T = 0, t h e n it is necessary to go to the first nonvanishing o d d - r derivative. In that case a much more complicated composite condensate containing more than one Cooper pair or more than one density fluctuation (or both) would enter. In our case, there is one of each. G.M. Eliashberg, Sov. Phys. J E T P 1 1 , 696 (1960). For example, J.R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1984). J.R. Schrieffer, A.V. Balatsky, E. A b r a h a m s , and D.J. Scalapino, J. Supercond. 7 501 (1994). J.R. Schrieffer and A.V. Balatsky (unpublished). B.D. Josephson, in Superconductivity, edited by R . D . Parks (Dekker, New York, 1969), Vol. 1, p. 423.

Journal of Low Temperature Physics, Vol 99, NOJ. 3/4, '995

Ward's Identity and the Suppression of Spin Fluctuation Superconductivity J . R Schrieffer National

High Magnetic Field Laboratory and Department of Physics, University, Tallahassee, Florida 32301

Florida

State

1. I n t r o d u c t i o n The remarkable success in explaining the properties of low Tc superconductors in terms of a simple one-phonon pairing potential rests on two features. 1 l)Tc is small compared to the Fermi temperature Tp of these materials, TC/TF ~ 1 0 - 4 , so that the BCS (mean field) approximation is valid, with small amplitude pairing fluctuations occurring about this state and 2)MigdaPs theorem holds, 2 namely the vertex function T^ — !•! where r » f is the ratio of the coupling gkq (of a phonon q with a dressed quasi-particle k) and the coupling jjj of a phonon and a bare electron, r » , = 9h,/g°kt

(i)

Migdal 3 proved that in conventional metals the vertex correction, T t , — l,is of order the ratio of the sound velocity and the Fermi velocity, s/vp ~ ^ ~ 10~ a . Thus, the small parameter in this case is not j j . , , which is of order unity when averaged over q in the Brilloiun zone, but rather the adiabatic parameter y/m/M, the square root of the electron to ion mass ratio. High T c materials are more complex in two ways. 4 l)Tc/Ep is not necessarily small, ~ 1 0 - 1 — 1 0 - 3 , with pairing fluctuation corrections to BCS mean field theory being on the scale of 1-10% and 2)Migdal's theorem Tj., ~ 1 holds for neither the phonon coupling g*. nor the coupling to the fluctuating antiferromagnetic (AF) order since neither uijj/Ep nor J/Ep are small compared to unity, where J is the exchange coupling which produces antiferromagnetism in the parent cuprates compound. In an attempt to extend the low Tc theory to incipient antiferromagnets, such as the cuprates, many authors have proposed schemes involving the exchange of a spin fluctuation (sf) between fermionic excitations (quasi holes) of the normal p h a s e . 5 - 1 5 By assuming a)the sf propagator, or dynamic spin susceptibility x(?i<*') is sharply peaked about q ~ Q - ( T , IT) with x(Q, 0)/x(0, 0) ~ 100 - 150, and 397 nm-nihVfKW-milSm.SOlO

« 1995 Plenum Publishing Corporation

398

J.R. Schneffer

b) that T'f ~ 1, these authors argue t h a t the gap parameter A* of the lowest free energy phase of the cuprates exhibits d-wave symmetry, with A* oc k* — h*, and that many of the anomalous properties of these materials can be accounted for in this manner. Here we prove that the spin fluctuation vertex is strongly suppressed, when XQ/XO becomes large. In fact, the actual one-spin fluctuation effective pairing potential is strongly reduced, rather than enhanced, since

n{ ~ rjfx, ~ hxt ~ -J- - o JLq

(2)

Kq

as q —• Q and L,/a becomes large. Hence, while V£* increases for very weak coupling, the correct pairing potential arising from the exchange of one-spin fluctuation quickly saturates and then sharply decreases as the bare coupling (coupling to bare holes) is increased. This result casts serious doubt on the correctness of present spin fluctuations schemes for high Tc, which neglect vertex corrections. To make these results plausible, we recall the well known deformation potential theorem which states that the electron-phonon coupling gkq vanishes as q —> 0 in a crystal. However, consider a crystal which undergoes a lattice distortion corresponding to a static amplitude for a zone boundary phonon mode of wave vector Q. Then the electronic eigenstates at in this broken symmetry phase are linear combinations of states ipk and ipk+Q of the undistorted crystal, 1 6 Qk = ukifik + vk
(3)

Clearly, the matrix elements g°q and gkq of the electrostatic potential V set up by a phonon when taken, between


(4)

for bare fermion energies ek = ek+9. This result leads to the abrupt drop of the acoustic attenuation rate below Tc in conventional superconductors. Similarly the interference of ipk and
Ward's Identity and the Suppression of Spin Fluctuation

399

2. T h e Ordered A n t i f e r r o m a g n e t For definiteness, we consider the spin-fermion model in which band electrons (holes') are coupled to localized spins, i.e., the periodic Kondo problem, ff =

ffjfff,

(5)

where

Ho =

$ 2 «*»»*., (« = ±1),

(6)

fc«

H1

=

-^^c\,S„.,Ci,.

Sit

(7)

ill'

ek

— -2t(cos kx + cos ky).

(8)

with the lattice spacing a — 1. In the ordered antiferromagnet with spins quantized along z we have within the mean-field approximation Si - (Si) = S 0 c o s Q

Rt

(9)

where Q = (ir.ir) The one-electron energy eigen-operators are 1 6 Ik.

-

UkCk, +- SVkCk + Q,.

(10)

7t+Q,j

=

aifcCfc+g,, - "fcC,.

(11)

«*

+

vl = l

(12)

where /fe is in the first magnetic Brillouin zone (MBZ) and k+Q is in the second MBZ, corresponding to the valence and conduction bands, respectively, with energies — Ek and 4-Sfc, where Ek

=

v/eJ + A j ,

(13)

A*

=

JS0.

(14)

Within this approximation, one can show that the vertex T^f" coupling a astate fermion to a spin wave is factor Vjb'krib,i<-/b = (M*ufc- - vkvk>)Vk-k - (ukvk.

- vkuk>)Vk>ik+Q

(15)

Since V^f" vanishes as q —* Q, i.e., k and k' = k + q approach the MZB, while T° for a bare (c-fermion) is unity, we see that vertex corrections are of crucial importance if the enhancement factor x(Q)/x{®) lB l a rge t a s required by the present spin fluctuation theories of H T S . 5 - 8 3. Fluctuating Antiferromagnetic O r d e r Having discussed the ordered antiferromagnet, we consider the weak coupling limit for which a self-consistent diagram scheme is appropriate. If the fermion selfenergy E is given by the one loop approximation, then Ward's identity requires that T is also given by the one loop approximation. One finds r f c f (w f w') = 1 - ( j )

Ga(k + Q,u)G .,(k + Q - q,u, - u')

(16)

J.R. Schnefler

400

As q —+ Q and the spin wave energy becomes small, u/' —» 0, rtO(w,0)=

2_J

—0

as u -

E fc .

(17)

This result reflects the fact that a q = Q magnon represents a uniform rotation of the spin quantization axes on the two sublattices and this leads to a zero matrix element as L, —* oo. The "spin deformation potential theorem" requires that T varies as (q —» —Q) as L, —» oo. Below we find that Ykq varies as [(q — Q ) 2 -f L~2]? for finite L,. The second approach is to cast the problem as a coherent spin state path integral, using the identity ^

J Midipi\ni)(ni\

= I,

(18)

where \Hi) is the spin eigenstate for Si quantized along the direction of the unit vector Hi. Tracing out the fermions F, the evolution function is given by

Z =TT t'pH

= Jv{m(T)}e-A

(19)

where r is the Feynman ordering parameter and e~A =Tr e~pa.

(20)

H is the above Hamiltonian with the operators Si replaced by the r-dependent c-number field hi(r)/2. A is the action of free fermions moving in an arbitrary r-dependent SU(2) gauge field, i.e., a magnetic field couplied to the spins. For large L, A can be expressed as a gradient expansion in the staggered order parameter Mi = hi cos Q Ri (21) In this approach, it is convenient to rotate the fermion spin quantization axis on each site i at Feynman time r to n.i(r) through a unitary transformation H = U+HU.

(22)

The exchange interaction (7) is thereby reduced to fermion spins coupled to a spacially uniform staggered 2-magnetic field J / 2 , as if the system were in the long-range ordered antiferromagnetic state. The hopping interaction also changes due to the product of the spin rotation SU(2) matrices U*Uj for hopping from site j to i entering the effective hopping in the twisted spin coordinates iii = UtUjUj.

(23)

Since U, is short ranged, e.g. nearest neighbor hopping a gradient expansion ( A M ) n can be carried out for large L,. This procedure is analogous to that used by G. Whitfield in the electron phonon problem in which a non-flat metric g is introduced to allow the coordinates to move with the ions (like the spin quantization axis moving with hi[r)). Then, the electron kinetic energy V 2 becomes V ( l / j ) V , with the electron phonon interaction arising from V j _ 1 . Clearly the coupling vanishes for uniform g since all of the scattering is already included in the basis states (i.e., the

Ward's Identity and the Suppression of Spin Fluctuation

401

4. Pairing Interaction The Green functions and therefore the effective pairing potential can be determined by adding appropriate source terms to H and differentiating Z with respect to these sources, so that Z becomes a generating functional. In this way the Green functions are expressed in terms of a nonlinear sigma model with an effective action A. Since the leading term in the gradient expansion of A is ( V M ) V it follows the dressed coupling T t , vanishes as q —» Q and L, —• oo. Using the results of Sachdev and Chubakov for the nonlinear sigma model, one finds TkqX[(q-Q)2

+ L;^.

(24)

Within the one spin fluctuation exchange approximation, the pairing potential is

K(1) - — r?

Y

(25)

Therefore, V is strongly suppressed by T precisely when xq >s large, i.e., large L, There remains the question of how large must L, be for this effect to be substantial. This is yet to be resolved, however, numerical evaluation of the one loop reduction of T already shows substantial for L, of order two lattice spacings. Monte Carlo calculations are also helpful in this regard. We note that because V^ ' is suppressed, the dominence of ti-wave pairing over s-wave in HTS is less clear than was concluded by previous a u t h o r s . 5 - 1 5 The importance of Vj," for n > 1, e.g. the spin bag potential V£ ' corresponding to two crossed spin fluctuation lines, should be further investigated. 18 The persistent difficulty of maintaining system stability over a range of doping with very large staggered spin enhancement factors x{Q)/x(®) ~ 100-150 may be resolved in that a far larger region of q about Q is in fact involved in HTS, with considerably smaller enhancement, i.e., the correct picture is closer to a real space description. In essence, the HTS problem is less like the Migdal's theorem dominated low Tc problem than was previously recognized, involving highly nonlinear (multi-magnon) quasi-local correlation effects rather than Bragg exchange scattering from antiferromagnetic spin fluctuations, as presumed in the present spin fluctuation approaches. 5. A c k n o w l e d g e m e n t s The author would like to express his appreciation to L. Lilly, N.E. Bonesteel, P. Monthoux, L.P. Gor'kov, D.J. Scalapino, D. Pines, and H. Schultz for useful discussions. This work was supported in part by NSF Grants 92-22-682 and 5024501-22. REFERENCES L. J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957). 2 J.R. Schrieffer, Superconductivity, p. 157-162, W.A. Benjamin Inc., New York (1964).

402

3. 4 5 6. 7 8. 9. 10. 11. 12. 13. 14. 15. 16. 17 18

J.R. Schrieffer A.B. Migdal, Soviet Physics, JETP 1, 996 (1958) High Temperature Superconductivity Proceedings, The Los Alamos Symposium, Addison Wesley (1989). A.J. Millis, H. Monien and D. Pines, Phys. Rev. B 42, ,167 (1990). H. Monieu, D. Pines and M. Takigawa, Phys. Rev. B 4 3 , 258 (1991). P. Monthoux and D. Pines, Phys. Rev. Lett. 69, 961 (1992); Phys, Rev B 46 14803 (1992); Phys. Rev. B. 47, 6069 (1993). D. Thelen, D. Pines and J.P. Lu, Phys. Rev. B 4 7 , 9151 (1993). D.J. Scalapino, E. Loh and J.E. Hirsch, Phys. Rev. B 34, 8190 (1086). N.E. Bickers, D.J. Scalapino and S.R. White, Phys. Rev. Lett. 6 2 , 961 (1989) N. Bulut and D.J. Scalapino, Phys. Rev. Lett. 6 7 , 2898 (1991). N. Bulut, D.J. Scalapino and S.R. White, Phys. Rev. B 4 7 , 2742, 4599 (1993) J.P. Lu, Mod. Phys. Lett. B. 6, 547 (1992). K. Ueda, et al., / . Phys. Chem. Solids 5 3 , 1515 (1992). R.J Radtke, et al., Phys. Rev. B 46, 11975 (1992); 48, 653 (1993). J.R. Schrieffer, X.-G. Wen and S.-C. Zhang, Phys. Rev. B. 3 9 , 11663 (1989) A.V Chubukov, S. Sachdev and H,-W. Ye, Phys. Rev. B. 4 9 , 11919 (1994). J R Schrieffer, X G Wen and S.-C Zhang, Physica Scripta T 2 7 , 99 (1988)

494 VOLUME 78, NUMBER 9

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LETTERS

3 MARCH 1997

Momentum, Temperature, and Doping Dependence of Photoemission Lineshape and Implications for the Nature of the Pairing Potential in High-rc Superconducting Materials

Z.-X. Shen1 and J. R. Schrieffer2 ^Department of Applied Physics and Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, California 94305 ^-National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306 (Received 27 September 1996) The anomalous momentum and temperature dependence of the spectral lineshape in data from underdoped Bi2Sr2CaCu20g+« (Bi2212) indicates that the quasiparticles are strongly coupled to collective excitations centered near Q = (TT, IT). The doping dependence of the spectral lineshape and its correlation with the size of the superconducting gap indicate these collective excitations are related to the pairing interaction in high-7"c superconductors, in analogy with phonon induced structures in tunneling spectra of low Tc materials. [S0031-9007(97)02505-2] PACS numbers: 74.62.Dh, 73.20.Dx, 74.72.Hs, 79.60.Bm Over the last decade, angle-resolved photoemission (overdoping) represents smaller (larger) hole concentraspectroscopy (ARPES) has contributed significantly to tion than those needed to achieve maximum Tc. the understanding of the high-7"c superconductors [1]. Figure 1 shows typical normal state photoemission specThe most recent example of this is the observation of tra recorded at 100 K from an underdoped and an overan anisotropic excitation pseudogap in the normal state doped Bi2212 sample with Tc of 84 and 80 K, respectively. of underdoped Bi2212 [2-4]. Despite the progress, (a) and (p) represent spectra taken at the Fermi surface the line shape of photoemission spectra remains poorly (FS) crossing along the (0,0) to (IT, TT) direction and the understood. This is particularly true for the underdoped (IT, 0) to (TT, IT) directions, respectively. The spectra from materials. In this paper, we analyze the momentum, the overdoped sample show Lorentzian-like peaks at the temperature, and doping dependence of the spectral line Fermi level. The spectrum from the underdoped sample shape, with emphasis on a broad feature at 100-200 meV recorded at (a) is very similar to that from the overdoped that is always present in spectra taken near (TT, 0) in unsample. In contrast, the spectrum at (/?) is dramatically derdoped samples. This feature is absent in spectra taken different. It is an edgelike structure with a very broad at the Fermi surface along the (0,0) to (IT, IT) direction maximum near 100-200 meV. Furthermore, the leading in underdoped samples and along both of these directions edge is pulled back from the Fermi level by about 20 meV, in overdoped materials. We argue that these results are reflecting the reported dxi—yi like normal state excitation due to a strong dressing of the photohole for k = (ir,0) pseudogap [2-4]. in the underdoped material, while the dressing is modest The difference between the spectra at (/?) from the in the other cases. This result is consistent with the hole underdoped and overdoped samples is generic. In the coupling strongly to collective modes of momentum q whose spectral function x"(q, a>) peaks near Q = (IT, IT) for the underdoped case, while the coupling is weak because x"(q, a>) is weak and broad in momentum for the overdoped case. The doping dependence of the ARPES spectrum and the superconducting gap suggests that these collective excitations are related to the pairing potential with momentum dependence consistent with d pairing symmetry [5]. Angle-resolved photoemission data were obtained using a Vacuum Science Workshop chamber attached to the beam line V-3 of the Stanford Synchrotron Radiation I I I I I I ll ' ' ' ' I L 400 200 0 400 200 0 Laboratory (SSRL) and a Scienta-200 analyzer using a He Binding Energy (meV) discharge lamp. Details of the experimental setup and the sample preparation are the same as those reported earlier FIG. 1. ARPES data from an underdoped Bi2212 (A) and an overdoped Bi2212 (B) sample with an energy resolution of [3]. The energy resolution of data discussed here is ei35 meV. The inset showsfc-spacelocations where the spectra ther 35 or 20 meV as specified later, and the momentum were recorded. The curve depicts the experimental Fermi resolution is ± 1 degree. Doping of the sample is accomsurface, (a) and (/3) denote the Fermi surface points along plished through oxygen content adjustment by annealing the (0,0) to (TT, IT) and (IT, 0) to (IT, TT) directions, respectively. under various conditions. The notation of underdoping (/?) is very close to (7r,0). 0031-9007/97/78(9)/1771(4)$10.00

© 1997 The American Physical Society

1771

495 VOLUME 78, NUMBER 9

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underdoped regime, the spectra are very insensitive to doping, although the broad maximum may shift with doping level and the details of the line shape may show small variations. Once in the overdoped regime, the spectra change rapidly from a broad feature with a normal state pseudogap to a much sharper peak at the Fermi level without the normal state gap. Our recent experimental investigations find that the superconducting gap size decreases rapidly with the increase of carrier density in the overdoped regime [6], This empirical correlation between the disappearance of the broad feature at 100-200 meV and the decrease of superconducting pairing strength suggests that these features are closely related to the pairing interaction. Although analysis of ARPES line shapes in high-r c cuprate is controversial [7-11], the doping, momentum, and temperature dependence provide new information to gain added insight. First, a few words about the experimental background due to inelastic scattering of photoelectrons. The background is usually a featureless signal that monotonically decreases towards lower binding energy with a steplike intensity drop near, but slightly below the Fermi level [3,8]. This background exists in all spectra, and our discussion in this paper is based on features observed above this background. Spectra taken from the overdoped and underdoped samples at (a) can be modeled with a broadened Lorentzian at the Fermi level, together with an experimental background. The spectrum from the underdoped sample at (/3) cannot be modeled in the same way unless one is willing to use a very large width and an asymmetric shape. On the other hand, since the spectrum is taken at the Fermi surface as determined by the relative intensity change as a function of angle, only a single Lorentzian at the gap edge is expected if the data solely reflect the excitation of a quasiparticle. Therefore, the broad feature at (/3) in Fig. 1(A) signals other excitation processes that are coupled to the process of creating a photohole. The fact that the ARPES data establish the existence of other excitations in addition to the quasiparticle is most evident in spectra from the superconducting state near (77,0) where two components are directly resolved. The top two sets of curves of Fig. 2 are from an underdoped sample recorded with 35 meV resolution, while the bottom set of curves is from a less underdoped sample with 20 meV resolution. In the superconducting state, the experimental data consist of a resolution limited peak, a broad feature near 100-200 meV that is separated from the peak by a dip in intensity [12,13] and the experimental background. This peak represents the excitation of a quasiparticle in the superconducting state, with its position at (/?) being determined by the superconducting gap. On the other hand, this peak only represents a fraction of the experimental spectral weight. A significant portion of the spectral weight remains as a broad feature which can only be the signal of additional excitations. It should be noted here that dispersion and polarization analysis sug1772

IEW

LETTERS

• • •'

400

i • • • • i • • • ' '

300

3 MARCH 1997

• •'

• * • • • •' • • • • i • ' '

200

• i • • • * i • • • • i • • • *

100

0

-100

Binding Energy (meV)

FIG. 2. ARPES data from normal and superconducting states of underdoped Bi2212 near (w, 0). As illustrated in the inset of Fig. l(/3) is the Fermi surface crossing point along the (7r,0) to (IT, -IT) line and it is very close to (77,0). The upper two sets of curves were recorded with 35 meV energy resolution while the low set of curves was recorded with 20 meV energy resolution. gest that the broad feature is not caused by one-electron excitation from a second band [14,15]. In Fig. 3 we propose an interpretation of the data discussed in Figs. 1 and 2. Figure 3(a) depicts the hole spectral function in the weak coupling case in which the quasiparticle peak is approximately a Lorentzian centered at e*, plus a weak incoherent background. Although the present system is far from a normal Fermi liquid, the spectra of the overdoped and of the underdoped samples at (a) resemble this picture, albeit the quasiparticle width is too large and does not vary as le^l2 which the Landau theory predicts. Figure 3(/J) depicts the spectrum of a hole in a system where it is strongly coupled to collective excitations. In addition to the quasiparticle peak at Ej, one has excitations at higher binding energy due to the excitation of collective modes. In this case the probability of creating a quasiparticle without creating collective excitations is small, resulting in a weaker spectral weight at e^. Thus, the creation of a photohole is more likely to produce collective excitations plus a hole in the quasiparticle band in this case. This loss feature, caused by collective excitations, will have a broad energy distribution because of the momentum dependence of the collective mode spectrum as well as the recoil energy of the hole when a collective excitation is emitted. In a simple boson model [16] of a localized hole coupled a collective excitation of frequency <wn, this crossover from weak to strong coupling is evident in the spectrum A(co), where u> is the negative of the binding energy. Thus,

496 PHYSICAL

VOLUME 78, NUMBER 9

(0,1)

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W,K)

-Zrrmn Zk-hwi

%

Superconducting state

•it-ha t*»c

tmc

FIG. 3. Illustration of photoemission process and spectral shape in systems with weak (a) and strong couplings [(/3) and (7r,0)]. The Fermi surface picture depicts the phase space considerations for the coupling between the quasiparticle and collective excitations near {-IT, n). The light shaded area indicates thefilledstates, and the dark shaded area indicates the flat band region near the Fermi level. A{co) = e~g Xr=o(§2/M•)£(&» - e + ncoo); where e is the quasiparticle energy, equal to the bare hole energy eo < 0 plus the self-energy g2coo. For g > 1, this model yields a spectrum in which the quasiparticle peak is comparable to or weaker than that of the collective excitations, and corresponds roughly to the normal state data. In Cgo and related compounds, broadening of photoemission features has been attributed to this effect [17,18]. Under favorable conditions, the individual phonon sidebands have been resolved, facilitating a realistic estimate of the electron-phonon coupling constant that is consistent with the Tc of doped C ^ compounds [17]. From the above discussion, we believe the broad feature in spectra from near (w, 0) in the underdoped sample is a manifestation of a strong coupling between the quasiparticle and the collective excitations. From a recent study of underdoped and overdoped samples with similar doping to those reported in Fig. 1 [6], it is found that the superconducting gap of the overdoped sample is about 35% smaller than the underdoped sample. This means that the pairing strength is significantly reduced in the overdoped sample. Hence, the presence and absence of the broad loss feature is empirically correlated to a stronger and weaker pairing. This fact points directly to the collective excitations being the glue that pairs the electrons. In conventional superconductors, the same role is played by phonons which were detected in tunneling experiments [19,20].

LETTERS

3 MARCH

1997

While the doping dependence of the loss feature in the photoemission spectra suggests that the collective excitations are a key feature of the pairing interaction, the k dependence of the loss spectra gives added weight to this conclusion. There are two possible routes to explain why the above loss feature is highly k dependent in the underdoped samples. The first is that the coupling constant gkq may be strongly dependent on the photohole momentum k, while the second is a phase space argument. We believe the second route is sufficient to explain the data. Based on neutron scattering data from other cuprates [21-23], we assume the collective modes have a wide energy distribution, with a characteristic energy COQ, and their spectral function x"(q, co) is peaked near Q = («-, TT). With this assumption, it follows that a photohole created at the Fermi surface near (0, ir) can emit a collective excitation q = Q, with the hole recoiling to a quasiparticle state in the vicinity of (7r, 0) near the Fermi energy. Thus, the energy loss in this case is dominated by the collective modes; i.e., it occurs at an energy scale of order coo- On the other hand, for photohole near the Fermi surface at (a), emission of a collective mode Q will lead to the hole recoiling to a state k-Q far from the Fermi energy, and the loss spectrum will be shifted to considerably larger binding energy due to the hole's recoil energy and be intrinsically weak being off resonance (see below) or is forbidden if the hole is scattered to an unfilled state. A further effect enhancing the strong loss spectrum near (/?) or (TT, 0) is the flatness of the quasiparticle bands in this vicinity [24]. Thus, even if the peak of ;r"(q, co) is spread over a sizable region around Q, the recoil energy of the quasiparticle is modest and the above conclusions continue to hold. To quantify these ideas consider a coupled fermionboson model with band structure corresponding to the Fermi surface in Fig. 1. The photohole spectral function is given by A(k, to) = X2(k, co)/{[co - ek - 2i(*, to)]2 + 2i(*, co)}, (1) where the self-energy is given within the rainbow approximation by X2(k, co) = fglqx"(q,

v)A(k - q, to')

X 8(co - co' - v)dvdco'd3q/(2Tr)\ 1i(k,a>) = P j[^2(k,co)/{to

-

co')]dco'/7r.

(2) (3)

The quasiparticle peak occurs when the resonance condition co - ek - 2i(fc, co) = 0 is satisfied and the level width 2,i(k, co) is small. The loss peaks occur when X2(k, co) is large. This leads to a strong loss spectrum if the hole energy e/t + 2i(&, co) is small compared to co0. However, the magnitude of these peaks are reduced 1773

497 VOLUME 78, NUMBER 9

PHYSICAL

RE

by the large energy denominator in (1) when the recoil hole is in a high energy state or is in an unfilled state above the Fermi surface. This is the case in the underdoped samples at (a). However, near (TT, 0) or (/3) point, the recoil quasiparticle is near the Fermi energy and the admixture of collective modes is large for coupling constant J > 1 . It is the geometry of phase space imposed by the peak in x"(q, w) coupled with the shape of the Fermi surface which enhances the loss spectrum at (IT, 0) or /? while suppressing it at (a). In overdoped specimens, x"(7-a, it is difficult to be more specific about the energy distribution of the collective excitations. Data from La2-xSrxCu04 with q = (IT, TT) do show a depressed scattering intensity at low frequency in the superconducting state [22]. The scattering intensity appears to be moved to an edgelike feature at a somewhat higher energy. This is qualitatively consistent with our picture. On the other hand, data from YBa2Qi307-« is somewhat different [21]. The scattering intensity shows a sharp peak in the superconducting state with weight from both the lower and higher energy regimes. To make a more quantitative analysis, we need to know ;f"(&),q) of Bi2212 and calculate A(k, a>) using (l)-(3). Aside from the change of the collective excitations, the observation of the resolution limited quasiparticle seen in the superconducting state is also due to the longer lifetime in the superconducting state [25,26]. The sharpening of the peak will also make the dip look stronger. In summary, momentum, temperature, and doping dependence of photoemission line shape yields critical information about the collective excitations which appear to be the glue to pair the electrons. Unlike the phonon anomalies in conventional superconductors that have their distinct characteristics in the energy axis, the collective excitations here have their important characteristics in both momentum and energy space. The collective exci1774

IEW

LETTERS

3 MARCH 1997

tations, and thus the pairing interactions, are peaked at Q = (TT, TT). Although collective excitations of similar properties are observed in neutron experiments, it is important to detect them in the single particle spectral function because it measures the combination giqx"(to,q) rather than ,y"(q, <*>) alone. Z.X. S. would like to thank Shou-cheng Zhang for stimulating discussion. Z. X. S. acknowledges support from DOE's Office of Basic Energy Science, Division of Material Sciences through SSRL, and NSF Grant No. DMR-9311566. J. R. S. acknowledges support from NSF Grant No. DMR-9629987.

[1] For a review, see Z.-X. Shen and D, S. Dessau, Phys. Rep. 253, 1-162(1995). [2] D. S. Marshall et al, Phys. Rev. Lett. 76, 4841 (1996). [3] A.G. Loeser et al, Science 273, 325 (1996). [4] H. Ding etal, Nature (London) 382, 51 (1996). [5] For reviews, see D. J. Scalapino, Phys. Rep. 250, 329-365 (1995); D. Pines and P. Monthoux, J. Phys. Chem. Solids 56, 1651 (1995). [6] P.J. White et al, Phys. Rev. B 54, R15669 (1996). [7] C. G. Olson et al, Phys. Rev. B 42, 381 (1990). [8] J. L. Liu, R. O. Anderson, and J. W. Allen, J. Phys. Chem. Solids 52, 1473 (1991). [9] P. W. Anderson and Y. Ren, in High Temperature Superconductivity, Proceedings of the Los Alamos Symposium, 1989, edited by K.S. Bedell, D. Coffey, D.E. Meltzer, D. Pines, and J. R. Schrieffer (Addison-Wesley, Reading, MA, 1990), p. 3. [10] P.B. Littlewood and CM. Varma, Phys. Rev. B 46, 405 (1992). [11] N.V. Smith, Comments Condens. Matter Phys. 15, 263 (1992). [12] D.S. Dessau etal, Phys. Rev. Lett. 66, 2160 (1991). [13] Y. Hwu et al., Phys. Rev. Lett. 67, 2573 (1991). [14] D. S. Dessau et al, Phys. Rev. B 45, 5095 (1992). [15] H. Ding etal, Phys. Rev. Lett. 76, 1533 (1996). [16] G. D. Mahan, Many-Particle Physics (Plenum, New York, 1981), 2nd ed., pp. 289-298. [17] O. Gunnarsson et al, Phys. Rev. Lett. 74, 1875 (1995). [18] J. Wu et al, Physica (Amsterdam) 197C, 251 (1992). [19] J.M. Rowell and L. Kopf, Phys. Rev. 137, 1910 (1965). [20] J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Lett. 10, 336 (1963). [21] H.F. Fong et al, Phys. Rev. Lett. 75, 316 (1995); H.F. Fong et al, (to be published). [22] S.M. Hayden et al, Phys. Rev. Lett. 76, 1344 (1996); H. A. Mook et al., Phys. Rev. Lett. 70, 3490 (1993). [23] G. Shirane et al, Physica (Amsterdam) 197B, 158 (1994); R.J. Birgeneau et al, J. Phys. Chem. Solids 56, 1913 (1995). [24] D.S. Dessau etal, Phys. Rev. Lett. 71, 2781 (1993). [25] K. Krishana, J. M. Harris, and N. P. Ong, Phys. Rev. Lett. 75, 3529 (1995). [26] D. A. Bonn et al, Phys. Rev. B 47, 11 314 (1993).

498

VOLUME 58, NUMBER 10

PHYSICAL REVIEW B

Unusual states of inhomogcneous dx2-y2+idxy

1 SEPTEMBER 1998-11

superconductors

M. I. Salkola and J. R. Schrieffer NHMFL and Department of Physics, Florida State University, Tallahassee, Florida 32310 (Received 19 May 1998) Superconductors whose order parameter violates time-reversal symmetry and parity have unusual properties that set them apart from conventional superconductors. In particular, if translation symmetry is also broken, the superconducting state generates spontaneously a current and a magnetic field. These phenomena are studied using a dx2~yi+idxy superconductor as a prototype. Some of the most striking consequences of the symmetry breaking are examined at boundaries and in the presence of poinUike impurities and quasiparticles. [S0163-1829(98)50634-0]

The question of the exact nature of the pairing state in high-temperature superconductors has been a focal point of many experimental and theoretical studies. Perhaps the most intriguing one is the suggestion that the superconducting ground state may break time-reversal symmetry and parity. Moreover, the elementary excitations relative to such a state should obey fractional statistics. Later on, it was shown that the anyon mechanism for superconductivity is formally related to a BCS state with dxi-yi+idxy symmetry.2 These ideas have gained further interest because of recent experimental findings,3'4 which have been interpreted as an evidence for a dx2-y2 + idxy wave order parameter.5,6 In this paper, we examine the various effects at boundaries and in the vicinity of impurities that arise in superconductors whose ground state violates time-reversal symmetry and two-dimensional reflection symmetry. We also consider quasiparticle excitations that form when they are allowed to relax to a self-consistent state. Interestingly, inhomogeneous states that develop are characterized with current-density and magnetic-field distributions which can be used for their identification. For example, the typical field strengths of these configurations are in the range of 0.1-60 G, when the ratio of superconducting energy gaps to the half bandwidth is about 1/10 and the hole doping —10%. These fields might be observed by various local probes with a high enough sensitivity—muon-spin relaxation (/uSR) is one method capable of detecting small inhomogeneous magnetic fields. Nevertheless, /*SR studies have established that any anomalous magnetic fields in high-temperature superconductors must be less than 1 G.7 Because quasiparticles and impurities in the superconducting phase may generate a magnetic-field pattern whose maximum strength is above the upper bound of this experiment, one might then look for the signatures of anyon superconductivity in systems doped by stronglyscattering nonmagnetic impurities8 or by injecting spinpolarized electrons into them. Impurity-induced magnetic properties in superconductors have been considered before in the context of an A -phase order parameter of superfluid 3 He. 9

is the contribution due to impurities. In the mean-field approximation and assuming that the pairing of electrons occurs in the spin-singlet channel, the BCS Hamiltonian is given by

# o = 2 ^ t (R)W(R.r)*(R+r)- / t2 ^ t (R)f 3 *(R), Rr

R

(1) with H(R,r)=-r(r)f3e^
" imp =2 ^(rMrWr).

(2)

where V(r)=T31nV„Srr and V„ is the interaction strength of the nth impurity potential located at r„. Given that the pairing of electrons is described by the spin-singlet state, the superconducting order parameter (amplitude) can be expressed in the form f(R,r)=xS f„(«R+r)WR)),

(3)

As a model for superconductivity, we consider here electrons moving on a square lattice with lattice spacing a and interacting via an effective two-particle interaction. The model is defined by the effective Hamiltonian, Hc!!=H0 +Him , where H0 describes a BCS superconductor and Him„

where ^av={iT-i)av is a totally antisymmetric Levi-Civita tensor. The relation between the order parameter and the gap function is given by the equation

0163-1829/98/58(10)/5952(4)/$15.00

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A(R,r) = - u ( r ) F ( R , r ) .

(4)

© 1998 The American Physical Society

499

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UNUSUAL STATES OF INHOMOGENEOUS dx: 2+id,.

Here, v(r) is the strength of the two-particle interaction, which is assumed to be instantaneous in time. Thus, the energy cutoff in the gap equation is set by the bandwidth. In a translationally invariant system, the BCS Hamiltonian reduces to (5)

k

where A k = 2 r A ( R , r ) e ~ ' k r and ¥k=(<*kT
j^T)

=

ina

-—-{^HK)Tie1^{R)T^(R+ali)-H.c), (6)

where ^ ( R ^ i p C R . a ^ ) . The discrete Maxwell equations can be readily written down in the time-independent case; namely, Ampere's law and the absence of magnetic monopoles can be stated as (7a) (7b) where /ti0 is the magnetic permeability of the vacuum. Choosing the gauge so that V-A-'(r) = V_ M /l^(r) = 0, the equation for the vector potential becomes -V-„V^4 J ^(r) = / u, 0 ; M (r). These forms are consistent with the continuity equation, V-j(r) = 0. Note that it enforces one-to-one correspondence between the current j ^ and the magnetic field H. The equation for A'^ can be solved in the form A^(r)

R5953

= 2 r .<5(r-r');'^(r'), where the lattice Green's function Q, essentially the inverse of V _ „V „, depends on the boundary conditions. Before actual numerical calculations, the values of the material-dependent parameters must be determined. They all can be combined to one, dimensionless quantity, r/ = Aa(atlhc), where a—1/137 is the fine-structure constant, which appears when the vector potential is solved with a given current density as a source. For W = 4 r = 0 . 5 eV and a — Ak, we have ??— 7 X 1 0 - 6 . Also, in the following examples, we assume that no external field is present, A**x = 0, although it would lead to quite interesting effects. Impurities. First, consider a single impurity in a dxi-yi + idxy superconductor. This problem can be qualitatively solved in the T-matrix approximation, because the orderparameter suppression gives only a small correction. For one impurity, the single-particle Green's function, G ( r , r ' ; r -t') = -i{T9(.r,t)V*(r',t')), can be written in the form G(r,r';w) = G ( 0 ) ( r - r ' , w ) + G <0) (r,w)f( ) , where the T matrix is f(to) = [V'~ 1 -G < 0 ) (r=0,&>)]" 1 , the single-particle Green's function of a clean system is G <0) X(k,)f 0 , because the energy gap averaged over the Fermi surface vanishes. Thus, there is always a bound state in the energy gap. For A ^ A j , the energy of the bound state can be shown to be n 0 = A 2 -4A 1 exp(-cA 1 /A 2 ), when c = (irNFVyl>\, and n 0 =cA 1 [ln(4A 1 /A 2 )]" 1 , when c<\. Note that the impurity state is twofold degenerate by virtue of PT symmetry. The spatial dependence of the bound state is found by computing the spectral density per site, A „ ( r , u ) = - (l/ir)Im Gm(r,T;a>). For | + no), where Z(f\r) describes the particle (hole) weight at positive (negative) energy with spin orientation l, the induced magnetic moment is mz = a f 2 / 0 / c , where f2 = ^t ; F/A 2 is the coherence length and 70 = eA2/8ft. For c<\, we find m z = 3 a f 2 / 0 . That the induced moment is independent of A 2 is understood once it is noticed thai j e(r)^((2''2)~1 e'2r/(:L. Figure 1 shows the current density and the magnetic field, whose spatial dependence reflect fourfold symmetry of the impurity-induced bound state. The maximum magnetic-field strength corresponds a flux * ~ 2 x l ( T 7 < t o per plaquette, where * 0 = hie is the flux quantum. It is clear that because there are always inhomogeneities in every system, there must also be

500

M. I. SALKOLA AND J. R. SCHRIEFFER

R5954

FIG. 1. The spectral weight Z^'O) and Z\~\T) (in units of NFAj) of the impurity-induced intragap quasiparticle state with the energy O 0 =A 2 /3 for (a) up spin and (b) down spin. Also shown are (c) the magnitude of the current distribution j(_r) (arrows show the direction of the current, which circulates the impurity site) and (d) the induced magnetic field H(r). The parameters are Aj/W = 0.2, A 2 /W=0.1, A * / W = - 0 . 0 8 , and V/W=5, where W=4f is the half bandwidth. Here, the unit current density is j0=I0/a2 = 10pA/A2. Note that electrons move on a 21x21 torus, and the total magnetic flux through its surface is zero. a nonzero current distribution. In fact, a disorder potential would create a current and magnetic-field distribution pattern everywhere in the superconductor that could be detected by various means, one example being /xSK. Boundaries. There are two zero-energy states associated with a horizontal (vertical) and diagonal boundary. The twofold degeneracy comes from the fact that the Hamiltonian has PT symmetry. As an example, consider a horizontal boundary. A quasiparticle whose angle of incidence is ir/4 will be reflected so that the sign of the order parameter is opposite for initial and final wave vectors. As a consequence, a zero-energy state results. Because of the finite energy gap Aj along these directions, the quasiparticle state is localized

_

v

0.0

^

FIG. 3. (a) The magnitude of the current distribution j(r) (the arrows show the direction of the current density) and (b) the induced magnetic field H(r), when there are two impurities at the boundary located at r=0 and r=a,,. The parameters are A[/W = 0.2, A 2 /W=0.1,/i/W=-0.08, and WW=5. to the boundary, the decay length being governed by the coherence length £2 • The density of states has then a peak at zero energy within the energy gap of magnitude A 2 . A similar situation is found when the boundary is oriented diagonally. The generation of a dxy component removes all but the two quasiparticle states in a dxi-yi superconductor. If the superconductor forms a strip whose width is comparable to f 2 , the localized states at the opposite edges hybridize, and the level repulsion causes the states to be shifted from the zero energy.

<»>

(a)

~\A,M •

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^A«

-

5.0

10.0

15.0

20.0

Distance y/it FIG. 2. (a) The dxi-yz and dxy gap functions, A^r) and A2(r), (b) the current distribution ;,(r) (Jy — 0) and the induced magnetic field H(r) as a function of the distance from the specular boundary. The parameters are A1/W=0.2, A 2 /W=0.1, and n/W=-0.08. Note that the maximum dx2^yi energy gap at the Fermi surface is A!=4 A d, where A d is the bulk value of A x (r). Electrons move on a strip.

FIG. 4. The magnitudes of the gap functions (a) A^r) and (b) A2(r) in the presence of 12 spin-polarized quasiparticles. Also shown are (c) the magnitude of the current distribution /'(r) (arrows show the direction of the current) and (d) the induced magnetic field H(r). Here, A,/W=0.2, A2/1V=0.1, and M-'W=-0.0&. Electrons move on a 30X 30 torus.

501

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UNUSUAL STATES OF INHOMOGENEOUS dr%-,i + id„

Figure 2 shows the order parameters and the current and magnetic-field distributions near a horizontal boundary. As has been pointed out previously, there will be a boundary current.511 In this example, the total current turns out to be 7 f l =/ 0 , where A 2 is the dxy energy gap in the bulk. The dx2~y2 component is defined as a field on the dual lattice, A i ( r ) = jt^JlvAir,^), where the forward summation operator is 2 „/(r) = / ( r + a„) + / ( r ) . The dxy component is defined as a field A 2 ( r ) = j [ A ( r , a , + a , ) - A ( r + a J , « I - a J ) ] . The dxy component is much more sensitive to the presence of the boundary than the dxi-yi component, which is slightly enhanced. Defects make the boundary nonspecular. Strongly scattering scalar impurities at the boundary will have a drastic effect on the current density which has the maximum at the outermost row of sites. The boundary current must now go around the impurities, distorting the current and magneticfield patterns; see Fig. 3. As a result of the squeezed current density next to the impurity sites, the magnetic field is enhanced. Quasiparticle excitations. Third, consider a quasiparticle excitation whose configuration is determined selfconsistently. We find that there is a quasiparticle state just below the energy gap A 2 . The appearance of the quasiparticle bag leads to a stronger suppression in the dxy component than in the dxi-yi component which oscillates around its bulk value. This is evident from the fact that, at low energies

'V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987); R. B. Laughlin, Science 242, 525 (1988). 2 D. S. Rokhsar, Phys. Rev. Lett. 70, 493 (1993). 3 K. Krishana et al, Science 277, 280 (1997). 4 R. Movshovich et al, Phys. Rev. Lett. 80, 1968 (1998). 5 R. B. Laughlin, Phys. Rev. Lett. 80, 5188 (1998). 6 A. Balatsky, Phys. Rev. Lett. 80, 1972 (1998). 7 R. F. Kiefl et al., Phys. Rev. Lett. 64, 2082 (1990). 8 A nonzero magnetic moment induced by many-body effects may complicate the interpretation; see, for example, A. Mahajan

R5955

and for A 2 < SA ] , the quasiparticle bag is mostly involved with the electronic degrees of freedom whose wave vectors are located near the points where A ^ k ) vanishes at the Fermi surface. For A2/Al = 1/2, the current distribution generates a magnetic field whose maximum strength is approximately 0.2 G—the maximum field is sensitive to the ratio of A 2 and Aj and can be much higher for a larger ratio. The current distribution reflects fourfold symmetry of the boundstate wave function associated with the quasiparticle bag and produces a nonzero magnetic moment mz. Perhaps even more interesting are the configurations that are obtained when there are many quasiparticles in the system; see Fig. 4. It shows that the smaller order-parameter component has antiphase domain walls into which the quasiparticles have localized (cf. Ref. 12). The induced current, producing a magnetic field with maximum strength ~4.7 G, circulates along the walls. The domain-wall structure in the dxy component leads to a local enhancement of the dxi-yi component. In conclusion, superconductors that break both timereversal symmetry and parity exhibit distinctive properties at boundaries and in the presence of impurities and quasiparticles, offering exciting possibilities for their investigation. In particular, quasiparticles may develop exotic configurations that in some cases resemble those of vortices. The support by the NSF under Grant Nos. DMR-9527035 and DMR-9629987 is gratefully acknowledged.

et al, Phys. Rev. Lett. 72, 3100 (1994). C. H. Choi and P. Muzikar, Phys. Rev. B 39, 9664 (1989), and references therein. 10 M. I. Salkola, A. V. Balatsky, and J. R. Schrieffer, Phys. Rev. B 55, 12 648 (1997); M. I. Salkola, A. V. Balatsky, and D. J. Scalapino, Phys. Rev. Lett. 77, 1841 (1996). 11 B. I. Halperin, J. March-Russel, and F. Wilczek, Phys. Rev. B 40, 8726 (1989). 12 M. I. Salkola and J. R. Schrieffer, Phys. Rev. B 57, 14 433 (1998). 9

502

PHYSICAL REVIEW B

VOLUME 58, NUMBER 10

1 SEPTEMBER 1998-11

Collective excitations in high-temperature superconductors M. I. Salkola and J. R. Schrieffer NHMFL and Department of Physics, Florida State University, Tallahassee, Florida 32310 (Received 3 December 1997) Collective, low-energy excitations in quasi-two-dimensional d-wave superconductors are explored. While the long-range Coulomb interaction shifts the charge-density wave and phase modes up to the plasma energy, the spin-density wave excitation that arises due to a strong local electron-electron repulsion can propagate as a damped collective mode within the superconducting energy gap. These excitations are suggested to be relevant to high-7"c superconductors, near the antiferromagnetic phase boundary, and may lead to distinctive signatures in spectral-density and neutron-scattering data. [S0163-1829(98)51134-4]

An important aspect of superconductivity is concerned with collective modes as they may modify low-energy properties of superconductors.1 In principle, these excitations are either attributed to a symmetry transformation under which the system is invariant or to a spontaneously broken continuous symmetry. One might then expect that, in addition to the usual charge and spin fluctuations, collective modes associated with the phase and the amplitude of the superconducting order parameter are important. Yet, in conventional superconductors with a momentum-independent energy gap, collective modes have essentially no practical significance regarding the low-energy properties of the superconductors mainly because the long-range Coulomb interaction causes the phase mode to appear at the plasma energy.2 The only mode that is not diffusive at long wavelengths is the amplitude mode, but it has no direct coupling to charge or spin degrees of freedom making its observation demanding.3 Moreover, excitonic states that are bound pairs of quasiparticles have been predicted to exist in superconductors.4 Such states appear in angular-momentum channels other than the one in which the Cooper pairing occurs. Theoretically, they should be present once the effective electron-electron interaction has an attractive partial-wave component with a given angular momentum. Nonetheless, there is no experimental evidence for these kind of states possibly because of their small binding energy.5,6 While for most phenomena collective excitations in conventional superconductors can be ignored — for instance, the superconducting energy gap in the electronic spectrum at the Fermi energy is not affected by these modes — there are both theoretical and experimental reasons to expect that high-Tc superconductors may behave differently in this respect. Theoretically, the energy gap has a strong momentum dependence and there is a large local electron-electron repulsion which may qualitatively change the nature of collective excitations and allow new ones to develop that are not related to the broken gauge symmetry. Experimentally, neutron-scattering studies in superconducting La,. 6 - I Nd,Sr J Cu0 4 (Refs. 7 and 8) and YBa2Cu30^ (Refs. 9 and 10) have established the existence of elastic and inelastic peaks in the magnetic structure factor in the vicinity of wave vector (7r/a,7r/a). In addition to commensurate peaks,10 which may be explained in terms of two-particle states,11 the 0163-1829/98/58(10)/5944(4)/$15.00

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experimental data7"9 provide a clear evidence for (diffusive) incommensurate spin-density wave fluctuations in high-7*c superconductors. These excitations can be regarded as a manifestation of fluctuating charge stripes and antiphase spin domains12 that have been shown to furnish a natural explanation for the unusual features observed in angle-resolved photoemission experiments.13 They may also account for a broad "bosonic" feature in the electronic spectral density near the (ir/aft) and (0,TT/O) points observed by angleresolved photoemission spectroscopy.14^16 Indeed, the presence of a collective mode appears to be able to explain qualitatively the data.17 In this paper, we examine under what conditions collective excitations develop in d-wave superconductors and what are their experimental implications, when the quasiparticle picture is appropriate and the generalized Ward's identity is ensured. We focus on the low-doping regime where only few experimental and theoretical studies exist. Our most important observation is that rf-wave superconductors close to an antiferromagnetic phase boundary support a low-energy spin-density wave mode. Its properties are governed by competing antiferromagnetic and superconducting correlations that are driven by an on-site repulsion, favoring local spin polarization, and by a nearest-neighbor attraction, favoring pairing. As a consequence of a superconducting energy gap, this mode can propagate coherently with a reduced damping, unlike its precursor in the normal state where only a diffusive spin-density wave is realized. It can be excited by magnetic processes making it observable, for instance, by inelastic neutron scattering. The mode is "massive" because rotational symmetry in spin space is not broken. Under specific conditions, however, the mass of the mode may vanish and may even become negative signaling an instability of the superconducting ground state against a spontaneous creation of a spin-density wave state. Our work is complementary to excellent studies on magnetic excitations in d-wave superconductors which, for example, consider the Hubbard model,18 utilize nonconserving approximation,19 or primarily examine two-particle, spin-triplet excitations.11 Consider, for example, the Hamiltonian ff=-2

R5944

' , - r ' * U r ' ( r + l 2 "r-r'' I r"r

(1)

1998 The American Physical Society

503

COLLECTIVE EXCITATIONS IN HIGH-TEMPERATURE

PRB 58

where nT=~S.ail/lail>ra is the electron number operator at site r, fr_r. is the tunneling-matrix element between sites r and r \ and vr is an instantaneous electron-electron interaction. In describing superconducting order, it is useful to express the Hamiltonian in the form H = HBCS + HiM, where the BCS and interaction Hamiltonians are #BCS=2 *k(*kT3-Ak?1)1'k

Hi «=2tf2

(2a)

i»(q)(*t+ ,T3*k)(*i.- q ?i*kO

kk'q

+ 2 A,,*!?,*,,

(2b)

Here, Vk=((k-p)Tr T^GCp.co), where G(p,fci) = l/(&)fo-e p r3 + A(lTi) denotes the Green's function of the BCS Hamiltonian. We use the notation in which 2,=Ar- 1 2 p /(do>/2ir) andp = (p,aj). In order to determine whether the system can support collective excitations, consider an effective two-particle interaction f that describes mutual scattering of two quasiparticles. Poles of the effective interaction then yield energies and lifetimes of two-particle collective excitations. In the ladder approximation, the Bethe-Salpeter equation for f may be written formally as

I

• X

which is equivalent to the equation f(k,k';q)

=

f0(k,k';q) + * 2 t0(k,p;q)A(p,q)f(p,k';q),

(3)

p

with k(p,q) = G(p + ql2)®G(p-ql2)T; here, q is the total four-momentum of a quasiparticle pair. The bare interaction

R5945

vertex, denoted as

X

> <

is r0(k,k';q) = v(k-k')T3®r3-v(q)P, with P = (f0 - T ^ e O . Moreover, the outer product, [A0B] a f t =.AyB H , where a = (i7) and b = (kl), is defined so that there is a one-to-one correspondence between the ordered index lists o , i e {1,2,3,4} and (i7),(W)e{(ll),(22),(12),(21)}. On a square lattice, it is convenient to define orthogonal functions r)JM) m terms of which the vertex functions and the electron-electron interaction can be expanded.21 For example, u ( k - k ' ) = U+ V^cosikx-k'Ja+cosiky—k'y)d\ may be written as v(k—k') = 2ava7!a(k)qa(k'), where, for a = 0, va=U is the on-site electron-electron interaction and, for a= 1 , . . . ,4, va= Vj/2 is the nearest-neighbor electronelectron interaction. Below, we will assume that the on-site interaction is repulsive, U>0, and the nearest-neighbor interaction is attractive, V^-CO, as should be appropriate for a phenomenological model describing high-7",. superconductors. Similarly, f(k,k';q) = 2,afifa(l(q)i?a(k)y/1(k') and f0(t,*';9) = 2 . ^ ( 9 ) 7 o ( k ) ^ ( k ' ) , with f<°>(0) = vadapT-i®Ti-v(q)SaO8p0P. Thus, the Bethe-Salpeter equation becomes

f^(?) = fL°i(9) + 2 tfy{q)kyy,(q)tylp{q), y y

(4)

i

where Kap(q)^iXpria(f,)k(p,q)r]^). To simplify the notation, define new matrices such that [T]ap=Tap, etc. Then, it is immediately clear that the effective interaction vertex is

f(4)=[i-f<°>((
(5)

where # = (q,ft + *'0 + ). Because, for q = 0, A(—p,q) — k(p,q), there is no mixing between even and odd parity sectors of tae{q): the subspaces {r/w^vL^} and {£ k ±} } describing scattering in the singlet and triplet channels decouple. In contrast, for q = Q , the subspaces {TJ^ .^j,*'} and {^ir'} a r e decoupled. When the system has particle-hole symmetry at the Fermi energy and f l ~ 0 , a further factorization can be shown to occur; namely, particle-hole and particle-particle channels decouple. This means that, for q = Q , the subspace {f k _> } decouples from {!?k0) ,f k + ) } and, f o r q = 0 , the same occurs for{i^ _ > } and {J7k0),?7k+)}. This is particularly convenient because r^~) determines the dwave gap function, A k =(A 0 /2)77 k _ ) . At zero temperature, the gap equation becomes 1 = — ( V 1 W ) 2 k [ j 7 k ' ) ] 2 / 4 £ k . Here, £ k = V* k +A k is the quasiparticle energy in the superconductor. We find A 0 =4We" 1 W F | V l 1 , where W is the half bandwidth and NF=N(fi) is the density of states at the Fermi energy in the normal state. Collective excitations are described by the poles of the effective two-particle interaction, Eq. (5), and they are conveniently classified by the symmetry properties of the system. The Hamiltonian (2) possesses a number of symmetries which lead to conserved currents. First, gauge symmetry,

504

R5946

M. I. SALKOLA AND J. R. SCHRIEFFER

C° T" °l

PRB 58

v

FIG. 1. The energy ftq of the spin-density wave mode as a function of wave vector q at zero temperature for UIW= 0.485 and V\IW= — 0.608. Here, the superconducting energy gap is A0/W = 0.1 and the chemical potential is chosen so that the density of holes equals 10% (relative to half filling). The excitation energy flq is computed numerically from the exact result, Eq. (7). The shaded regions indicate wave vectors of those modes whose lifetimes are infinite (ft q ,i'k = e'*' r ° 1 i' k , leads to a new mode in the particle-hole sector, which is driven by the on-site Coulomb repulsion. As an example, consider a cylindrical Fermi surface and a wave vector q = Q spanning two given k points with vanishing quasiparticle energies and locating at the opposite sides of the Fermi surface. For U \Vi\, the energy O Q of the collective excitation is

"(Si-* u

-KQ,

(6)

-1

when /3>0. For &0,\v\<W, V0 = (l/A02 k (l/2£ k ) H ^ i l - 1 - The parameter /3=-[A0<JlogW(e)/
where *Q" -2(V,/W0 2 (/?/27r) 2 and /fJj 2 ) ~(Vi/V) 2 (n Q /irA 0 ) 2 . For /3<0, the collective excitation remains massive ( f l Q > 0 ) for all values of V0/U, down to the point where the superconductor becomes unstable. The excitation is damped, because it overlaps with the quasiparticle continuum. For the other values of q, the continuum does not necessarily start from zero but at some finite value d q =min k (£ , k +£' k + q ). It is then possible to have collective excitations with a very long lifetime, when flq< i>q. The exact nature of the collective excitation is deduced by examining its coupling to the spin operator 5 z (q) = 2 k * k - y ^ 0 ) * k + q , where ^ 0 ) = | T 0 . That the mode

FIG. 2. The imaginary part of the spin-correlation function x"(q,CL) as a function of (a) the wave vector q with ft/2A0 = 0.13 and (b) ft and qx with qy=ttla at zero temperature for f//W=0.485 and VJW= -0.608. A small broadening of resonances is obtained by surmising a finite energy resolution of magnitude y/W=3X10" 3 due to quasiparticle lifetime effects. Here, the superconducting energy gap is Ao/W=0.1 and the chemical potential is chosen so that the density of holes equals 10% (relative to half filling). is a spin-density wave becomes evident by computing the spin correlation function, ^ ( q , r ) = (7' T 5 i ,(q,r)5 z ( — q,0)),

Specifically, its temporal Fourier transform is given by the formula

^(«)=,

(7)

where the column vector \yz) is defined as (aa\yz) = 2[^o]i;^ao- Ifl the u m i t fl—>0, our result reduces to the form obtained by nonconserving approximation,19 when the energy spectrum has particle-hole symmetry at the Fermi energy. However, for energies i l s A 0 or in the absence of particle-hole symmetry, one must use the general result, Eq. (7). Note that, by incorporating the mixing of the particlehole and particle-particle degrees of freedom, KQ accounts for, for example, the effect of any two-particle, spin-triplet excitations11 at the momentum Q. Similar calculation shows that no resonance develops for the charge response near q =Q. The most favorable conditions for observing these excitations are most likely found in underdoped high-Tc superconductors close to the antifeiromagnetic phase boundary. In the antiferromagnetic phase, the above collective mode is re-

505

PRB 58

COLLECTIVE EXCITATIONS IN HIGH-TEMPERATURE . . .

placed by the Goldstone mode of the antiferromagnet. Interestingly, for t / s V0, the system is unstable against a spontaneous creation of quasiparticle-quasihole virtual bound states at q = Q . This implies a phase transition to an antiferromagnetic state. The competition between the antiferromagnetic and superconducting orders is driven by the on-site repulsion, favoring local spin polarization, and the nearestneighbor attraction, favoring pairing. To obtain a quantitative understanding of the dispersion relation of the collective mode, we resort to numerical methods. Figure 1 shows the energy of the spin-density wave as a function of wave vector near half filling with hole density equal to 10%. The minimum value of the excitation energy, A q /2A 0 =0.08, is located at the wave vector q ~(0.95v/a,tr/a), implying that the superconductor becomes unstable against a spontaneous creation of an antiferromagnetically ordered state with vertical (horizontal) antiphase domain walls. In contrast to neutron-scattering data8 in superconducting La16_^Ndo4Srj.Cu04 which show that the mean separation / between antiphase domain walls should scale with the hole density x as l—al2x, we find that / is more than by a factor of 2 longer than the experimental one. (In the present approximation, / also depends on A 0 .) The failure to predict correctly the domain-wall periodicity is similar to the problem of describing the static, incommensurate stripe order in the Hartree-Fock approximation23 and may suggest that longer range interactions are important.12 Although at zero temperature exciting vertical spin-density wave fluctuations requires the least amount of energy, the rotation of their orientation relative to the underlying lattice constitutes a relatively soft mode; see Fig. 1. The excitation energy at the saddle point q=0.98Q 0 is about 60% larger than the minimum energy required to excite a vertical mode. The form of the dispersion relation fl q is affected by the

1

Historically, collective excitations had an important role in establishing the gauge-invariant theory of superconductivity; for example, see Ref. 20 below. 2 P. W. Anderson, Phys. Rev. 112, 1900 (1958). 3 P. B. Littlewood and C. M. Varma, Phys. Rev. B 26,4883 (1982). 4 A. Bardasis and J. R. Schrieffer, Phys. Rev. 121, 1050 (1961), and references therein. 5 P. C. Martin, in Superconductivity, edited by R. D. Parks (Dekker, New York, 1969), Vol. I, p. 371. 6 One may ask whether excitonic states of bound pairs of quasiparticles are feasible in d-wave superconductors. While the effective interaction has an attractive partial wave in the (extended) s-wave channel, it is too weak near half filling to support excitons with the same angular quantum number. It is only far away from half filling that these states might appear as virtual bound states because of the proximity to a superconducting state with extended s-wave symmetry. 7 T. E. Mason et al., Phys. Rev. Lett. 77, 1604 (1996). 8 J. M. Tranquada et al, Phys. Rev. Lett. 78, 338 (1997). 9 P. Dai, H. A. Mook, and F. Dogan, Phys. Rev. Lett. 80, 1738 (1998). 10 H. F. Fong et at., Phys. Rev. Lett. 78, 713 (1997). "E. Demler and S. C. Zhang, Phys. Rev. Lett. 75, 4126 (1995); E. Demler et al. (unpublished).

R5947

lifetime effects: the troughs in Fig. 1 coincide qualitatively with the boundaries between the regions of damped (ft q > # q ) and undamped ( n q < # q ) modes. Note that the mode with q = Q 0 and 0.94Q0 is undamped whereas the one with q=0.97Qo is damped. Figure 2 illustrates the behavior of the imaginary part of the spin correlation function, Eq. (7). The collective excitation produces a distinctive resonance structure in ^"(q,fl) at energies specified by the dispersion relation, £l = flq. At low energies, the resonance becomes narrower as the excitation energy flq decreases. However, in the vicinity of the (ir/a,Tr/a) point, for example, the collective excitation appears below the onset energy of the two-particle continuum # q . Accordingly, the collective excitation would acquire an infinite lifetime yielding a resolution-limited peak in ;f"(q,ft) — also, below this onset energy, the usual quasiparticle contribution to x"(q,ft) would vanish — if the lifetime of quasiparticles were infinite. Clearly the collective mode is an important feature describing the low-energy spincorrelation function.24 In conclusion, we have shown that d-wave superconductors can support propagating collective modes that are best described as spin-density waves. It is then natural to anticipate that the most favorable conditions for detecting them are found close to the antiferromagnetic phase boundary. Furthermore, superconducting fluctuations couple particle-hole and particle-particle excitations allowing the latter ones to be probed by the usual means. This coupling is particularly important if the spin-density wave and two-particle excitations are (nearly) degenerate. We would like to thank D. Scalapino, S. Trugman, E. Demler, A. Fetter, S. Rabello, and S. C. Zhang for useful discussions. This work was supported by the NSF under Grant Nos. DMR-9527035 and DMR-9629987.

12

V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993). M. I. Salkola, V. J. Emery, and S. A. Kivelson, Phys. Rev. Lett. 77, 155 (1996). 14 Z.-X. Shen and D. S. Dessau, Phys. Rep. 253, 1 (1995). 15 Z.-X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997). 16 M. R. Norman et al, Phys. Rev. Lett. 79, 3506 (1997). 17 M. R. Norman and H. Ding (unpublished). I8 C.-H. Pao and N. E. Bickers, Phys. Rev. B 51, 16 310 (1995). 19 N. Bulut and D. J. Scalapino, Phys. Rev. B 53, 5149 (1996). 20 I. R. Schrieffer, Theory of Superconductivity (BenjaminCummings, Reading, 1983). 21 They are defined as Vo(k)mV*)=lJ?i(k)=7k+) = c o s * i a +cos£/i, jj2W=rj\l~'> = cosk1fl-coskya, %(k) = £k+) = sink:fl+smkya, ^(kjsfj^^sinfc^-sinjyi, etc. 22 Y. Nambu, Phys. Rev. 117, 648 (1960). 23 H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990). 24 Further contact with experiments (Ref. 8) can be made by noting that impurity moments pin the spin-density-wave mode leading to a frozen spin order. The resulting incommensurate, elastic peaks in magnetic scattering are expected to disappear at a temperature Tp, which scales with the binding energy and therefore decreases as fi Q increases. Thus, Tp decreases with the increasing hole density. 13

506 VOLUME 80, NUMBER 15

PHYSICAL

REVIEW

LETTERS

13 APRIL 1998

de H a a s - v a n Alphen Effect in Anisotropic Superconductors i n M a g n e t i c Fields Well B e l o w Hci L. P. Gor'kov1-2 and J. R. Schrieffer1 l

NationaI High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306 L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences. 117334 Moscow, Russia (Received 21 October 1997)

2

We develop a quasiclassical approach to the energy spectrum of an anisotropic superconductor in a magnetic field, B, such that Hci « : B « Hc2. Low temperature de Haas-van Alphen oscillations are considered for two cases: (1) the extremal electron orbit may coincide with a symmetry line and (2) the orbit crosses points where the superconducting order parameter has zeros. The signal is shown to be small in both cases. [S0031 -9007(98)05873-6] PACS numbers: 74.20.Fg, 72.15.Gd, 74.60.-w

The de Haas-van Alphen (dHvA) effect in superconductors continues to attract theoretical and experimental efforts since its observation in Nb2Se at fields B below the critical field Hc2 [1], The effect was later seen in V3SL NbsGe, and borocarbides, for which Hc2(T = 0) is high enough to satisfy the criterion COCT ~» 1 (see [2] for recent review). Theoretical efforts [3-9] have mostly been concentrated on the field range Hc2 — B «. Hc2. Nonzero dHvA effect is considered to be due to a finite normal density of states at the Fermi level, which survives below Hc2. The normal dHvA effect is a quasiclassical feature: the total number of states occupied by electrons in the magnetic field being large, N — -^ » 1, a minor field variation -g—- ^f leads to crossing the chemical potential by a single level and changing the magnetization in a steplike manner. We investigate the possibility of a low temperature dHvA for a superconductor in the field regime Hc\ «. B « : Hc2, where distance between vortices d is large enough to separately treat the "bulk" of the superconductor, where the gap value, |A(r)|, is a constant, and the vortex cores, occupying only a fraction, (£o/d)2
O031-9007/98/80(15)/3360(4)$15.00

We investigated the problem for two situations mentioned above. Our conclusion in the first case is that the dHvA in the regime, Hcy < B < Hc2, is suppressed by the electron's scattering on periodic superconducting currents in the vortex state, leading to Dingle's temperature of order of A(£o/d). The excitation spectrum in the second situation possesses a topologic property which excludes level crossing the chemical potential. We developed below the approach which allows a general treatment of a superconductor in the mixed state. In the chosen field range the spatial dependence of the gap is of the form A(r) = Ae'* ( r ) .

(1)

Superconducting currents outside the vortex cores are

j(r) = - ^ ( A - £ v ^ - ^ - Q . (2) The topological singularity of the phase at each point, r,, where the gap passes through zero, results in the following equations [12]: curlQ = B - 0o X S(r ~ r>)• ' B - 5£AB = tf>02_.S(r ~ r,).

(3)

i

For the square lattice [r, = d(n,m)],

«->-$? *ff7T

(»->»)•<«>

From (2) and (4) it follows that B = 0d-\

|Q| ~
Afi ~ B(d/SLf,

(5)

where fi = fi is the spatial field average, and Afi the weak deviation of B from B. For simplicity we assume two dimensionality of the Fermi surface. Quite generally, the oscillatory part of the magnetization M is contained in the expression M = --^Spur(r "

V

Y. z=i(2n

Gii(z)J.

+ l)irT

© 1998 The American Physical Society

/

(6)

507 PHYSICAL

VOLUME 80, NUMBER 15

REVIEW

Spur(' • •) means trace over all states (including spin variables) of the component Gu(z) of the matrix G{z)

+

Qiz)

J(z) 1 -Q(z)\

~\T (z)

-px.

h,H^^{pi

(8)

2

(9)

pi}

SPxQx + PyQy), (13)

^-i^f^) A(r) = A 0 (r) + Q(r)

(14)

The bare Hamiltonian, jm^Pi + Py)' n a s eigenvalues En = eoc(N + 5). with real wave functions, which may be used for calculating Spurf • •} in Eq. (6). The terms linear in Q in (13), and the gap terms in (8) both result in nondiagonal transitions between free-electron eigenstates for the matrix Q(Z)N,N

of the form

X A;y,jv'.7>ft,

2^ AJVJV/ QN'N ,

N<

A"

(15)

(16)

The only nonzero matrix elements of operator py are

and use below the Landau gauge A*o = "By. [A different choice of Ao(r) would redefine Q(r) without affecting its topological singularities in (3).] Expanding in (9)

(Pyhji-i

= (Pyh-ij/

= \JNeB/2c.

(17)

One immediately obtains [A (rf) (p)k,v ± 2 = (l/2)Ao.

(11)

(18)

[The diagonal elements are zero, as expected from the broken gap symmetry in (16). We have previously used the fact that at the Fermi surface N = pF/2ma>c.] From (17) and (18) it is straightforward to conclude that the matrix elements, Ajyjy. i n (15), for arbitrary gap anisotropy are of the form

to first order in Q, comparing the linear term with the order of magnitude of the gap in (8), it follows from (5) (d » f 0 ) •

-iB(d/dpy).

A(rf)(p) = A 0 cos2,p - A0(l - 2p2y/p2F).

(10)

-Q(r)

=

if A(p) is anisotropic. In the weak coupling limit the gap dependence on distance to the Fermi surface may be neglected. Only the angular dependence of the gap, A(p), on position of the electron momenta along the Fermi surface is relevant. For instance, for d-wave gap, A(rf' a px — p^, one has

Equations (3) and (4) properly determine the magnetic field outside the cores and the part of the vector A(r) [or Q(r)] responsible for the periodic variations, leaving unspecified the potential Ao(r) which corresponds to the average field, B = curlA0(r). We write for A(r) in (9)

~ vF/d « A;

+

A0x

In (8) the spatial gap dependence (1), of the phase (r), is removed by an appropriate gauge transformation. On the other hand, the gap itself, A = A(p), may be anisotropic and depend on the position along the Fermi surface. For simplicity we consider a spherical Fermi surface. Two Hamiltonians H and H in Eq. (8) act on operators Iji and
(evF/c)Q\

After px is excluded, we have

where px, py are the kinetic momenta operators, [px,Py] = ie/cB. Here and below all calculations were done in the momentum representation. Correspondingly,

A G(z) = 1. z ~ (H ~ n)

# , # = ^ - f p ± - A 0 ( r ) ) ± -yF 2m V c / c

13 APRIL 1998

(7)

which satisfies the system of the Gor'kov equations, ' z - (H - fi) -A

LETTERS

(12)

To start with, we omit the Q term in (11). In the Aw.v» = Aw-. N-N' Landau gauge px is conserved. This is also true for Eqs. (8), for the gap couples together momenta px and i Equation (8) then becomes [z ~ o)c(N + 5) + M I ^ ' . O ;

[-z - coc(N

!\

lc

IGN-N',N(Z)

— (1)N,N

(19)

•

(80

j ) + MJOJV'.O/

Introducing the notation for the chemical potential /j, = aicNf, + p.,

(20)

where No is the integer part of p./u>c, one may rewrite (8') for any function / participating in (8'), in the new representation

/(*) = ^V'-V.; with n = N

fn

^hl-S9f{
(21)

N0 in (8'). As a result, one arrives at the following system of equations: -A((p)

-z - itocl^

+ p.

G(z;
(22) 3361

508 PHYSICAL REVIEW

VOLUME 80, NUMBER 15

with the periodicity, f(


13 APRIL

1998

substitution E => E + h(
• Q(r).

(31)

The dependence on


Ex\vA(
9{u\(v>'),vW)),

LETTERS

(23)

f —» ro + (-77. cos o)ct, ri sin coct),

(32)

where the BCS notations are adopted for the normalized eigenfunction [ux(
and \F = r,

cjj + £)"(
e-'W^fav),

(26')

- iz);

v = l/2(z-iy),

z = - y [ ( » c / + A(?)y],

(27)

n = 0,1,....

(29)

It is clear that (29) is not applicable at too low energy En because the energy scale (£ucAo)1/2 is small, if compared to the £ terms in Eq. (11): (30) 1. {d/vF){a>cAo)U1 ~ (PFto)-1'2 To incorporate the Q terms into the scheme of Eqs. (22) and (24) we apply the quasiclassical approach developed in [13]. Omitting details of the derivation, the resulting equation can be obtained from (22) and (24) by the 3362

^[\ui\MEx)]dX (33)

Correspondingly, the two terms arising from differentiation of the expression in the bold parentheses, present oscillations due to a level crossing the chemical potential (at T = 0), lirikX'

(28)

which describes global features for motion of a quasiparticle along the Fermi surface in the magnetic field B. In particular, at \E\ < AQ in (16) excitations are confined in the four "potential wells" formed by A2(cp) around the points
kX

-00

(26)

in terms of which the problem (25) reduces to the Schrodinger equation, E2y = -w2cy" + [A 2 (?) - a>cA'(
2

e"

/

and come over to the functions (y, z), ii = l/2(y

'•<£).

+OO

/:

ux\2S(Ei)d\,

(33')

or a singularity due to a specific structure of the excitations spectrum in the superconducting state, +

°°e2"-kX^(\utf)n(Ei)dX.

/ . -00

(33")

a A

Transition from summation over A to integration in (33) poses the problem of expressing the energy Ex as a function of the continuous variable A. To implement this approach to the two situations discussed, consider first a symmetry line of zeros, and the field direction such that the extremal electron orbit would lie on it. In (22) and (24) the gap equals zero identically, so the electron (hole) wave functions are of the form exp[±iS(p)];

S(
f*[E

+ h(
Jo

(34)

509 PHYSICAL

VOLUME 80, NUMBER 15

REVIEW

The eigenvalue problem follows from the periodicity of (34): 2TTA = S(2TT) = — f Oic Jo

[Ex + h(
(34')

Comparison with (33) and (33') leads to the relation between the k components of Mosc in the superconducting and normal states, 2w

Ms0Jk) = M?Jk)Uxp JL [


h(ip')d
(35)

The average in (35) is taken over electronic orbits. By making use of (3), (31), and (32), the exponent may be presented as the integral over the orbit area — [ (»c Jo

h(0) cJ

fj\B-4,0Y.S(T-r,)

dS. (36)

Here, the factor in exponent (35) measures fluctuations in number of fluxes encircled by electronic orbits nv. For a periodic vortex lattice we evaluate (36) in terms of the Gaussian distribution for nv with the average, n^, as TV = (2irrLd)/d2

= 2ir(rL/d).

(37)

For the Dingle factor, expf-2ircm/erB\ we get 1/r ~ Ao(f 0 /d).

(38)

Note that (37) is but an estimate. In any case, the dHvA signal will be very weak: (TC)~1 ~ pfd » 1. Turn now to the other case when the orbits of normal electrons would cross points at the Fermi surface where the gap changes sign and passes through zero, as in Eq. (16). Although (29) is not applicable at small n, its structure indicates that the dHvA contribution (33') is to be zero identically. In fact, the dHvA effect consists in pushing up an energy level through the chemical potential at small field variation. Equation (25) is qualitatively different from the spectrum of normal electrons in that, while in the latter case levels are pushed up from the bottom of the band, d-wave superconductivity fixes levels structure in the vicinity of the chemical potential. The (+) and ( - ) branches in (29) are moving up and down, respectively, with the field increase. The existence of levels in the vicinity of E = 0 is a topological property for a solitonlike problem [14] at each point
(39)

Equation (27) becomes the familiar Schrodinger equation with the S-function potential. There is only one bound state (for each
LETTERS

13 APRIL 1998

function exponentially decays at \

A 2 . Their dependence on magnetic field B shows up through dependence of \h\ ~ vp/d ~ B1?2, i.e., is the same as in (29). Averaging over trajectories smears each level into a band, \\E\ ~ vF/d. Recall now that one level has the capacity (25). In the 2D case it produces the density of states of order [15] 1/2 C Vp

BF

\HC2

To summarize, we developed a rigorous approach to quantization of motion of superconducting excitations in the presence of magnetic field B «. HC2. The dHvA effect is formulated in terms of a singularity produced either by the Fermi function or by a modified structure of levels in the superconducting state. For an order parameter with a line of zeros there is a field direction at which dHvA oscillations may become observable. In case of points of zeros on a 2D Fermi surface the effect is expected to be weaker. Analysis of dHvA effects produced by singularities in the energy spectrum of the superconducting state will be published elsewhere. This work was supported by the NHMFL through NSF cooperative agreement No. DMR-9016241 and the State of Florida, and No. DMR 9629987 (J. R. S.).

[1] J.E. Graebner and M. Robbins, Phys. Rev. Lett. 36, 422 (1976). [2] R. Corcoran, N. Harrison, C.J. Haworth, S.M. Hayden, P. Meeson, M. Springford, and P. J. Van der Wei, Physica (Amsterdam) 206-207B, 534 (1995). [3] K. Maki, Phys. Rev. B 44, 2861 (1991). [4] M.J. Stephen, Phys. Rev. B 45, 5481 (1992). [5] T. Maniv, A.I. Rom, I.D. Vagner, and P. Wyder, Phys. Rev. B 46, 8360 (1992). [6] S. Dukan and Z. Tesanovic, Phys. Rev. Lett. 74, 2311 (1995). [7] P. Miller and B. L. Gyorffy, J. Phys. Condens. Matter 7, 5579 (1993). [8] M.R. Norman and A.H. MacDonald, Phys. Rev. B 54, 4239 (1996). [9] M. G. Vavilov and V. P. Mineev, Sov. Phys. JETP (to be published). [10] K. Miyake, Physica (Amsterdam) 186B, 115 (1993); [11] G.E. Volovik, JETP Lett. 58, 469 (1993). [12] A. A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957). [13] E. A. Shapoval, Sov. Phys. JETP 20, 675 (1965); see also L. P. Gor'kov and G. M. Eliashberg, Sov. Phys. JETP 21, 940 (1965). [14] T.L. Ho, J.R. Fulco, J.R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 52, 1524 (1984). [15] The factor A / E F <sc 1 is the artifact of (39), to disappear for the smooth gap (16), as in Ref. [11]. 3363

***V

r receive bachelor's degree from the Massachusetts Institute of Technology in 1953 and his Ph.D. from the University of Illinois in 1957. In addition, he holds honorary D.Sc. degrees from universities in Germany, Switzerland, and Israel, as well as from the University of Pennsylvania, the University of Cincinnati, and the ersity of Alabama.

Since 1992, Dr Schrieffer has been a professor of physics at Florida State : "irsity and the University of Florida, rt from being the Chief Scientist of the National High Magnetic Field Laboratory. He also holds the FSU Eminent Scholar Chair in Physics, ore moving to Florida in 1991, he lical Physics from 1984 to 1989 and _.ity of California in Santa Barbara from 1984 to 1991. He was Mary Amanda Wood Professor of Physics at the University of Pennsylvania from 1964 to 1979. In 1968. Dr Schrieffer received the Oliver E. Buckley Solid State Physics Prize and the Comstock Prize of the National Academy of Sciences. In 1972, he won the Nobel Prize in Physics, jointly with John Bardeen and Leon Cooper, for the microscopic theory of superconductivity. In 1984, he received the National Medal of Science. Dr Schrieffer served as President of the American Physical Society in 1996, and he is a member of the academies of sciences of the United States. Denmark, Russia, and Korea. Presently Dr Schrieffer's research is focused on the theory of high temperature superconductivity and magnetism in condensed matter systems.

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