Theory Of Superconductivity, A Primer

THEORY OF SUPERCONDUCTIVITY A Primer by

Helmut Es hrig

Dire tor of the Institute for Solid State and Materials Resear h and

Professor for Solid State Physi s, Dresden University of Te hnology

These le ture notes are dedi ated, on the o

assion of his eighties birthday on 4 July 2001, to Musik Kaganov who has done me the honor and delight of his friendship over de ades and whom I owe a great part of an attitude towards Theoreti al Physi s.

2

Contents

1 INTRODUCTION

4

2 PHENOMENA, LONDON THEORY

5

2.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Gauge symmetry, London gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 THE THERMODYNAMICS OF THE PHASE TRANSITION

3.1 3.2 3.3 3.4

The Free Energy . . . . . . . . . The Free Enthalpy . . . . . . . . The thermodynami riti al eld Heat apa ity jump . . . . . . . .

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4 THE GINSBURG-LANDAU THEORY; TYPES OF SUPERCONDUCTORS

4.1 4.2 4.3 4.4 4.5

The Landau theory . . . . . . . . . . The Ginsburg-Landau equations . . The Ginsburg-Landau parameter . . The phase boundary . . . . . . . . . The energy of the phase boundary .

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13

13 14 15 16

17

17 18 20 21 23

5 INTERMEDIATE STATE, MIXED STATE

25

6 JOSEPHSON EFFECTS

32

5.1 The intermediate state of a type I super ondu tor . . . . . . . . . . . . . . . . . . . . 25 5.2 Mixed state of a type II super ondu tor . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 The ux line in a type II super ondu tor . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.1 The d. . Josephson ee t, quantum interferen e . . . . . . . . . . . . . . . . . . . . . . 33 6.2 The a. . Josephson ee t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 MICROSCOPIC THEORY: THE FOCK SPACE

7.1 7.2 7.3 7.4

Slater determinants . . . . . . . . . . The Fo k spa e . . . . . . . . . . . . O

upation number representation . Field operators . . . . . . . . . . . .

.. .. .. ..

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8 MICROSCOPIC THEORY: THE BCS MODEL

8.1 8.2 8.3 8.4

The normal Fermi liquid as a quasi-parti le gas . The Cooper problem . . . . . . . . . . . . . . . . The BCS Hamiltonian . . . . . . . . . . . . . . . The Bogoliubov-Valatin transformation . . . . .

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9 MICROSCOPIC THEORY: PAIR STATES

37

37 38 39 40

42

42 44 46 47

51

9.1 The BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 The pair fun tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9.3 Non-zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10 MICROSCOPIC THEORY: COHERENCE FACTORS

10.1 10.2 10.3 10.4

The thermodynami state . . . . . . . . The harge and spin moment densities . Ultrasoni attenuation . . . . . . . . . . The spin sus eptibility . . . . . . . . . .

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3

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55 55 56 57

1

INTRODUCTION

These le ture notes introdu e into the phenomenologi al and qualitative theory of super ondu tivity. Nowhere any spe i assumption on the mi ros opi me hanism of super ondu tivity is made although on a few o

asions ele tron-phonon intera tion is mentioned as an example. The theoreti al presuppositions are ex lusively guided by phenomena and kept to a minimum in order to arrive at results in a reasonably simple manner. At present there are indi ations of non-phonon me hanisms of super ondu tivity, yet there is no hard proof up to now. The whole of this treatise would apply to any me hanism, possibly with indi ated modi ations, for instan e a symmetry of the order parameter dierent from isotropy whi h has been hosen for the sake of simpli ity. This is a primer. For ea h onsidered phenomenon, only the simplest ase is treated. Referen es are given basi ally to the most important seminal original papers. Despite the above mentioned stri t phenomenologi al approa h the te hni al presentation is standard throughout, so that it readily

ompares to the existing literature.1 More advan ed theoreti al tools as eld quantization and the quasi-parti le on ept are introdu ed to the needed level before they are used. Basi notions of Quantum Theory and of Thermodynami s (as well as of Statisti al Physi s in a few o

asions) are presupposed as known. In Chapter 2, after a short enumeration of the essential phenomena of super ondu tivity, the London theory is derived from the sole assumption that the super urrent as an ele tri al urrent is a property of the quantum ground state. Thermoele tri s, ele trodynami s and gauge properties are dis ussed. With the help of simple thermodynami relations, the ondensation energy, the thermodynami

riti al eld and the spe i heat are onsidered in Chapter 3. In Chapter 4, the Ginsburg-Landau theory is introdu ed for spatially inhomogeneous situations, leading to the lassi ation of all super ondu tors into types I and II. The simplest phase diagram of an isotropi type II super ondu tor is obtained in Chapter 5. The Josephson ee ts are qualitatively onsidered on the basis of the Ginsburg-Landau theory in Chapter 6. Both, d. . and a. . ee ts are treated. The remaining four hapters are devoted to the simplest phenomenologi al weak oupling theory of super ondu tivity on a mi ros opi level, the BCS theory whi h gave the rst quantum theoreti al understanding of super ondu tivity, 46 years after the experimental dis overy of the phenomenon. For this purpose, in Chapter 7 the Fo k spa e and the on ept of eld quantization is introdu ed. Then, in Chapter 8, the Cooper theorem and the BCS model are treated with o

upation number operators of quasi-parti le states whi h latter are introdu ed as a working approximation in Solid State Physi s. The nature of the harged bosoni ondensate, phenomenologi ally introdu ed in Chapter 2, is derived in Chapter 9 as the ondensate of Cooper pairs. The ex itation gap as a fun tion of temperature is here the essential result. The treatise is losed with a onsideration of basi examples of the important notion of oheren e fa tors. By spe ifying more details as lower point symmetry, real stru ture features of the solid (for instan e

ausing pinning of vortex lines) and many more, a lot of additional theoreti al onsiderations would be possible without spe ifying the mi ros opi me hanism of the attra tive intera tion leading to super ondu tivity. However, these are just the notes of a one-term two-hours le ture to introdu e into the spirit of this kind of theoreti al approa h, not only addressing theorists. In our days of lively spe ulations on possible auses of super ondu tivity it should provide the new ommer to the eld (again not just the theorist) with a safe ground to start out.

1 Two lassi s are re ommended for more details: J. R. S hrieer, Theory of Super ondu tivity, Benjamin, New York, 1964, and R. D. Parks (ed.), Super ondu tivity, vol. I and II, Dekker, New York, 1969.

4

Figure 1: Resistan e in ohms of a spe imen of mer ury versus absolute temperature. This plot by Kamerlingh Onnes marked the dis overy of super ondu tivity. (Taken from: Ch. Kittel, Introdu tion to Solid State Physi s, Wiley, New York, 1986, Chap. 12.) 2

PHENOMENA, LONDON THEORY

Helium was rst lique ed by Kammerling Onnes at Leiden in 1908. By exhausting the helium vapor above the liquid the temperature ould soon be lowered down to 1.5K. Shortly afterwards, in the year 1911, it was found in the same laboratory1 that in pure mer ury the ele tri al resistan e disappeared abruptly below a riti al temperature, T = 4.2K. Deliberately in reasing ele tron s attering by making the mer ury impure did not ae t the phenomenon. Shortly thereafter, the same ee t was found in indium (3.4K), tin (3.72K) and in lead (7.19K). In 1930, super ondu tivity was found in niobium (T = 9.2K) and in 1940 in the metalli

ompound NbN (T = 17.3K), and this remained the highest T until the 50's, when super ondu tivity in the A15 ompounds was found and higher T -values appeared up to T = 23.2K in Nb3Ge, in 1973. These materials were all normal metals and more or less good ondu tors. In 1964, Marvin L. Cohen made theoreti al predi tions of T -values as high as 0.1K for ertain doped semi ondu tors, and in the same year and the following years, super ondu tivity was found in GeTe, SnTe (T  0.1K, ne  1021 m 3 ) and in SrTiO3 (T = 0.38K at ne  1021 m 3, T  0.1K at ne  1018 m 3). In the early 80's, super ondu tivity was found in several ondu ting polymers as well as in \heavy fermion systems" like UBe13 (T  1K in both ases). The year 2000 Nobel pri e in Chemistry was dedi ated tho the predi tion and realization of ondu ting polymers (syntheti metals) in the late 70's. 1 H. K. Onnes, Commun. Phys. Lab. Univ. Leiden, No124 (1911); H. K. Onnes, Akad. van Wetens happen (Amsterdam) 14, 818 (1911).

5

September, 1993 (under pressure)

160 Liquid CF4

Hg−1223

140 Tl−2223

[

[

August, 1993 (under pressure) April, 1993 September, 1992 February, 1988

Tc(K)

120 100

Shuttle

Bi−2223

January, 1988

Y−123

January, 1987

Liquid N2

80 60

(La−Ba)−214

40

[

December, 1986 (under pressure) December, 1986 (under pressure) April, 1986

Liquid H2

20

Liquid He

1900

NbC

Pb Hg

Nb3Ge

Nb3Sn

NbN V3Si

Nb−Al−Ge

Nb

1920

1940

1960

1980

2000

YEAR

Figure 2: The evolution of T with time (from C. W. Chu, Super ondu tivity Above 90K and Beyond in: B. Batlogg, C. W. Chu, W. K. Chu D. U. Gubser and K. A. Muller (eds.) Pro . HTS Workshop on Physi s, Materials and Appli ations, World S ienti , Singapore, 1996.). In 1986, Georg Bednorz and Alex Muller found super ondu tivity in (La,Sr)2CuO4 with T = 36K, an in redible new re ord.1 Within months, T -values in uprates were shooting up, and the re ord is now at T  135K. 2.1 Phenomena

Zero resistan e2

No resistan e is dete table even for high s attering rates of ondu tion ele trons. Persistent urrents magneti ally indu ed in a oil of Nb0:75 Zr0:25 and wat hed with NMR yielded an estimate of the de ay time greater than 105 years! (From theoreti al estimates the de ay time may 10 10 be as large as 10 years!) Absen e of thermoele tri ee ts3 No Seebe k voltage, no Peltier heat, no Thomson heat is dete table (see next se tion). Ideal diamagnetism m = 1. Weak magneti elds are ompletely s reened away from the bulk of a super ondu tor. Meissner ee t4 If a super ondu tor is ooled down in the presen e of a weak magneti eld, below T the eld is ompletely expelled from the bulk of the super ondu tor. Flux quantization5 The magneti ux through a super ondu ting ring is quantized and onstant in time. This phenomenon was theoreti ally predi ted by F. London in 1950 and experimentally veri ed 1961. 1 J. G. Bednorz and K. A. M uller, Z. Phys. B64, 189 (1986). 2 J. File and R. G. Mills, Phys. Rev. Lett. 10, 93 (1963). 3 W. Meissner, Z. Ges. K alteindustrie 34, 197 (1927). 4 W. Meissner and R. O hsenfeld, Naturwiss. 21, 787 (1933). 5 B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961);

51 (1961).

6

R. Doll and M. Nabauer, Phys. Rev. Lett. 7,

2.2 London theory1

Phenomena (a) and (b) learly indi ate that the super urrent (at T = 0) is a property of the quantum ground state: There must be an ele tri ally harged ( harge quantum q), hen e omplex bosoni eld whi h ondenses in the ground state into a ma ros opi amplitude: (1) nB = j j2 ; where nB means the bosoni density, and is the orresponding eld amplitude. Sin e the eld is ele tri ally harged, it is subje t to ele tromagneti elds (E ; B) whi h are usually des ribed by potentials (U; A):  A U ; (2a) E= t  r B=

  A: r

(2b)

The eld amplitude should obey a S hrodinger equation 1  ~  qA2 + qU = E B
; (3) 2mB i  r where the energy is reasonably measured from the hemi al potential B of the boson eld, sin e what is measured in a voltmeter is rather the ele tro hemi al potential  = B + qU (4) than the external potential U , or the ee tive ele tri eld  A 1  : (5) Ee = t

q r

As usual in Quantum Me hani s, i~= r is the anoni al momentum and ( the me hani al momentum. The super urrent density is then 



i~= r qA) = p^m is

2

= 2iqm~  r r  mq  A: (6) B B It onsists as usual of a `paramagneti urrent' ( rst term) and a `diamagneti urrent' (se ond term).2 In a homogeneous super ondu tor, where nB = onst., we may write (7) (r; t) = pnB ei(r;t); and have js = ~q r A;  = nmBq2 : (8) B Sin e in the ground state E = , and E = i~ =t, we also have js = q

pm n mB B

=

q  p^m
< mB

 t

= : (9) The London theory derives from (8) and (9). It is valid in the London limit, where nB = onst. in spa e an be assumed. ~

1 F. London and H. London, Pro . Roy. So . A149, 71 (1935); F. London, Pro . Roy. So . A152, 24 (1935); F. London, Super uids, Wiley, London, 1950. 2 These are formal names: sin e the splitting into the two urrent ontributions depends on the gauge, it has no deeper physi al meaning. Physi ally, paramagneti means a positive response on an external magneti eld (enhan ing the eld inside the material) and diamagneti means a negative response.

7

The time derivative of (8) yields with (9)
 js = or

A t

t

 js t




1  ; q r

= Ee

(10)

This is the rst London equation: A super urrent is freely a

elerated by an applied voltage, or, in a bulk super ondu tor with no super urrent or with a stationary super urrent there is no ee tive ele tri eld ( onstant ele tro hemi al potential). The rst London equation yields the absen e of thermoele tri ee ts, if the ele tro hemi al potentials of ondu tion ele trons, el, and of the super urrent, , are oupled. The thermoele tri ee ts are sket hy illustrated in Fig. 3. The rst London equation auses the ele tro hemi al potential of the super urrent arrying eld to be onstant in every stationary situation. If the super urrent arrying eld rea ts with the ondu tion ele tron eld with n ele trons forming a eld quantum with

harge q, then the ele tro hemi al potentials must be related as nel = . Hen e the ele tro hemi al potential of the ondu tion ele trons must also be onstant: no thermopower (Seebe k voltage) may develop in a super ondu tor. The thermoele tri urrent owing due to the temperature dieren e is an eled by a ba k owing super urrent, with a ontinuous transformation of ondu tion ele trons into super urrent density at the one end of the sample and a ba k transformation at the other end. If a loop of two dierent normal ondu tors is formed with the jun tions kept at dierent temperatures, then a thermoele tri urrent develops together with a dieren e of the ele tro hemi al potentials of the two jun tions, and several forms of heat are produ ed, everything depending on the

ombination of the two metals. If there is no temperature dieren e at the beginning, but a urrent is maintained in the ring (by inserting a power supply into one of the metal halfs), then a temperature dieren e between the jun tions will develop. This is how a Peltier ooler works. In a loop of two super ondu tors non of those phenomena an appear sin e a dieren e of ele tro hemi al potentials

annot be maintained. Every normal urrent is lo ally short- ir uited by super urrents. If, however, a normal metal A is ombined with a super ondu tor B in a loop, a thermoele tri

urrent will ow in the normal half without developing an ele tro hemi al potential dieren e of the jun tions be ause of the presen e of the super ondu tor on the other side. This yields a dire t absolute measurement of the thermoele tri oeÆ ients of a single material A. The url of Eq. (8) yields (with r  r = 0)     js = B : r

(11)

This is the se ond London equation. It yields the ideal diamagnetism, the Meissner ee t, and the

ux quantization. Take the url of Maxwell's equation (Ampere's law) and onsider r  r  B
= r Br
r22 B:
  B = 0 js + j ; r
    B r r 2 B  r2 2 B  r2

= = = 8

B r

= 0;


  js + j ; r
  0  js + j ; r 0   B 0  r  j :

0

(12)

normal ondu tor

super ondu tor

 T1 1;el

<

T1

T2

normal ondu tor

n1;el = 

2;el

ne ! q

Seebe k voltage

2  1  T 2 T 1 fT1 ( 1;el )

super urrent

due to the rst London equation.

loop of two metals A and B: Ohm's heat + Thomson heat A (
T; J )

no total urrent no heat A

A T1

<

T2

B

super ondu tor

T1

Peltier heat AB (T2 ; J )

Ohm's heat + Thomson heat B (
T; J ) normal ondu tor

n2;el = 

Sin e j (t) = onst.

The dieren e of the Fermi distribution fun tions fT in onne tion with a non- onstant density of states results in a dieren e of ele tro hemi al potentials  due to the detailed balan e of urrents.

Peltier heat BA(T1; J )

T2

j

) js (t) = onst. ) Ee =
=q = 0;
 = 0

fT2 ( 2;el )

thermoele tri

urrent J

<

super ondu tor thermoele tri urrent j 
T

<

B

T2

B

A T1

<

J T2 Js

Figure 3: Thermoele tri phenomena in normal ondu tors and super ondu tors. 9

q ! ne

external eld B0

js

B

e

z=L

z L

1111 0000

y x

super ondu tor

z

Figure 4: Penetration of an external magneti eld into a super ondu tor. If j = 0 or r  j = 0 for the normal urrent inside the super ondu tor, then with solutions

2 B B = 2 ; L = 2 r L

s

 = r mB  n  q2 0

B

0

n2 = 1; n
B0 = 0

(13)

(14) several of whi h with appropriate unit ve tors n may be superimposed to ful ll boundary onditions. L is London's penetration depth. Any external eld B is s reened to zero inside a bulk super ondu ting state within a surfa e layer of thi kness L. It is important that (11) does not ontain time derivatives of the eld but the eld B itself: If a metal in an applied eld B0 is ooled down below T , the eld is expelled. Consider a super ondu ting ring with magneti ux  passing through it (Fig. 5). Be ause of B (14) and (12), js = 0 deep inside the ring on the ontour C . Hen e, from (10), Ee = E = 0 there. From Faraday's law, (= r)E = B, Z I d d = B dS = E dl = 0; dt dt A C (15) C where A is a surfa e with boundd   ary C , and  is the magneti ux through A. Figure 5: Flux through a super ondu ting ring. Even if the super urrent in a surfa e layer of the ring is hanging with time (for instan e, if an applied magneti eld is hanging with time), the ux  is not: B = B0 e

n
r=L ;

The ux through a super ondu ting ring is trapped.

Integrate Eq. (8) along the ontour C : I

C




A + js
dl =

10

~

q

I C


dl: r

The integral on the right hand side is the total hange of the phase  of the wavefun tion (7) around the ontour, whi h must be an integer multiple of 2 sin e the wavefun tion itself must be unique. Hen e, I
~ A + js
dl = 2n: (16) q C

The left hand integral has been named the uxoid by F. London. In the situation of our ring we nd  = ~q 2n:

(17)

By dire tly measuring the ux quantum 0 the absolute value of the super ondu ting harge was measured: (18) jqj = 2e; 0 = 2he : (The sign of the ux quantum may be de ned arbitrarily; e is the proton harge.) If the super urrent js along the ontour C is non-zero, then the ux  is not quantized any more, the uxoid (16), however, is always quantized. In order to determine the sign of q, onsider a super ondu ting sample whi h rotates with the angular velo ity !. Sin e the sample is neutral, its super ondu ting harge density qnB is neutralized by the harge density qnB of the remainder of the material. Ampere's law (in the absen e of a normal urrent density j inside the sample) yields now
  B = 0 js qnB v ; r

where v = !  r is the lo al velo ity of the sample, and js is the super urrent with respe t to the rest oordinates. Taking again the url and onsidering
  v = r  !  r = !  rr r

leads to We de ne the London eld



!


2  B = 0  js  r2 r BL 

 r = 3! ! = 2! r

20qnB !:

22L0qnB ! = 2mq B !

(19)

and onsider the se ond London equation (11) to obtain B BL 2 B=  r2 2L

:

(20)

Deep inside a rotating super ondu tor the magneti eld is not zero but equal to the homogeneous London eld. Independent measurements of the ux quantum and the London eld result in q = 2e; mB = 2me : (21) The bosoni eld is omposed of pairs of ele trons. 11

2.3 Gauge symmetry, London gauge

If (r; t) is an arbitrary dierentiable single-valued fun tion, then the ele tromagneti eld (2) is invariant under the gauge transformation A

! U

 ; r  : t

! 

 : t

!

A+

(22a) Sin e potentials in ele trodynami s an only indire tly be measured through elds, ele trodynami s is symmetri with respe t to gauge transformations (22a). Eqs. (8, 9), and hen e the London theory are ovariant under lo al gauge transformations, if (22a) is supplemented by 2e ;  !  U



~

(22b)

From (8), the super urrent js is still gauge invariant, and so are the ele tromagneti properties of a super ondu tor. However, the ele tro hemi al potential  is dire tly observable in thermodynami s by making onta t to a bath. The thermodynami super ondu ting state breaks gauge symmetry. For theoreti al onsiderations a spe ial gauge is often advantageous. The London gauge hooses  in (22b) su h that the phase   0: Then, from (8), js = A; (23) whi h is onvenient for omputing patterns of super urrents and elds.

12

3

THE THERMODYNAMICS OF THE

1

PHASE TRANSITION

Up to here we onsidered super ondu tivity as a property of a bosoni ondensate. From experiment we know, that the onsidered phenomena are present up to the riti al temperature, T , of the transition from the super ondu ting state, indexed by s, into the normal ondu ting state, indexed by n, as temperature rises. The parameters of the theory, nB and L , are to be expe ted temperature dependent: nB must vanish at T . In this and the next hapters we onsider the vi inity of the phase transition, T

T  T .

3.1 The Free Energy

Experiments are normally done at given temperature T , pressure p, and magneti eld B. Sin e a

ording to the rst London equation (10) there is no stationary state at E =6 0, we must keep E = 0 in a thermodynami equilibrium state. Hen e, we onsider the (Helmholtz) Free Energy (24) Fs (T; V; B ); Fn (T; V; B ); F F F = S; = p; = V m; T V B m is the magnetization density. First, the

where S is the entropy, and determined from the fa t that in the bulk of a super ondu tor Bext + Bm = B + 0 m = 0 as it follows from the se ond London equation (11). Hen e, Fs B

(25) dependen e of Fs on B is (26)

2

= + VB0 =) Fs (B) = Fs (0) + V2B0 :

The magneti sus eptibility of a normal (non-magneti ) metal is jm;n j  1 = jm;s j; hen e it may be negle ted here: Fn (B )  Fn (0): Eq. (27) implies ( f. (25)) Fs (T; V; B ) p(T; V; B )

=

V B2 20 ; B2 20 :

(27) (28) (29)

Fs (T; V; 0) +

= p(T; V; 0)

(30)

The pressure a super ondu tor exerts on its surroundings redu es in an external eld B: The eld B implies a for e per area B2 F= n (31) 2

on the surfa e of the super ondu tor with normal n. 1 L.

0

D. Landau and E. M. Lifshits, Ele trodynami s of Continuous Media, Chap. VI, Pergamon, Oxford, 1960.

13

3.2 The Free Enthalpy

The relations between the Free Energy F and the Free Enthalpy (Gibbs Free Energy) G at B = 0 and B 6= 0 read Fs (T; V; 0) = Gs (T; p(T; V; 0); 0) p(T; V; 0)V and Fs (T; V; 0) +

V B2 20

These relations ombine to

= = =

Fs (T; V; B )

=

Gs (T; p(T; V; B ); B ) p(T; V; B )V = B2 Gs (T; p(T; V; 0) 20 ; B) p(T; V; 0)V

Gs (T; p(T; V; 0); 0) = Gs (T; p(T; V; 0)

or

2 : 0

+ V2B

B2 20 ; B);

B2 (32) 20 ; 0): In a

ord with (31), the ee t of an external magneti eld B on the Free Enthalpy is a redu tion of the pressure exerted on the surroundings, by B2=20. In the normal state, from (29), Gs (T; p; B ) = Gs (T; p +

Gn (T; p; B ) = Gn (T; p; 0):

(33)

The riti al temperature T (p; B) is given by Likewise B (T; p) from

Gs (T ; p +

B2 20 ; 0) = Gn(T ; p; 0):

(34a)

Gs (T; p +

B 2 20 ; 0) = Gn(T; p; 0):

(34b)

B B (T ) T (B )

B T

T

Figure 6: The riti al temperature as a fun tion of the applied magneti eld and the thermodynami

riti al eld as a fun tion of temperature. 14

3.3 The thermodynami riti al eld

The Free Enthalpy dieren e between the normal and super ondu ting states is usually small, so that at T < T (B = 0) the thermodynami riti al eld B (T ) for whi h (34) holds is also small. Taylor expansion of the left hand side of (34) yields Gn (T; p) = Gs (T; p) +

B 2 Gs 20 p

=

Gs (T; p) +

B 2 20 V (T; p; B = 0):

Experiment shows that at B = 0 the phase transition is se ond order,
Gn (T; p) Gs (T; p) = a T (p) T 2 : Hen e,
B (T; p) = b T (p) T ; p where a is a onstant, and b = 20a=V . T (p) is meant for B = 0. 0 M B

B T

= V Bm

B (T )

(37)

B

normal state Meissner ee t

b(T (p) T ) m =

P

(36)

B (T )

0
G

=0

(35)

1

T (p) T FIG. 7: The thermodynami riti al eld.

FIG. 8: The magnetization urve of a super ondu tor.

We onsider all thermodynami parameters T; p; B at the phase transition point P of Fig. 7. From (32), Ss (T; p; B )

=

Vs (T; p; B )

=

B2 Gs = Ss (T; p + T 20 ; 0); Gs B2 = Vs (T; p + p 20 ; 0):

Dierentiating (34b) with respe t to T yields, with (38),   B 2 (T; p) ; 0) = Gn (T; p; 0 or B ); Gs (T; p + T 20 T V (T; p; B )  2 Ss (T; p; B ) + s 20 T B (T; p) = Sn(T; p; B ); (T; p) :
S (T; p; B ) = Ss(T; p; B ) Sn(T; p; B ) = Vs (T;p; B ) B (T; p) B T 0

15

(38)

(39)

A

ording to (37) this dieren e is non-zero for B 6= 0 (T < T (p)): For B 6= 0 the phase transition is rst order with a latent heat (40) Q = T
S (T; p; B ): For T ! 0, Nernst's theorem demands Ss = Sn = 0, and hen e lim B (T; p) = 0: (41) T !0

T

3.4 Heat apa ity jump For B  0, T  T (p) we an use (35). Applying T  2=T 2 yields


Cp = Cp;s

Cp;n

=

T

 2  G ( T; p ) G ( T; p ) s n T 2

=

T V (T; p)  2 2 20 T 2 B (T; p):

(42)

The thermal expansion V=T gives a small ontribution whi h has been negle ted. With 2 2  B B = 2B 2 T T T

we nd


Cp =

TV 0

"

B T

=2

B T

!2

For T ! T (p), B ! 0 the jump in the spe i heat is
Cp =

T V B 0 T

!2

2 + B TB2

!2

2

+ 2B TB2 #

(43)

= T V b2:

(44)

0

It is given by the slope of B (T ) at T (p). S

Cp B=0

B=0


Cp Sn Ss T (p)

T (p)

T

FIG. 10: The heat apa ity of a super ondu tor.

FIG. 9: The entropy of a super ondu tor.

16

T

4

THE GINSBURG-LANDAU THEORY;

1

TYPES OF SUPERCONDUCTORS

A

ording to the Landau theory of se ond order phase transitions with symmetry redu tion2 there is a thermodynami quantity, alled an order parameter, whi h is zero in the symmetri (high temperature) phase, and be omes ontinuously non-zero in the less symmetri phase. 4.1 The Landau theory

The quantity whi h be omes non-zero in the super ondu ting state is nB = j j2 : (45) For nB > 0, the ele tro hemi al potential  has a ertain value whi h breaks the global gauge symmetry by xing the time-derivative of the phase  of ( f. (22b)). A

ording to the Landau theory, the Free Energy is the minimum of a \Free Energy fun tion" of the order parameter with respe t to variations of the latter: (46) F (T; V ) = min F (T; V; j j2): F

T > T (t > 0) T

= T (t = 0)

T < T (t < 0)

0110 0min n1

j j2

B;

Figure 11: The Free Energy fun tion. Close to the transition, for

t=

T

T

T

;

jtj  1;

the order parameter j j2 is small, and F may be Taylor expanded (for xed V ): F (t; j j2 ) = Fn (t) + A(t)j j2 + 21 B (t)j j4 +


From the gure we see that A(t) T 0 for t T 0; B (t) > 0: 1 V. L. 2 L. D.

Ginsburg and L. D. Landau, Zh. Eksp. Teor. Fiz. (Russ.) 20, 1064 (1950). Landau, Zh. Eksp. Teor. Fiz. (Russ.) 7, 627 (1937).

17

(47) (48)

Sin e jtj  1, we put Then we have and

A(t)  tV; B (t)  V:

(49)

Fn (t) = Fn (t) for t  0;

(50)

1 F V  j j2

= t + j j2 = 0; that is,

2 t2 ; Fs (t) = Fn (t) j j2 = t 2 V for t < 0:

(51)

Re alling that small hanges in the Free Energy and Free Enthalpy are equal and omparing to (35) yields r 2 t2 B 2 0 (52) 2 = 20 =) B (t) = jtj : From (43),


Cp =

T V B 0 T t

!2

2



= TV

(53)

follows. While
Cp an be measured, this is not always the ase for the thermodynami riti al eld, B , as we will later see. Eqs. (51) and (52) may be rewritten as nB (t) =

hen e,

=

 2 t2 0 j tj; B 2 (t) = ;

B 2 (t) B 2 (t) ; = : 2 0 nB (t) 0 jtjnB (t)

Sin e a

ording to (37) B  t, it follows

nB  t:

The bosoni density tends to zero linearly in T

T.

(54) (55)

4.2 The Ginsburg-Landau equations

If we want to in orporate a magneti eld B into the \Free Energy fun tion" (48), we have to realize that B auses super urrents js   = r, and these reate an internal eld, whi h was alled Bm in (26). The energy ontribution of must be related to (3). Ginsburg and Landau wrote it in the form F (t; B ; ) = Fn (t) + ) (   2 Z Z 1 2 2 2 ie B ~  + d3r 2m0 + d3r 4m  r + ~ A + tj j2 + 2 j j4 ; (56a) V

where also (21) was onsidered. The rst orre tion term is the eld energy of the eld Bm reated by , in luding the stray eld outside of the volume V while 6= 0 inside V only. A is the ve tor potential of the total eld a ting on :   A = B + Bm : r

18

(56b)

The Free Energy is obtained by minimizing (56a) with respe t to (r) and (r). To prepare for a variation of , the se ond integral in (56a) is integrated by parts: "  #"  # Z  2 ie 2 ie  d3 r + A A  = V

r

r

~

2



 2ie d3 r  + A r ~ V

Z

=

~

+



Z V



 2ie + A d2 n  r ~

:

(56 )

From the rst integral on the right we see that (56a) indeed orresponds to (3). The preferen e of the writing in (56a) derives from that kineti energy expression being manifestly positive de nite in any partial volume. Now, the variation  !  + Æ  yields ( )   Z ~2  2 ie 2 ! 3  2 0 = ÆF = d rÆ 4m  r + ~ A + t + j j + V   Z 2 ie ~2  2  + d nÆ 4m  r + ~ A : V

F is stationary for any variation Æ (r), if

1  ~  + 2eA2 4m i  r and



jtj + j j2 = 0

(57)



~ 

+ 2eA = 0: (58) The onne tion of with Bm must be that of Ampere's law: (= r)  Bm = 0js with js given by (6). Sin e in thermodynami equilibrium there are no urrents besides js in the super ondu tor, (= r)  B = 0 there. Hen e, we also have n

i r

  Btot = 0 js ; r  ie~   js = 2m  r

 Btot = B + Bm =  r  2 e2    r A : m

 A;

(59)

It is interesting to see that (59) is also obtained from (56a), if , and A are varied independently: The variation of A on the left hand side of (56 ) yields 2ie Z d3rÆA
   2ie A    + 2ie A : ~

r

V

r

~

~

With ÆBm = (= r)  ÆA the variation of the rst integral of (56a) yields Æ

Z

1

= =

2 =2 d3 rBm 1

z

Z

1

d3 rÆBm
Bm = 2

}| {

Z

1





 d3 r  Æ A
Bm = r

1 2 d3r r
ÆA Bm = 2 d3 rÆA
r  Bm   Z 2 d3 rÆA
r  Btot +


: V Z



Z





= (60)

In the fourth equality an integration per parts was performed, and a
(b  ) = b
(a  ) was used. (The over bra e indi ates the range of the dierential operator.) Finally, the integral over the in nite 19

spa e is split into an integral over the super ondu tor (volume V ), where (= r)Bm = (= r)Btot, and the integral over the volume outside of the super ondu tor, indi ated by dots, sin e we do not need it. Now, after adding the prefa tors from (56a) we see that stationarity of (56a) with respe t to a variation ÆA inside the volume V again leeds to (59). This situation is no a

ident. From a more general point of view the Ginsburg-Landau fun tional (56a) may be onsidered as an ee tive Hamiltonian for the u tuations of the elds and A near the phase transition.1 This is pre isely the meaning of relating (56a) to (3). Eqs. (57) and (59) form the omplete system of the Ginsburg-Landau equations. The boundary ondition (58) omes about by the spe ial writing of (56a) without additional surfa e terms. This is orre t for a boundary super ondu tor/va uum or super ondu tor/semi ondu tor. A

areful analysis on a mi ros opi theory level yields the more general boundary ondition n






i + 2 eA = ; i r b

~ 

(61)

where b depends on the outside material: b = 1 for va uum or a non-metal, b = 0 for a ferromagnet, b nite and non-zero for a normal metal.2 In all ases, multiplying (61) by  and taking the real part yields n
js = 0 (62) as it must: there is no super urrent passing through the surfa e of a super ondu tor into the nonsuper ondu ting volume. Btot must be ontinuous on the boundary be ause, a

ording to  Btot = r = 0 and (59), its derivatives are all nite. 4.3 The Ginsburg-Landau parameter

Taking the url of (59) yields, like in (13),  2 Btot  r2

= Btot2 ;

m m (63) 20e2j j2 = 20e2jtj ; where (51) was taken into a

ount in the last expression.  is the Ginsburg-Landau penetration depth; it diverges at T like   jtj 1: if T is approa hed from below, the external eld penetrates more and more, and eventually, at T , the diamagnetism vanishes. Eq. (57) ontains a se ond length parameter: In the absen e of an external eld, A = 0, and for small , j j2  jtj= , one is left with 2 2 = ~2 : ;  = (64)  r2  2 4mjtj This equation des ribes spatial modulations of the order parameter j j2 lose to T .  is the GinsburgLandau oheren e length of su h order parameter u tuations. It has the same temperature dependen e as , and their ratio, s 2m2 ;  (65) = =  ~2  e2 2 =

0

is the elebrated Ginsburg-Landau parameter. 1 L. 2 P.

D. Landau and E. M. Lifshits, Statisti al Physi s, Part I, x147, Pergamon, London, 1980. G. De Gennes, Super ondu tivity in metals and alloys, New York 1966, p. 225 .

20

Introdu tion of dimensionless quantities x = r=;

=

,s

p

jtj ;

r   2B (t) = B jtj 2 0 ; .p   is = js 0 2B (t) ; .p  2B (t) a=A

b=B

.

(66)

yields the dimensionless Ginsburg-Landau equations 2  + a + j j2 = 0; i  x  i    b = is ; is = 2  x x





1

  x



(67)

a

whi h ontain the only parameter . 4.4 The phase boundary

We onsider a homogeneous super ondu tor at T . T in an homogeneous external eld B  B (T ) in z -dire tion. We assume a plane phase boundary in the y z -plane so that for x ! 1 the material is still super ondu ting, and the magneti eld is expelled, but for x ! 1 the material is in the normal state with the eld penetrating. z

b=0

=1

b = 1=

=0

b

super ondu ting

2

normal x

y

phase boundary

is

Figure 12: Geometry of a plane phase boundary. We put

p

= (x);

bz = b(x); bx = by = 0;

21

ay = a(x); ax = az = 0; b(x) = a0 (x):

Then, the super urrent is ows in the y-dire tion, and hen e the phase of depends on y. We

onsider y = 0 and may then hoose real. Further, by xing another gauge onstant, we may hoose a(0) = 0: Then, Eqs. (67) redu e to 1

00 + a2

2

+ 3 = 0;

a00 = a 2 :

(68)

Let us rst onsider   1: For large enough negative x we have a  0 and  1. We put = 1 (x), and get from the rst equation (68) 00  2

1

1 + 3
= 22;



p   e 2x ; x .  1 :

On the other hand, for large enough positive x we have b = 1=p2; a = xp2;  1; hen e, again from the rst equation (68), 2 2 00   x ;

2

e

x2 =2

p2

; x2  1:

1 0 ; where 0 denotes the value of

The se ond Eq. (68) yields a penetration depth  eld drops:

1

1

1

p

e 2x

p1

b

2

e

(x) where the

p

0 

0x

e

x2 =2

p2

1=p > 1

1= =  Figure 13: The phase boundary of a type I super ondu tor. In the opposite ase   1; falls o for x & 1; where b  1=p2; a  x=p2; and for x  1; 00  2 x2 =2 : 22

1

1

p1

b

2 e

x2 =2

p2

1=p < 1 1= Figure 14: The phase boundary of a type II super ondu tor. 4.5 The energy of the phase boundary

For B = B (T ); b = 1 in our units, the Free Energy of the normal phase is just equal to the Free Energy of the super ondu ting phase in whi h b = 0; = 1: If we integrate the Free Energy density variation (per unit area of the y z-plane), we obtain the energy of the phase boundary per area: s=n =

Z

1 1

(

dx

(B

B )2 20

~2

+ 4m

2



j 0 j2 + 4e2 A2 j j2 ~



)

jtjj j2 + j j4 :

2

(69)

Sin e the external eld is B , we have used Bm = Btot Bext = B B : In our dimensionless quantities this is (x is now measured in units of )  Z 4
2
2 1 B 2 1 1 2 0 2 s=n = dx b p + 2 + a 1 + 0  2 = 2 1  2Z 1 1
1 00 + a2 1
2 + 4 =

= B dx a0 p 2 0 2 2 2  1  Z 1 4
B 2 1 (70) =  dx a0 p 2 2 : 2 0 1 First an integration per parts of 0 2 was performed, and then (68) was inserted. We see that s=n an have both signs: 2  4 2
1 0 ? 2 or p 7 p1 a0 : s=n ? 0 for a p 2 2 2

Sin e b must de rease if 2 in reases and = 0 at b = 1=p2, (b 1=p2) = (a0 1=p2) and have opposite signs whi h leads to the last ondition. If 2 p1 a0 = p 2 2 would be a solution of (68), it would orrespond to s=n = 0: 23

2 must

We now show that this is indeed the ase for 2 = 1=2: First we nd a rst integral of (68): 2

=

2 a2

0 00

=

2

02

1 2 and have from (71)

2 = ;

p1

2



2





1
+ 0 a2

3 ; 0+2 3 0

2



=



2 0+2 3 0 = 0 by the se ond Eq. (67)   4 = 2 2 a2 a02 2 + 2 + onst. sin e 0 = = 0 for a0 = p12 =) onst. = 12 :

=

Now we use



00

2

a0 =

02 =

1 2 2

2

p

2

02

2

0 a2 + 2 2 aa0 2a0 a00 | {z }

)

a00 =

a0 2

1

p

p

2

0= a 2

2a0
+ p12

0 = ap

)

(71)

2




1 a0 2 + ;

2

whi h is indeed an identity. Sin e 02=2 > 0 enters the integral for s=n in the rst line of (70), it is lear that s=n is positive for 2 ! 0: Therefore, the nal result is 1 s=n ? 0 for  7 p : type III (72) 2 The names \type I" and \type II" for super ondu tors were oined by Abrikosov,1 and it was the existen e of type II super ondu tors and a theoreti al predi tion by Abrikosov, whi h paved the way for te hni al appli ations of super ondu tivity.

1 A.

A. Abrikosov, Sov. Phys.{JETP 5, 1174 (1957).

24

5

INTERMEDIATE STATE, MIXED STATE

In Chapter 2 we onsidered a super ondu tor in a suÆ iently weak magneti eld, B < B ; where we found the ideal diamagnetism, the Meissner ee t, and the ux quantization. In Chapter 3 we found that the dieren e between the thermodynami potentials in the normal and the super ondu ting homogeneous phases per volume without magneti elds may be expressed as ( f. (35)) 1 hGn(p; T ) Gs (p; T )i = 1 hFn (V; T ) Fs (V; T )i = B 2(T ) (73) V V 2 0

by a thermodynami riti al eld B (T ): (We negle t here again the ee ts of pressure or of orresponding volume hanges on B :) If a magneti eld B is applied to some volume part of a super ondu tor, it may be expelled (Meissner ee t) byR reating an internal eld Bm = B through super urrents, on the ost of an additional energy d3rBm2 =20 for the super ondu ting phase ( f. (56a)) and of a kineti energy density (~2 =4m)j(= r + 2ieA=~) j2 in the surfa e where the super urrents ow. If Bm > B ; the Free Energy of the super ondu ting state be omes larger than that of the normal state in a homogeneous situation. However, B itself may ontain a part reated by urrents in another volume of the super ondu tor, and phase boundary energies must also be onsidered. There are therefore long range intera tions like in ferroele tri s and in ferromagnets, and orresponding domain patterns

orrespond to thermodynami stable states. The external eld B at whi h the phase transition appears depends on the geometry and on the phase boundary energy. 5.1 The intermediate state of a type I super ondu tor

Apply a homogeneous external eld Bext to a super ondu tor. B = Bext + Bm depends on the shape of the super ondu tor. There is a ertain point, at whi h B = Bmax > Bext (Fig. 15). If Bmax > B ; the super ondu ting state be omes instable there. On ould think of a normal-state on ave island forming (Fig. 16).

Bext

Bmax

Bext + Bm

SC SC

n

Bmax

FIG. 15: Total (external plus indu ed) magneti eld around a type I super ondu tor.

FIG. 16.

25

This, however, annot be stable either: the point of Bmax = B has now moved into the super ondu tor to a point of the phase boundary between the normal and super ondu ting phases, whi h means that in the shaded normal area B < B ; this area must be ome super ondu ting again (Fig. 16). Forming of a onvex island would ause the same problem (Fig. 17).

Bmax

n

SC

SC

FIG. 17.

What really forms is a ompli ated lamellous or lamentous stru ture of alternating super ondu ting and normal phases through whi h the eld penetrates (Fig. 18). The true magnetization

urve of a type I super ondu tor in dierent geometries is shown on Fig. 19. It depends on the geometry be ause the eld reated by the shielding super urrents does. In Se tion 3.C, for B 6= 0 the phase transition was obtained to be rst order. Generally, the movement of phase boundaries is hindered by defe ts, hen e there is hysteresis around B .

FIG. 18.

2 3 B

Bm

B

sphere

Bext

long rod rst order transition

0
fenergy of the stray eld +

+ energy of phase boundariesg Figure 19:

26

5.2 Mixed state of a type II super ondu tor1

If the phase boundary energy is negative, germs of normal phase may form well below B ; at the lower

riti al eld B 1 ; and germs of super ondu ting phase may form well above B ; at the upper riti al eld B 2 ; in both ases by gaining phase boundary energy. At B / B 2; in small germs  1 and b  onst. The dimensionless Ginsburg-Landau equations (67) may be linearized:  1  + a2 = : (74) i  x We apply the external B- eld in z-dire tion (Fig. 20), bx = by = 0; bz = b; ax = by; ay = az = 0; b and assume germ laments along the eld lines: = (x; y): a Then, (74) is ast into . " # y 1  by2 1  2 = : i x

2 y2

With (x;2y) =2 eipx2 (y); p=(b)2 = y0 this equation simpli es x to [ (1= )(d =dy )+ b(y y0 ) ℄ = ; or, after multiplying with =2 and by de ning u2  (y y0)2 : # " Figure 20: 1 d2 + 1 b2u2  =  : (75) 2 du2 2 2 This is the S hrodinger equation for the ground state of a harmoni os illator with p ! = b =  =) B = 2B : Hen e, p B 2 (T ) = 2B (T ): (76) If a germ lose to the surfa e of a super ondu tor at y= onst. is onsidered, then the u- oordinate must be ut at some nite value. There, the boundary ondition (58) yields d=du = 0 (sin e n
a = 0). Therefore, instead of the boundary problem, the symmetri ground state in a double os illator with a mirror plane may be onsidered (Fig. 21). 0 (0) = 0

u0

u

FIG. 21: The ground state of a double os illator.

1 A.

A. Abrikosov, unpublished 1955; Sov. Phys.{JETP 5, 1174 (1957).

27

~! min E = 0:59 u0 0 2

Hen e, at

B 2 0:59

= 1:7B 2 = 2:4B (77) super ondu tivity may set in in a surfa e layer of thi kness  . In a type I super ondu tor, B 2 < B . However, only below B 2 germs with arbitrarily small -values ould form where the super ondu ting phase is already absolutely stable in a type I super ondu tor. Here, for B 2 < B < B germs an only form with a non-zero minimal -value, whi h is inhibited pby positive surfa e energy. In this region the super ooled normal phase is metastable. For  > 0:59= 2; B 3 > B ; and surfa e super ondu tivity may exist above B : B 3 =

5.3 The ux line in a type II super ondu tor

To determine B 1; the opposite situation is onsidered. In the homogeneous super ondu ting state with = 1; the external magneti eld is tuned up until the rst normal state germ is forming. Again we may suppose that the germ is forming along a eld line in z-dire tion. Sin e the phase boundary energy is negative, there must be a tenden y to form many fa e boundaries. However, the normal germs annot form arbitrarily small sin e we know from (17) that the ux onne ted with a normal germ in a super ondu tor is quantized and annot be smaller than 2 p p (78) 0 = 22e~ = 2 2B  = 2 2B  : We onsider a ux line of total ux 0 along the z-dire tion (Fig. 22): Bz 

j j

= j jei ; B = ez B(); 2 = x2 + y2

 

FIG. 22: An isolated ux line.

One ould try to solve the Ginsburg-Landau equations for that ase. However, there is no general analyti solution, and the equations are valid lose to T only, where j j is small. Instead we assume   1; that is,   ; and onsider only the region   ; where j j = onst. Then, from Ampere's law and (6), 2   B = 0 js = 0 e~ j j2  20 e j j2 A = r

hen e,

m

r

 2 0e2 j j2 ~  = m 2e  r

A + 2

A



m

 1 = 2 20 r



A ;

  B = 20 r : r

We integrate this equation along a ir le around the ux line with radius    and use Stokes' theorem, H dsA = R d2n
( r  A) : Z

d2 n
B + 2

I

ds


28



 B r



= 0 :

(79)

The phase  must in rease by 2 on a ir le around one uxoid. Now, we take     : Then, the rst term in (79) may be negle ted. We nd = 0 22 dB d

or

(80)

   0 0 (81) B () = 22 ln  ; 0  : The integration onstant 0 was hosen su h that (81) vanishes for  & ; where a more a

urate analysis of (79) is ne essary to get the orre t asymptoti s. By applying Stokes' theorem also to the se ond integral of (79), we have for all    Z





  d2 n
B + 2  B r r



=

Z



d2 n
B  2

2 B  r2



= 0 :

Sin e the right hand side does not hange if we vary the area of integration, 2

B

must hold. In ylindri oordinates,

2  r2

hen e,

  1 2 2 = 1   + 2 2 + 2;    z B + 2

or

2 B=0  r2

1    B=0   

1

This equation is of the Bessel type.

1

B 00 + B 0 B = 0:  2 Its for large  de aying solution is

(82)

0 K0   ! p 0 e = ; (83) 22  !1 82 where K0 is M Donald's fun tion (Hankel's fun tion with imaginary argument), and the oeÆ ient has been hosen to meet (80) for   : The energy per length  of the ux line onsists of eld energy and kineti energy of the super urrent: B () =



=



= = = =

B 2 mB + 2 nB v2 ; js = 2enB v; nB = j j2; d2 r 2 0  2  Z b m 2 d2 r + j 20 4e2nB s =  2   2 Z m  B 2 dr 20 + 40e2j j2  rB =   2 Z B 2  d2 r +  B
r  B = 2 0 20  r 1 Z d2 rB
B + 2      B 20 r r !    I I  2  B : ds
B  20  !1 r Z





29

The rst integrand was shown to vanish for   ; and the last ontour integral vanishes for  ! 1. We negle t the ontribution from  . ; and nd with (80)) dB  B ( ) 0 B (0) 2 2 B  0 (84)  20 d 20  20 : With (81), the nal result with logarithmi a

ura y (ln(=)  1) is   20 20 ln :   =  ln (85) 402  402 This result also proves that the total energy is minimum for ux lines ontaining one uxoid ea h: For a ux line ontaining n uxoids the energy would be n2 while for n ux lines it would only be n ( > 0). In this analysis, B was the eld reated by the super urrent around the vertex line. Its intera tion energy per length with a homogeneous external eld in the same dire tion is Z B BB (86) d2 r ext = 0 ext : 0

0

(85) and (86) are equal at the lower riti al eld Bext = B 1:  ln  B 1 = 0 2 ln  = B p ;   1: (87) 4 2 The phase diagram of a type II super ondu tor is shown in Fig. 23. (There might be another phase with spatially modulated order parameter under ertain onditions, theoreti ally predi ted independently by Fulde and Ferell and by Larkin and Ovshinnikov; this FFLO phase has not yet been learly observed experimentally.) B

surfa e s

FFLO?

B 3

mixed phase B 1

n

B 2

B

s

T

T

Figure 23: The phase diagram of a type II super ondu tor. A more detailed numeri al al ulation shows that in an isotropi material the energy is minimum for a regular triangular latti e of the ux lines2in the plane perpendi ular to them. From (87), at B p= B 1 the density of ux lines is ln =4 ; that is, the latti e onstant a1 is obtained from a21 3=2 = 42 = ln : s 8 (88) a1 = p 3ln   & : 30

The lines (of thi kness ) indeed form nearly individually (Fig. 24). Sin e B 2 = B 122= ln ; the latti e onstant a2 at B 2 is s s r  ln  4  4 a2 = a 1 (89) 22 = p3  = p3  & : The ores of the ux lines (of thi kness ) tou h ea h other while the eld is already quite homogeneous (Fig. 25). Sin e j j  1 in the ore, the Ginsburg-Landau equations apply, and j j may rise

ontinuously from zero: the phase transition at B 2 is se ond order. 

B

j j B

j j a2 & 

a1 &  FIG. 24: Mixed phase for Bext  B 1 .

solution of (4.3)

FIG. 25: Mixed phase for Bext

 B 2 .

For a long ylindri rod the stray eld reated by super urrents outside of the rod may be negle ted, and one may express the eld inside the rod as B = Bext 0 M by a magnetization density M: The hange in Free Energy at xed T and V by tuning up the external magneti eld is dF = MdBext Z B 2 B2 Fn F s = dBextM = : 2 0

0

0 M

B 1

B

B 2 Bext

111111111 000000000 00000000 11111111 000 111 000000000 111111111 00000000 11111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 jump or 000 111 000000000 111111111 in nite slope 000 111 000 111

Figure 26: Magnetization urve of a type II super ondu tor. The dierently dashed areas in Fig. 26 are equal. For type II super ondu tors, B is only a theoreti al quantity. 31

6

JOSEPHSON EFFECTS

1

The quantitative des ription of Josephson ee ts at T  T (the usual ase in appli ations) needs a mi ros opi treatment. However, qualitatively they are the same at all temperatures T < T , hen e qualitatively they may be treated within the Ginsburg-Landau theory. Consider a very thin weak link between two halfs of a super ondu tor (Fig. 27). 2 S1

S2 d x1

1

x2

2 =  1 2



(x1)  1 = j jei1 (x2)  2 = j jei2 x

2 = 1 +  FIG. 27: A weak link between two halfs of a super ondu tor.

The order parameter has its thermodynami value on both sides x < x1 ; x > x2; but is exponentially small at x = 0: Hen e, any super urrent through the weak link is small, and may be

onsidered onstant in both bulks of super ondu tor. In the weak link, not only j j is small, also its phase may hange rapidly (e.g. from 2 = 1 to 2 = 1 +  by a very small perturbation). Without the right half, the boundary ondition (58) would hold at x1 :    2ie + A = 0: x ~ x x1 In the presen e of the right half, this ondition must be modi ed to slightly depending on the value 2 :    2ie (90) + Ax = 2; x

1 B.

x1

~

D. Josephson, Phys. Lett. 1, 251 (1962).

32

where is a small number depending on the properties of the weak link. Time inversion symmetry demands that (90) remains valid for ! ; A ! A; hen e must be real as long as the phase of does not depend on A. For the moment we hoose a gauge in whi h Ax = 0: Then, the super urrent density at x1 is " # ie~     js;x (x1 ) = 2m 1 x x1 1 x x1 = h i = 2iem~ 1 2 1 2 = jm





= jm sin 2 1
: = 2 We generalize the argument of the sine fun tion by a general gauge transformation (22): 2e ;  ! 

2  1

ei(2 1 )

A+

A

!



! 

~



 ; r  t

! =  2 1

ei(1 2 )

Ax = 0 ! A x =

2e ~

2  1




=

2



 ; x Z 2 e 2 1 + dxAx ;

= 2~e 2 1
: Re all that  is the ele tro hemi al potential measured by a voltmeter. Now, the general Josephson equation reads js = jm sin : d dt

(91)

~

1

(92) (93) (94)

6.1 The d. . Josephson ee t, quantum interferen e

A

ording to (94), a d. . super urrent of any value between jm and jm may ow through the jun tion, and a

ording to (93) the potential dieren e 2 1 is zero in that ase. Now, onsider a jun tion in the y z-plane with a magneti eld applied in z-dire tion. There are super urrents s reening the eld away from the bulks of the super ondu ting halfs. y

b

S1

11 00 B 00 11

(y)11 00 4 11 3 00 00 1 0 11 00 2 11 d  2

S2 x

Figure 28: A Josephson jun tion in an external magneti eld B. 33

Let us onsider (y); and let (0) = 0 at the edge y = 0: We have Byd =

Z

1234

ds
A =

Z

1

2

dxAx +

Z

In the jun tion we hoose the gauge Ax =

2

3

dyAy +

4

Z

3

dxAx +

Z

4

1

dyAy :

By; Ay = 0: Then, the y-integrals vanish, and from (92), 2 Z 3 dxBy = 0 + 2Bd y:

(0) = 2 1 = 0 ; (y) = 0 + 0 4 0 The d. . Josephson urrent density through the jun tion os illates with y a

ording to !

2Bd y : js (y) = jm sin 0 + 0 In experiment, at B = 0 one always starts from a biased situation with js = jm; hen e 0 = =2; and 2Bd y: js (y) = jm os (95) 0 The total urrent through the jun tion is Z b 2Bd y; Is = jm dy os 0 0 where is the thi kness in z-dire tion. F = b is the area of the jun tion. With Z b Z b ei b 1 sin( b) dy os( y) = < dyei y = < = 0

i

0



we nd that, depending on the phase 0; the maximal urrent at a given eld B is j sin(2Bbd=0 )j : Is;max = F jm 2Bbd=0

(96)

An even simpler situation appears, if one splits the jun tion into a double jun tion: Now, H ds
A =  is the magneti ux through the ut-out, and I  2 

a b =  ds
A = 2  : 0

0

11 00 00 11 00 11 00 11 00B 11 00 11 00 11 00 11 00 11

b

S1

S2 s

a

Figure 29: A simple SQUID geometry. 34

Hen e, the maximal d. . Josephson urrent is "

F jm 2 = x

 2  sin a + sin a +  Is;max = F jm max

a 0

2=0

a

I s;m

#



a

F jm

Is;max = 2

Figure 30: Phase relations in Eq.(97).



 

os   : 0 

:

(97)

This is the basis to experimentally ount ux quanta with a devi e alled super ondu ting quantum interferometer (SQUID).

6.2 The a. . Josephson ee t

We return to (93) and (94), and apply a voltage 2 An a. . Josephson urrent js (t) = jm sin !J t;

1 = V !J

to the jun tion, that is, = (2e=~)V t:

= 2eV=~

(98)

results, although a onstant voltage is applied. For a voltage of 10 V a frequen y !J =2 = 4.8 GHz is obtained: the a. . Josephson ee t is in the mi rowave region. If one overlays a radio frequen y voltage over the onstant voltage, 2  2 = V

+ Vr os(!r t);

(99)

one obtains a frequen y modulation of the a. . Josephson urrent: js

= =





2eV jm sin !J t + r sin(!r t) = ~!r ! 1 X 2 eVr sin(!J + n!r )t: jm Jjnj ~!r n= 1

(100)

is the Bessel fun tion of integer index. To interpret the experiments one must take into a

ount that a non-zero voltage a ross the jun tion

auses also a dissipative normal urrent In = V=R; where R is the resistan e of the jun tion for normal ele trons. The experimental issue depends on the oupling in of the radio frequen y.1 If the impedan e of the radio sour e is small ompared to that of the jun tion, we have a voltage-sour e situation, and the total urrent through the jun tion averaged over radio frequen ies is Jjnj

I = Is + V=R; Is = js F: 1 S.

Shapiro, Phys. Rev. Lett. 11, 80 (1963).

35

(101)

I ~!r =2e

d. . Josephson spike, Im

V=R

V

Shapiro spikes for !J + n!r = 0 Figure 31: Josephson urrent vs. voltage in the voltage-sour e situation. If the impedan e of the radio sour e is large ompared to the impedan e of the Josephson jun tion, as is usually the ase, we have a urrent-sour e situation, where the fed-in total urrent determines the voltage a ross the jun tion: V

= R(I = 2~e d dt

Is ) = R(I

Im sin ):

(102)

I

Shapiro steps

V =R

Im

V

Figure 32: Josephson urrent vs. voltage in the urrent-sour e situation. The a. . Josephson ee t yields a possibility of pre ise measurements of h=e. 36

7

MICROSCOPIC THEORY: THE FOCK SPACE

The S hrodinger wavefun tion of an (isolated) ele tron is a fun tion of its position, r; and of the dis rete spin variable, s :  = (r; s): Sin e there are only two independent spin states for an ele tron | the spin omponent with respe t to any (single) hosen axis may be either up (") or down (#) |, s takes on only two values, + and , hen e  may be thought as onsisting of two fun tions   (r; s) = ((rr;; +)) (103) forming a spinor fun tion of r: The expe tation value of any (usually lo al) one-parti le operator A(r; s; r0 ; s0 ) = Æ(r r0 )A^(r; s; s0) is XZ d3 r (r; s)A^(r; s; s0 )(r; s0 ): (104) hAi = s;s0

We will often use a short-hand notation x  (r; s). The S hrodinger wavefun tion (x1 : : : xN ) of a (fermioni ) many-parti le quantum state must be totally antisymmetri with respe t to parti le ex hange (Pauli prin iple): (: : : xi : : : xk : : :) = (: : : xk : : : xi : : :): (105) For a pie e of a solid, N  1023, this fun tion is totally in omprehensible and pra ti ally ina

essible: For any set of N values, + or for ea h si; it is a fun tion of 3N positional oordinates. Although by far not all of those 2N fun tions are independent | with the symmetry property (105) the +- and -values an always be brought to an order that all -values pre ede all +-values, hen e we have only to distinguish 0, 1, 2, . . . , N -values, that is (N + 1) ases | and although ea h of those (N + 1) fun tions need only be given on a ertain se tor of the 3N -dimensional position spa e | again with the symmetry property (105) ea h fun tion need only be given for a ertain order of the parti le

oordinates for all si = and for all si = + parti les |, it is lear that even if we would be ontent with 10 grid points along ea h oordinate axis we would need  101023 grid points for a very rude numeri al representation of that -fun tion. Nevertheless, for formal manipulations, we an introdu e in a systemati manner a fun tional basis in the fun tional spa e of those horrible -fun tions. 7.1 Slater determinants

Consider some (for the moment arbitrarily hosen) omplete orthonormal set of one-parti le spinor fun tions , l (x);

(l jl0 ) = X l

XZ s

d3 rl (r; s)l0 (r; s) = Æll0 ;

l (x)l (x0 ) = Æ(x x0 ) = Æss0 Æ(r

r0 ):

(106)

They are ommonly alled (spinor-)orbitals. The quantum number l refers to both the spatial and the spin state and is usually already a multi-index (for instan e (nlm) for an atomi orbital or (k) for a plane wave), and we agree upon a ertain on e and forever given linear order of those l-indi es. Choose N of those orbitals, l1 ; l2 ; : : : ; lN ; in as ending order of the li and form the determinant (107) L(x1 : : : xN ) = p1N ! det kli (xk )k: L = (l1 : : : lN ) is a new (hyper-)multi-index whi h labels an orbital on guration. This determinant of a matrix aik = li (xk ) for every point (x1 : : : xN ) in the spin-position spa e has the proper symmetry property (105). In view of (106) it is normalized, if all li are dierent, and it would be identi ally 37

zero, if at least two of the li would be equal (determinant with two equal raws): Two fermions annot be in the same spinor-orbital. This is the ompared to (105) very spe ial ase of the Pauli prin iple (whi h is the ommonly known ase). Now, given a omplete set of orbitals (106), we mention without proof that all possible orbital

on gurations of N orbitals (107) form a omplete set of N -fermion wavefun tions (105), that is, any wavefun tion (105) may be represented as X (x1 : : : xN ) = CLL(x1 : : : xN ) (108) L

jCL j2 = 1 (` on guration intera tion'). with ertain oeÆ ients For any operator whi h an be expanded into its one-parti le, two-parti le- and so on parts as for instan e a Hamiltonian with (possibly spin-dependent) pair intera tions, X 1 X w 0 0 (ri ; rj ); H^ = h^ si s0i (ri ) + (109) 2 i=6 j si si;sj sj i P CL ; L

the matrix with Slater determinants is HLL0 = hL jH^ jL0 i =

XX i

P

(lP i jhjli0 ) ( 1)jPj

Y j (6=i)

ÆlPj l0j +

XX Y + 21 ÆlPk l0k ; (lP ilP j jwjlj0 li0 ) ( 1)jPj i6=j P k(6=i;j )

(110)

where P is any permutation of the subs ripts i; j; k; and jPj is its order. The matrix elements are de ned as XZ 0 (li jhjli) = (111) d3 rli (r; s)h^ ss0 (r)l0i (r; s0 ); and (li lj jwjlj0 li0 ) =

ss0

X si s0i ;sj s0j

Z

d3 ri d3 rj li (ri ; si )lj (rj ; sj ) wsi s0i ;sj s0j (ri ; rj ) l0j (rj ; s0j )l0i (ri ; s0i ):

(112)

(Note our onvention on the order of indi es whi h may dier from that in other textbooks but leads to a ertain anoni al way of writing of formulas later on.) The rst line on the r.h.s. of (110) is non-zero only if the two on gurations L and L0 dier at most in one orbital, and the sum over all permutations P has only one non-zero term in this ase, determining the sign fa tor for that matrix element. The se ond line is non-zero only if the two on gurations dier at most in two orbitals, and the sum over all permutations has two non-zero terms in that ase: if P is a perturbation with lP k = lk0 for all k 6= i; j , then the orresponding ontribution is (1=2)[(lP ilP j jwjlj0 li0 ) ((lP j lP ijwjlj0 li0 )℄( 1)jPj. For L = L0 and P = identity (and sometimes also in the general ase) the rst matrix element is

alled dire t intera tion and the se ond one ex hange intera tion. 7.2 The Fo k spa e

Up to here we onsidered representations of quantum me hani s by wavefun tions with the parti le number N of the system xed. If this number is ma ros opi ally large, it annot be xed at a single de nite value in experiment. Zero mass bosons as e.g. photons may be emitted or absorbed in systems of any s ale. (In a relativisti des ription any parti le may be reated or annihilated, possibly together with its antiparti le, in a va uum region just by applying energy.) From a mere te hni al point of view, quantum statisti s of identi al parti les is mu h simpler to formulate with the grand anoni al ensemble with varying parti le number, than with the anoni al one. Hen e there are many good reasons to onsider quantum dynami s with hanges in parti le number. 38

In order to do so, we start with building the Hilbert spa e of quantum states of this wider frame: the Fo k spa e. The onsidered up to now Hilbert spa e of all N -parti le states having the appropriate symmetry with respe t to parti le ex hange will be denoted by HN . In the last subse tion we introdu ed a basis fLg in HN . Instead of spe ifying the multi-index L as a raw of N indi es li we may denote a basis state by spe ifying the o

upation numbers ni (being either 0 or 1) of all orbitals i: X ni = N: (113) jn1 : : : ni : : :i; i

Our previous determinantal state (107) is now represented as jL i = j0 : : : 01l1 0 : : : 01l2 0 : : : 01lN 0 : : :i: Two states (113) not oin iding in all o

upation numbers ni are orthogonal. HN is the omplete linear spa e spanned by the basis ve tors (113), i.e. the states of HN are either linear ombinations P jLiCL of states (113) (with the sum of the squared absolute values of the oeÆ ients CL equal to unity) or limits of Cau hy sequen es of su h linear ombinations. (A Cau hy sequen e is a sequen e fj n ig with limm;n!1 h m n j m n i = 0. The in lusion of all limits of su h sequen es into HN means realizing the topologi al ompleteness property of the Hilbert spa e, being extremely important in all onsiderations of limits. This ompleteness of the spa e is not to be onfused with the ompleteness of a basis set fi g. The extended Hilbert spa e F (Fo k spa e) of all states with the parti le number N not xed is now de ned as the ompleted dire t sum of all HN . It is spanned by all state ve tors (113) for all N with the above given de nition of orthogonality retained, and is ompleted by orresponding Cau hy sequen es, just as the real line is obtained from the rational line by ompleting it with the help of Cau hy sequen es of rational numbers. Note that F now ontains not only quantum states whi h are linear ombinations with varying ni so that ni does not have a de nite value in the quantum state (o

upation number u tuations), but also linear ombinations with varying N so that now quantum u tuations of the total parti le number are allowed too. (For bosoni elds as e.g. laser light those quantum u tuations an be ome important experimentally even for ma ros opi N .) 7.3 O

upation number representation

We now ompletely abandon the awful wavefun tions (105) and will ex lusively work with the o

upation number eigenstates (113) and matrix elements between them. The simplest operators are those whi h provide just a transition between basis states (113) whi h are as lose to ea h other as possible: those whi h dier in one o

upation number only. The de nition of these reation and annihilation operators for fermions must have regard to the antisymmetry of the quantum states and to Pauli's ex lusion prin iple following from this antisymmetry. They are de ned as P (114)

^i j : : : ni : : :i = j : : : ni 1 : : :i ni ( 1) j
^yi j : : : ni : : :i = j : : : ni + 1 : : :i (1 ni ) ( 1) j
upation number eigenstates (113), it is easily seen that these operators have all the needed properties, do parti ularly not reate non-fermioni states (that is, states with o

upation numbers ni dierenty from 0 or 1 do not appear: appli ation of ^i to a statey with ni = 0 gives zero, and appli ation of ^i to a state with ni = 1 gives zero as well). The ^i and ^i are mutually Hermitian

onjugate, obey the key relations n^ i j : : : ni : : :i  ^yi ^i j : : : ni : : :i = j : : : ni : : :i ni (116) and (117) [^ i ; ^yj ℄+ = Æij ; [^ i; ^j ℄+ = 0 = [^ yi ; ^yj ℄+ 39

with the anti ommutator [^ i ; ^yj ℄+ = ^i ^yj + ^yj ^i de ned in standard way. Reversely, the anoni al anti ommutation relations (117) de ne all the algebrai properties of the ^-operators and moreover de ne up to unitary equivalen e the Fo k-spa e representation (114, 115). (There are, however, vast

lasses of further representations of those algebrai relations with a dierent stru ture and not unitary equivalent to the Fo k-spa e representation.) The basis (113) of the Fo k spa e is systemati ally generated out of a single basis ve tor, the va uum state ji  j0 : : : 0i (with N =0) by applying ^y -operators: jn1 : : : ni : : :i = : : : ^yi : : : ^y1 ji: (118) Observe again the order of operators de ning a sign fa tor in view of (117) in agreement with the sign fa tors of (114, 115). Sin e produ ts of a given set of N ^y-operators written in any order agree with ea h other up to possibly a sign, all possible expressions (118) do not generate more dierent basis ve tors than those of (107) with the onvention on the order of the li as agreed upon there. Hen eforth, by using (118) we need not bother any more about the given linear order of the orbital indi es. With the help of the ^-operators, any linear operator in the Fo k spa e may be expressed. It is not diÆ ult to demonstrate that the Hamiltonian X 1 X ^y ^y(ij jwjkl)^ k ^l

^yi (ijhjj )^ j + H^ = (119) i j 2 ij

ijkl

has the same matrix elements with o

upation number eigenstates (113) as the Hamiltonian (109)) has with determinantal states in (110). Be ause of the one-to-one orresponden e between the determinantal states (107) and the o

upation number eigenstates and be ause both span the Fo k spa e, by linearity the Hamiltonians (109) and (119) are equivalent. The building prin iple of the equivalent of any linear operator given in the S hrodinger representation is evident from (119). The S hrodinger wavefun tion of a bosoni many-parti le quantum state must be totally symmetri with respe t to parti le ex hange (omission of the minus sign in (105)). The determinants are then to be repla ed by symmetrized produ ts (permanents), with a slightly more involved normalization fa tor. The orbitals may now be o

upied with arbitrary many parti les: ni = 0; 1; 2; : : : : This ase may be realized with bosoni reation and annihilation operators ^bi j : : : ni : : :i = j : : : ni 1 : : :ipni; (120) p ^byi j : : : ni : : :i = j : : : ni + 1 : : :i ni + 1; (121) y n^ i j : : : ni : : :i  ^bi ^bi j : : : ni : : :i = j : : : ni : : :ini : (122) with the anoni al ommutation relations [^bi; ^byj ℄ = Æij ; [^bi; ^bj ℄ = 0 = [^byi ; ^byj ℄ : (123) The basis states of the Fo k spa e are reated out of the va uum a

ording to ^by1
n1 ^byi
ni jn1 : : : ni : : :i = pn !


pn !


ji: (124) 1 i The order of these operators in the produ t does not make any dieren e. The hoi e of fa tors on the r.h.s. of (120, 121) not only ensures that (122) holds but also ensure the mutual Hermitian

onjugation of ^bi and ^byi . 7.4 Field operators

A spatial representation may be introdu ed in the Fo k spa e by de ning eld operators ^(x) = X i (x)^ai ; ^y (x) = X i (x)^ayi ; i

i

40

(125)

where the a^i mean either fermioni operators ^i or bosoni operators ^bi . The eld operators ^(x) and ^y (x) obey the anoni al (anti-) ommutation relations [ ^(x); ^y (x0 )℄ = Æ(x x0 ); [ ^(x); ^(x0 )℄ = 0 = [ ^y (x); ^y (x0 )℄: (126) They provide a spatial parti le density operator X ^y (r; s) ^(r; s) n^ (r) = (127) s

having the properties hn(r)i =

XX ij

s

i (r; s)ha^yi a^j ij (r; s);

Z

d3 r n^(r) =

X i

a^yi a^i :

(128)

These relations are readily obtained from those of the reation and annihilation operators, and by taking into a

ount the ompleteness and orthonormality (106) of the orbitals i . In terms of eld operators, the Hamiltonian (109) or (119) reads XZ H^ = d3 r ^y (r; s) h^ ss0 (r) ^(r; s0 ) + ss0

+ 21

X s1 s01 s2 s02

Z

d3 r1 d3 r2 ^y (r1 ; s1 ) ^y (r2 ; s2 ) ws1 s01 ;s2 s02 (r1 ; r2 )

It is obtained by ombining (119) with (125) and (111, 112).

41

^(r2; s02) ^(r1; s01): (129)

8

MICROSCOPIC THEORY: THE BCS MODEL

The great advantage of the use of reation and annihilation or eld operators lies in the fa t that we

an use them to manipulate quantum states in a physi ally omprehensible way without expli itly knowing the wavefun tion. We even an think of modi ed operators of whi h we know little more than their algebrai properties. The point is that the o

upation number formalism applies for every orbital set (106). The transition from one set of operators obeying anoni al (anti-) ommutation relations to another su h set is alled a anoni al transformation in quantum theory. 8.1 The normal Fermi liquid as a quasi-parti le gas

A normal ondu ting Fermi liquid has a fermioni quasi-parti le ex itation spe trum whi h behaves very mu h like a gas of independent parti les with energies k (for the sake of simpli ity we assume it isotropi in k-spa e although this assumption is not essential here). Non-intera ting fermions would have a ground state with all orbitals with  <  o

upied and all orbitals with  >  empty;  is the hemi al potential. By adding or removing a fermion with  =  new ground states with N  1 fermions are obtained. By adding a fermion with  >  an ex ited state is obtained with ex itation energy  . By removing a fermion with  <  | that is, reating a hole in the original ground state | an ex ited state is obtained with ex itation energy j j: rst lift the fermion to the level  and then remove it without hanging the hara ter of the state any more (Figs. 33 and 34).  

k

quasi-ele tron



jk j

0

k

kF

quasi-hole kF FIG. 33: Creation of an ex ited ele tron and of a hole, resp.

k

FIG. 34: Ex itation spe trum of a Fermi gas.

The ground state j0i of a normal metal has mu h the same properties: ondu tion ele trons with and holes with  <  may be ex ited with ex itation energies as above. These are not the original ele trons making up the metal together with the atomi nu lei. Rather they are ele trons or missing ele trons surrounded by polarization louds of other ele trons and nu lei in whi h nearly all the Coulomb intera tion is absorbed. We do not pre isely know these ex itations nor do we know the ground state j0i (although a quite elaborate theory exists for them whi h we ignore here). We just assume that they may be represented by fermioni operators with properties like those in the gas: (130) k <  :

^yk j0i = 0; ^yk ^k j0i = j0i;

>

k >  :

^k j0i = 0; ^k ^yk j0i = j0i;

42

(131)

k is the waveve tor and 

the spin state of the quasi-parti le. Sin e all intera tions present in the ground state j0i are already absorbed in the quasi-parti le energies k , only ex ited ondu tion ele trons or holes exert a remainder intera tion. Hen e, we may write down an ee tive Hamiltonian  >;  > ^H~ = X ^yk (k )^ k + 1 k Xk0 ^yk+q ^yk0 q0 wkk0 q ^k0 0 ^k + 2 k; k0 0 ; q k +

k >; k0 < X

^yk+q ^k0

k; k0 0 ; q k <; k0 < X

+ 21

k; k0 0 ; q

q0

^k+q ^k0

wkk0 q ^yk0 0 ^k

q0

+

wkk0 q ^yk0 0 ^yk :

(132)

The matrix elements in the three lines are qualitatively dierent: they are predominantly repulsive in the rst and last line and attra tive in the se ond line; ele trons and holes have opposite harges. With the relations (130, 131) one nds easily h0jH^~ j0i =

X k <

k

(k

H^~ j0i = j0iE~0 :

) = E~0

(133)

We subtra t this onstant energy from the Hamiltonian and have H^  H^~ E~0 ; h0jH^ j0i = 0 H^ j0i = 0: (134) We may reate a quasi-parti le above  in this ground state: (135) jk1 1 i = ^yk1 1 j0i; H^ jk1 1 i = jk1 1 i (k1 ): k1 >  : The last relation is easily veri ed with our previous formulas. For k 6= k1 1, we have ^yk ^k ^yk11 =

^yk1 1 ^yk ^k , and together with ^k1 1 ^yk1 1 j0i = j0i one nds the above result. Likewise (136) jk1 1 i = ^k1 1 j0i; H^ jk1 1 i = jk1 1 i jk1 j k1 <  : is obtained. Here, ^yk11 ^k1 1 ^k11 = 0, so that one term has to be removed from the sum of (133). Hen e, the single-parti le ex itation spe trum of our ee tive Hamiltonian above the state j0i is just k = jk j; (137) and the ex ited states are jk11 i of (135, 136), whatever the wavefun tion of j0i might be. To say the truth, this all is only approximately right. There are no fermioni operators for whi h the relations (130, 131) hold true exa tly for the true ground state j0i. Therefore, the rst relation (133) and the last relations (135, 136) are also not rigorous. The quasi-parti les have a nite lifetime whi h may be expressed by omplex energies k . However, for jk j   the approximation is quite good in normal, weakly orrelated metals. Consider now a state with two ex ited parti les: (138) jk1 1 k2 2 i = ^yk1 1 ^yk2 2 j0i: To be spe i we onsider two ex ited ele trons, the ases with holes or with an ele tron and a hole are ompletely analogous. The appli ation of the ee tive Hamiltonian yields X H^ ^yk1 1 ^yk2 2 j0i = ^yk1 1 ^yk2 2 j0i (k1 + k2 ) + (139)

^yk1 +q1 ^yk2 q2 j0i wk1 k2 q : q

43

The intera tion term is obtained with the rule kii ykii j0i = j0i. One ontribution appears from k = k1 1 ; k0 0 = k2 2 ; and another ontribution ^yk2 +q2 ^yk1 q1 j0i wk2 k1 q from k0 0 = k1 1 ; k = k2 2 : The minus sign in this ontribution is removed by anti ommuting the two ^y -operators, and then, by repla ing q with q under the q-sum and observing wk2 k1 q = wk1 k2 q whi h derives from w(r1 ; r2) = w(r2 ; r1), this se ond ontribution is equal to the rst one, when e omitting the fa tor (1/2) in front. (This is how ex hange terms appear automati ally with ^-operators sin e their anti ommutation rules automati ally retain the antisymmetry of states.) For the simpli ity of writing we omitted here and in (132) the spin dependen e of the intera tion matrix element. It is always present in the ee tive quasi-parti le intera tion, and it may always be added afterwards without

onfusion. (We might introdu e a short-hand notation k for k.) The ee tive intera tion of two quasi-parti les with equal spin diers from that of two quasi-parti les with opposite spin. The spin ip s attering of quasi-parti les | an intera tion with hanging 1 and 2 into 10 and 20 | may often be negle ted. Then, the q-sum of (139) need not be ompleted by additional spin sums. 8.2 The Cooper problem1

From (139) it an be seen that the state (138) is not any more an eigenstate of the ee tive Hamiltonian, not even within the approximations made in the previous subse tion. The two ex ited quasi-parti les intera t and thus form a orrelated pair state. We try to nd this pair state of lowest energy for two ele trons ( > 0) within our approximate approa h. Sin e we expe t that this state is formed out of quasi-parti le ex itations with energies   0 our approximations annot be riti al. (The quasi-parti le lifetime be omes in nite for jj ! 0:) We expe t that the state lowest in energy has zero total momentum, hen e we build it out of quasi-parti le pairs with k2 = k1: X j i = ak jk k0 i: (140) k

where we assume a xed ombination of  and 0 and the yet unknown expansion oeÆ ients to depend on k only, be ause the sought state is to be expe ted to have a de nite total spin. (Re all that we are onsidering an isotropi metal in this hapter.) We want that this pair state j i is an eigenstate of H^ : X X H^ j i = j iE; H^ j i =

^yk ^y k0 j0i 2k ak +

^yk+q ^y k q0 j0i wk kq ak : (141) k

kq

Multiply the last relation with h0j ^ k0 0 ^k0  and observe h0j ^ k0 0 ^k0  ^yk ^y k0 j0i = h0j ^ k00 (Ækk0

^yk ^k0  )^ y k0 j0i = Ækk0 h0j(Ækk0 ^y k0 ^ k0 0 )j0i h0j ^ k0 0 ^yk (Æ kk0 Æ0 ^y k0 ^k0  )j0i = Ækk0 h0j(Æ kk0 Æ0 ^yk ^ k0 0 )Æ kk0 Æ0 j0i = Ækk0 Æ kk0 Æ0 to obtain E ak0

a k0 Æ0




= 2k0

ak0

a k0 Æ0




+

X

q

wk0

q; k0 +q;q

a k0

q




a k0 +q Æ0 :

(142)

In the last term, we also used again w k0 q;k0+q;q = wk0 +q; k0 q; q and then repla ed the sum over q by a sum over q. Due to the isotropy of our problem we expe t the solution to be an angular momentum eigenstate, hen e ak should have a de nite parity. It is immediately seen that a non-trivial solution with even parity a k = ak (even angular momentum) is only possible, if Æ0 = 0, that is for a singlet 0 = . For a spin triplet 0 =  only a non-trivial solution with odd parity (odd angular momentum) is possible. To be spe i , onsider the singlet ase. (The triplet ase is analogous.) Assume (143) ak = ak Ylm (k=k) 1 L.

N. Cooper, Phys. Rev. 104, 1189 (1956).

44

with even l: In (142), rename k0 ! k, k0 q ! k0. The matrix element wk0 ; k0 ;k s attering amplitude from states k; k into states k0; k0. we use an expansion X  (k0 ): wk0 ; k0 ;k k0 = l wkl wkl0 Ylm (k)Ylm lm

This redu es (142) to

ak =

X l wkl C ; C= wkl0 ak0 : Elm 2k 0 k

k0

determines the (144) (145)

Inserting the left relation into the right one yields 1 = l

X k

jwkl j2 E 1 2 = l F (Elm ): lm k

(146)

The sought lowest pair energy orresponds to the lowest solution of F (Elm ) = 1=l. The fun tion F (Elm ) has poles for Elm = 2k where it jumps from 1 F to +1. Re all that the k -values are all positive and start from zero. For Elm ! 1, F (Elm ) approa hes zero from negative values. Hen e, if l > 0, then the lowest solution Elm of (146) is positive and the ground state j0i of the normal metal 1 l > 0 is stable. If at least one l -value is negative (at0 E tra tive intera tion), then 1 there is unavoidably a negl < 0 ative solution Elm of (146): Elm the `ex ited pair' has negative energy and the normal 2k ground state j0i is unstable against forming of pairs of bound quasi-parti les, no matter how small jl j is (how weak the attra tive intera tion is). Pairs are spontaneously formed and the ground state re onFigure 35: The fun tion F (E ) from Eq. (146). stru ts. This is the ontent of Cooper's theorem. If the intera tion is ut of at some energy ! ,  0 <  k < ! ; wkl = 10 for (147) elsewhere and the density of states for k is nearly onstant in this interval, N () = N (0); then, with negative Elm , 45

X

hen e,

k

j j2 wkl

Elm

1

2k =

N (0)

Z !

0

jElm j =

This yields

8 > <

1 = d jElm j + 2 h

exp

2!

2 N (0)jl j

"

i

1

N (0)

"

#

jElm + 2! ; ln 2 jElm j

(148)

:

#

2 (149) jElm j  > 2! exp N (0)jl j for N (0)jl j   1 : N (0)jl j! in the weak and strong oupling limits. In this hapter we only onsider the weak oupling limit where jElm j is exponentially small. The whole analysis may be repeated for the ase where the pair has a non-zero total momentum q. In that ase the denominator of (146) is to be repla ed with Elm (q) k+q=2  k+q=2 where now jk  q=2j must be larger than kF . For small q; this ondition redu es the density of states in ee t in an interval of thi kness j= kjq=2 = vF q=2 at the lower -integration limit; vF is the Fermi velo ity. The result is Elm (q)  Elm + vF q=2: (150) In the weak oupling limit, Elm(q) an only be negative for exponentially small q. We performed the analysis with a pair of parti les. It an likewise be done with a pair of holes with an analogous result. 8.3 The BCS Hamiltonian

Frohli h1 was the rst to point out that the ele tron-phonon intera tion is apable of providing an ee tive attra tion between ondu tion ele trons in the energy range of phonon energies. From Cooper's analysis it follows that, if there is a weak attra tion, it an only be ee tive for pairs with zero total momentum, that is, between k and k. With the assumption that the attra tion is in the l = 0 spin singlet hannel, this led Bardeen, Cooper and S hrieer2 to the simple model Hamiltonian H^ BCS =

X

k

^yk (k

)^ k

g V

 ! <X k ;k0 <+!

kk0

^yk0 " ^y k0 # ^ k# ^k" :

(151)

Here, g > 0 is the BCS oupling onstant, and V is the normalization volume. Sin e the density of plane-wave states in k-spa e is V=(2)3: Pk = V=(2)3 R d3 k, the matrix element of an n-parti le intera tion (appearing in an n-fold k-sum) must be proportional to V (n 1) in order that the Hamiltonian is extensive ( V ). The modeled attra tive intera tion is assumed in an energy range of width 2! around the hemi al potential (Fermi level in the ase T = 0), where ! is a hara teristi phonon energy for whi h the Debye energy of the latti e an be taken. The state j0i of (130, 131) annot any more be the ground state of this Hamiltonian sin e Cooper's theorem tells us that this state is unstable against spontaneous formation of bound pairs with the gain of their binding energy. The problem to solve is now to nd the ground state and the quasi-parti le spe trum of the BCS-Hamiltonian. This problem was solved by Bardeen, Cooper and S hrieer, and, shortly thereafter and independently by means of a anoni al transformation, by Bogoliubov and Valatin. Bardeen, Cooper and S hrieer thus provided the rst mi ros opi theory of super ondu tivity, 46 years after the dis overy of the phenomenon. 1 H. Fr ohli h, Phys. Rev. 79, 845 (1950). 2 J. Bardeen, L. N. Cooper, and J. R. S hrieer,

Phys. Rev. 108, 1175 (1957).

46

8.4 The Bogoliubov-Valatin transformation1

Suppose the ground state ontains a bound pair. Ex iting one parti le of that pair leaves its partner behind, and hen e also in an ex ited state. If one wants to ex ite only one parti le, one must annihilate simultaneously its partner. Led by this onsideration, for the quasi-parti le operators in the ground state of (151) an ansatz ^bk" = uk ^k" vk ^y k#; ^bk# = uk ^k# + vk ^y k" is made. uk and vk are variational parameters. Again we onsider the isotropi problem and hen e their dependen e on k = jkj only. The reason of the dierent signs in the two relations be omes

lear in a minute. Sin e for ea h orbital annihilated by ^k" ; ^ k#; ^bk"; ^b k# an r-independent phase fa tor may be arbitrarily hosen, uk and vk may be assumed real without loss of generality. These Bogoliubov-Valatin transformations together with their Hermitian onjugate may be summarized as ^bk = uk ^k vk ^y k  ; ^byk = uk ^yk vk ^ k  : (152) We want these transformations to be anoni al, that is, we want the new operators ^bk ; ^byk again to be fermioni operators. One easily al ulates   [^bk ; ^bk0 0 ℄+ = uk vk0 0[^ k ; ^y k0 0 ℄+ + [^ y k  ; ^k0 0 ℄+ = = uk vk0 0Æk k0 Æ 0 + Æ kk0 Æ 0
= uk vk Æ kk0 (  + ) = 0: In the rst equality it was already onsidered that annihilation and reation operators ^ and ^y , respe tively, anti ommute among themselves. The analogous result for the ^by-operators is obtained in the same way. The sign fa tor  in the transformation ensures that the anti ommutation is retained for the ^b- and ^by-operators, respe tively. Analogously, [^bk ; ^byk0 0 ℄+ = u2k [^ k ; ^yk0 0 ℄+ + 0 vk2 [^ y k  ; ^ k0 0 ℄+ = u2k + vk2
Ækk0 Æ0 ; hen e the ondition u2k + vk2 = 1 (153) ensures that the transformation is anoni al and the new operators are again fermioni operators. Multiplying the rst relation (152) by uk , repla ing in the se ond one k with k , multiplying it with vk , and then adding both results yields with (153) the inverse transformation (154)

^k = uk^bk + vk ^by k  ; ^yk = uk^byk + vk^b k  : Observe the reversed sign fa tor. The next step is to transform the Hamiltonian (151). With   

^yk ^k = uk^byk + vk^b k  uk^bk + vk^by k  =   = u2k^byk^bk + vk2^b k ^by k  + uk vk ^byk^by k  + ^b k ^bk and the anti ommutation rules it is easily seen that the single-parti le part of the BCS-Hamiltonian transforms into   X X X X 2 (k )vk2 + (k )(u2k vk2 ) ^byk^bk + 2 (k )uk vk ^byk"^by k# + ^b k#^bk" : k

k



k

It has also been used that under the k-sum k may be repla ed by k: Further, with    B^k = ^ k# ^k" = uk^b k# vk^byk" uk^bk" + vk^by k# =   = u2k^b k#^bk" vk2^byk"^by k# + uk vk ^b k#^by k# ^byk"^bk" ; 1 N.

N. Bogoliubov, Nuovo Cimento 7, 794 (1958); J. G. Valatin, Nuovo Cimento 7, 843 (1958).

47

(155)

the full transformed BCS-Hamiltonian reads X X H^ BCS = 2 (k )vk2 + (k k X

)(u2k

k  uk vk byk" by k#

+2 (k k

+ ^b k#^bk"

^ ^

)

vk2 )

X  

^byk^bk + g V

X

kk0

B^ky 0 B^k ;

(156)

where for brevity we omitted the bounds of the last sum. Now, we introdu e the o

upation number operators n^ k = ^byk ^bk

(157)

of the ^b-operators and assume analogous to (130, 131) that for properly hosen ^b-operators the ground state j 0i of the BCS-Hamiltonian is an o

upation number eigenstate. Then, its energy is found to be E=2

X

k

(k

g V

X )vk2 + (k )(u2k vk2 )(nk" + nk# )

k

"

X

k

#2

uk vk (1 nk" nk# ) :

(158)

The nk are the eigenvalues (0 or 1) of the o

upation number operators (157) in the ground state

j 0 i.

This energy expression still ontains the variational parameters uk and vk whi h are onne ted by (153), when e vk =uk = uk =vk . For given o

upation numbers, (158) has its minimum for E uk

"

= 4(k

u2 v2 )uk + 2
k k vk


= Vg

X

k

uk vk

#

1

nk" nk#




= 0;




1

(159)

nk" nk# :

Hen e, uk and vk are determined by (153) and 2(k

)uk vk =
u2k




vk2 :

(160)

Their ombination yields a biquadrati equation with the solution u2k vk2



"

= 21 1  p (k 2 ) 2 (k ) +


#

;

2uk vk = p

Insertion into (159) results in the self- onsisten y ondition X 1 = 2gV p1 nk" 2 nk# 2 ; (k ) +
k

(k


: )2 +
2

(161)

(162)

whi h determines
as a fun tion of the BCS oupling onstant g, the dispersion relation k of the normal state ^-quasi-parti les (in essen e the Fermi velo ity), and the o

upation numbers nk of the ^b-quasi-parti les of the super ondu ting state (in essen e the temperature as seen later). 48

The parameters and vk are de2
pi ted in the gure vk uk on the right. For1 mally, inter hanging uk with vk would also be a solution of the biquadrati equation. It is, however, easily seen that k  this would not lead to a minimum of (158). With (161) it is readily seen that Figure 36: The fun tions u and v. the se ond sum of (158) is positive definite. Hen e, the absolute minimum of energy (ground state) is attained, if all o

upation numbers of the ^b-orbitals are zero. For the ground state, the self- onsisten y ondition (162) redu es to uk

1=

g 2V

X

p

(k

k

1

)2 +
20

gN (0)

= 2

p

1 = gN (0) ln p! 2 +
20 + !  d! p 2 2 ! +
20 ! 2 +
20 ! !

Z !

2 0

 gN2(0) ln 4
!2

resulting in



 1
0 = 2! exp gN (0) (163) for the value of
in the ground state (atPzero temperature). If one repla es the last termP (g=V )  B^ky 0 B^k of the transformed BCS-Hamiltonian (156) by the mean- eld approximation
k B^k +B^ky (re all that
was introdu ed as
= (g=V )h 0j Pk B^k j 0i,

f. (155, 159)), than it is readily seen that the relation (160) makes the anomalous terms (terms ^by^by or ^b^b) of this Hamiltonian vanish: In mean- eld approximation the BCS-Hamiltonian is diagonalized by the Bogoliubov-Valatin transformation, resulting in

H^ m-f

X

= 2 (k k

= onst. + = onst. +

X )vk2 + (k Xh

k X k

k

vk2 )^byk ^bk

)(u2k

2


i )(u2k vk2 ) + 2
uk vk ^byk^bk =

(k k ^byk^bk

X

k

uk vk



1

X 



^byk^bk = (164)

with the ^b-quasi-parti le energy dispersion relation k =

p

(k

)2 +
2

obtained by inserting (161) into the se ond line of (164). 49

(165)

The dispersion relation k together with the normal state dispersion relation k  and the normal state ex itation energy dispersion k jk j is depi ted on the right. It is seen that the physi al meaning of
is the gap in the ^b-quasiparti le ex itation spe trum of the jk j super ondu ting state. The name bogolons is
 often used for these quasi-parti les. fF k The fa t that the Bogoliubov-Valatin transformation diagonalizes the BCS-Hamiltonian at least in mean- eld approximation justi es a posteriori our assumption that the ground k  state may be found as an eigenstate of the ^bo

upation number operators. In the litera- Figure 37: Quasi-parti le dispersion relation in the ture, the key relations (160) are often derived as super ondu ting state. diagonalizing the mean- eld BCS-Hamiltonian instead of minimizing the energy expression (158). In fa t both onne tions are equally important and provide only together the solution of that Hamiltonian. Clearly, the BCS-theory based on that solution is a mean- eld theory. From (152) one ould arrive at the on lusion that a bogolon onsists partially of a normal-state ele tron and partially of a hole, and hen e would not arry an integer harge quantum. However, as rst was pointed out by Josephson, the true ^b-quasi-parti le annihilation and reation operators are (166) ^k = uk ^k vk P^ ^y k  ; ^ky  = uk ^yk vk ^ k  P^ y ; where P^ annihilates and P^y reates a bound pair with zero momentum and zero spin as onsidered in the Cooper problem. Its wavefun tion will be onsidered in the next hapter. Now, a bogolon is annihilated, that is, a hole-bogolon is reated, by partially reating a normal-state hole and partially annihilating an ele tron pair and repla ing it with a normal-state ele tron. The se ond part of the pro ess also reates a positive harge, when e the hole-bogolon (with jkj < kF ) arries an integer positive harge quantum. Likewise, an ele tron-bogolon (with jkj > kF ) is reated partially by reating a normal-state ele tron and partially by reating an ele tron pair and simultaneously annihilating a normal-state ele tron. Again, the bogolon arries an integer (negative) harge quantum. The P^ operators make the ^b-bogolon be surrounded by a super ondu ting ba k ow of harge whi h ensures that an integer harge quantum travels with the bogolon. It is positive for jkj < kF and negative for jkj > kF . Of ourse, it remains to show that the new ground state j 0i is indeed super ondu ting.

50

9

MICROSCOPIC THEORY: PAIR STATES

In mean- eld BCS theory, the ground state is determined by the omplete absen e of quasi-parti les. With the properties of the Bogoliubov-Valatin transformation, this ground state is found to be the

ondensate of Cooper pairs in plane-wave K = 0 states of their enters of gravity. By o

upying quasi-parti le states a

ording to Fermi statisti s, thermodynami states of the BCS super ondu tor are obtained. 9.1 The BCS ground state

In Se tion 8.D the BCS ground state j 0i was assumed to be an o

upation number eigenstate of n^ k = ^byk ^bk ; and the ^bk were determined a

ordingly. Then, it was found after (162) that all o

upation numbers n^k are zero in the ground state. This implies that h 0j^byk^bk j 0i = 0; and hen e ^bk j 0i = 0 (167) for all k: The (mean- eld) BCS ground state is the (uniquely de ned) ^b-va uum. We show that j 0 i =

Y

k


uk + vk ^yk" ^y k# ji

(168)

is the properly normalized state with properties (167). The normal metal ground state j0i of (130, 131) is j0i =

Y k <

k

^yk ji =

Y k <

k

^yk" ^y k# ji

(169)

and hen e has the form (168) too, with uk = 0 for k <  and uk = 1 for k >  and the opposite behavior for vk . We now demonstrate the properties of (168). Sin e





hj uk + vk ^ k# ^k" uk + vk ^yk" ^y k# ji = u2k + vk2 = 1; j 0 i of (168) is properly normalized. Moreover,

^bk0 "j 0i = = =

"

Y

k(6=k0 ) " Y "

k(6=k0 ) Y

k(6=k0 )

#


uk + vk ^yk" ^y k# uk0 ^k0 " #



vk0 ^y k0 # uk0 + vk0 ^yk0 " ^y k0 # ji =


uk + vk ^yk" ^y k# uk0 vk0 ^k0 " ^yk0 " ^y k0 # #


 uk + vk ^yk" ^y k# uk0 vk0

^y k0 #




ji = 

1 ^yk0 " ^k0 "
^y k0# ^y k0 # ji = 0

(170)

and analogously ^bk0 #j 0i = 0. This ompletes our proof. Histori ally, Bardeen, Cooper and S hrieer solved the BCS model with the ansatz (168), before the work of Bogoliubov and Valatin. 51

9.2 The pair fun tion

Next we nd the wavefun tions ontained in j 0i: Re all, that ^yk reates a ondu tion ele tron in the plane-wave state  exp(ik r) (s) so that the eld operator ^y (rs) is ^y(rs) = X ^yk exp(ik r) (s): (171)





k

(That is, rs denotes the enter of gravity and the spin of the ele tron with its polarization loud whi h together make up the ` ondu tion ele tron'.) The N -parti le wavefun tion ontained in j 0i and depending on these variables is Y 0(x1 ; : : : ; xN ) = hx1 : : : xN j 0i = hj ^(xN )


^(x1 ) uk + vk ^yk" ^y k#
ji: (172) k

For the sake of brevity we suppress the spinors , and nd 0(x1 ; : : : ; xN ) = Y Y X 1 ^kN 1N 1


eik1 r1 ^k1}1 1 + gk ^yk" ^y k# uk ji = = hj e|ikN rN ^kN N eikN 1 rN {z k1 :::kN

=  

with

X

k1 :::kN X0

fk2i g A(r1










k

=0; if not ki i 6=kj j for i6=j

p p

q q

N= Y2 0 i(k1
r1 +


+kN
rN ) hj ^kN N


^k1 1 gk ^yk" ^y k# ji e

k

{z

|

Y

k

}

sum over all possible ontra tions

eik2
(r1

r2 ) g

k2 Æ1 2

r2 )singlet (r3




eikN (rN


1

rN ) g Æ kN  N

r4 )singlet


(rN 1

() 

X

k

gk eik
 ;

!

uk



!

1

N




=

rN )singlet

gk =

k

vk : uk

(173) (174)

In the se ond line of (173), (171) was inserted for the ^(xi ) of (172), and the uk s were fa tored out of the produ t of (172) (leaving gk = vk =uk behind in the se ond item of the fa tors). The k-produ ts run over all grid points of the (in nite) k-mesh, e.g. determined by periodi boundary onditions for the sample volume V , while the sum runs over all possible produ ts for sets of N disjun t k-values out of that mesh. This disjun t nature of the k-sums is indi ated by a dash at the sum in the following lines. Expansion of the rst k-produ t yields terms with 0, 2, 4, . . . ^y -operators of whi h only the terms with exa tly those N ^y -operators that orrespond to the N ^-operators of an item of the k-sum left to the produ t produ e a non-zero result between the ^-va uum states hj


ji. These results are most easily obtained by anti ommuting all annihilation operators to the right of all reation operators and are usually alled ontra tions; depending on the original order of the operators, ea h result is 1. The produ t over the uk , whi h multiplies ea h ontribution and whi h as previously runs over all in nitely many k-values of the full mesh, yields a normalizing fa tor whi h is independent of the values of k1; : : : ; kN of the sums. Like ea h individual fa tor uk , it depends on the hemi al potential  and on the gap
0 . At this point one must realize that the gk are essentially non-zero inside of the Fermi surfa e ( f. (161)). Hen e, the ontribution to 0(x1 ; : : : ; xN ) has a non-negligible value only for all ki-ve tors inside the Fermi surfa e, and this value in reases with an in reasing number of su h ki -ve tors and de reases again, if an appre iable number of ki -ve tors falls outside of the Fermi surfa e: the norm of (172) is maximal for N -values su h that the k2i o

upy essentially all mesh points inside the Fermi surfa e. That is, this norm is non-negligible only for those N -values orresponding to 52

the ele tron number in the original normal Fermi liquid: j 0i is a grand- anoni al state with a sharp parti le-number maximum at the anoni al N -value. We did not tra e normalizing fa tors in (173) and use a sign of approximation in (174) again, assumig () to be normalized. The result is a state j 0i onsisting of pairs of ele trons in the geminal (pair-orbital) ()singlet . In order to analyze what this pair-orbital () looks like, re all that uk and vk may be written as fun tions of
0 and (k )=
0  ~vF k=
0 ( f. (161)). Hen e, gk  g~(~vF k=
0), and ()

 

Z 1 Z 1 Z 1 eik e ik  d3 kgk eik
  dkk2 gk deik = dkk2 gk ik 0 1 0     1 Z 1 dkkg~ ~vF k sin k  1 Z 1 dxxg~(x) sin
0x  f (=0);  0
0  0 ~vF 1 ~vF 0  = : Æk 
0 Z

(175)

By omparison with (173),  is the distan e ve tor of the two ele trons in the pair, their distan e on average being of the order of 0 , while the pair-orbital  does not depend on the position R of the enter of gravity of the pair: with respe t to the enter of gravity the pair is delo alized, it is a plane wave with wave ve tor K = 0. Moreover, again due to (173), all N=2  1023 ele tron pairs o

upy the same delo alized pair-orbital  in the BCS ground state j 0i : this ma ros opi ally o

upied delo alized (that is, onstant in R-spa e) pair-orbital is the ondensate wavefun tion of the super ondu ting state, and hen e the stru ture (173) of j 0i ensures that the solution of the BCS model is a super ondu tor. For a real super ondu tor, the gap
0 an be measured (for instan e by measuring thermodynami quantities whi h depend on the ex itation spe trum or dire tly by tunneling spe tros opy. With the independently determined Fermi velo ity vF of ele trons in the normal state, this measurement yields dire tly the average distan e 0 of the ele trons in a pair whi h an be ompared to the average distan e rs of to arbitrary ondu tion ele trons in the solid given by the ele tron density. For a weakly oupled type I super ondu tor this ratio is typi ally 0 rs

 103 : : : 104:

(176)

In this ase, there are 109 : : : 1012 ele trons of other pairs in the volume between a given pair: There is a pair orrelation resulting in a ondensation of all ele trons into one and the same delo alized pair orbital in the super ondu ting state, however, the pi ture of ele trons grouped into individual pairs would be by far misleading. In order to reate a super urrent, the ondensate wavefun tion, that is, the pair orbital must be provided with a phase fa tor (; R)  eiK R (177) by repla ing the reation operators in (168) with ^yk+K=2" ^y k+K=2#: Obviously, it must be K0  1 in order not to deform (and thus destroy) the pair orbital itself. Hen e, 0 has the meaning of the


oheren e length of the super ondu tor at zero temperature.

9.3 Non-zero temperature

The transformed Hamiltonian (164) shows that the bogolons reated by operators ^ky  of (166) and having an energy dispersion law k of (165) are fermioni ex itations with harge e and spin  above the BCS ground state j 0i: Sin e they may re ombine into Cooper pairs , the hemi al potential of the Cooper pairs must be 2 where  is the hemi al potential of bogolons. Hen e, at temperature T the distribution of bogolons is 1 : (178) nk" = nk# = k =kT e +1 53

The gap equation (162) then yields ( f. the analysis leading to (163)) p
1 Z d! ep!2 +
2 =kT 1 gN (0) ! g X 1 2 ek =kT + 1 p = 2 1 = 2V = 2 2 k ! ! 2 +
2 e ! +
=kT + 1 k p 2 2! Z ! d! (179) = gN (0) p!2 +
2 tanh !2kT+
0 where in the last equation the symmetry of the integrand with respe t to a sign hange of ! was used. For T ! 0, with the limes tanh x ! 1 for x ! 1, (163) is reprodu ed. For in reasing temperature, the numerator of (162) de reases, and hen e
must also de rease. It vanishes at the transition temperature T , when e Z ! =2kT

Z !

dx ! = gN (0) tanh tanh x: (180) 1 = gN (0) d! 2kT x 0 0 ! Integration of the last integral by parts yields Z ! =2kT Z ! =2kT dx ln x + ln ! tanh !  dx tanh x= 2x x 2kT 2kT

osh 0 0 Z 1 ln x + ln ! = ln 4 + ln ! = ln 2 ! ;  dx  2kT kT

osh2 x 2kT 0 where ln = C  0:577 is Euler's onstant. The se ond line is valid in the weak oupling ase kT  ! and the nal result for that ase is ) (

2

1 = kT = ! exp
(181)  gN (0)  0 with 2 =  1:13; and 2
0=kT = 2=  3:52: The gap
as a fun tion of
temperature between zero and T
0 must be al ulated numeri ally from (178). However, the simple expression s


(T ) 
0 1



 T 3 T

(182)

is a very good approximation. From the energy expression (158), the thermodynami quantities an be al ulated, on e
(T ) and hen e k (T ) is given. The main results are the ondensation energy at T = 0,

0

T

T

Figure 38: The gap as a fun tion of T . B (0)2 20

= 12 N (0)
20; yielding the thermodynami riti al eld at zero temperature, the spe i heat jump at T , Cs C n  1:43; Cn and the exponential behavior of the spe i heat at low temperatures, Cs (T )  T 3=2 e
0=kT for T  T : 54

(183) (184) (185)

10

MICROSCOPIC THEORY: COHERENCE FACTORS

As already done in the last se tion, thermodynami states of a super ondu tor are obtained by o

upying quasi-parti le states with Fermi o

upation numbers. External elds, however, in most ases

ouple to the ^-operator elds. Due to the oupling of ^-ex itations in a super ondu tor interferen e terms appear in the response to su h elds. 10.1 The thermodynami state Let fkg be any given disjun t set of quasi-parti le quantum numbers. Then, Y ky  j 0i j fkg i =

(186)

k2fkg

is a state with those quasi-parti les ex ited above the super ondu ting ground state j 0i: If there are many quasi-parti le ex itations present, they intera t with ea h other and with the ondensate in the ground state (the latter intera tion is in terms of the operators P^ ^yk and ^kpP^y), and this intera tion leads to the temperature dependen e of their energy dispersion law k = (k )2 +
2 via the temperature dependen e
=
(T ): The thermodynami state is X j fkg i P =

Q

k2fkg fk

Z

fkg

h fkg j;

(187)

where the summation is over all possible sets of quasi-parti le quantum numbers, 1 fk = exp(k =kT ) + 1

(188)

is the Fermi o

upation number, and Z = Z (T; ) is the partition fun tion determined by X trP =

Q

k2fkg fk

Z

fkg

= 1:

(189)

As usual, the thermodynami expe tation value of any operator A^ is obtained as trA^P: For instan e the average o

upation number itself of a quasi-parti le in the state k00 is Q   X j fkg i fk h fkg j = nk0 0 = tr ^ky 0 0 ^k0 0 =

X

fkg

h fkg

fkg j ^y 0 0 ^k0 0 j k

Z

Q

fk fkg i Z

=

fk0 trP = fk0 :

(190)

The state ^k00 j fkgi is obtained from the state j fkg i by removing the quasi-parti le k00 : Hen e, if one fa tors out fk0 , the remaining sum is again P. 10.2 The harge and spin moment densities

The operator of the q Fourier omponent of the harge density of ele trons in a solid is  X X n^ (q) = e ^yk+q ^k = e

^yk+q" ^k" + ^y k# ^ k q# :

(191)

That of the spin moment density (in z-dire tion) is X X m ^ (q) = B ^yk+q  ^k = B ^yk+q" ^k" ^y k# ^

(192)

k

k

k

k

55

k q#



:

The statisti al operator P of a normal metalli state is omposed in analogy to (187) from eigenstates j fkg i of ^-operators. Then, in al ulating thermodynami averages, ea h item of the k-sum of (191) and (192) is averaged independently. In the super ondu ting state, the items in parentheses of the last of those expressions are oupled and hen e they are not any more averaged independently: there appear ontributions due to their oherent interferen e in the super ondu ting states j fkg i: Those ontributions appear in the response of the super ondu ting state to external elds whi h ouple to harge and spin densities. Performing the Bogoliubov-Valatin transformation for the harge density operator yields  X

^yk+q" ^k" + ^y k# ^ k q# = k

= =

 X 

k X

k

vk^bk" ujk+qj^b

ujk+qjuk

+

ujk+qjvk + uk vjk+qj

=



vk ujk+qj^bk"^b

uk vjk+qj^by k#^byk+q"

 vjk+qj vk ^byk+q"^bk" + ^by k#^b





k q# b k#

bk"^byk+q" q# + vk vjk+qj^

k

vjk+qj^bk+q"

k q#

^y + ujk+qjvk^byk+q"^by k# + vjk+qjuk^b

ujk+qjuk^byk+q"^bk" + vjk+qj vk^b

 X 

k

k 



+ uk^by k#

+uk ujk+qj^by k#^b =

y + ^ ^ b b + v u k k# k k" q# 

ujk+qj^byk+q" + vjk+qj^b

k q#



^byk+q"^by k# ^bk"^b

^ +

k q# bk" 

k q#

=

+ k q#



In obtaining the last equality some operator pairs were anti ommuted whi h leads to the nal result n^ (q)

=

e

 X 

k

 vjk+qjvk ^byk+q"^bk" + ^by k#^b

ujk+qjuk 

 ujk+qj vk + uk vjk+qj ^byk+q"^by k#

+

k q#

^bk"^b



+

k q#



:

(193)

An analogous al ulation yields m ^ (q)

=

B

 X 

k

 ujk+qj uk + vjk+qj vk ^byk+q"^bk" 

+

ujk+qjvk

^by k#^b

k q#

 uk vjk+qj ^byk+q"^by k# + ^bk"^b



+

k q#



:

(194)

The rst line of these relations re e ts the above mentioned oupling between ^-states, and the se ond re e ts the oupling to the ondensate. Both lines ontain oheren e fa tors omposed of u and v. 10.3 Ultrasoni attenuation

As an example of a eld (external to the ele tron system) oupling to the harge density we onsider the ele tri eld aused by a latti e phonon. The orresponding intera tion term of the Hamiltonian is
g Xp !q a^y q + a^q n^(q); (195) H^ I = p V q

where g is a oupling onstant relating the ele tri eld of the phonon to its amplitude, V is the volume, ! is the phonon frequen y and a^y its reation operator. 56

We onsider the attenuation of ultrasound with ~!q <
; then, in lowest order pair pro esses do not ontribute. A

ording to Fermi's golden rule the phonon absorption rate may be written as 2 tr H^ I Æ(Ef Ei )H^ I P
= Ra (q) = ~

2 X = 4~gV !q nq ujk+qjuk k

2 vjk+qj vk fk (1 fjk+qj)Æ(jk+qj

k

~!q ):

(196)

Here, Ef and Ei are the total energies of the states forming the H^ I -matrix elements and nq is the phonon o

upation number of the thermodynami state whi h in this ase in extension of (187) also

ontains phononi ex itations in thermi equilibrium. Half of the result0 of the last line is obtained from the rst term in the rst line of (193). After renaming k q ! k ; the se ond term yields the same result. The phonon emission rate is analogously 2 tr H^ I Æ(Ef Ei )H^ I P
= R (q) = e

~

2 X = 4~gV !q nq ujk+qjuk k

With fk (1 dnq dt

fjk+qj ) fjk+qj(1 fk ) = fk

2 X = 4~gV !q nq ujk+qjuk k

2 vjk+qj vk fjk+qj(1 fk )Æ(jk+qj

~V

~!q ):

(197)

fjk+qj, the attenuation rate is vjk+qj vk

The attenuation in the normal state is 2 X dnq = 4g !q nq (fk dt

k

k

2

(fk

fjk+qj )Æ(jk+qj

fjk+qj)Æ(jk+qj

k

k

~!q ):

~!q ):

(198) (199)

From a measurement of the dieren e,
(T ) an be inferred. 10.4 The spin sus eptibility

With the intera tion Hamiltonian

H^ I

= H ( q)m^ (q) + : :; (200) where H is an external magneti eld, the expe tation value of the energy perturbation is obtained as  
E (q) = tr H^ I (Ef Ei) 1 H^ I P : (201) The sus eptibility is (q)

=

d2
E (q) dH ( q)

=

"

2 f X  f 22B ujk+qjuk + vjk+qj vk jk+qj k +   jk+qj k k #  2 1 f k fjk+qj : + ujk+qjvk uk vjk+qj k +  jk+qj

(202)

The sus eptibility drops down below T and vanishes exponentially for T ! 0. Coheren e fa tors appear in similar manners in many more response fun tions as in the nu lear relaxation time, in the diamagneti response, in the mi rowave absorption, and so on. 57

Aknowledgement: I thank Dr. V. D. P. Servedio for preparing the gures ele troni ally with great

are.

58