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=1V. Though deviations from this magnitude may be expected to occur at the internal surface of GaN and the nucleation layer we will assume that the Fermi level remains deep in the forbidden gap, in accord with observation [10]. At the lower boundary of the 2D gas the transition from quantum well to bulk will be relatively rapid. For simplicity we assume that the 2D gas acts like a sheet charge of zero width exactly like all the other charges in the problem. Beginning on the left, at the surface, we assume without loss of generality, the existence of a surface charge +0"sl (adsorbed ions/surface state population charge) in response to the polarization charge of the Ga face AIGaN, -0"pl (spontaneous plus piezoelectric), and there will also be a surface-state charge -OQ\ arising from the electrons from the donors in the barrier. If el is the permittivity of the barrier, the field Eu immediately in the barrier is given by: e^En = <Jsl - apl -
118
E^
at the
interface:
Transport in a Polarization-Induced
2D Electron Gas
481
f]£ I 2 = £ I £ J I + enDax, where nrj> is the donor density and ax is the barrier width. The field E2 in the buffer layer is then given by: e2E2 = £\El2 + op\ - ap2 - aD2 - an.
AIGaN
GaN
'F-
+o»s1
-°s2
Figure 1. Schematic conduction-band energy dependence through a AlGaN/GaN heterostructure. The AIGaN barrier is doped n-type and completely depleted, and the GaN is free of mobile charge. The sheet-charge densities are defined in the text. Here, e 2 is the permittivity in the buffer layer, o"p2 is the magnitude of the polarization charge in GaN, cD2 is the charge contributed by the donors, and rj n is the magnitude of the induced charge. Assuming zero field beyond the buffer layer, we obtain another equation for E2, i.e.: -£2E2 = op2 -
+0.5enDai -(e2
2
(1)
where r\ - EyJth Ie m*a\ and a~ £\a2 I t2a\. This simple equation predicts that half of the doping electrons appear in the 2D gas and that the barrier width must exceed a minimum, given by the zero of Eq. (1), for a 119
482
B. K. Ridley & N. A.
Zakhleniuk
2D gas to be established. Figure 2 illustrates the barrier-width- and (^-dependences for doped and undoped, pseudomorphic AIQ^GOQ-JN barrier on 2iim GaN with el = 10.3£ 0 , ex =10.4£ 0 , apl = apl(spon) + CFpl(piezo) = (0.045 + 0.ll)Cm~2,
op2=0.029Cm~2.
2.0-1
100
-nD=1018cm"3
200
300
400
Barrier Width (A) Figure 2. Dependence of electron density on barrier width and Fermi level. The effect of doping with 10'°cm~3 donors is shown for the case
120
Transport in a Polarization-Induced
2D Electron Gas
483
occupation probability, f(k), of the state with wavevector k to the volume rate and to the divergence of the probability current:
i f f i l J i m ] -v.(v(k,/(k,), d t
d t
\
m
Jvol
where v(k) is the group velocity. The last term vanishes in the absence of spatial nonuniformity and:
(im) Jim) Jim) Jim)
.
m
\ dt Jvol V Ot Jfields V Ot Jsca[ V Ot Jgellirec We can dispense with the generation and recombination rates in the present context. The field term is obtained from the probability current flow in k-space and the acceleration theorem, dk/dt= eF/h, where F is the electric field and e carries the sign of the charge: (df(k)\
„
,
„
= Vk jk= Vk
i—j^ - 'fields
( d k \
- -u
/(k) =
dk
J -^- V k / ( k ) -
(4)
The scattering rate has the form:
= ]£$•= £($"+#).
(5)
where the sum is over all scattering processes and S is the scattering-out rate i.e. the rate at which an electron is scattered out of the state k, and S+ is the scattering-in rate. The Fermi Golden Rule applied to phonon scattering, in which the absorption or emission of a phonon causes a transition from state k to k' and vice versa, leads to the form: St = J W,(k' ,k)[{/7«U9) + l}/(k' ){1 - /(k)} - n(0)q)f (k){l - /(k')}] x<5(£k.
-Ek-hcoq)dk'
(6)
+ J Wt (k", k)[n(a), )/(k" ){1 - /(k)} - {«(©,) + l}/(k){l - /(k")}] x<5(Ek.. -Ek+hco^dk", where co„ is the phonon frequency, n(coq) is the phonon occupation factor, and q is the phonon wavevector (q= k'- k or q = k - k" in the first or second integral, respectively). There are three rates with this form, those associated with optical-phonon, piezoelectric and deformation-potential acoustic-phonon scattering. (There will be a fourth if intervalley scattering is relevant.) Electron-electron scattering is different from the mechanisms just considered in that four electron states are involved: an incident electron with wavevector k j collides with a target electron with wavevector kj and after scattering they occupy states with wavevectors k i ' and k2'. If electrons were classical particles we would think of k i going to k i ' and k2 going to k2* and that would be it. But electrons are not distinguishable particles: the scattering event would look the same, ignoring spin, if the end states were exchanged, and this extra, possible process adds to the rate. But the exchange of fermions with the same spin changes the sign of the wavefunction, with the result that interference occurs between the two processes. The squared matrix element that enters the Fermi Golden Rule then consists of four components:
M2 =^[(M?21)2 +(M 2 V) 2 +(Mi12T - M J / ) 2 ] ,
121
(7)
484
B. K. Ridley & N. A.
Zakhleniuk
where the factor 1/2 takes into account the fact that in half of the collisions the spins are aligned and otherwise they are not aligned. The scattering rate is of the form:
5«/(k1) = JJJ[W(k,1,k,2,k„k2)/(k',)/(k,2){l-/(k1)}{l-/(k2)} -W(kj,k 2 , k' i, k' 2 )/(kj ) / ( k 2 ){1 - /(k 1 ! )}{1 - /(k 1 2 )}]dk2dk', dk' 2 . At the steady state the distribution function is to be obtained from: * V dt heat Under the influence of the field the distribution function consists of the sum of a symmetric part, / + ' and an antisymmetric part, / ~ , and Eq. (9) becomes:
£F.v(k)^Q^ = 5/-(k), dt
(10) +
^ • V k / - ( k ) = 5/ (k), n where v(k) is the electron group velocity. Further progress requires the explicit scattering rates for the e-e and phonon processes. The general complexity of the problem has driven most workers in this area to resort to purely numerical techniques, in particular, Monte Carlo methods. While adopting such an approach is inevitable if a particular experimental situation is to be modelled, it is not useful if a detailed understanding of the operation of the various processes involved is required. A particular problem is the incorporation of e-e scattering into the Monte Carlo simulation. Unlike other scattering mechanisms which can be regarded usually as one-electron processes, e-e scattering is a two-electron process that entails keeping information about two electron trajectories. Furthermore, taking into account the long range coulomb interaction and the simulation of screening processes calls for extremely sophisticated molecular-dynamic techniques. In the light of such numerical complexities it is useful to adopt a complementary analytical approach even if this involves the sacrifice of some accuracy, and we take this approach. In order to highlight the new physics as clearly and as briefly as possible it is necessary to adopt some approximations. Accordingly, we assume that the conduction band is spherical and parabolic and, to avoid the complexity of intersubband transitions, we assume that the electrons are all in the lowest subband of the quantum well. We will further assume that the electron gas is non-degenerate, which will be reasonably valid for hot electrons even for relatively high electron densities. All interactions will be dynamically screened by the dielectric response of the polar lattice and of the electron gas. Screening is a complicated process that deserves a chapter on its own. We will return to this topic later, but for simplicity we will ignore screening entirely. The scattering rates for phonons are reasonably straightforward provided the phonons are regarded as bulk-like and their interaction with electrons taken in an elastically isotropic, spherical approximation. The e-e scattering rates of interest are less familiar, and we begin with these. 4. Electron-Electron Scattering Poisson's equation describes the interaction potential: VV(R) = - ^ 1 = - 5 ( R - R , ) . £
If this is expanded in Fourier series we get:
122
£
(11)
Transport in a Polarization-Induced
2D Electron Gas
V(R)=fnQ)e'Q(R-R»)dQ-^T, J (In? 5(R-R,)=fe'
Q(R
R
485
(12)
)
- » dQ—L-
Inserting into Poisson's equation, we obtain:
V(Q) = - - J - , £(2 £2
,Q.(R-R,)
f
V(R) = - e
= £(22
J
(13)
,
dQ
=-. (2TT) 3
The energy of interaction with an electron at R=R2 is: .
iQ-CRj-Rx)
-eV(R2)= e2
3 J
EQ
,
(14)
dQ—4-.
2
(2JT)
3
Introduce in-plane vectors r and q, Q=(q, qz)- Then: -eV(R2) = e2
=
= - dqrf^
3-,
(15)
where r 1 2 = r 2 - rj and z\2 = 22 ~~ z lThere is no conservation of crystal momentum in the z direction, so there is no special restriction on q z . Integration over q z can be carried out using: dq=^e-q^y (16) 2 . 2 ^z °°q +qz Thus:
I
-eV(R2) = -fT
dq. (17) 8;r J eq Let the incident electron have in-plane wavevector k i and wavefunction: Vr1(r,z) = A _ 1 / 2 e i k l J '>0 1 (z). (18) A is the area of the plane. The electron collides with an electron with wavevector k2. After the collision the wavevectors are k i ' and k2'. Following the Born approximation, we assume that the wavefunctions after collision are the unperturbed ones. For the moment, exchange and spin interference are ignored. The matrix element is: M
i 2 = J v / ' 2 ' ( r 2 . Z 2 ) ^ i ; ( r i , z i ) { - ^ ( R 2 ) } ^ 2 ( r 2 . Z 2 ) ^ i ( r i , z i ) d r 2 d r 1 & 2 * i - (19)
Transforming to centre-of-mass and relative coordinates gives: r
cm = - (ri + r 2 ). r 12 = r 2 - r t ,
A/, 2
=
dr 2 dr x = dr c m dr 1 2 .
(20)
_ _ £ _ f e '(ki + ^ - ^ , - ^ 2 J r ^ ^ - i C k ! - k j + k i - k ^ - r n ^
8^ 2 A 2 J
(21)
x —2
dr cm dr 12 dq. &7
The form-factor is given by: F(q) = j
123
•
(22)
486 B. K. Ridley & N. A. Zakhleniuk
The integration over r c m leads to the conservation of in-plane crystal momentum in the case of N-processes (which are the only ones we need consider). The integration over rj2 specifies q. Thus: ki + k 2 - k, - k 2 = 0,
(23) k1-k1+k2-k2=2q. Since q is defined, the integration over q yields a factor 471.2/A and the other integrations yield a factor A 2 . The matrix element becomes:
We consider the case in which both electrons, before and after colliding, are in the lowest subband: 2 W(k1,k2) = ^{\Ml2\28(E2+E{-E2-El)dk\dk2-^-T. (25) h J (in) When the electron population is degenerate the integrand would contain the factor {1 - / ( k j )}{1 - / ( k 2 ) } . We take k2* to be fixed by momentum conservation, and so: W(k1,k2)
e 4 f F2(a) = ——\-^-8(E)dk1. tSTtnAJ e q
(26)
where, for a parabolic band, 8(E) = sl{ti2 12m *)(*22 + k? ~kl~
k
\ )}•
4.1. The scattering rate At this point in the calculation it is usual to go on to obtain the bare scattering rate which is what is required as input to Monte Carlo simulations and to provide estimates of the dephasing rate. This is not the most useful rate for our purposes - we require energy and momentum exchange rates - but, for completeness, we continue along the usual lines. We now define relative wavevectors: 812 = 2 (ki " k 2 ) and g 1 2 = ~ (k'i - k 2 ),
(27)
and observe that: Sl2 =~(kl
+k
2 -2*1*2 COS0),
I k l + k 2 | 2 = *12 + k2 + 2 M 2
cos
0. (28)
Similarly, ••>
1 ( '•)
512 = 2 p
+
">
11 •
• |2
*22-^|ki+k2|
(29)
whence it follows that the 8-function in Eq. (26) can be represented by: 8(E) = 8[(h2 I m*)(g\22 -g22)}.
(30)
Furthermore, g 12 = — {k, - (kt + k 2 - kj)} = k x - k c m , and, since k c m is a constant of the motion, we can change variables and replace k i ' by gi2':
124
Transport in a Polarization-Induced
w
2D Electron Gas 487
4
k )
s isA s ds M TI , \£ -M » » *>- > --<^l7^ **
A
( 3 1 )
,
. 3 , J ,2.2 de,
16TTAJA J e V 0
9
'9
9
'
where 8 is the angle between gi2* and gi2- Since g = gl2 + g\2 -2g\2gx2cos9 ' 9 9
9
9
and
9
^12 = 5i2. we obtain q - 4g 12 sin (0/2). The total rate is obtained by integrating over the target-electron states, taking into account spin degeneracy, weighted by the probability of occupancy. 9
9
9
In the case of the exchange process, q = 4g 12 cos (9/2). Strong interference effects, where the spins are parallel, will be confined to scattering angles around 7t/2. Ignoring interference altogether would double the rate in Eq. (31). However, when screening reduces the dependence on q, interference will be more important. Furthermore, many-body effects involving exclusion and correlation will limit the interaction between like spins. In view of these considerations, the contribution from collisions with like spins 9
9
is often neglected, and it is assumed that M = M ] 2 , in which case Eq. (31) is the total rate including exchange. In what follows we focus on the squared matrix element Mj 2 , keeping in mind that the total rate subsequently deduced will be a factor of two larger when interference effects are negligible. 4.2. Energy exchange The theory so far has focused on the total rate of scattering, given the wavevectors of the incident and target electrons. It tells us nothing, however, about the rate at which energy is exchanged in electron-electron collisions. This rate is important for understanding how fast thermalization occurs in a quasi-2D electron gas. In order to calculate this rate we have to focus on the wavevectors of the incident and scattered electrons rather than on the wavevectors of the incident and target electrons, which means integrating over k2 rather than k j \ Given k i and k i ' , we must sum over all possible target states weighted by the probability that a target state is occupied. This statistical factor introduces the distribution function into the problem. We will continue to assume non-degenerate statistics and take the distribution function to be isotropic and Maxwellian described by an electron temperature T e : f(E) = ^ - e - E ' ^ , N d = ^ ^ , (32) Nd 7thz where n is the areal electron density and Nd is the effective areal density of states taking account of spin degeneracy. For intrasubband processes, Eq. (26) is replaced by: W ( k l ' k ' l ) = TZniT \e-Ei'k*T< ^ 2 ^ 5 ( E ) 2 d k 2 , (33) $7tnANd J eq where the factor 2 is included to account for spin degeneracy. The delta function conserving energy must now be expressed in a form convenient for integration over k2This can be done by noting that Eq. (23) implies that: q = k2-k2, •2
h
2
=k2+q
2
-
+2k2qcos<S>lq,
and so: 125
<34>
488
B. K. Ridley & N. A. Zakhleniuk
8(E) = Sl(h2 12m *)(^ 2 - k2 +q2+ 2k2qcos
1
Integrating over the angle, and noting that I dd = 2 I J J sin 9 e nm * 27th3
2
ANd
( ,'2
sin
i2
2 min
3
dk2,
sm02q
2
(36)
2k2q
_k'2-k2+q2
K
, we obtain:
e q 2\
(35)
—
„
' *2 max
—
°°-
Note that q = k ] - k ] and so it is independent of k2- Integration over k2 is straightforward: wn,'
i, \
mkM=
2
e4n(7Qii*)U2 2
eX
h2
F2(q)
(k'2-k2+q2^
2^H {kBTef AlV T^kBTe
(37)
This rate is dependent on the angle between k j ' and k j through q. What is required is an average over angle: K
W(k\,kl) = -[ W(kltki)d
(38)
Clearly, the angle dependence is not straightforward, especially as screening, in general, is angle-dependent. Esipov and Levinson [11] approach the problem by putting F(q)=l, assuming static screening by the lattice (e = es), and defining the variable u as follows: 9 (39) (*, -*;2)"2 and we consider the case ki>ki' for the present. The relation of u to the angle is obtained from: q2=k2+k\2-2kxk\co%(p = (kx-k\)2+Akxk\s\Ti2(
. 1/2 U
=y
+
,2 ,'2sin2(ff/2), kx - kx
X=- J —+ kx + kx
•
(41)
The energy difference can be embodied in the symbol cr = (Ex - Ex) / kBTe. Substitution into Eq. (38) and noting that: 4k\kx={k2-k\2)(y-2-y2\ gives:
126
(42)
Transport in a Polarization-Induced
W(k
GT,3/2
e nh
w =
2D Electron Gas
489
-"(c-^Hr
(43)
Snv2e2m*(kBTe)2A "o So far, G7 > 0. The rate for GT < 0 can readily be obtained by using the principle of detailed balance (consistent with the equilibrium implied by the existence of an electron temperature). The expression for both cases is then: , Mf
W(k\,kx)=WQ J" or
3/2
2
1 "l ,1/2
2
(44)
" ¥-4^-"''
Following Esipov and Levinson, we have put F(q)=l and assumed only static lattice screening. We note that for strictly 2D electrons the form-factor is unity, but it will be close to unity for quasi-2D electrons for quasi-elastic collisions. The rate becomes: /-' (45) W(k\,kl) = W0 [w(u)du, Y
where:
W(u) =
N
\m
exp
m/2
«2+-
1
3/2
,1/2 •
(46)
•'F-'-'Hr- 1
In such a case, Esipov and Levinson have shown that the integral in Eq. (44) can be obtained in terms of a modified Bessel function of the second kind, provided that y
«1
2
and y / | n r | « l , „C7/2
W(kl,kl) =
W0—m-yKl(\uj\/2).
(47)
In terms of energy, the parameter y is given by: i„il/2 V2
(E/kBTe) +(E/kBTe)V2 Thus, for quasi-elastic scattering, | c t | « l , KI(|GJ|/2)=2/|GJ|, and Eq. (47) becomes:
"^•^"^^IT^'
(48)
(49)
In the case of a lower energy incident electron, the rate increases rapidly. For a thermal electron (E/kBTe=l), the quasi-elastic rate is obtained from an integrand that is concentrated around u=y, where, under these conditions, y =|GT|/(4£ / kBTe)«1. If,
127
490
B. K. Ridley & N. A.
Zakhleniuk
once again, we take the 2D limit of the integral in Eq. (38) and ignore screening, we obtain the result of Esipov and Levinson: ^
W(kl,kl) = 2W0-—re-E/k'>T'
^ \e'~dt.
(50)
As in the case of a fast electron, the rate diverges as \w\ . However, quasi-elastic scattering does little for energy and momentum exchange. Rates for these are going to be associated with strongly inelastic scattering in which | n j | » l . In this case y is no longer small. When c r ( l - y ) » 1 , most of the integral comes from the region around u=l and when E, Ei", |E-Ei'|»kBT e : F2(a \Jn/2-\m\'2) jA12 1 W(k\,k) = W0 7 / — \erf[ujm(Y-l-l)] + e,f[uju2(l-Y)}L (51)
M
(y
where qx=^m(2m*kBT
~Y)
2
L
J
I h ), so for the downward transition (w > 0): 1/2
W(k\,k,) = W0 ° and for the upward transition (tn<0):
W
^
W
F
,., 2 n"2
l,
F2(gx)
e*>
= o—W
x
,l<7'; r^, \rn\V\ExlkBTe)vl
T i
^r-
(52)
(53)
Note that downward transitions are significantly emphasised via the Maxwellian factor. The form-factor has been taken to be determined by the condition u = 1. The energy-relaxation rate, Q, is obtained by integrating Eq. (45) over ki': 4 2 (54) - / ( £ , - Ex')W<,k\,kx)k\dk\AI2n=—^-F (q0, 32 esh where we have introduced a mean value qx of qx for the form-factor in order to facilitate comparison with the result of Esipov and Levinson. Since small energy-transfers are favoured, the form-factor is very roughly unity. An energy-relaxation time can be defined by Q = -E/zee:
ree 32 efhE With a dielectric constant equal to 9, and taking the form-factor to be unity, the rate is 2.09n/E s" , where n is the areal number of electrons per cm^ and E is the electron energy ineV. 4.3. Momentum exchange Esipov and Levinson do not discuss momentum relaxation, but this is readily derived. Returning to Eq. (44), we can define a momentum-relaxation rate, W m , by weighting the integrand by (l-cos9), where, from Eq. (40): l-cos9 = 2(" " V , Y -Y so that:
128
y « l , y2/|nr|«l,
(56)
Transport in a Polarization-Induced
.mil W,A
2D Electron Gas 491
|CT|
-oo j
~-\U
+ —
Jo u riable to: The integral can be evaluated by changing the variable to: If 2 1 ^ and = + 2
(57)
|nr|
H
21
r-f+n
„
Jo
whence:
,
(58)
Jo Ji
,„i±
GT/2
Wm=2WoY3-pjK0(\m\/2)^*->W0 ) m j ^ (59) |nr| 4{E/kBTe) where Krj(x) is the modified Bessel function. This is the momentum-relaxation rate for quasi-elastic scattering which, in the absence of screening, diverges logarithmically. With screening, quasi-elastic collisions will be inhibited. For strongly inelastic scattering, less affected by screening, the momentum-relaxation rate coincides with the energyrelaxation rate given by Eq. (55). Thus, we take the Esipov-Levinson expression of Eq. (55) to quantify both energy and momentum exchange rates. 5. Phonon Scattering The strength of e-e scattering plays an important role in determining the form of the distribution function under the influence of an electric field, but e-e scattering cannot relax momentum and energy of the whole electron system. These relaxation processes depend on there being other scattering mechanisms. Whereas all scattering mechanisms (apart from e-e scattering) can relax momentum, only inelastic processes can relax energy and, of these, phonon processes are in nearly all cases the most important. Here, we limit attention to the momentum and energy relaxation rates associated with scattering by phonons. There is first the problem of describing phonons in a quantum well, where the discontinuity of electrical and mechanical properties across the heterojunction between well and barrier materials affects the spectrum of lattice vibrations. This is a well-known problem that has been discussed at length elsewhere [12]. It has been found that, provided the well is not too narrow, an approximation can be adopted based on the assumption that the scattering rates are given without much error if the spectrum of phonons in the well is taken to that of the bulk material. Such an assumption would be wholly invalid for describing situations where individual phonon modes are observed, as in Raman scattering, but it works quite well for calculating scattering rates which entails summing over all modes. In what follows we exploit this approximate sum rule and work with bulk phonon modes. The net scattering rates that affect the occupation probability of a given electron state depend upon the occupation probabilities of states involved in the processes of absorption and emission of phonons. For the moment we will ignore this dependence and we will only consider scattering-out events. 5.1. Polar optical phonons Because of the large optical-phonon energy, scattering by optical phonons is highly inelastic. Moreover, the net scattering rate associated with a particular electron state depends on scattering rates of states a phonon energy above and below. A ladder of scattering with rungs a phonon energy apart is therefore involved in order to calculate the momentum and energy relaxation rates associated with a given state, and this requires a knowledge of the distribution function. In the presence of an electric field, the distribution 129
492
B. K. Ridley & N. A.
Zakhleniuk
function, in general, cannot be found without talcing into account momentum ana energy relaxation, so the problem is one of finding a self-consistent solution. This problem is usually solved by using Monte Carlo methods, but in some cases it is possible to use an analytical approach, and this will be adopted here. The scattering rate associated with an electron in the state with wavevector k and energy E is: li
1/2
"44^1
1*
(»(»>+i)f-^*+»<»>f-^*
2 \ E J J qsmd+ where n(co) is the optical phonon occupation factor and: 0
<7l
h )
4nH
=k
,-f,-f)
[e^
1/2
1/2 92
,
-T)
sin0+ = J l -
J qsmd_
es
1/2 <73
(60)
q4=k
,•(,-*)
Kr-
(61)
m * ft) q hkq
2k
The form factor is given by: a/2
F(q)=
a/2
jdzdz'V2(zW2(z)e- •dz-z'l
J" -a/2
(62)
-a/2
where i|/(z) is the electron wavefunction, and we make the assumption that scattering is confined to the lowest subband in a square quantum well of width a. The estimation of the corresponding momentum relaxation rate can be obtained by weighting the emission integrand by (q/k)cos8+ and the absorption integrand by -(q/k)cos8_, thus:
(B(fll) +!)? J32L( J. + »^L V
') qsin9+{2k
hkq )
g F(q) I q i.
H
+ n(co)
J ,qsind_\2k
(63) m*(0 dq hkq
93
The energy-loss rate is:
to
.1/2
VHT)
J qsmu+
J qsmu_
(64)
<73
These integrals must be evaluated numerically. They involve scattering-out rates only, and they are strictly valid only for a drifted Maxwellian distribution. 5.2. Piezoelectric scattering In strongly polar materials the most powerful interaction with acoustic phonons at low energies is via the piezoelectric effect. The phonon energy in this case is small and 130
Transport in a Polarization-Induced
2D Electron Gas
493
in most cases we can adopt a quasi-elastic approximation, and except at very low temperatures we can assume that equipartition holds for the phonon occupation factor. The scattering rate is then given by: W =
e2K2m
Ik
* kBT
27IEsh3k
f
dq,
JJ q^\-{ql2k)2{\ 0Wi-(
+ qs
(65)
Iq)2
where K is the electromechanical coupling coefficient averaged over direction, T is the lattice temperature, F(q) is the form factor of Eq. (62) and q s is the static screening factor, which for non-degenerate statistics is: e1nF(q) (66)
e2K2m*kRT 8nesh3k
Ik
qF{q)
\J ^\-(q/2k)2(l
-dq.
(67)
+ qs I q):
In order to obtain the energy relaxation rate we must take into account the small but finite energy of the acoustic phonon, which means treating the emission and absorption integrals separately. Thus: ~2k-n hco{n(0)) + l}J(q,qT) -dq ^jl-[(q/2k)+J1}2(l + qs I q)2 e2K2m* 0 (68) WP =
h "rh -h
Ane£-k
2k+r]
hO)n(0))J(q,qz)
^l-[(q/2k)-t1}2(l
-dq +
qs/q):
In this equation J(q, q z ) is given by: G2(qz),
(69)
\ir2(z)eiq*zdz
(70)
J(q,qz): a/2 G\qz)
=
J -a/2
and r|=2m*v s /7i, where vs is the averaged velocity of the acoustic waves. Once more one is faced with numerical integrations. For quasi-elastic processes such as scattering by acoustic phonons, involving only the scattering-out rates in the derivation of momentum and energy relaxation rates is more generally justified than it is for optical phonon scattering. 5.3. Deformation-potential scattering In the case of non-polar scattering by acoustic phonons it is possible to obtain analytical solutions. The scattering rate is:
131
494 B. K. Ridley & N. A. Zakhleniuk
W
_
~ Ik
2
^Z fm*k. T f f G'W dqzdq, 71 h Vspk 1 2k L o V - (?1 ? o+is'
(71)
where p is the mass density of the lattice, E is the deformation potential. For a deep well such that \|f(z)=cos(rcz/a):
I
G2(qz)dqzl2n
=
(72)
^-, la
and the scattering rate is: 3E2m * kBT
W
2h\2spa
where a=q s /2k and L{a) •
I-a2
nk
ln(Vl-a2 +1)1/VT
I-a2
(73)
a
The momentum relaxation rate, calculated as before, is: 3E2m*kBT
1
2
+ 6a •
2h\ pa n For the energy relaxation rate we need: ,9„9, „, (q + qz)2G2{qz)dqzl2n
4aJ z
(L(a){4-a2}-l)
77(1-a )
,_
(74)
3q 2TC = ^ - + —T la a (75)
(q + qz)G2(qz)dqz/27l
= ^-. la
Then the energy relaxation rate is: 3E2m*2
„. W
E
= —4
X
h pa E + -EQ-—[
3
8aJ 7r
r
n\
4E +
3
-EQ\L{a)-L{a\l\-2kBT\+6a2E
[
1-cr J
^^BRr
£ 3 L(a) + i ^ l ^ 2 [
\-a2
J
J
4-a
I
4^(l-a2) I 1-a
L
(76)
i-Vi-«2
r («>- i +
where Eo=ft^Jt^/2m*a^ is the subband energy. 6. Electron-Electron Scattering Dominated Transport of 2D Gas in GaN/AlGaN Quantum Wells In this section we will consider behaviour of a 2D electron gas in a square infinite quantum well (QW) in the presence of strong longitudinal electric field F. In line with previous discussions we deal with an ideal GaN/AIN Q W as this double heterostructure is best suited in order to observe the some new physical effects which have not been discussed or studied before. Since the depth of the GaN/AIN QW is about 2 eV it can accommodate a high density electron gas and the free electrons can be supplied not only via modulation doping but also via the doping effect of the intrinsic polarisation fields. Apart from this doping effect, polarisation fields will be ignored. The latter 132
Transport in a Polarization-Induced
2D Electron Gas
495
property creates an exceptional situation when there is no direct correlation between the number of free electrons in the QW and the number of background and remote charged impurities. In such circumstances the scattering by the deformation acoustic (DA) phonons, the piezoacoustic (PA) phonons, and the polar optical (PO) phonons are the main scattering mechanisms besides the e-e scattering. In real structures it would be necessary to include scattering by background impurities, charged dislocations and interface roughness which usually determine the low-field electron mobility, but as here we mostly will focus on the high-field transport we do not include them. Also each one of these can be eliminated, whereas the scattering mechanisms we consider cannot be eliminated, although inclusion of the these mechanisms in our theory has no principal difficulties. It is useful to appreciate the magnitude of each of the main intrasubband scattering rates. The most rapid is that for PO phonon emission ( - 10 4 s~l) when the electron energy is above the PO phonon energy; below the PO phonon energy, however, the rate is determined by the PO phonon absorption which becomes weak toward low lattice temperatures kBT « h(0o. Due to high polarity of the III-V nitrides the PO phonons mediated scattering rate is about an order of magnitude higher in GaN than in GaAs. Because of this the PO phonon absorption considerably contributes to the electron mobility even at relatively low temperatures when the above condition is satisfied, for example at room temperature [13]. We will include the PO phonon absorption in calculating the PO phonon mobility. PA phonon scattering is always significant in wurtzite GaN especially at low temperatures (at T=50 K the rate is about 5x10 { for energies about kBT, decreasing with increasing energy). DA phonon scattering is slightly weaker, the corresponding rate being about 10 s ). The e-e scattering rate depends on the electron density. At energy equal to the PO phonon energy (the worst case in the range we consider) the rate in a gas of density 1011 cm~2 is about 4 x l 0 1 2 s~l. In the range of the electron energies 0 - 100 meV it is easy for the e-e scattering to dominate both the energy and momentum rates randomisation at densities above 10 cm . Such densities are easy obtained in GaN/AlGaN structures [1, 2]. Here we will deal with the case in which the electrons occupy only the ground QW state with the quantization energy E0, and the lattice temperature T is small in comparison with the PO phonon energy h(0o, kBT «h(0o <3E0. These conditions are easily satisfied within a wide range of lattice temperature in GaN-based QWs, where hco0 = 92.8 meV, and for the QW thickness d=70 A. We ignore the electron gas degeneracy. Of course, at high electron densities this effect is important, but with increasing electric field its importance will not be so significant as the electron gas will occupy the high energy states, and therefore the degeneracy will be partially or completely removed. The strength of the electron-electron (e-e) interaction is a key parameter which defines the distinctively different regimes of energy and momentum relaxation and nonequilibrium electron kinetics in semiconductors [14]. Because the integral operator of the inter-electron scattering Iee\f(k),f(.k'
) \ is a bilinear functional of the electron
distribution function f(k) of the interacting electrons in the states with the wavevectors k, k', the magnitude of the transition probability Wee(k,k' )is proportional to the electron density n , as it can be seen from Eq. (33). Screening can be expected to modify the linear dependence. Indeed, for collisions involving small energy exchanges, which can be taken to be screened statistically, an increase of rate with density is countered by an increase of screening, so little dependence on density occurs. But, as discussed in 133
496
B. K. Ridley & N. A.
Zakhleniuk
discussed in Section 4.2, small quasi-elastic collisions are not important for relaxing energy. Highly inelastic collisions are the more significant, and for these, screening is no longer a static process but rather a dynamic one. To give full treatment of the dynamic screening of the e-e interaction is beyond the scope of this work, but preliminary indicators are that anti-screening effects become important at energy exchange of order of the optical phonon energy. The interplay of screening and anti-screening is therefore complex in the dynamic regime, which is just the regime we are interested in. Pending a full study we will assume that there is sufficient balance between screening and antiscreening for us to ignore the screening of the e-e interaction altogether. Therefore the e-e scattering contribution to the relaxation processes can be ignored only when the electron density is very low, otherwise e-e scattering will control the electron energy relaxation (at intermediate electron densities) or both the energy and momentum relaxations (at high electron densities). In the last two cases the electron system can be described by means of the electron temperature Te. Due to dependence of the e-e scattering operator on the electron density it is possible in principle to distinguish between three physically different situations [15]. Case I, which can be called the partial energy control case, takes place at the intermediate electron densities when the e-e scattering controls the energy relaxation only within the passive energy region E < ho)0. In the active energy region E > hco0 the PO phonon scattering is stronger than the e-e scattering. Case II, which can be called the full energy control, takes place at higher electron densities when the e-e scattering is responsible for the energy relaxation at all energies, but the electron momentum relaxation in the active region is still controlled by the PO phonon scattering. Case III, which can be called the electron momentum-energy control, takes place at yet higher electron densities when the e-e scattering controls both the electron energy and the momentum relaxation at all energies. The most interesting physical situation belongs to the case HI which is characterised by a unique strongly non-linear regime with a non-monotonous behaviour of the electron temperature Te as a function of an applied electric field F. But for completeness we carry out below the kinetic equation based analysis of the all above three cases. 6.1. Electron kinetics and the electron temperature at the partial energy relaxation control by the e-e scattering In this case in the passive energy region the energy relaxation rate due to the e-e scattering is higher than the energy relaxation rate due to any other scattering mechanism, but the electron momentum relaxation is controlled by other mechanisms (in our case by the DA and PA phonon scattering). It is obvious physically that the distribution function in this region will be very close to the Maxwellian function, f(E) = FT(E) = A0exp(-E/kBTe), with the electron temperature Te (A0 is a normalisation constant, see Eq. (32)). Since the deviation from the Maxwellian function will take place near the PO phonon energy only, as the PO scattering dominates the e-e scattering at this energies, it may appear that since this region is quite small it would be possible to neglect the effect of the PO phonon scattering altogether. But such a neglect would be wrong. This is because the interaction with PO phonons is extremely inelastic process which results in large change in the electron energy. It is physically obvious that the intensity of this interaction depends strongly on the electron population (electron distribution function) at the energies near the threshold energy E = h(0o. Therefore much care should be taken in calculating the distribution function at these energies even if the majority of the electrons still are in the passive energy region. The emission of the PO phonons even by the relatively small number of the electrons could be very effective
134
Transport in a Polarization-Induced
2D Electron Gas
497
channel of the electron energy loss for the whole electron gas due to large magnitude of the PO phonon energy. Although the emission of the PO phonons takes place only at the energies E > hco0 this process will effect the electron distribution function not only at these energies but it will also have a profound effect on the electron distribution function at the energies E < hco0 just below the PO phonon energy. This is because any electron which is in the state just below the threshold energy may acquire the necessary excess of the energy from the other electrons due to the e-e scattering. In this case this "lucky" electron will leave the passive energy region and will never come back into this region by means of losing the excess of the energy due to e-e scattering. This is because the PO scattering is stronger than the e-e scattering in the active energy region and any electron is transferred from the active energy region into the passive energy region due to the PO phonon scattering. Therefore, the e-e scattering acts as some kind of pump which supplies the electron into the active energy region but which does not transfer them back. As a result of this asymmetry the electron distribution function will be depleted at the energies just below the PO phonon energy. This case was analysed in detail by Levinson and Esipov [11, 16] for photoexcitation but in the absence of the electric field. They shown that the competition between the e-e and the PO phonon scattering near E = hco0 can be described by the parameter X0 which is proportional to the ratio of the characteristic e-e scattering rate and the PO phonon scattering rate near the threshold, X0 = 2^jhco0 I nkBTe vee(h(0o)/ vpo(tiu>0), where vee(hco0)= n2e4n/ e2ti20)o is the e-e scattering frequency (e s is the static dielectric constant), vpo(hco0) = na.p(D0 is the PO phonon collision frequency ( a F is the Frohlich constant). The regime in question takes place if X0 «1. For a GaN QW A0 = 0.02 when n = 1011 cm'2. The determination of the distribution function / ( £ ) in this case requires the solution of an integral kinetic equation [16] which explicitly includes the e-e and the PO phonon scattering operators: oo
(dENs(E)\w{E,£)/(£)J 0
W(E,E )/(£)]
L
*
= vpo(hco0)(N0 +
(77)
\)f(Eye{E-tuo0),
where W(E\ E) is given in Eq. (44), A^(E) = m* A12nti2 is a 2D density of states (A is the area of the QW), N0 = f exp(h(0o I kBT) - ll is the PO phonon distribution function at equilibrium, and 0(x) is the step-function. It is necessary to note that in deriving the above kinetic equation we assume that the rate of the electron transfer from the passive into the active energy region due to gain of the energy directly from the electric field is small in comparison with the rate of the transfer due to the e-e scattering near the threshold energy. This assumption imposes an additional limit on the strength of the electric field F. By the direct comparison of the corresponding terms in the kinetic equation we obtain the following approximate criterion for the electric field: e2F2zp(h(o0)ree(hco0) _f j-
«1.
(lo)
m no)0 where Tp(hco0) and tee(hco0} are the electron momentum scattering time due to DA and PA phonons and the e-e scattering time, respectively, at the PO phonon energy.
135
498
B. K. Ridley & N. A.
Zakhleniuk
A considerable complication in a 2D case in comparison with the 3D case arise because the e-e scattering in a 2D gas cannot be considered as a diffusion in the energy axis [16]. As a result of this the above integral equation cannot be transformed to the differential form as it was the case in a 3D electron gas. The integral equation (77) has been solved analytically in Ref. 16 using the Wiener-Hopf method. The obtained solution is: (2A 0 ) , / 2 , f(E) = At* In
-1/2
\t\«Xot
X0t • " V ,
(79)
/ > 0, t » Xc
e 'enr/(VR),
r<0,
\t\»Xn
where t = (E-hco0)/lcBTe, and A* = FT(E = tico0) = Aoexp(-h(0o I kBTe). It follows from Eq. (79) (the third line) that deep into the passive energy region (|t| » 1) the distribution function is indeed equal to the Maxwellian function Fj-(E) as we pointed out earlier. In the important part of the active energy region the distribution function can be presented as (second line in Eq. (79)) F+(e) = FT(e)2A0^(e - hco0) I nk0Te . The distribution functions obtained above should be used in the energy balance equation for the electron temperature Te and for the calculation of the electron drift velocity vd. The energy balance equation is derived from the kinetic equation (9) by multiplying it by the electron energy E(k) and summing up over all k. For a 2D electron gas interacting with unscreened DA, PA, and PO phonons in the square QW the balance equation is
F2MTe) = ?f 1--L -K, 2>n
vop(tuo0)ka
1+
2 En 3 kRT T. eJ
Q
5
Aa ST sL sTTpa Te (80)
f>(Oo
kBT P(Wo)(N0+l) 2m sL o * 2
V
I
1 , hco0 2 kB'eJ RT.
kKT,
kBT
As we see the last term in Eq. (80) is proportional to the parameter K0 which in its turn is proportional to the rate of the e-e scattering, i.e. the e-e scattering rate is explicitly enters the balance equation. Here we introduced the following notations for the dimensionless electric field F and dimensionless electron mobility p., and the form-factor P(w0) for interaction between the 2D electrons and the PO phonons, respectively: 2 _ (eFXa)2 ;MTe)-. Fz = * 2, 9nTstE 0
P{W0):
il+ropYlj
wn l+ <
pBTe R(Te) + x1'2'
1 1- -2mvn nwla + w2.)2
( M
~ 3
AEn
1 + ypo
(81)
(82)
The electron mobility is: H(Te) = (l + Y,op )^MTe). m
136
(83)
Transport in a Polarization-Induced
2D Electron Gas
499
600 To=10K 500
o= 20K T o = 50K
400
T o =100K
T
n o = 10 11 cm" 2
TQ = 300K ~
300 200
100
200
400 600 F (V cm 1 )
800
1000
800
1000
10 8
n =10 11 cnrf 2 0
1000 200
T =10K o T =20K o T =50K o T =100K o T =300K o
400 600 F (V cm'1)
Figure 3. Variation of (a) electron temperature Te and (b) drift velocity Vj with electric field F in a GaN square QW at the intermediate electron densities n0 corresponding to the partial control of the electron energy relaxation by the e-e scattering for different lattice temperatures T0.
137
500
B. K. Ridley & N. A.
Zakhleniuk
In Eq. (83) r^ = 3vQ 12Xa is the DA phonon scattering time for the 2D electrons, Xa = 7thAps\ I m*2E2kBT is the electron mean free path in a bulk material due to DA phonon scattering, p is the material density, E is the DA potential constant, s^ is the longitudinal (A. = L) or transverse (k = T) acoustic velocity, Yop = 2N0vop(ha)0)Aa
13ve, and ypa = (kBTka I E0sTrpa)(l6
VQ=^2E0/
in ,
+ I2s$ I s2L)/ 35 are the
dimensionless coefficients, tpa = 2nph sT I m e hu is the characteristic scattering time for the bulk material due to PA phonons, h\^ is the piezoelectric constant. The balance equation (80) has been solved numerically in order to obtain the electric field dependences of the electron temperature Te(F) and the drift velocity v rf (F) = p(Te)F using the following parameters for the electrons in a square GaN/AIN QW (the well width was 70 A with the ground state energy Eo=10 meV): m* = 0.2lmo, E=10.1eV, / i 1 4 = 4 . 2 4 x l 0 7 VIcm, p = 6.1 glcm3,
aF = 0.45, co0 = 1.41 xlO 1 4 s~l,
sL=4.57xl05 cml s,znd sT =2.68x10 cm/ s. The obtained dependences are shown in Figures 3 (a, b). The most remarkable feature of these dependences is the sharp increase of Te and vd with F at low lattice temperatures (T0 <50 K) and the smooth behaviour of these dependences at higher temperatures. This is the result of competition between PA phonon and DA phonon scattering where the PA scattering dominates at low temperatures and the DA scattering dominates at high temperatures. 6.2. Electron temperature of the hot electrons for the case of full energy control by the e-e scattering This case means that the e-e scattering dominates the energy relaxation at all i •y
>y
electron energies. This takes place in a QW when n0 > 10 cm . The distribution function is F{E) = FT(E) in the whole energy region. It is necessary to note however, that the real material parameters of the GaN QW are such that probably this regime cannot be realised in practice. The necessary increase in the electron density leads to fast increase of the e-e scattering rate in such a way that at the above densities the e-e scattering will control not only the energy relaxation but also the momentum relaxation as well. (This case is considered in the next Section). For consider this case here for completeness only. The energy balance equation corresponding to this case can be obtained in the same way as the Eq. (80) in the previous section. The only difference is that in this case the e-e scattering rate does not enter the balance equation. Direct calculation shows that the balance equation in this case has the same form as the balance Eq. (80) with only one difference: the parameter X0 in the last term in Eq. (80) has to be formally substituted by the factor (1/2). The electron temperature and the drift velocity are shown in Figures 4 (a, b) as functions of F (solid lines). For comparison, we also show the Te(F) and vd(F) dependences for the previous case (A 0 « 1 ) for two different densities n0 = 10
cm
and n0 = 5 • 10 cm . The main difference between cases I and II is that the electron temperature and the drift velocity in case I do depend explicitly on the electron density n0 while in case II there is no explicit dependence on n0.
138
Transport in a Polarization-Induced
350
r~
300 -
n
;
n
250
o= o=
i
11 10
>
i
'
2D Electron Gas
501
i
i
3
cm-2 ./'
5x10 11 cm" 2
•
^'
•
^ 200 -- ~~
150 100 •
T = 20K
50 ~
0
i
0
,
200
i
,
400
i
i
600
800
1000
F (V cm"1)
II
3
•
i
•
i
'
i
'
10 11 cm"2
_
5x10 1 1
-2
.
"S^Z-^^
b ^ •
i
i
o
3
i i i . 11
o
10 7
II
i
i T ^
(0
£ o
10 6 r
'
:
1 n5
-
T0=20K
/
'
•
1
200
,
1
,
400
1
.
600
1
:
,
800
1000
F(Vcm"1) Figure 4. Variation of (a) electron temperature Te and (b) drift velocity vd with electricfieldF in a GaN square QW at the electron densities corresponding to the full control of the electron energy relaxation by the e-e scattering for different lattice temperatures T0 (solid lines). The rest of the curves correspond to the case of the partial energy control.
139
502
B. K. Ridley & N. A.
Zakhleniuk
6.3. The 2D electron gas cooling and squeezed electron distributions Physically most interesting and unique case arises at high electron densities which are sufficient o allow the e-e collisions to control both the energy and the momentum relaxation. Our calculation shows that this takes place if n0 > 5 • 10 cm The distribution function in this case is a Maxwellian drifted function F(k) = A0exp[-E(k - K)lk0Te], where K is the displacement wavevector. It is necessary to note that usually the drifted distributions are inhibited by the presence of impurity and other scattering mechanisms which tend to control the electron momentum relaxation. This is because in order to obtain the necessary high electron density one has to use the highly doped materials. In this case the high electron density comes together with the high ionised impurities density and both the electron-electron and the electronimpurity scattering rates will have the same order of the magnitude. As result of this the e-e scattering alone is unable to dominate the electron momentum relaxation. (At the same time the e-e scattering is able to control the energy relaxation due to considerably lower rate of the energy relaxation in comparison with the momentum relaxation rate. Physically this is because it is enough only a few collisions in order to change the electron momentum direction, but it is necessary considerably more collisions in order to change the electron's energy if the collisions are quasielastic.) So far the only experimental possibility to obtain the drifted distributions were realised at the intensive photoexcitation conditions where the high electron density can be obtained through the interband excitation. This usually correspond to the transient regime. Here we want to point out that the piezoelectric field doping in III-V nitride-based heterostructures open physically new possibility to establish the electron drifted distributions at the steady-state conditions. And this leads to novel transport properties, including absolute cooling and a squeezing of the distribution in the direction of drift. Due to strong interaction with the PO phonons the electrons encounter the strong phonon emission barrier when the average kinetic energy approaches the PO phonon energy. In order to investigate the non-equilibrium electron kinetics with drifted distribution it is necessary to derive two balance equation (the energy conservation and the momentum conservation). In the presence of the external electric field F the electron system gains from the electric field both the momentum and the energy. It is important to point out that since the e-e scattering is the fastest scattering mechanism in the system, the energy and the momentum gained from the electric field by each individual electron will be first distributed within the whole electron gas without substantial loss to the other scatterers. At some point the balance will be established between the whole electron gas and the thermal bath. As a result the electron gas will acquire the drifted (macroscopic) momentum hie. The average kinetic energy of the gas, which is described by the electron temperature Te, will also change. Usually both these parameters increase when the electric field increases. Here we want to point out that this is not the case any more if a strong inelastic scattering mechanism is present. In the case considered such a mechanism is mediated by the interaction with the PO phonons. Due to large magnitude of the PO phonon energy in GaN, the majority of the electrons will interact with the PO phonons only at relatively high electric field (-1 kV/cm). Until these fields will be reached the electron gas will interact mainly with the PA and DA phonons. This interaction is quasielastic [17] and it does not prevent both hk and Te to grow when F increases. At higher electric fields, when the interaction with the PO phonons dominates over the PA and DA phonon interaction, the electron gas looses its energy by large portions ( - h(00). This effectively hinders the further growth of the electron temperature. Due to the large coupling constant in GaN the optical phonon energy level acts as a "hard wall" for the electrons, which prevent the electrons from penetrating to the higher energy. In general 140
Transport in a Polarization-Induced 2D Electron Gas 503
the above parameters K and Te are found from the system of two balance equations which describe the momentum and the energy conservation. Using the above drifted Maxwellian function we have derived two balance equations for 2D electrons. The only assumption that has been made is that k0Te «ti(o0. The equations in question are: 1
V^
da
T
eF T
pa
kBT
^l(K)
^2m"sjkBTt
(84) +
(N0 + l)P(wa) e 1
eh
—
m
+x¥2(K)e
keT
'
po * 1
„
KF =
keT
Am SrE L sn --
3kBTe
J_ + A^L 10 r
r
^ da
I^TABTL
pa 1 '" SLE0
T
1
h(0o+EK
+
ha0{N0 + l)P(w0)
keT
Y3(ff)c
'
-
e
kBTe
(85)
ft(0„'\
-e
ksT
*po 0
Here £K=h
0
*
—1
K 12m is the electron drift energy and T 0 ~ vpo(h(0o).
The function
i
Y,-(K-) in Eqs. (84), (85) is defined as VjiK) = 4 / n\ 9 ( (u)V 1 -u2du,
where cp^u) are
o given by the expressions
IkBTe),
cpji") = ch\2u^eKhco0
IkBTA,
and (Pi(u) = q>2(u)12(1 - u ). The above equations have been solved numerically with the same parameters which we used in the previous sections in order to obtain K and Te as a function of F. First we calculate the electric field dependence of the electron temperature Te and the drift velocity vd = tiK I in* of 2D electrons which are shown in Figure 5 for different lattice temperatures T. The most interesting feature of these dependences is that the electron temperature Te is a non-monotonous function of the electric field. Another interesting results concerns the field dependence of the drift velocity: vd(F) has the regions which obey an S-type dependence. These regions exist only at low lattice temperatures (T~ 10-20 K) and they disappear when T increases. This behaviour is a result of a complicated Independent competition between PA and DA phonon scattering [18]. At higher lattice temperatures the DA scattering dominates over the PA scattering and the S-type regions disappear. Another interesting feature is a saturation of vd at high electric field (F order of 1-10 kV/cm). This effect is completely due to the e-e and PO phonon scattering. The PO phonon scattering effectively limits any further increase of the drift electron momentum since every time that an electron emits the optical phonon it loses almost all the energy and the momentum.
141
504
B. K. Ridley & N. A.
200
—
1
Zakhleniuk
1
1
1 —— i
1—i—r i i "|
1—i—i—r-r
•'T
•
•
1 1 1 1 _
1 = 1 OK
T =20K 0
150
a .
T =50K. 0
.
T=100K 0
^ " ^ Vv.. -"' N.
100 -
/
'
\
1
/
\
\
•:
•
I/;
i-
l--'i 50 -
'/
/
*>
10
100
,1
i:
\
:"
1
I
1 ...
j i
,
.-•
,
,
,
,
i
i
i
t
i
1000
F (V cm1) i i r |
1 U
:
'
i
i
i
i
.
0
T =20K
.......
0
T=100K 0
'
f
<"
J
,' y
'
•
•
•
y,' .--" y )' y y
6
. ^ ^ X X
r r r
"
'
)
' -
y
" .
y^ / / V
y^y
i
b ;
/y /
£
10
,
/
CO
^ l O
,
f
—
0
i
i
T=10K
T=50K
10 7
i
•
5 i
10
i
i , ,
,1
,
,
,
,
,
100
,
,
,1
1000 1
104
F(Vcm' ) Figure 5. Variation of (a) electron temperature Te with drift energy eK =h2K2 12tn* and (b) drift velocity vd with electricfieldF for different lattice temperatures T0.
142
Transport in a Polarization-Induced
2D Electron Gas 505
The existence of the S-type regions and the saturation of the drift velocity is evident also from the electric field dependence of the electron mobility /u(F) = vd(F) I F , which is shown in Figure 6.
-1
1
r~
~rlrT
'
' ' ' <'I
T=10K o T=20K o
I l
T=50K o w CM
E o
T=100K o
10 4
\
y
-I
1000 10
\
100
1
1—i—' ' ' 1
F (V cm'1)
_i
1
i_
1000
Figure 6. Mobility as a function of the electricfieldfor different lattice temperatures T0. It is necessary to note that usually the drift velocity saturation and decrease of the mobility at high electric field take place in the streaming regime [19], when the electrons move ballistically in the momentum space until they reach the optical phonon energy, emit the optical phonon and repeat the ballistic motion again. But in our case the streaming regime does not take place because for the range of electric field considered the acceleration time [19] rF = ^2m*hco0 I eF necessary to reach the PO phonon energy, is much longer ( r f - 5 x 10~ 1 2 i _ 1 ) in comparison with the e-e scattering time. The electric field dependence of the total mean electron energy <E>,is shown in Figure 7. The total mean energy of the electron is a sum of the mean kinetic energy < Ek >= kBTe and the drift energy eK: <E>= kBTe + £K. We see that at low T the electric field dependence of <E> has more complicated character than at higher T. At low T the PA phonon scattering is very strong in GaN and it suppresses increase of <E>. When F increases the electrons penetrates into the higher energy region where the PA scattering is weak. This results in a steep increase of <E> when F increases. At higher T the intensity of the PA scattering is small in comparison with the DA scattering and the region of steep increase of <E> disappears. Note that this region corresponds to the same range of F where the drift velocity obeys the S-type dependence as was shown in Figure 5 (b), and Figure 6. 143
506 B. K. Ridley & N. A. Zakhleniuk -i
1
"Tl
r-
T=10K o T =20K o T =50K o T=100K o
100
> E.
1—I I I
/
/ y
A LU V
-i—^
10
i
• • • i
-J
100
1 I
L-
1000
F (V cm-1)
Figure 7. Variation of the total mean electron energy <E> with electric field F for different lattice temperatures T0.
200
T
r—T-TTTT
100
0
U_LJ
0.01
_J
1—I I I 1 111
0.1
1
1—» ' i t i l l
1 e (meV)
•
i
i i i • i .1
10
Figure 8. Variation of the electron temperature Te with the drift energy tK temperatures T0 .
144
i
i
i . i
100 for different lattice
Transport in a Polarization-Induced
2D Electron Gas
507
Increase in the total energy <E> does not mean that the electron temperature Te increases as well when F increases. Figure 8 shows variation of the electron temperature Te with the drift energy EK for different lattice temperatures T. As we see this dependence is a non-monotonous function which has a region where the electron temperature decreases. This region corresponds to the electron cooling effect because the electron temperature Te decreases with increase of the electric field. It is even possible to obtain at high electric field an electron temperature Te which is smaller than the lattice temperature T - the absolute cooling effect. Of course, the total energy of the electron gas increases, as it should be, due to increase of the drift energy eK. The physical reason of the electron gas cooling is the intensive emission of the optical phonons when the total energy of the majority of the electrons is close to the PO phonon energy hco0. It is interesting to investigate behaviour of the electron distribution function with the increase of the electric field F. This is shown in Figure 9 for two different lattice temperature T=10 K and T=100 K. The numbers near each curve are the values of the drift energy eK. As the drift energy is a monotonous function of the electric field the higher drift energy corresponds to the higher electric field. Figure 9 shows that at very small electric fields the electron distribution function is close to the Maxwellian equilibrium distribution function which is a maximum at zero kinetic energy. When the electric field increases the distribution becomes wider in the momentum space. This corresponds to an increase of the electron temperature Te. At the same time the distribution function is no longer centered at zero energy but has shifted along the electric field, a shift that corresponds to the drift of the electron gas as a whole. This behaviour continues with increase of the electric field until the electrons start to penetrate to the optical phonon energy. Strong inelastic scattering prevents the electrons from any further increase of their kinetic energy. As a result the electron distribution becomes more narrow or squeezed. This corresponds to a decrease of the electron temperature. At the same time the centre of the distribution function continues its shift when the electric field increases, which means increase of the electron drift energy. The most interesting physical consequence of this behaviour is that the electron distribution function is inverted in the momentum space ft a majority of the electrons populate the high-energy region. Another interesting consequence of the decrease of the electron temperature with increase of the electric field is that the non-equilibrium electron gas becomes "less randomized". This should give, for example, a decrease of the electron noise temperature.
145
508 B. K. Ridley & N. A. Zakhleniuk :
1
:
_ w
0.1 -
'c
:
1
1
1
1
1—
!
j
,
j
K = 0 . 0 1 meV
-
K=10meV
K = 0 . 2 0 meV
-—
K=50
meV
K=90
meV
K=1.00meV-
-
r
i
i
1
'
0
"
•
/
1
~:
i
i
i
•
y
V /]•
./"
j'j
o
\
0.001 t-
/ \ / /
0.0001 -0.6
/ .i
.A ,
1
,
. \ ,
\, 1
;. \ .
'
(
1
J**
y<^
/ S
.-'
I
"^, «— ,.-*"
.
I
f
\
/ ... /
.•~r>»»__ .
\
/
.^ ^ i - - . - ^ -
: \ .
.
i
•-.
•
\:
o
b
/
I
J2
i
<,
*'
:/'
^—s.
\
*
f'
'
•
/
.'.
\
•
K\ \ N
/
/ i
-0.6
>:
•
w-. \
0.001 \
i
\\'f.
/
Li.
: 1.2
T =100K o -
i4
0.6
(e / haQ) 1/2
+-»
!
/ /
1
itrary Un
i
! *
T = 10K
co ~ 0.01 k
0.1
i
i
.' .1
1
0
1
1
1
I
I
\
,/, °-
\ . \
I
6
1 .
\ 1 \
t
1.2
Figure 9. Squeezing of the distribution function F0 (k) of hot electron gas for two different lattice temperatures T o =10 K and JTo=100 K at different values of the drift electron energy EK (numbers near each curve).
146
Transport in a Polarization-Induced
2D Electron Gas
509
7. Summary AlGaN/GaN structures constitute a new class of 2D systems in that a large population of electrons can be produced without doping as a result of spontaneous and strain-induced polarization. We have shown how a simple electrostatic model can describe the dependence of the induced electron density on barrier width in a AlGaN/GaN heterostructure. Large electron densities mean that a complete description of electron transport must include the effects of degeneracy, electron-electron scattering and dynamic screening. Such a description does not exist as yet, but an approach that ignores degeneracy and screening has a certain validity in the hot-electron regime. The effect of electron-electron scattering, in the absence of scattering by impurities and other defects, can then be regarded as establishing a drifted Maxwellian distribution. Accordingly, we have illustrated some consequences of the possibility of impurity-free hot-electron transport in perfect AlGaN/GaN heterostructures. These include S-type negative differential resistance, carrier cooling and squeezed electrons, novel properties that appear most strongly in the temperature range 100K and below. These properties should become accessible to experiment as material quality improves. 8. Acknowledgements We wish to thank the U.K. Engineering and Physical Sciences Research Council (GR/L/56725) and the U.S. Office of Naval Research (N00014-99-1-0014) for their support. 9. References 1. R. Oberhuber, G. Zandler, and P. Vogl, Appl. Phys. Lett. 73 818 (1998). 2. O. Ambacher, J. Smart, J.R. Shealy, N.G. Weimann, K. Chu, M. Murphy, W.J. Schaff, L. Eastman, R. Dimitrov, L. Wittmer, M. Stitzmann, W. Reiger, and J. Hilsenbeck, J. Appl. Phys. 85 3222 (1999). 3. J. Burm, W.J. Schaff, L.F. Eastman, H. Amano, and I. Akasaki, Appl. Phys. Lett. 68 2849 (1996). 4. Y-F. Wu, B.P. Keller, S. Keller, D. Kapolnek, P. Kozodoy, S.P. Denbaars, and U.K. Mishra, Appl. Phys. Lett. 69 1438 (1996). 5. B.K. Ridley, O. Ambacher, and L.F. Eastman, Semicond. Sci. Technol. 15 270 (2000). 6. B.K. Ridley, Appl. Phys. Lett. 77 1 (2000). 7. J.P. Ibbetson, P.T. Fini, K.D. Ness, S.P. Denbaars, J.S. Speck, and U.K. Mishra, Appl. Phys. Lett. 77 250 (2000). 8. N.G. Weimann, L.F. Eastman, D. Doppalapudi, H.M. Ng, and T.D. Moustakas, J. Appl. Phys. 83 3656 (1998). 9. H.M. Ng, D. Doppalapudi, T.D. Moustakas, N.G. Weimann, and L.F. Eastman, Appl. Phys. Lett. 73 821 (1998). 10. M.J. Murphy, K. Chu, H. Wu, W. Yeo, W.J. Schaff, O. Ambacher, L.F. Eastman, T.J. Eustis, J. Silcox, R. Dimitrov, and M.S tutzmann, Appl. Phys. Lett. 75 3653 (1999). 11. S.E. Esipov and I.B. Levinson, Zh. Eksp. Teor. Fiz. 90 330 (1986); (Sov. Phys. JETP. 63 191 (1986). 12. B.K. Ridley, Electrons and Phonons in Semiconductor Multilayers (Cambridge University Press, Cambridge, 1997). 13. N.A. Zakhleniuk, C.R. Bennett, B.K. Ridley, and M. Babiker, Appl. Phys. Lett. 73 2485 (1998). 14. B.K. Ridley, J. Phys. C: Solid State Phys. 17 5357 (1984). 15. N.A. Zakhleniuk, B.K. Ridley, M. Babiker, and C.R. Bennett, Physica B 272 309 (1999). 16. S.E. Esipov and Y.B. Levinson, Advances in Physics 36 331 (1987). 17. B.K. Ridley, Quantum Processes in Semiconductors, 4th edition, Oxford Press, 1999. 18. N.A. Zakhleniuk, B.K. Ridley, M. Babiker, and C.R. Bennett, Physica B272 309 (1999). 19. V.E. Gantmakher and Y.B. Levinson, Carrier Scattering in Metals and Semiconductors, NorthHolland, 1987.
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International Journal of High Speed Electronics and Systems, Vol. 11, No. 2 (2001) 511-524 © World Scientific Publishing Company
IMPACT IONIZATION A N D HIGH FIELD EFFECTS IN W I D E B A N D G A P S E M I C O N D U C T O R S
M. REIGROTZKI, J. R. MADUREIRA^A. KULIGK, N. FITZER, R. R E D M E R Fachbereich Physik, Universitat Rostock D-18051 Rostock,
Germany
and
Department
S. M. GOODNICK, M. DUR+ of Electrical Engineering, Arizona State Tempe, Arizona 85287-5706, USA
University
Impact ionization plays a crucial role for electron transport in wide-bandgap semiconductors at high electric fields. Therefore, a realistic band structure has to be used in calculations of the microscopic scattering rate, as well as high field quantum corrections such as the intercollisional field effect. Here we consider both, and evaluate the impact ionization rate for wide-bandgap materials such as ZnS. A pronounced softening of the impact ionization threshold is obtained, as found earlier for materials like Si and GaAs. This field dependent impact ionization rate is included within a full-band ensemble Monte Carlo simulation of high field transport in ZnS. Although the impact ionization rate itself is strongly affected, little effect is observed on measurable quantities such as the impact ionization coefficient or the electron distribution function itself.
1. Introduction High field transport in semiconductors has long been a concern in relation to the performance of semiconductor electronic and optoelectronic devices for over three decades. 1 To a high degree of success, nonequilibrium transport has been described within the context of the semi-classical Boltzmann Transport Equation (BTE), with instantaneous scattering events described by Fermi's Golden rule, and uncorrelated scattering processes of carriers with the lattice and one another. Such a framework is the basis for semiconductor device simulation tools based on moments of the BTE, or through direct solution of the BTE via particle based techniques such as the Ensemble Monte Carlo (EMC) method. 2 However, the question has always remained as to the limitations of this approach in terms of the underlying quantum transport equation, and the role of corrections to the BTE such as collision broadening (CB) •permanent address: Instituto de Fisica, Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, Sao Paulo, Brazil t permanent address: Siemens GmbH, An der Untergeis 8 D-36251 Bad Hersfeld, Germany 149
512
M. Reigrotzki
et al.
and the intercollisional field effect (ICFE), which become more pronounced at high fields. In particular, the role of such effects in the performance of submicron Si and GaAs devices has been uncertain at best. In recent years, interest has developed in wide-bandgap semiconductors such as ZnS, GaN and SiC for optoelectronic and high-power, high-frequency electronic applications. In such wide-bandgap materials, fields in excess of 1 MV/cm are common, which necessitates re-evaluation of the validity of the BTE for such systems. A high-field process of particular interest in such wide-bandgap systems is the impact ionization rate associated with electron-hole pair excitation due to energetic hot carriers in the conduction or valence bands. In ZnS and SrS thin film electroluminescent devices for example, this mechanism is responsible for space charge formation which may result in field clamping and suppression of luminescence.3 We have previously calculated the impact ionization rate for Si and GaAs, 4 ZnS, 5 ' 6 ' 7 GaN, 8 and SrS 9 using a full band structure approach, but neglecting the influence of the electric field on the collision term through the ICFE. In these studies, we have used the local and nonlocal empirical pseudopotential method to calculate the band structure, which has a pronounced influence on the numerical results for the impact ionization rate compared to analytical approximations for the electronic dispersion. The ICFE should also affect the behavior of the impact ionization rate in the high-field regime. For instance, the evaluation of the Barker-Ferry kinetic equation 10 for Si 11 has indicated that the threshold energy for impact ionization is lowered due to the field because the impacting electron is further accelerated during the collision. This results in a higher ionization rate near the threshold, whereas for higher energies of the impacting electron the field influence vanishes. Quade et al.12 applied a density matrix approach to carrier generation in semiconductors. Within the parabolic band approximation, they were able to give an essentially analytical result for the field-assisted impact ionization rate which was evaluated for GaAs and Si. Again, systematic lowering of the threshold energy with the field strength has been shown. The field-dependence of the collision integral was also studied by means of the Green function technique, solving the Kadanoff-Baym equations 13 in various approximations. Avoiding the conventional gradient expansion or delta-function approximation for the spectral density, an integral equation was derived for the EDF taking into account the ICFE. 1 4 ' 1 5 The Levinson16 or Barker-Ferry transport equation 10 was evaluated within a saddle-point approximation for GaAs at high field strengths, taking into account electron-phonon interactions. 17 Alternatively, a gauge-invariant formulation of the Airy representation of the Kadanoff-Baym theory was developed. 18 The Mori projection operator technique was applied to study nonlinear transport in semiconductors including the ICFE and collision broadening. 19 The method of the nonequilibrium statistical operator as developed by Zubarev 20 was applied to study both steady-state and transient properties in hot-electron transport. 21 ' 22 In the present work, we derive quantum kinetic equations for the EDF in
150
Impact Ionization
and High Field Effects in Wide Band Gap Semiconductors
513
semiconductors using the Zubarev approach, and take into account the full fielddependence of the collision integral. We then focus on impact ionization processes and re-derive the general, field-dependent impact ionization rate given by Quade et al.12 within the parabolic band approximation, and the Keldysh formula 23 valid for energies near the threshold. As previously found, a softening of the threshold is obtained due to the ICFE, which is much more pronounced in the wide-bandgap materials due to the much higher onset fields required for impact ionization to occur. These field-dependent ionization rates for ZnS are incorporated into an EMC simulation for high field transport to evaluate the effect on observable quantities such as the electron distribution function, and the impact ionization coefficient. In both cases, the role of the ICFE is found to be minimal even though the effect on the impact ionization rate itself is substantial. 2. I m p a c t Ionization R a t e 2 . 1 . Quantum
kinetic
equation
Quantum kinetic equations for a semiconductor in a homogeneous electric field Eo(t) can be derived within response theory taking into account the full fielddependence of the collision term. 24 The distribution function for electrons in the band vy with momentum k\ is determined from jrf^kAt) Ot
+ eE0(t) • —^ fVlkl(t) dk\
= Jei^k^t).
(1)
The collision term Je{vik\,t) contains electron-phonon, electron-impurity and electron-electron scattering via a respective Hamilton operator. We restrict ourselves to electron-electron collisions since we are especially interested in the impact ionization rate. Defining {vi,ki} = i, we get within second-order perturbation theory a Markovian form for the electron-electron collision term: 25 .£'(M)
=
- 4 f ] T | M t o t ( l , 2 ; 3 , 4 ) | 2 l i£i r v f n
di'ee<*'-f> cos[n°(l,2;3,4)(t,t')]
-»°./-oo
2J3J4
x {/i(*)/2(*)[l " /s(*)][l - hit)] - h(t)h(t)[l
- A(i)][l - f2(t)}} • (2)
Aftot is the total matrix element including direct, exchange, and umklapp processes. The collision term is given by an integral over all former times t' which is usually interpreted as memory effect. In the limit of constant electric fields Eo, canonical momenta transform the distribution functions to a gauge-invariant form: fi°(l,2;3,4)(M')
=
\(ez + £* ~ £i - £2)(t - t')
_eEofkL+kj__kL_h\{t_tl)2+l3{t_tl)3 2h
\m3
rrn
mi
m2 J
3 (3)
151
514 M. Reigrotzki et al.
The kinetic energies e* are given by the band structure, and irn is the effective mass of an electron in the band Vi at wave vector ki. The quantity WF is denned by 3
e2El (I 2n \rri3
1 TO4
1 mi
1 m^
(4)
In the process of impact ionization, a conduction band electron impact ionizes a valence band electron, i. e. 1 + 2 - • 3 + 4, see Fig. 1. The band indices and energies 1,3,4 run over the conduction bands, while 2 belongs to the valence bands. Supposing that the semiconductor is not highly excited, the conduction bands are almost empty so that the Pauli blocking factors are unity, i. e. (1 — /j) « 1. Furthermore, the (second) in-scattering term in Eq. (2) can be neglected compared with the (first) out-scattering term in the balance for the population of states with momentum fci. The collision integral is then simply given by a field-dependent impact ionization rate ru(l,E0) via J^(l,E0,t) = -ru(l,E0,t)fVl(ki - eE0t) with
ru(l,Eo,t)
= - ^ ]T|M tot (l,2;3,4)| 2 h
2?A ^'e £ ( t '- t ) cos[n°(l,2;3,4)(t,t')]-
xlimf
(5)
J —OO
The wave-vector dependent impact ionization rate has to be evaluated considering the full field dependence and a realistic band structure in the cosine term as well as the full momentum dependence of the matrix element Mtot including an appropriate screening function for the Coulomb interaction.
valence bands Fig. 1. Schematic impact ionization process for electrons.
2.2. Electron
initiated
impact
ionization
at zero
fields
The numerical evaluation of Eq. (5) is rather complex. We first review results for the impact ionization rate within simple approximations. Neglecting the influence of the electric field in the collision term and considering the Markov limit t -> oo, the integral over the time t' gives the energy conserving delta function and we have r « ( l , 0 ) = - ~ Y, |M t0 t(l,2;3,4)| 2 (5(e3 + e4 - e i - £ 2 ) . 2,3,4
152
(6)
Impact Ionization
and High. Field Effects in Wide Band Gap Semiconductors
515
The integration over the momenta can easily be performed supposing a constant matrix element and spherical parabolic bands with effective masses for the valence (m2 = mv) and conduction bands (mi = 7713 — im = mc). Denning the effective mass ratio a = mc/mv and the parameter \i = ( l + 2 a ) / ( l + a ) , the threshold energy Eth = l*Eg is related to the fundamental band gap Eg, and the famous Keldysh formula 23 for impact ionization is derived: rlK)(£l,0)=P0[£l-Eth]2.
(7)
The prefactor PQ is often used as fit parameter for the energy-dependent impact ionization rate in simulations of high field transport in semiconductors. 26 e\ = h2ki/(2mi) is the kinetic energy of the impacting electron. However, previous calculations for Si and GaAs, 4 ' 27 ZnS, 5 ' 6 ' 7 GaN, 8 ' 28 - 29 ' 30 SrS, 8 ' 9 InN, 31 and SiC 32 have shown that the full band structure has to be considered when calculating the impact ionization rate via Eq. (6). Pronounced contributions arise from higher conduction bands, especially in wide-bandgap materials like ZnS, GaN, or SrS. The empirical pseudopotential method (EPM) is the standard tool to determine the band structure of a semiconductor material. Four to six conduction bands (and four valence bands) are usually considered for the complete numerical evaluation of the zero-field impact ionization rate (6). The influence of nonlocal pseudopotentials has been studied for ZnS 7 and ab initio band structures have been used in calculations of the impact ionization rate of SrS. 9 However, the EPM represents in most cases a reasonable compromise between desired accuracy and available computer capacity. The integrals in Eq. (6) extend over the entire Brillouin zone and are evaluated using an efficient numerical procedure developed by Sano and Yoshii.27 Making extensive use of symmetry relations imposed by the crystal structure, the integrations can be restricted to the irreducible wedge of the Brillouin zone where a large number of points can be taken into account for the numerical evaluation. For instance, using a uniform grid in wave vector space with 152 points in the irreducible wedge corresponds to 4481 points across the whole Brillouin zone; 5 ' 6 ' 7 ' 8 ' 9 228 points in the irreducible wedge were also considered. 28 ' 29 ' 30 ' 31 ' 32 A further increase of the number of grid points does not affect the calculated rate significantly, except in the threshold region. The interaction between the conduction and valence electrons is described by a wave-vector dependent dielectric function derived by Levine and Louie. 33 The frequency dependence of the dielectric function becomes more important as the energy of charge carriers increases. At high energies, we observe that primarily umklapp processes (which we take into account up to the sixth order) contribute to the calculated rate and, therefore, due to the large momentum transfer, the influence of the carrier energy on the screening function is less important. Jung et al.34 have employed a wave-vector and frequency-dependent dielectric function within the random phase approximation for the calculation of the impact ionization rate in GaAs using a Monte Carlo integration technique. Their results agree well
153
516
M. Reigrotzki et al.
with our findings which were obtained using a static dielectric function.4 The wave-vector dependent impact ionization rate ra(l,0) is usually averaged over the entire Brillouin zone to obtain an energy-dependent rate R(E) via
R(E) = J26(E~ ei)r«(l,0)/5>(E - £i), 1
(8)
I
which is shown in Fig. 2 for the wide-bandgap materials ZnS, GaN, and SrS. The general behavior is almost the same. The threshold energy relative to the conduction band minimum is given approximately by the gap energy. The numerical results according to Eq. (8) are well fit by a power law relation of the form R(E) = C[E - Eth]a,
(9)
which can easily be implemented in full-band Monte Carlo simulations of electron transport in semiconductors. The prefactor C, the threshold energy Eth, and the power a are given in Table 1 for a variety of semiconductor materials. Obviously, the influence of the band structure manifests itself in values a > 2 compared with the original Keldysh formula (7) derived for spherical parabolic bands.
electron energy [eV] Fig. 2. Electron impact ionization rate for GaN, ZnS, and SrS.
2.3. Hole initiated
impact
ionization
at zero
fields
A second possible electron-electron scattering process contributing to impact ionization is the relaxation of a valence band electron, passing enough energy to a second valence band electron to be ionized across the gap into the conduction band and generate an additional free carrier. This scattering process can be understood as the scattering of two holes in which a hole residing in the conduction band is ionized across the gap into the valence band, leading to an additional positive charge in the valence band and, correspondingly, to a negative free charge in the conduction
154
Impact Ionization
and High Field Effects in Wide Band Gap Semiconductors
517
Table 1. Parameters for the interpolation formula (8) for the electron and hole initiated (see below) impact ionization rate for various semiconductor materials. 3 5 ' 3 6
c
a
36.22 93.659 0.00949
3.683 4.743 7.434
3.8 4.0
5.935 59.723
5.073 3.182
3.4 3.8
0.35 0.71
5.33 C.23
Eth [eV]
[KPeV^s- 1 ]
GaAs GaN
0.8 1.8 3.6
ZnS SrS holes: GaN ZnS
electrons: Si
band, see Fig. 3. This scattering process is interpreted as the hole initiated impact ionization, and due to the above explanations, the corresponding ionization rate can be determined using the very same numerical scheme as in the calculation of the ionization rate of electrons described above. The sole difference is the inversion of the dispersion relation. conduction band 4,
3
valence band
Fig. 3. Schematic impact ionization process for holes.
The respective impact ionization rates are calculated for holes in all four valence bands. Holes in the upper valence band can not initiate ionization processes. In GaN and ZnS the main contribution arises from the lowest and second lowest band. In SrS only holes from the lowest valence band are able to initiate ionization events. The corresponding energy-dependent impact ionization rate is shown in Fig. 4 for the wide-bandgap materials ZnS, GaN, and SrS. 35 Previous theoretical results of Oguzman et al.29 for the threshold region of GaN using another band structure (B) are included for comparison. The shaded area covers their non-averaged, kdependent rates according to their band structure (B) via e(fc). For comparison, the inset shows our non-averaged, fc-dependent rates displayed in the same manner using the EPM band structures A and B (full and open circles). As can be seen, the calculated rate is relatively insensitive to modest changes in the band structure itself. The rates of Oguzman et al. are up to two orders of magnitude higher than ours.
155
518
M. Reigrotzki et al.
This difference is probably due to different integration schemes or screening models, and using different numbers of k points in the irreducible wedge. The corresponding parameters of the interpolation formula according to Eq. (9) are given in Table l. 36 10" 10 ,a 10"
10 10': 10" 10"
2 10" c g
w% 3.5
4.5
5.5
6.5
N 10 c o
I 101'
•—-ZnS •—• GaN (A) °—° QaN (B) —— SrS + GaN (Ref. 29)
10° 10 6
Fig. 4.
8
10 12 14 16 hole energy [eV]
18
20
22
Hole impact ionization rate for GaN, ZnS, and SrS.'
Compared with the electron initiated impact ionization rates displayed in Fig. 2, two special features have to be noticed for the hole rates. First, the threshold energy for hole ionization is in general slightly lower than for electron ionization. 9 This result is due to the flat shape of the top of the upper valence bands, which allows holes to initiate ionization events as soon as they have reached a kinetic energy equal to the gap energy. In the case of SrS, the width of the upper valence band is too narrow to allow holes to gain enough kinetic energy to initiate ionization events due to energy conservation. For this reason, only holes in the lowest valence band contribute to the ionization rate and, therefore, SrS does not show the usual threshold behavior near the gap energy. Instead, the ionization rate sets in only above 14.7 eV and immediately jumps to very high values. Second, the hole initiated impact ionization rates vanish in a certain energy range completely, i. e. between about 5 eV and 12 eV for ZnS and 7 eV and 19.5 eV for GaN. For higher energies, they set in again immediately as discussed already for SrS. This special behavior is due to the occurrence of an energy gap between the upper three valence bands and the lowest one and, therefore, no states are available for energetic holes. Thus, the hole initiated impact ionization rate shows a very different behavior compared with the respective electron rate except for the region near the threshold energy, if hole ionization occurs there at all. Since the hole scattering rates are as low as the electron rates near the threshold energy, and vanish completely in the medium energy range, we conclude that hole initiated impact ionization processes can be neglected compared with the electron processes for both ZnS and SrS.
156
Impact Ionization
2.4.
Field-dependent
and High Field Effects in Wide Band Gap Semiconductors
impact
ionization
519
rate
The evaluation of the field-dependent impact ionization rate (5) is important for high field strenghts when the electrons are further accelerated during the collision. This is usually denoted as intra-collisional field effect (ICFE). Quade et oZ.12 were able to evaluate the integrations over the momenta in Eq. (5) essentially analytically in the Markov limit t -» oo considering the full, statically screened Coulomb matrix element, but performing the effective mass approximation. We obtain with the definition of the Airy function
(3.
^Ai(±(3^)=rdrc°s(ar3±:cr)
ao)
the general result ^(ei,Eo)
=
P
Q
^°°dE5(
£ l
,
£ l
-£)^Ai(
£ ; t h
-j
+ £
),
(11)
(l + a X e E o ) 2 1 1 7 3 8mch Up is the electro-optical frequency. We follow the definition of the other quantities in Eq. (11) given in Quade et al.12 Performing the constant matrix element approximation but taking into account the full field-dependence of the collision integral (5), a modified impact ionization rate can be derived:
r?>
P
<-*> = *f ^ V ^ F ) '
<12
»
The zero-field limit yields the original Keldysh formula for impact ionization (7). Comparing Eqs. (7), (9), and (12), we have proposed recently 24 a new fit formula for impact ionization that considers the influence of an applied electric field and the full band structure according to
r
<*•*> • ° r < * { • £ ) ' ^ i * * ^ -
»
The parameters C, Etu, and a are already given in Table 1 for the zero-field electron initiated impact ionization rates. We show as an example the field dependent impact ionization rate for the widebandgap material ZnS in Fig. 5; the behavior of GaN and SrS is very similar. The ICFE influences only the direct threshold region and leads to a systematic lowering of the threshold energy with increasing field strength. The sensitivity of this lowering is related to the effective masses which are introduced as material parameters in the quantity EF' in Eq. (11) for the parabolic band result. The analytical form of Eq. (11) is also adapted for the fit formula (13) that reflects the full band structure via the parameter a and, again, the quantity EF\ About 0.5
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&•
9
. 2.0 MV/cm MV/cm — 1.5 MV/cm — 1.0 0.5 MV/cm
CD
•
10 7
-
I
//IP. ,u
3.0
• i l l /i / ' i
3.5
I
4.0
I
4.5 5.0 energy [eV]
0.1 MV/cm Zero Field
5.5
6.0
Fig. 5. Field-dependent electron impact ionization rate for ZnS. 2 4
eV (1 eV) above the threshold energy, the impact ionization rate becomes already independent of the field for the narrow (wide) band gap materials. 3. Ensemble Monte Carlo Simulation and Impact Ionization Coefficient In order to understand the role that the ICFE plays in high field transport, we include the field dependent rates of the previous section into a full band EMC simulation for ZnS described in detail elsewhere.7 Basically, an EMC simulation is a particle based simulation technique where the particle trajectories under the influence of external fields and random scattering events are tracked. 2 Instantaneous random scattering events in the crystal are generated stochastically using the random number generator. 2,37 As such, the EMC provides a direct solution to the Boltzmann equation for the one-particle distribution functions for electrons and holes and macroscopic averages derived from them. The full band dispersion relation E(k) for ZnS is taken into account using the EPM band structure. In the present work we only simulate electron transport and neglect the role of any holes generated by impact ionization processes. The Monte Carlo simulation includes polar optical phonon scattering, scattering due to acoustic phonons, optical deformation potential scattering, ionized impurity scattering, and band-to-band impact ionization using the modified rates of the previous section. As discussed elsewhere,7 we employ deformation potential scattering only above the intervalley threshold, where the density of states is used to renormalize the scattering rate to account for full band effects in the scattering rate. Two deformation potential constants are used to model the scattering rate due to optical phonons: D(l,2) = (1 x 10 9 ,9 x 108) eV/cm. 36 Simulations are performed for an electron concentration of 1016 c m - 3 in the conduction band and a temperature of 300 K. 158
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Fig. 6 compares the electron distribution function versus energy calculated from the EMC simulation for three different field strengths above 1 MV/cm. As can be seen, there appears to be no significant effect due to the inclusion of the ICFE at any field strength. T
'
i
'
i
'
i
•
i
'
i
'
r
electron energy [eV] Fig. 6. Electron distribution function for three different electric field strengths (in MV/cm) with (solid lines) and without (broken lines) the inclusion of the ICFE.
A measurable quantity directly associated with the impact ionization rate is the impact ionization coefficient which represents the number of electron-hole pairs created by an energetic particle per unit path length in the crystal. The impact ionization coefficient is directly related to carrier multiplication in a reverse biased junction, and hence can be obtained from current multiplication versus bias data. In the EMC simulation, this quantity is calculated by tabulating the number of impact ionization events per carrier traversing a 0.5 /xm thick layer of ZnS. Fig. 7 shows the calculated result with and without the inclusion of the ICFE. The result without the ICFE corresponds to the results reported earlier by us for this quantity. 35 As can be seen from this figure, there is only little difference in the impact ionization coefficient despite the rather dramatic effect on the impact ionization rate near the threshold energy shown in Fig. 5. 4. Discussion and Conclusions The main results from the present study are the calculated effect of the ICFE on the impact ionization rate in ZnS and other wide-bandgap materials, and its subsequent effect on the electron distribution function and impact ionization coefficient. While the effect on the bare impact ionization rate is substantial, the resulting effect on the distribution function itself and the associated impact ionization coefficient is negligible. The reason for the lack of any substantial influence on transport may be understood from Fig. 5. When the field is relatively low (below 0.1 MV/cm) the ICFE is negligible. At high fields close to the threshold for impact ionization to occur, 159
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et al.
11
0.50
i
I
0.52
i
l
i
I
i
l
.
1
i
0.54 0.56 0.58 0.60 inverse electric field [cm/MV]
I
0.62
i
1
0.64
Fig. 7. Calculated impact ionization coefficient versus inverse electric field strength for EMC simulations with (solid line) and without (broken line) the inclusion of the ICFE.
the threshold energy is broadened, but only for the region of energy over which the scattering rate is relatively low, below 10 10 s _ 1 . The strongest contributions to the total scattering must come from the higher energy tail of the distribution where the impact ionization rate is comparable to the electron-phonon scattering rate, for any effect to be observed in the distribution function itself. However, the effect of the ICFE in this high energy region is small. Likewise, in terms of the impact ionization coefficient itself, the significant contributions to band-to-band impact ionization come from electrons well above threshold where the scattering rate is large. Conversely, in this region, there is little effect again due to the ICFE, hence the influence on the impact ionization coefficient is negligible. The role of collision broadening on the impact ionization rate has been studied recently using nonequilibrium Green functions for the derivation of an appropriate kinetic equation. 38 It has been shown that collisions between the particles lead to a broadened one-particle spectral function which is of non-Lorentzian shape in contrast to former assumptions. 11 A further lowering of the threshold energy for impact ionization is obtained, and the rate itself is increased in the threshold region. Collision broadening is already effective for field strengths below 0.5 MV/cm, while the ICFE determines the behavior of the impact ionization rate for higher fields. For impact energies of about 1 eV above threshold, the collision broadening is almost negligible similar to the behavior found for the ICFE, see Fig. 5. Therefore, collision broadening has also no pronounced influence on the EDF and the ionization coefficient. Acknowledgements We thank K. Brennan (Atlanta), D. Ferry (Tempe), M. Fischetti (Yorktown Heights), T. Kuhn (Miinster), V. Morozov (Moscow), W. Schattke (Kiel), E. Scholl (Berlin) 160
Impact Ionization and High Field Effects in Wide Band Gap Semiconductors
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and P. Vogl (Miinchen) for stimulating discussions. This work was supported by the Deutsche Forschungsgemeinschaft under contract No. R E 882/9-2 and by DARPA under t h e P h o s p h o r Technology Center of Excellence, G r a n t No. MDA 972-93-10030.
1. Quantum Transport in Semiconductors, edited by C. Jacoboni, L. Reggiani, and D. K. Ferry, Plenum Press, New York, 1992. 2. C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer-Verlag, Berlin, 1989. 3. W. M. Ang, S. S. Pennathur, L. Pham, J. F. Wager, S. M. Goodnick, and A. A. Douglas, "Evidence for band-to-band impact ionization in evaporated ZnS:Mn alternatingcurrent thin-film electroluminescent devices", J. Appl. Phys. 7 7 , 2719 (1995). 4. M. Stobbe, R. Redmer, and W. Schattke, "Impact ionization rate in GaAs", Phys. Rev. B 4 9 , 4494 (1994). 5. M. Reigrotzki, M. Stobbe, R. Redmer, and W. Schattke, "Impact ionization rate in ZnS", Phys. Rev. B 52, 1456 (1995). 6. M. Reigrotzki, R. Redmer, I. Lee, S. S. Pennathur, M. Diir, J. F. Wager, S. M. Goodnick, P. Vogl, H. Eckstein, and W. Schattke, "Impact ionization rate and high-field transport in ZnS with nonlocal band structure", J. Appl. Phys. 8 0 , 5054 (1996). 7. M. Diir, S. M. Goodnick, S. S. Pennathur, J. F. Wager, M. Reigrotzki, and R. Redmer, "High-field transport and electroluminescence in ZnS phosphor layer", J. Appl. Phys. 8 3 , 3176 (1998). 8. M. Reigrotzki, M. Diir, W. Schattke, N. Fitzer, R. Redmer, and S. M. Goodnick, "HighField Transport and Impact Ionization in Wide Bandgap Semiconductors", phys. stat. sol. (b) 2 0 4 , 528 (1997). 9. M. Diir, S. M. Goodnick, R. Redmer, M. Reigrotzki, M. Stadele, and P. Vogl, Journal of the Society for Information Displays (accepted for publication). 10. J. R. Barker and D. K. Ferry, "Self-Scattering Path-Variable Formulation of highField, time-Dependent, Quantum Kinetic Equations for Semiconductors in the FiniteCollision-Duration Regime", Phys. Rev. Lett. 4 2 , 1779 (1979); see also D. K. Ferry, Semiconductors, Macmillan, New York, 1991, Chapter 15. 11. J. Bude, K. Hess, and G. J. Iafrate, "Impact ionization in semiconductors: Effects of high electric fields and high scattering rates", Phys. Rev. B 4 5 , 10958 (1992). 12. W. Quade, E. Scholl, F. Rossi, and C. Jacoboni, "Quantum theory of impact ionization in coherent high-field semiconductor transport", Phys. Rev. B 50, 7398 (1994). 13. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, Benjamin, New York, 1962. 14. A. P. Jauho and J. W. Wilkins, "Theory of high-electric-field quantum transport for electron-resonant impurity systems", Phys. Rev. B 2 9 , 1919 (1984). 15. S. K. Sarker, "Quantum transport theory for high electric fields", Phys. Rev. B 3 2 , 743 (1985). 16. I. B. Levinson, Zh. Eksp. Teor. Fiz. 4 7 , 660 (1969) [Sov. Phys.-JETP 3 0 , 362 (1970)]. 17. P. Lipavsky, F. S. Khan, F. Abdosalami, and J. W. Wilkins, "High-field transport in semiconductors. I. Absence of the intra-collisional-field effect", Phys. Rev. B 4 3 , 4885 (1991). 18. R. Bertoncini and A. P. Jauho, "Quantum transport theory for electron-phonon systems in strong electric fields", Phys. Rev. Lett. 6 8 , 2826 (1992). 19. J. Y. Ryu and S. D. Choi, "Quantum-statistical theory of high-field transport phenomena", Phys. Rev. B 4 4 , 11 328 (1991). 20. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bureau, New
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York, 1974. 21. D. Y. Xing and C. S. Ting, "Green's-function approach to transient hot-electron transport in semiconductors under a uniform electric field", Phys. Rev. B 3 5 , 3971 (1987). 22. D. Y. Xing, P. Hu, and C. S. Ting, "Balance equations for steady-state hot-electron transport in the approach of the nonequilibrium statistical operator", Phys. Rev. B 3 5 , 6379 (1987). 23. L. V. Keldysh, "Kinetic theory of impact ionization in semiconductors", Zh. Exp. Theor. Phys. 3 7 , 713 (1959) [Sov. Phys.-JETP 3 7 , 509 (I960)]. 24. R. Redmer, J. R. Madureira, N. Fitzer, S. M. Goodnick, W. Schattke, and E. Scholl, "Field effect on the impact ionization rate in semiconductors", J. Appl. Phys. 8 7 , 781 (2000). 25. D. N. Zubarev, V. Morozov, and G. Ropke, Statistical Mechanics of Nonequilibrium Processes, Vol. 1: Basic Concepts, Kinetic Theory, Akademie-Verlag, Berlin, 1996. 26. M. V. Fischetti and S. E. Laux, "Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects", Phys. Rev. B 3 8 , 9721 (1988). 27. N. Sano and A. Yoshii, "Impact-ionization theory consistent with a realistic band structure of silicon", Phys. Rev. B 4 5 , 4171 (1992). 28. J. Kolnik, I. H. Oguzman, K. F. Brennan, R. Wang, and P. P. Ruden, "Calculation of the wave-vector-dependent interband impact-ionization transition rate in wurtzite and zinc-blende phases of bulk GaN", J. Appl. Phys. 7 9 , 8838 (1996). 29. I. H. Oguzman, E. Bellotti, K. F. Brennan, J. Kolnik, R. Wang, and P. P. Ruden, "Theory of hole initiated impact ionization in bulk zincblende and wurtzite GaN", J. Appl. Phys. 8 1 , 7827 (1997). 30. E. Bellotti, K. F. Brennan, R. Wang, and P. P. Ruden, "Calculation of the electron initiated impact ionization transition rate in cubic and hexagonal phase ZnS", J. Appl. Phys. 8 2 , 2961 (1997). 31. E. Bellotti, B. K. Doshi, K. F. Brennan, J. D. Albrecht, and P. P. Ruden, "Ensemble Monte Carlo study of electron transport in wurtzite InN", J. Appl. Phys. 8 5 , 916 (1999). 32. E. Bellotti, H.-E. Nilsson, K. F. Brennan, P. P. Ruden, and R. Trew, "Monte Carlo calculation of hole initiated impact ionization in 4H phase SiC", J. Appl. Phys. 8 7 , 3864 (2000). 33. Z. H. Levine and S. G. Louie, "New model dielectric function and exchange-correlation potential for semiconductors and insulators", Phys. Rev. B 2 5 , 6310 (1982); see also M. S. Hybertson and S. G. Louie, "First-principles theory of quasiparticles: Calculation of band gaps in semiconductors and insulators", Phys. Rev. Lett. 5 5 , 1418 (1985); "Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies", Phys. Rev. B 3 4 , 5390 (1986). 34. H. K. Jung, K. Taniguchi, and C. Hamaguchi, "Impact ionization model for full band Monte Carlo simulation in GaAs", J. Appl. Phys. 7 9 , 2473 (1996). 35. M. Reigrotzki, R. Redmer, N. Fitzer, S. M. Goodnick, M. Diir, and W. Schattke, "Hole initiated impact ionization in wide band gap semiconductors", J. Appl. Phys. 8 6 , 4458 (1999). 36. M. Reigrotzki, PhD thesis, University of Rostock, 1998. 37. C. Moglestue, Monte Carlo Simulations of Semiconductor Devices, Chapman and Hall, New York, 1993. 38. J. R. Madureira, D. Semkat, M. Bonitz, and R. Redmer, "Impact ionization rates of semiconductors in an electric field: The effect of collisional broadening", in preparation.
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International Journal of High Speed Electronics and Systems, Vol. 11, No. 2 (2001) 525-584 © World Scientific Publishing Company
Simulation of Carrier Transport in Wide Band Gap Semiconductors E. Bellotti1, M. Farahmand2, M. Goano3, E. Ghillino3, C. Garetto3, G. Ghione3, H.-E. Nilsson4, K. F. Brennan5 and P. P. Ruden6 'Dept. of Electrical Engineering, Boston University, Boston, MA. 02215. 2 Movaz Networks, 5445 Triangle Parkway, Norcross, GA 30092 3 Dipartimento di Elettronica, Politecnico di Torino, corso Duca degli Abruzzi 24,110129 Torino, Italy 4 Dept. of Information Technology, Mid-Sweden University, S-851 70 Sundsvall, Sweden. 5 School of Electrical and Computer Engineering, Georgia Tech, Atlanta, GA, 303320250 6 Dept. of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, 55455.
1.
Introduction
Future semiconductor device applications are expected to emerge in high frequency, high power electronics, high temperature and radiation tolerant electronics, short wavelength detection and emission, and chemically corrosive environments. The insertion of traditional semiconductor technologies based on silicon and GaAs in these environments will certainly encounter difficulty due to the relatively small energy band gaps in these materials. Though silicon and GaAs are not generally characterized as narrow band gap materials, their band gaps are sufficiently small that at temperatures much above 250 C they act like intrinsic semiconductors thereby limiting their usefulness in high temperature device applications. The relatively small band gap of these materials further limits their utility due to their low breakdown voltage. In addition, the low thermal conductivity of silicon limits the packing density of silicon integrated circuits and requires sophisticated packaging schemes for high levels of circuit integration. Finally, owing to its indirect energy band gap, silicon is of limited usefulness in a wide variety of optoelectronic device applications. For the reasons stated above, alternative semiconductor materials that have a wider energy band gap have become of interest as possible candidates for microelectronic devices operating in harsh environments and at high power and high frequencies. Wide band gap semiconductors greatly extend the reach of conventional electronics since these materials can maintain performance at elevated temperatures, function satisfactorily under high applied voltages, are radiation hard and can emit short 163
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wavelength radiation. Many wide band gap semiconductors have additional advantageous properties. Among these are relatively high thermal conductivities, high saturation drift velocities, small dielectric constants, high breakdown voltages, and very low thermally generated leakage currents. The combination of all of these features makes the wide band gap semiconductors highly attractive in many emerging high power, high temperature, and high frequency device applications. There are many different wide band gap semiconductor materials. Of these the most attractive candidate materials for high power electronics are SiC and the IllNitrides. The Ill-Nitride family is constituted by the binary compounds A1N, InN and GaN and their associated ternary and quaternary alloys, AlxGa!_xN, InxGa!.xN, AlxIn!_ XN, and InxGayAV^yN. These compounds typically crystallize in the wurtzite structure, but in some instances the zincblende structure occurs as well. SiC crystallizes in many different polytypes. Each polytype has a different band structure and consequendy different transport properties that greatly affect their device potential. Therefore, just among the Ill-Nitrides and SiC compounds there exist a wide variety of candidate materials that can be considered as potential replacements for conventional silicon or GaAs based electronics. Given the large number of options, the selection of which material composition and polytype that is most suitable for a given application becomes challenging. The experimental characterization of these possibilities is further hampered by the relative technological difficulty encountered in reliably and reproducibly growing each material. This is particularly important for the Ill-Nitrides since the lack of suitable substrate materials has greatly impeded their technological maturation. Therefore, it is highly useful to employ a theoretical scheme that can proceed relatively independently of experimental work to assess the transport and device potential of candidate materials. In this way, a judicious assessment of each material can be made in a cost effective and timely manner. To this end, we have developed a comprehensive scheme called materials theory based modeling.1'2 Materials theory based modeling consists of a series of hierarchical modeling tools that extend from a fundamental physics based, microscopic analysis to macroscopic, engineering based device models. The key ingredient within the materials theory based modeling method is the full band, ensemble Monte Carlo technique.3'4 The full band Monte Carlo model has less reliance on extensive empirical information than other simulation techniques, making it particularly attractive for studying technologically immature materials and their related devices where a lack of definitive information about the transport properties of the material exists. Additionally, the inherent flexibility of the Monte Carlo model makes it suitable for adaptation to a wide variety of materials and device types. The fall band Monte Carlo model has mostly been applied to the study of cubic symmetry semiconductors, principally silicon, GaAs and other compound semiconductor materials that crystallize in the zincblende phase. However, the wide band gap semiconductors mainly crystallize in polytypes that have different symmetry from that of the zincblende phase. These different polytypes present new challenges in the formulation of the Monte Carlo algorithm. Standard Monte Carlo algorithms then must be adapted to handle the different symmetries
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presented by the various polytypes of the wide band gap semiconductors. In this chapter, we will present a discussion of the principal challenges presented in numerically modeling semiconductor materials that crystallize with noncubic symmetry. Specifically, we will examine what modifications must be made to the materials theory based modeling approach in general, and the ensemble Monte Carlo simulation, in particular. With these modifications, the materials theory based modeling approach can then be utilized to study and compare various materials and their different polytypes to assess their overall suitability in device applications.
2.
Transport Model for the Wide Band Gap Semiconductors
The study of the physics and transport properties of wide band gap semiconductors requires knowledge of much information about their intrinsic properties. First of all, both electrons and holes move under the effect of the driving forces in the band structure, which determines the relation between the energy and momentum of the particle in the crystal. The computation of the band structure is not a trivial task in itself5. A possible simplification widely employed is the use of model or analytical forms. Even though this approach has provided valuable insight, it breaks down when the transport regime is no longer ohmic or when nonlinear transport becomes important. In this work, the full band structure approach will generally be used4. For this purpose, the realistic energy-momentum dispersion for the material under investigation will be determined by using suitable methods. The other piece of information needed concerns the interaction of the carriers with the surrounding environment, in particular with phonons. The scattering mechanisms between the electrons and the elementary excitations of the crystal lattice, phonons, provide an energy relaxation mechanism. The information relevant to the determination of the scattering rates is the phonon dispersion relation and the coupling constants between the phonons and the carriers. The phonon dispersion relation can be computed using different techniques 6'7. The general approach relies on the use of a model or simplified description of the lattice dynamics. Different attempts have been made8'9 to use the full lattice dynamics in transport studies, particularly for silicon. The possibility of applying the same technique to wide band gap semiconductors has not yet been implemented10. The difficulty is related to the complex structure of the lattice dynamics in the hexagonal polytypes and the even more complex interaction between the phonons and carriers in such materials. A simplified picture will be used in this work. The determination of the relevant coupling constants is certainly the most challenging task in a transport study. Their first-principles determination is in general quite involved. On the other hand, the experimental results are useful only in the warm carrier regime. In fact, from the measured carrier drift velocity and mobility, some of these coupling constants can be determined. Unfortunately, when the high-field regime is considered, the commonly measurable quantity, e.g., the ionization coefficients, is insufficient to determine all of the coupling constants required. For this reason, first-
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principles calculations of some relevant quantities have to be performed. In particular, the evaluation of the impact ionization transition rates will be presented. Once all the relevant quantities are known or estimated, the transport study can be performed. This will be carried out for low- and high-field strengths. The goal is to determine the relevant transport parameters, such as the drift velocity, diffusion coefficient, carrier energy, and high-field ionization coefficients.
2.1.
Band structure calculation
Band theory is the main tool used in classifying crystalline solids in terms of their electrical conductivity. The classification of materials as insulators, semiconductors, semimetals, and metals is based on the presence of bands that overlap or are separated by energy gaps. The study of the transport properties of semiconductors relies on the determination of each material's band structure and its properties. The basic problem that needs to be solved is the motion of a many-electron system in a periodic potential generated by the lattice ions. In principle, this problem can be solved based on quantum mechanics. Unfortunately, from a computational point of view, no closed form solution can be practically found. Some drastic simplifications have to be introduced, in particular, regarding the way the electron-electron interaction is treated. Many methods have been developed and include a different amount of physics. In this work we look at two of them: the empirical pseudo-potential method (EPM) n ' 12 and the linear augmented plane wave method 13 (LAPW) within the framework of the density functional theory 14 (DFT). They can be placed at the opposite ends of the spectrum of the possible methods available to solve the band-structure problem.
2.1.1. Pseudopotentials and the empirical pseudopotential method The pseudopotential method has opened the way to calculation of the electronic structure of a solid using a very simple technique compared with the complexity of the problem. The Phillips-Kleinman cancellation theorem15 can be considered the milestone in the development of the pseudopotential technique. This theorem is based on the observation that, by computing the expansion coefficients for the crystal wavefunction, a particular potential can also be built. This potential is such that it repels electrons away from the core region. This acts in the opposite way of the attractive ionic potential and compensates it, leaving as a result a weak potential, or pseudopotential. Such a potential can then be used, for example, in the context of the nearly free electron picture to solve equation the Schroedinger equation for the crystal, and obtain pseudo-wavefunctions. Since the pseudo-potential reflects in general the symmetry of the core states, because of their angular momentum t, a certain degree of nonlocality is always associated with it. As a consequence, the cancellation theorem works best in all those cases where the symmetry of the core states is similar to the symmetry of the valence states. As an example, the Carbon atom has a (Is) 2 core type and no p-like core states in the repulsive potential. In this case, the repulsive potential will not compensate for the
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p-valence electrons and they will have a higher probability of being found closer to the core than they do in reality. The cancellation theorem provides a guideline for determining the repulsive potential and consequently the weak pseudo-potential itself. However, this is not a practical way from a computational standpoint. Two classes of techniques have found wide application in building the pseudopotential: ab-initio13 and empirical pseudo-potential12. The ab-initio methods rely on the fact that the pseudopotential can be built in a self-consistent way by solving simultaneously the Poisson and Schrodinger equations for the solid. The empirical method follows a different path. The pseudopotential is built directly by fitting available experimental data such as the measured energy gaps, dielectric response, and effective masses. The great advantage of EPM is the relative simplicity of the inputs compared to the relative accuracy of the band-structure obtained. The basic steps involved in the EPM techniques are now briefly outlined. Let us suppose the pseudopotential can be written as
VpAr)=YJVa{r-R-f)
(1)
R,f
where R is a direct lattice vector and ^ is a basis vector. If we expand the pseudopotential in reciprocal lattice vectors, we obtain Vpse ( ? ) =
Va(p)s(G)e'6"
£
(2)
where the structure factors are given by
S(G)=
- J - S e-'*ir> N
(3)
,
iV.®
and a V ' are the atomic Fourier components of the pseudopotential. The pseudofunction and the band energy values are given by the equation
•— 2m
V2+V{r) *,-(?) = * . ( * ) * , - ( ? )
(4)
Now by using (2) we can rewrite the equivalent of (11) in Fourier space:
H66,{k)a,{G)-E(k}l,(G)
=0
(5)
The Hamiltonian in its simplest form can be written using a local version of the pseudopotential, that is
*'
" '',tobe
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H
^)^(k
+
G)2Sdd,
+ V(\G-G'\)
(6)
Expression (6) is a homogeneous system of linear equations where the unknowns are the expansion coefficients of the pseudo-wave-functions and the eigenvalues the band energies. The Fourier components
a
(o) of the pseudopotential are determined in such a way that the band structure characteristics from (6) agree with the experimental data. For many materials, the simple use of a local pseudopotential is not sufficient to reproduce correctly the bandstructure features. As seen in the case of Carbon atoms a non local correction may be needed because of the lack of p-type core states. The simplest way of modeling the non-locality is to introduce a square well potential that gives a non-local correction in the Hamiltonian as follows12: "
0
i#
'
(7)
a
p c p Where ^' a ' ^ are respectively the Legendre's polynomial of order /, the structure factor, and the Fourier transform of the non-local potential well. e ( ' is an energydependent depth for the potential well used. The next step involves the use of EPM to compute the band structures of wide band-gap semiconductors.
2.1.2. Some examples A good starting point for the determination of the pseudopotential for GaN are the pseudopotentials for the constituent atoms, gallium, and nitrogen16 as shown in Figure 1. Using these pseudopotentials as a first guess new values for the Fourier components can be found by an optimization process so that better agreement to experimental data is reached. Alternatively some analytic form of the effective potential can be assumed17 and the characteristic parameters adjusted to get the best fitting with the available experimental values. For the cubic phase there are seven parameters (form-factors) that define the pseudo-potential. To reach the best fitting, it is then necessary to vary a subset or all the seven parameters by using a suitable optimization algorithm. From Figure 2 it can be seen that the GaN wurtzite phase is direct with the conduction band minimum at the T point and a 3.5eV energy gap. Table 1 provides the calculated values of effective masses and energy gaps of wurtzite phase GaN. Table 1. Calculated values of effective masses and energy gaps of wurtzite phase GaN.
GaN
EPM
Longitudinal Eff. Mass, mi
0.18
Transverse Eff. Mass, m,
0.23
168
Egap (eV)
3.52
Crystal-Field Splitting (eV) 0.042
Simulation of Carrier Transport in Wide Band Gap Semiconductors
531
0.4 0.2
0.2
A
n
•
I-
•
§-0.4
o Bloom, Ref. 24 o Yeo et al„ Ref. 42 - - - Pughetal.,Ref. 45 present work
AIN GaN InN
•
10 15 q2 (2it/a)2
10
25
15
q^ (2nlaf
Figurel. Effective pseudopotentials for Ga and N [16].
v\
CKl
\y
I
V—N
15 <si^~~'
-
K
)
en
Energy (eV)
10
-
/
-
0
^3Af ~\
^ -5 •
\
-s
M
y
K
^x
\ \
r
yf
V y^-. L
A
H
Figure 2. Wurtzite GaN band structure obtained from EPM.
169
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Using the same technique, the band structure of other materials can be computed as well. The SiC wurtzite phase can be computed18 starting from the pseudopotentials of silicon and carbon. Since very little experimental information is known about SiC-2H the fitting of the EPM band structure can only be done using information obtained from first principles techniques such as the LAPW19 or quasi-particle (QP)20 methods. SiC2H is indirect with a band gap of 3.33eV and the conduction band minimum is located at the K point. The density of states effective mass at K is 0.377mo. Table 2. 2H-SiC Band Structure Data.
EPM
MKr
HIKM
mKH
0.43
0.43
0.29
Exp.
K2C-r6v (eV)
3.33 3.33
2.1.3. Screening, dielectric function and density of states From the EPM band structure some important information can be extracted. Most relevant to the transport study are the dielectric function and the density of states (DOS). The former is essential for a correct formulation of the electron-electron interaction and the latter for treatment of phonons scattering. The dielectric function describes how the various charged species move in response to external, or internal potentials and provides a screening for those potentials. In semiconductor materials, electrons and holes and the atoms themselves, due to the ionic bonding if present, are the charged species. The dielectric function can be written as21
e(q1o))=£eo+SeL + See
(8)
The explicit dependence of \"' ' on the wave-vector " provides a major effect in the screening process. The second and the third terms of equation (8) are respectively the lattice and electronic contribution to the dielectric function. At sufficiently high frequency, neither the free electrons nor the ions can follow any time dependent perturbation. Consequently their contribution vanishes. The only contribution left is due to the valence electrons and it is normally called the optical dielectric constant £M. Consequently the electronic contribution to the dielectric function can be divided into a high frequency component due to the valence electrons, and free carrier screening. At low frequency, in absence of free carriers, the lattice and the valence electrons contribute to the static dielectric constant. In case of purely covalent materials, such as silicon or germanium there is no lattice contribution to the static dielectric function. For III-V, II-VI semiconductors and SiC, even though it is a IV-IV compound, the polar
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contribution is relevant and the difference between the optical e„ and static dielectric constant Eo can be written as:
Se L = [e0-
e„ ] -
®TO 2
'(0
T0
-CO2
(9)
As expected, it goes to zero when the frequency approaches infinity. By using the random phase approximation (RPA)22 an expression for the electronic, including the high frequency part, contribution can be computed as
=e , g 2 V ° q2YEc(k
£(gco)
/(*)-/(* + $) + q)-Ev(k) + hQ) + ia
(10)
where ^ ' and ^ " ' are the electron distribution functions. The summation includes valence electrons and free carriers. It is possible to separate out the contribution of the valence electrons and then explicitly define the optical dielectric constant "A"' /, such that it now includes the contribution from the valence electrons in the following way: £(q,C0) = £„(q,Ct)) +
M„)-«,]-isL-+4s 2
co
2
TO-0)
_/(*>-/(*+«>—
2
q ^Ec(k+q)-Ev(k)
<">
+ hco + ia
where the last sum runs only over the free carriers. The optical dielectric constant, due to the valence electrons can be computed directly from the EPM band structure as follows23:
£^{q,0)) = \ +
2e2 l l
nq
2
lim
v
a -> 0Atr,v
(MY (k,c k+q,vj [Ec(k +q)-Ev(k)f
[Ec(k + q) - Ev(k)]
(12)
+ [hco + iaf
The valence-conduction band matrix element is computed using the pseudo wave function from the EPM Hamiltonian. The optical wave-vector dependent dielectric functions for wurtzite GaN are shown in Figure 3. Figure 4 presents the calculated dielectric function for SiC-4H. The role of the optical and free carrier dielectric function is essential in determining the electron-electron interaction and it will be examined in the next chapters. The second important information obtained from the band structure is the density of states.
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8
0
0.5
1.0
1.5
2.0
q (2*00 Figure 3. Wave-vector dependent dielectric function for GaN wurtzite phase 24 .
Since in general the bands are not parabolic, simple expressions for the density of states can not be used. It is then necessary to determine the DOS directly from the bandstructure. If N(E)dE is the number of electrons per unit volume between the energy E and E+dE, then we can compute N(E) from:
d'k
lif E<E(k)<E+dE
AT?
0 otherwise
(13)
As an example, the density of states for 4H SiC is shown in Figure 5. The plot shows the DOS for valence bands obtained from the EPM band structure and another band structure calculation based on the DFT-LDA approach.
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Figure 4. Wave-vector dependent dielectric function for 2H-SiC.
14
Total valence band density of states, 4H-SiC
x10 1
i
12--
\
-
10--
\ // '•ft
-
jl
i
>
/J
i
\A"
tf
\
\ /*
\
> v- — — -v
// //// //
CD
c HI
-
-
// Local EP band structure DFT-LDA band structure
jf tf
, 0.5
1.5
i
2 2.5 Energy [eV]
i
i
3.5
Figure 5. Valence bands density of states for 4H-SiC. Calculated using both the empirical pseudopotential method and the density functional theory - local density approximation technique.
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2.2.
Carrier transport
Within the framework of the semiclassical transport theory25, the Boltzmann Transport Equation (BTE) describes the motion of carriers in phase space and can be considered as a continuity equation for the distribution function in phase space. When the force field does not depend on k and the velocity is not functionally dependent on the space coordinate, that is in bulk material, the BTE has the form21
df(r,k,t) , - ^ r , , F +v-yf(U,t)+--Vrf(r,k,t) dt h k-
=
df(rjc,t) dt V
(14)
Jc
where f(r, k, t)is the particle distribution function in phase space, and F is the applied electric field. The BTE treats electrons and holes as classical point particles with definite position and momentum. The carriers are also considered to be uncorrelated and the single particle distribution functions are valid. The collision term is given by (15)
dt
,I=A
Jc
(15)
it S(k k')
v
v'
v
' ' is the scattering rate from the state ^ to ^ due to several mechanisms, such as phonons, impurities, impact ionization and others. The collision integral contains the quantum description of the system, since the scattering rates are evaluated using quantum mechanical techniques. Clearly the description of the band structure enters in the BTE through the velocity computed as ~~ * ' . and the momentumE(k} energy relation for the particles as given by v '. The task of solving the BTE is made challenging by the presence of many obstacles. Among these are the complicated scattering rates description, the nonlinear response, and boundary conditions in the devices. Some analytical techniques are available. Assumptions can be made about the form of the distribution function. In this case a possible way is to use a drifted Maxwellian formulation. Another possibility relies on the functional expansion of the distribution function in terms of spherical harmonics, the same technique used for neutron transport problems. Although useful, these techniques provide only a limited answer to the study of transport in realistic semiconductors. Monte Carlo Methods26,27,28,29,30 certainly represent the most powerful tool in determining the transport properties. Rather than directly solve the BTE, the distribution function is built through direct simulation of the motion of an ensemble of particles in phase space. The ensemble is subjected to the action of external forces, such as an electric field, and
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scattering phenomena. In order to limit the computational requirements the motion is studied in the irreducible wedge of the Brillouin zone. In this way the band structure data and scattering rates have to be computed for a limited number of points inside the irreducible wedge. Interpolation and symmetry operations enable calculation of the information needed outside the irreducible wedge. For wide band gap semiconductors, that have several polytypes, the transport is studied by using two different formulations for the cubic and hexagonal polytypes. Before describing the results obtained for the material being studied a brief review of the scattering mechanisms is presented.
2.2.1. Carrier-phonon and ionized impurity scattering The adiabatic approximation decouples the motion of the ions from the electronic part of the Hamiltonian25. In this case the interaction with the phonons has to be introduced as a perturbation that forces a transition, from a one-electron Bloch state to another. The simplest approach is based on a first order treatment25'31'32 using Fermi's Golden Rule which gives the probability per unit time that the transition may occur
s(U')^e\Hp\^,)f 441-4)1 The total scattering rate out of the Bloch state over the Brillouin zone as:
1 r(k)
k
(16)
can then be computed integrating
V js(k,k')dk' (in)3 BZ
(17)
Fermi's Golden Rule is derived under the assumption that the scattering event is instantaneous. This assumption breaks down in a few important cases. In case of very fast transients, where the dynamics of the system have to be studied on a time scale comparable to the scattering rate, then the scattering rates have to be described in a different way33. In case of very high scattering rates, the initial state may decay appreciably in the time the scattering is completed, and the determination of the rate has to be done taking this into account. Lastly, in case of very high electric fields, the field can transfer a significant amount of energy to the particle during the scattering events. Different attempts have been made to include such phenomena in transport simulators but no unified treatment is presently available. In cases where these phenomena can be neglected, the first order transition rate provides a good description of the scattering events. Each scattering process can then be described by specializing (16) using an appropriate expression for the perturbation Hamiltonian. The scattering events considered in the transport study of wide band gap semiconductor materials can be divided into three categories, polar scattering, deformation potential scattering, and scattering with impurities. Scattering with polar optical phonons can be described by using the Frolich formulation as
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(
l
m-- ^(^h- A\'^M')-4)^J ATN^
2~2
K£~
v
L w
' '
w
opi
(18>
Where I(k',k) is the overlap integral between the initial k' and final k, Bloch state, Nq the phonon occupation number and hcoop is the longitudinal optical phonon energy. A common approximation is the use of a constant value for the polar optical energy associated with the longitudinal optical mode. The dependence of the matrix element on the inverse square of the phonon wave vector, and the angular dependence of the overlap integral makes this scattering mechanism anisotropic. In case of hexagonal materials the presence of many optical modes makes things more complicated. At this moment no complete treatment has been performed for this case and the cubic approximation is still used. Intra-valley and inter-valley scattering rates are evaluated by using the deformation potential formulation. Intra-valley scattering with acoustic phonons is modeled by using an acoustic deformation potential and the phonon energy is taken to be proportional to the product of the phonon wave-vector and the sound velocity. The resulting expression for the acoustic phonon deformation potential transition probability is given as,
S
(^')=^{N,
f
+U^i(k\k)^if
)-4^nqu]
(19)
Scattering with non-polar optical phonons is modeled using a suitable deformation potential as,
4.*')-^gf{». 4 ±^}4'M4')- 4> »»j
(20)
A similar expression can be used for inter-valley scattering, where an intervalley deformation potential has to be used and the number of equivalent final valleys has to be taken into account. In an attempt to gain better understanding of the intervalley scattering process an alternative formulation has been proposed8. By including in the computation of the matrix elements the full description of the lattice dynamics it is possible to overcome the need of the empirically determined deformation potential. Even though this approach has given very good results for silicon and gallium arsenide, it has not been applied yet to any wide band gap semiconductor10. Scattering with ionized impurities is modeled using the Ruch and Fawcett approach34 as
HV 47te
P'+q
(21)
where 3 is the screening length which depends on the free carrier density, and nT is the ionized impurity density. Because of the difficulties in evaluating (21) numerically for a realistic band structure, an energy dependent rate has been used for all the different materials studied. The total rates for each scattering mechanism can be evaluated by
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computing the integral over the Brillouin zone numerically using the full band description, or analytically by assuming a suitable analytical expression for the bandstructure.
2.1.2. Alloy scattering The problem of carrier transport in a disordered or partially ordered alloy is very difficult to treat. A widely used approach to the simulation of bulk alloys is the inclusion of an alloy scattering rate expressed as
Wall (E) = -—V0U2Alloyx(l
- x)N(E)
(22)
where V0 is the volume of the primitive cell of the direct lattice, U^toy is the random alloy potential, x is the alloy mole fraction, and N(E) is the density of states. Different interpretations for the choice of the random alloy potential have been given in the past. Littlejohn et. al.35 have used the conduction band offset (equal to the difference in the affinities) to represent the random scattering potential. Chin et. al.36 have determined the random potential from band gap bowing parameters and applying Phillips' theory of electronegativity differences.37 In accordance with our previous work,38 we have made two sets of simulations to bracket the effect of alloy scattering. One set of simulations has been made using the alloy scattering potential calculated from whichever method that produces the highest value, to account for the worst case, i.e., strongest alloy scattering. In the other set of simulations the random alloy potential has been set to zero in order to account for the best case, i.e., no alloy scattering. The conduction band offsets have been calculated from the difference of the energy gaps using the known valence band offsets.39 It was observed that calculation of the random potential from the conduction band offsets results in higher potentials than those calculated from the electronegativity differences. Therefore, the conduction band offsets were used to represent the random alloy potential in the first set of simulations.
2.2.3. Impact ionization The study of the transport properties of wide band gap semiconductors under high-field conditions requires an accurate description of the impact ionization process. Impact ionization limits the maximum operating voltage of electronic devices because of the possible self-destruction caused by avalanche breakdown. Consequently, a correct description of impact ionization is instrumental in determining the breakdown properties of electronic devices. On the other hand, in some devices such as avalanche photodiodes (APDs), controlled impact 'ionization is used to govern device performance. In the case of APDs, impact ionization provides an internal gain mechanism and improves the signal to noise ratio at the device terminals. Since impact ionization occurs for carriers with a substantial energy, equal or greater than the energy
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gap, it becomes crucial to use a correct description of the band structure to characterize the semiconductor material. This is even more important for wide band gap semiconductors, where the energy gap can be as large as several electron-volts. As opposed to narrow gap materials the use of simple analytical models40 is no longer possible. Many wide band gap semiconductors present a high degree of anisotropy in their transport characteristics. This means that even impact ionization phenomena can present a directional dependence, i.e., the ionization coefficients will be different when the electric field is applied along different directions. Calculation of the ionization coefficients poses a most challenging task both from a computational standpoint and the level of details in the physical description of the material system. The problem will be treated at the beginning from first principles and subsequently some simplifications will be considered aimed to reduce the computational complexity.
2.2.3.1. Impact ionization transition rate calculation The band to band impact ionization process is the generation of electron-hole pairs due to a high-energy electron (hole) that, by scattering with an electron in the valence band, promotes it to the conduction band. A threshold is clearly involved in this process since the initiating particle must have an energy higher than the energy gap. This process can be described by a pair production rate. The calculation of this rate can be performed to first order by using Fermi's Golden Rule. In this case the initial and final states of the system are described by a properly symmetrized two-electron wave function. The total rate for a point
1 belonging to band nj in the Brillouin zone is given by31: 2tf V2
viffl.j2^„
„\ „-„(23)
Because of the fact that particles are indistinguishable, it is necessary to take into account direct and exchange processes when computing the matrix elements,
|M|2=2|MD|2+2|M£|2-(M;M£+MDM;)
(24)
The direct MD and exchanged ME matrix elements are computed using pseudo wavefuctions from the EPM band structure. The general form of the matrix elements is given by
MD
2 1 \fc\?l\ s ^ i ^ i ) * V.^2^2 ' 2 ^ 2 / ^/- \2 2
e{qD)q
D
/ n n MF =e 2 /(fc 1 n 1 ,^ 2 n2)^(^2 2'^i' i) e{qE)q\
(25)
C»)
where the expressions of the form ' ! •' 1 I'are overlap integrals between the initial and final states of the direct and exchange processes.
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541
The other important part of the matrix element expression is the choice of the dielectric function. The difficulties in the evaluation of (23) are of both a physical and numerical nature. Physically, the major difficulty lies in choosing an accurate yet manageable expression for the dielectric matrix. Within the random phase approximation (RPA), the dielectric function is given by the usual Lindhard expression (12). It is customary to consider only the longitudinal response (i.e., the diagonal terms of the dielectric function) and it is convenient to separate (12) into the two terms representing the response of the valence electrons, and of the free carriers. Kane41, Bude, Hess, and Iafrate42, and Kamakura and coworkers43 have ignored the contribution of the free carriers, and have employed the RPA and the pseudopotential band structure to evaluate the valence-band dielectric function, while retaining its frequency dependence23. In general, though, in treating the response of the carriers in the valence bands, the static approximation is a satisfactory simplification. Since the energy exchanged in a pair-production process at the fields of interest here is relatively small (< 4 eV or so), the static value of the dielectric function is adequate, as a look at the figures of Ref. 23 clearly shows. This is also demonstrated by the fact that the holeinitiated ionization rate in Si we have obtained using static screening is identical to the result obtained by Kunikiyo and co-workers44, who have used dynamic screening. Stobbe et al.45 have used the static, wavevector dependent expression for the valenceband contribution, as given by Levine and Louie46. Note also that dynamic corrections, in principle important in treating the response of the free carriers whose plasma energy is much lower than the typical energy exchanged in ionization events, become negligible at long wavelengths, so that the use of a static approximation appears justified in this context. Here we have made use of an analytical fitting for the valenceband contribution. Choosing a free carrier concentration of 1017 cm"3, the ThomasFermi expression reduces to its non-degenerate Debye-Hiickel form. The calculated ionization rates do not depend dramatically, in general, on the particular choice made for the value of the free carrier concentration. Indeed, the free-carrier density plays essentially a two-fold role. It is a cut-off required to avoid the singularity of the Coulomb matrix element at q = 0. Since in practically all cases the volume of the phasespace associated with q = 0 transitions is very small, the results will not depend on the value of the free carrier density, as long as it is small enough to avoid overscreening the interaction. Secondly, if the carrier density is large enough to affect the Coulomb matrix element, typically also carrier-carrier scattering will play a significant role. These are situations which, despite their arguable interest in practical applications, do not correspond to the experimental situations we shall compare our results to. Thus, we must require the free carrier density to be low enough to ensure that inter-particle Coulomb scattering can be neglected, so that the only additional scattering mechanism of importance is carrier-phonon scattering. In this case, the optical dielectric constant is used, the static valence band screening is computed by using (12) and the free,carrier screening is evaluated by the Debye-Huckel screening. The values of • / r « (»!• *i) depend on the k-vectors. In general it would be useful to have energy dependent rates. By averaging the k-dependent rates over constant energy surfaces an energy dependent rate can be obtained as follows:
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.l
XJ TU(E)
fS[E-En(k)]dk (ZI)
^J5[E-En(k)]dk n
Care must be exercised in the way this isotropic rate is evaluated. As previously shown47, a correct (logarithmic) interpolation must be used in performing the sum over points in the BZ implied in (27), as an incorrect energy-dependence of the isotropic rate may result, particularly near the all-important near-threshold region. In addition to the evaluation of the full k-dependent ionization rate expressed by (23), which we shall refer to as ab initio, it is interesting to consider two approximations of the ab-initio expression. The first one, the 'constant matrix element' (CME) approximation48,49 emphasizes the role played (or, more likely, not played) by the matrix element in determining the energy dependence of the ionization rate. Accordingly, we assume that the Coulomb matrix element M in (23) is a constant, equal to its average value over all k-points in the whole BZ, over all bands, and over both normal, N, and umklapp, U, processes. Thus the interband impact ionization transition rate becomes,
The actual value of the average matrix element <M2> can be determined, whenever (23) and (28) exhibit the same isotropic energy dependence via (27), by adjusting its value in (28) in order to match the result of (23). An even stronger approximation, motivated by its success in the case of electron-initiated ionization in Si41, consists in assuming that the kinematics of the ionization process is not controlled by momentum conservation, but mainly by the (joint) density of states available to the final (recoil and ionized) particles. This is expected to be approximately true when momentum randomization via U-processes is the most probable pair-production channel. In this case, many nonzero G-vectors contribute in the sum in (28). Thus, it is sensible to make the approximation, known as the 'random-k approximation"41, of replacing the many spikes of the momentum-conserving delta function with a uniform sampling of the entire Brillouin Zone. The quality of this approximation can be judged by whether its energy dependence matches that of the isotropic versions via (27). Finally, we report here the popular Keldysh form of the ionization rate,
-JT(E)
h
= e(E-Eth)p
Ku
Eth
'"
180
(29)
Simulation of Carrier Transport in Wide Band Gap Semiconductors
543
where E,h is some ionization-threshold energy, P a coupling parameter related to the Coulomb matrix element, and is either the carrier-phonon scattering rate at threshold, according to the original formulation by Keldysh50, or the energy relaxation rate at threshold, according to Ridley's lucky drift theory51. This expression has been used extensively in the past, treating the parameters P and Eth as fitting quantities.
2.2.4. K-dependent formulation of carrier phonon scattering Fundamental transport studies of semiconductor material systems, in particular, high field transport, are important in determining the parameters useful for device analysis. Although the methodology used for cubic materials is well established52'48 '53, very few groups have tackled the problem of transport in non-cubic materials54'55,56,57. The investigation of high-field carrier transport in such systems presents problems that can not be solved with an approach similar to the methodology employed for cubic III-V semiconductors. The study of carrier transport in 4H-SiC is the most evident example of this new situation. The aim of this work is to highlight the difficulties and new issues associated with this problem, and evaluate high field hole transport in SiC. Theoretical investigations over the last ten years have clearly shown that high field transport studies have to be performed using a full band approach58, and that simplified models fail to correctly predict the high field transport parameters. For this reason, the high field transport study presented in this chapter employs a full band approach. A particularly important example is the study of hole transport in 4H-SiC. The 4H polytype of SiC has hexagonal crystal structure and eight atoms per unit cell. Theoretical calculation of the 4H-SiC band structure has been performed with a variety of methods19'59,20. For this study we have chosen to compute the 4H-SiC band structure using the empirical pseudopotential method5. Several reasons underlie this choice. First, the method is computationally efficient. Furthermore, a careful optimization of the band structure can include all of the most important features derived either from experiment or first principles calculation. Finally, an accurate selection of the plane wave sets used for the wavefunction expansion enables the use of a symmetry transformation to reduce the storage requirements and simplify overlap integral calculations used in the transport analysis. Several important features of the band structure should be highlighted. The total width of the valence bands is slightly larger than 8 eV, that is, less than three times the fundamental gap. From the results obtained studying other wide-band gap materials57, it is likely that at high field strengths a substantial number of particles occupy the highest energy bands. For this reason the model includes the first twelve valence bands. Bands above the twelfth are separated by a large energy gap and do not contribute to the transport process. A second important feature is the presence of numerous mixing and crossing points. This particular characteristic is present in the valence band structure of other noncubic symmetry materials systems57. The problem was first addressed in the study of hole transport in GaN.60 The effect of such points on the transport characteristics in 4H-SiC is more dramatic. This problem will be addressed in the next section.
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Traditional transport models ' , employ energy dependent carrier-phonon scattering rates. Some attempts61'62 have been made to extend this approach to include the momentum dependence of the scattering rate. The motivation behind the model in Ref. 62 was the necessity of describing the warping of the valence bands. The scattering rate for each mechanism was interpolated using the energy dependent values along three principal directions. Because of the complexity of the present problem the previous approach is not as attractive, and a full k-dependent model is used. It is also important to appreciate that the energy dependent formulation of the scattering can be seen as an average of the k-dependent rate over constant energy surfaces. Then, not only the angular dependence of the rate is discarded, but the particular features of different parts of the Brillouin zone are removed as well. This is an important point since for non-cubic materials the anisotropy of the Brillouin zone introduces substantial differences in the transport characteristics in different directions. This means that, although the energy dependent model works for cubic materials, its validity becomes questionable for 4H-SiC. A simple but effective way to appreciate this new situation is to compare the distribution of the value of the scattering rate as a function of energy. Figure 6 presents the calculated scattering rate for the non-polar optical phonon interaction in 13095 k-points of the irreducible wedge. Each dot represents the rate for a given k-point corresponding to a given energy and band.
Figure 6. Calculated scattering rate for the non-polar optical phonon interaction in 13095 k-points of the irreducible wedge.
A total of twelve valence bands are included in the plot. Notice the large spread of values, at least three orders of magnitude, for the rates corresponding to a fixed energy value, implying a large variation in the scattering rate as a function of the k-vector. It is instructive to compare this result to a similar plot for a cubic material, for example
182
Simulation of Carrier Transport in Wide Band Gap Semiconductors
545
Figure 2 of reference 62. The spread in the value of the scattering rates at a fixed energy for the cubic material is much less than one order of magnitude, implying little variation with k-vector. Hence, an energy dependent averaging of the scattering rate in 4H-SiC would not properly reflect the strong k-dependence of the scattering mechanisms. The model includes hole-phonon interactions with acoustic, non-polar, polar optical modes, and ionized impurity scattering. The rate for each mechanism is computed for twelve valence bands on a grid of 13095 k-points in the irreducible wedge. For each mechanism, the intra-band and inter-band emission and absorption rates are computed and stored. The parameters employed in the phonon dispersion have been estimated from a lattice dynamics calculation63. Ionized impurity scattering is also included in the model using an energy dependent formulation28. The scattering rates are then interpolated for each k-point outside the grid using a suitable logarithmic or linear approach to ensure minimum interpolation error. The peculiarity of the 4H-SiC band structure also requires that the selection of the final state after a scattering event be carefully considered. The problem is twofold. The numerically computed scattering rates and the large number of bands, already present at low energy, make the use of a final state selection based on an analytical expression very difficult. On the other hand, the strong anisotropy of the Brillouin zone requires a final state selection that does not introduce any bias that would artificially remove the effect of the anisotropy of the transport parameters. Two different methods have been employed in the simulator to determine the final band and final k-point after a scattering. Since the overlap integrals can be evaluated for each initial and possible final state in different bands they can be used directly in the simulator. A large database, more than a hundred million values of overlap integrals between a set of kpoints, covering all twelve bands, inside the irreducible wedge to all the other points in the Brillouin zone has been pre-computed and stored. Every time a scattering event is selected, a group of final k-states satisfying the energy conservation criteria is collected for all twelve valence bands. A vector of values is then built, in which each entry is the product of the matrix element for the scattering mechanism selected and the overlap integral between the initial state and the specific final k-point. The final state, that is, the final k-point and band are selected randomly from the vector containing all the possibilities. Since both the total and inter-band scattering rates are known, an alternative selection technique can be used. The final band can be determined at the same time the scattering mechanism is by a secondary random selection of the interband scattering rate values. Once the final band is determined, the final k-point within the band is selected using the same method employed as in the previous case. The advantage of this second option lies in the fact that the final state has to be searched over a smaller set of k-states, and as a result the selection process is computationally more efficient. Both approaches have been tested and have been found to yield nearly identical results. Finally, particular attention has been taken in the correct treatment of the band degeneracy at the top of the Brillouin zone. Since the height of the zone along the caxis is much smaller than the dimension in the basal plane, the particles can drift beyond the zone edge when a high electric field strength is applied to the material.
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Following such an event, the particle has to be assigned to the correct band after it has been transferred back to the first zone.
2.3.
Multiband transport models
Monte Carlo (MC) simulations are an effective and popular tool to theoretically analyze the charge carrier transport properties of semiconductors.28 A particular version of these simulations that is best suited for the investigation of very high field effects is based on full Brillouin zone (BZ) bandstructures, i.e. continuous bands throughout the BZ. This technique has yielded many useful results on carrier drift velocities in high electric fields and impact ionization coefficients in materials such as silicon, gallium arsenide, etc.64 Recently, full BZ Monte Carlo simulations have been performed for electrons and holes in GaN65 66 67 SiC68 and other materials that typically have more atoms per primitive cell than the conventional semiconductors, which crystallize in the diamond or zincblende structure. The larger primitive cell in real space implies a smaller primitive cell (smaller BZ) in k-space and typically leads to a greater number of bands in the energy interval of interest in a transport study. As a consequence of this crystal structure, transport simulations for these novel materials require extensions of the standard paradigm underpinning MC transport simulations, which is essentially classical, i.e. it is assumed that the charge carrier wave vector, band index, and its position can be specified simultaneously with arbitrary precision. The corresponding quantum mechanical picture is to view the carriers as represented by a wave packet, i.e. by linear combinations of Bloch states, that have reasonably sharply defined average wave vector and average position. If the constituent Bloch states of a wave packet all originate from a single band at time t = 0 then this will also be true at all later times, provided that no coupling between different bands exists and that no scattering event induces a transition to another band. The possibility that an electron may tunnel to another band between collision events is typically not modeled in standard MC simulations. For conventional semiconductors this is an adequate picture. Typically, at any given point in k-space, the bands are well separated in energy and the electric fields considered in the transport simulation are usually sufficiently small so that the band structure can be viewed as unaffected by the applied field. This is not necessarily the case in the wide band gap materials. Due to the small size of the BZ and the relatively small bandwidths, multiple conduction bands or valence bands may exist in relatively narrow ranges of energy. Thus, even though the fundamental band gaps in these materials are quite large, an electron may undergo band-to-band transitions between different conduction bands (or a hole between different valence bands) while being accelerated by an applied electric field. In order to model this type of situation in a MC simulation of charge carrier transport in a homogeneous applied electric field the standard approach is generalized such as to allow for possible tunneling transitions between bands, typically in limited regions of k-space.
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2.3.1.
547
Modeling of band-to-band transitions in transport calculations
Ruden et al. 69 have shown that the trajectory of the carrier when tunneling transitions between bands can occur can be estimated stochastically by evaluating the magnitude of the overlap integral between the cell periodic part of the initial state and the cell periodic part of each of the possible final states. The probability of finding the carrier in either band 1 or band 2 after a drift of At in duration assuming it was in band 1 at t=0 initially is given as,
42k,+Ak -+M(A'K*.+A*WH*J U hk0 +M ^hk0
+M ^=l-hk0
+Ak^2k0
(30)
+M ( A 0
W
X12k is the interband transition matrix element between band 1 at ko and band 2 at ko+Ak during an accelerated drift between t = 0 and t = At. The probability for a transition to band 2 during an accelarated drift during At for a carrier initially in band 1 can then be written as,
2 2k +M o
[d r u - W (r) u ~ -(r) i \k 2k +M V A
2k +M
U
o
o
(32)
o
This integral is readily calculated using the wavefunctions generated by the bandstructure calculation. An alternative expression based on k'P theory is,
Ifl m„
iik'
K~E2i.f
(Mf
(33)
Here, P^k- is the interband momentum matrix element and m0 is the free electron mass. In the standard MC simulation, phonon, impurity, impact ionization related scattering events take the electron instantaneously from an initial state of definite band index and k-vector to a final state, again of definite band index and k-vector. A first approximation of taking possible band-to-band transitions during the accelerated drift between successive scattering events into account may be implemented as follows. Prior to each scattering event the band index of the initial state is selected by a stochastic process based on the relative probabilities of equations (30) and (31). This procedure complicates the MC simulation because of the necessity to calculate the
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appropriate overlap integrals. Operationally however, this is necessary only if there is a reasonable chance that tunneling actually occurs, i.e. if the electron is in a region of the BZ where two or more bands are closely spaced. This is the scheme implemented in the example to be discussed below and a graphic representation of the single step approach is shown in Figure 7a. In some cases, for example under strong applied fields, it is questionable that the Ak associated with accelerated flights between successive real scattering events are sufficiently small to expect that the transition probability and the bands can be adequately described by evaluating them for a single k-point. One may then subdivide At = A^+.^+At,, by adding n "self-scattering" events. This effectively subdivides the interval Ak = Ak! +.. .+Akn, as is illustrated in Figure 7b. The probability amplitudes can then be calculated iteratively for each step. Finally, the probability for a band transition is again evaluated stochastically based on the relative probabilities |
Figure 7. Schematic representation of k-space where two bands are nearly degenerate.
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2.3.2. Effect of band-to-band tunneling on the transport characteristics of holes in 4HSiC To illustrate the importance of including the band-to-band tunneling during the drift of a charge carrier, it is useful to perform MC calculations of the transport properties of carriers within a material wherein numerous bands coexist in a relatively narrow range of energy relevant to carrier transport. The ensemble MC simulations discussed here are based on a bandstructure calculated using local empirical pseudopotentials. As can be seen in Figure 8, valence bands cross (or anti-cross) at multiple points along the main symmetry axes. (In fact, the distinction between band crossing and anti-crossing is not easily made in the context of a bandstructure calculation that treats each k-point separately.) The dominant high-field scattering mechanisms are implemented using numerical results for the phonon scattering and impact ionization rates. Calculations are performed under two different conditions. First the simple, single step overlap integral scheme described above to determine the band occupancy of a hole after a drift is used. The results obtained are then compared to a simulation neglecting band-to-band tunneling during drift. The calculated hole average energies as a function of the electric field strength applied along the c-axis at high electric field strengths shows there is a significant difference in the calculated average energy between the two models.
r
M K
TA
L H
AMLKH
Figure 8. Empirical pseudopotential calculated 4H-SiC valence bands.
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1
2 3 Energy (eV)
Figure 9. Calculated hole number density function for an applied field of 4MV/cm parallel to the c-axis. The plot on the left represents the calculated number density function when the band to band tunneling is not allowed. The plot on the right is obtained when the band to band tunneling is allowed.
The difference in the average hole energies can be understood from the calculated number density functions with and without tunneling effects shown in Figure 9. Inspection of Figure 9 clearly shows that there is a large difference in the highenergy tails of the two distributions. In the absence of interband tunneling in the neighborhood of band crossing points, the hole distribution remains relatively cold since very few holes move to high energy states. However, if tunneling in the vicinity of band crossing points is included in the model, a much hotter hole distribution is obtained. The hole initiated impact ionization coefficient provides some measure of the temperature of the hole distribution. Therefore, comparison of the calculated hole initiated impact ionization coefficient between the two models and to experimental data provides a means of assessing their relative validity. Comparison of the calculated hole ionization coefficients for the two models for fields applied parallel to the c-axis in 4HSiC to the experimental results of Raghunathan and Baliga70 clearly show the importance of this phenomenon. The calculated ionization coefficients without interband tunneling, are lower than the experimental data by about two orders of magnitude. This clearly shows that inclusion of band-to-band tunneling during drift is crucial for obtaining the correct physical picture of high field hole transport in 4H-SiC and more generally in materials in which numerous bands cross or anti-cross.
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2.3.3.
of Carrier Transport in Wide Band Gap Semiconductors
551
Discussion
The method outlined above to include tunneling assisted transitions between bands during carrier accelerated drift preserves the essentially classical paradigm of traditional MC transport simulations for semiconductors. Each initial state of a phonon, impurity or impact ionization related scattering event has a definite k-vector and band index. However, band-to-band transitions may occur not only through scattering but also through direct tunneling during the accelerated drift of a charge carrier. The scheme presented here is readily extended to more than two bands (trivial in the case of the single step approximation). In fact, in many cases this generalization will be necessary because of the multiplicity of bands in relatively narrow energy ranges. Obviously, the inclusion of more bands also implies additional computation of overlap integrals. The consideration of interband transitions during the drift phase of a carrier is very important in obtaining the correct physical behavior in the transport simulation as is evident from the example of hole transport in 4H SiC. Of course, as pointed out above, the simple single step procedure may not always be adequate. It should be considered only as a first order approximation of the interband tunneling effect. A more comprehensive approach is to treat the drift as a collection of subdrifts and to sum up the tunneling transition rates after each subdrift to obtain the probability of transferring between bands. If the subdivision of an accelerated flight is used by introducing the form of self-scattering in this way, it is important to determine an adequate time step that is a good compromise between the accuracy desired and the computational effort that is acceptable. In addition to the interband matrix elements of the coordinate operator, x, intraband matrix elements need to be evaluated if, as is typically the case, the crystal lacks inversion symmetry. The subsequent iterative calculation of the probability amplitudes up to the end of the carrier drift is straightforward but is expected to slow the simulation.
3.
Bulk Material Results
We have applied the model described in Section 2 to the study of the transport physics in bulk Ill-nitride and SiC material. In this section, we discuss the results for both low and high applied electric field strength.
3.1.
Low applied field strength
One of the most important parameters that is used to characterize a material is the mobility. Experimentally, the mobility is usually measured using the Hall effect, resulting in what is typically referred to as the Hall mobility. The Hall mobility though differs from the drift mobility, which is simply defined as the ratio of the drift velocity to the applied electric field. The drift mobility is the more commonly calculated quantity using Monte Carlo simulation since it can be readily determined from the slope of the velocity field curve. In this section, we discuss the calculated results for the drift
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mobility of GaN and related ternary Ill-nitride compounds extracted using the ensemble Monte Carlo simulation. We present both zero field and field dependent mobility models. We have employed two general approaches in calculating the drift mobility. These are a direct technique in which the mobility is determined from the slope of the velocity-field curve at low applied electric field strength, and indirectly through the diffusion coefficient. We have utilized both general techniques to calculate the zero field mobility. In the direct approach, we assume an applied field strength of 1 kV/cm (which is considered quite low for the Ill-nitride compounds). The mobility is then determined from the ratio of the steady-state velocity to the applied electric field strength. Though this approach is quite simple, owing to the relatively low applied field strength, the calculated results are noisy. Therefore, the calculated mobility needs to be averaged over a long duration to reduce the effects of random noise. For this reason, the simulation is performed for 10,000 electrons for 50 ps. A representative result is shown in Figure 10.
20
25
30
Time [ps] Figure 10. Calculated electron mobility using the direct method for an applied field of 1 KV/cm.
In the indirect method the low field mobility is calculated from the diffusion coefficient based on the Einstein relation
M=
qD
(34)
TJ »
where q is the electronic charge, kB is the Boltzmann constant, T is the absolute temperature, and D is the diffusion coefficient. The diffusion coefficient can be calculated from the velocity-fluctuation noise spectrum. The noise spectrum of velocity fluctuations is defined as
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553
foMt)e""dt (35)
Sv(a» = lim
Where GO is the angular frequency, T is the simulation time, and 5v is the electron velocity fluctuation from its average value. The velocity-fluctuation noise spectrum is related to the steady-state velocity autocorrelation function through
Sv(co) = 2r C{t)ei0* dt
(36)
JO
where C(t), the autocorrelation function of velocity fluctuations is defined as (37)
C(t) = {Sv(t')-Mt' + t)) The diffusivity is related to velocity autocorrelation through
(38)
at) dt
D=T
Jo From Eqs. (36) and (38), the diffusivity can be calculated as (39)
D = ±Sv(
0
5
10
15
20
25
30
35
40
45
50
T[ps] Figure 11. Calculated electron mobility from the indirect method.
Figure 11 shows a typical zero-field mobility calculated from the velocityfluctuation noise spectrum versus time. The calculation has been performed with an analytic-band ensemble Monte Carlo simulation of electrons in wurtzite GaN, under zero applied field. Ten thousand electrons were simulated for a real time of 50 ps. It is
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observed that if the simulation is long enough, both methods give approximately the same value of mobility, as expected. It should be noted that there is always some inherent randomness-noise associated with the results of the Monte Carlo simulation. The low field electron mobility was calculated for the wurtzite-phase binary compounds, GaN, InN, and A1N, and the ternaries, AlxGa!.xN and InxGa!_xN (x = 0.2, 0.5, 0.8). In order to bracket the effect of alloy scattering we have included in our simulations of the ternary nitride alloys both methods discussed in Sec. 2.1.2. For AlxGai_xN and InxGai.xN the two bracketing conditions are no alloy scattering and alloy scattering with an alloy potential equal to the affinity differences. Table 3 lists the calculated low field mobilities for the binary and ternary compounds at a lattice temperature of 300 K, and a set of ionized impurity concentrations of 1016 cm"3, 1017 cm"3, and 1018 cm"3. As expected, as the impurity concentration increases, the mobility decreases monotonically. In addition, the mobility decreases monotonically from InN to GaN, and from GaN to A1N in the absence of alloy scattering. However, when alloy scattering is included, the minimum mobility can be at the intermediate ternary composition, where alloy scattering is at its maximum. Table 4 lists the calculated low field mobilities determined with ionized impurity concentrations of 1017 cm'3, and a set of lattice temperatures of 300 K, 450 K, and 600 K. For comparison, the mobilities corresponding to a lattice temperature of 300 K and an ionized impurity concentration of 1017 cm"3 have been repeated from Table 3. As can be seen from Table 3, the mobility decreases monotonically with increasing temperature due to the enhanced scattering rate. Again it is observed that for AlxGai_xN and InxGa!_xN, the inclusion of alloy scattering can have a major effect on the low field mobility for the temperature range studied here. It is useful to formulate an expression for the mobility of the binaries and ternary compounds that can be used for device simulation. Generally, in drift diffusion simulators, the mobility is modeled as a function of the local electric field, temperature 71
and doping density as {1 = M{M0(T,N),E}
(40)
where n is the electron mobility, E is the local electric field, T is the lattice temperature, N is the total doping density, and |io, which is called the low-field mobility, describes the temperature and doping dependency of the mobility. The general expression for the low field mobility that is typically used in drift-diffusion simulations is given as71'73 V
-
(
H0{T,N) = fi„
{^-
(41)
300
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of Carrier Transport in Wide Band Gap Semiconductors
555
where T is the lattice temperature, N is the total doping density and a, p l ; |32, P3, P
ND+ = 1016 cm"3
ND+ = 1017 cm'3
ND+ = 1018 cm"3
3060
2284
1252
Ualloy - AE C
1146
973
701
Ualloy = 0
2974
2185
1202
Ualloy = AEc
713
624
454
Ualloy = 0
2516
1787
954
Ualloy = AE C
648
538
399
Ualloy = 0
1803
1317
697
1371
990
544
Ualloy = AEc
243
223
180
Uaiioy = 0
1347
978
525
Ualloy = AE C
140
130
110
Ualloy = 0
1139
856
454
Ualloy = AE C
141
128
110
Ualloy = 0
839
658
366
657
533
313
InN Ino.8Gao.2N
In0.5Gao.5N
In0.2Gao.8N
GaN Al0.2Gao.8N
Alo.5Gao.5N
Alo.8Ga0.2N
A1N
In this case, due to lack of experimental data, Monte Carlo simulation is the only way of determining these parameters. Using our Monte Carlo simulator, we calculated the low-field mobility for a grid of ionized impurity concentration and temperature values. Ionized impurity concentrations were chosen from the range of 1015 cm"3 to 1018 cm"3 and lattice temperatures were chosen from the range of 300 K to 600 K. These calculated data where then used in a least squares fitting procedure74 to determine the required parameters in the low field mobility model. As an example of this procedure, Figure 12 shows the results of fitting the low field mobility model for GaN. As observed from this figure, the mobility model can be nicely fit to the calculated mobility values.
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Table 4. Calculated low field mobility [ in cm2/V.s ] for ionized impurity concentration of 1017 cm'3, and a set of lattice temperature of 300K, 450K and 600K. Uaii0y stands for the random alloy potential. AEC is the conduction band offset. T = 300 K
T = 450 K
T = 600K
2284
1022
632
Ualloy - AEc
973
589
405
Ualloy = 0
2185
963
594
Ualloy = AEc
624
399
273
Ualloy = 0
1787
784
456
Ualloy = AEc
538
320
212
Ualloy = 0
1317
537
305
990
391
215
Ualloy = AEc
223
149
107
Ualloy = 0
978
390
211
Ualloy - AEc
130
93
73
Ualloy = 0
856
332
183
Ualloy = AEc
128
91
68
Ualloy = 0
658
266
141
533
219
119
InN In0.8Gao.2N
In0.5Gao.5N
In0.2Gao.8N
GaN Alo.2Gao.8N
Alo.5Gao.5N
Al0.8Gao.2N
A1N
The extracted parameters for the low field mobility model for GaN, InN, A1N, AlGaN, and InGaN are shown in wurtzite phase GaN. For the ternary compounds, the bracketing cases of alloy scattering are included.
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GaN 1500
T
1
1
1—i—I
1
H— o
1
1—i—I
i
1
r-
Low-Field Mobility Model Monte Carlo
16Crrf3
IN=
100C-
S o
:8 a
50018 r m -3
w
300
400
500
600
300
400
500
600
300
400
500
600
300
400
500
600
Lattice Temperature [K] Figure 12. Least squares fitting of low-field mobility model to Monte Carlo calculations in wurtzite phase GaN.
3.2.
High-field mobility calculation and modeling
No preexisting mobility model provides a satisfactory description of the field dependence of the Monte Carlo calculated GaN mobility. Therefore, a new field dependent mobility model has been developed such that it can model the velocity field dependence of GaN, as calculated from Monte Carlo simulation. This new model is given in Eq. 42 as, ju0(T,N)
M:
+ vs ,n2
(42)
l+a where |io is the low field mobility as expressed in Eq. (41). There are five parameters in the new model, which are determined from a least squares fit to the results of our Monte Carlo simulation. These parameters are vsat, Ec, a, nl, and n2. The values of these parameters, extracted for GaN, A1N, InN and both ternary compounds for the two bracketing cases of alloy scattering, are shown in Table 5. As an example of this procedure, Figure 13 shows the results of fitting the high field mobility model for GaN.
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Table 5. Extracted parameters for the proposed high field mobility model. y sat
InN In0.8Gao.2N
U a lioy = A E C U a lloy - 0
Ino.5Gao.5N
U a lloy = A E C Ualloy = 0
In0.2Gao.8N
U a iioy = A E C Ualloy - 0
GaN Alo.2Gao.8N
Ualloy = A E C Ualloy - 0
Alo.5Gao.5N
Ualloy = A E C Ualloy - 0
Alo.8Gao.2N
Ualloy = A E C Ualloy = 0
A1N
(107 cm/s) 1.3595 0.8714 1.4812 0.7973 1.6652 1.0428 1.8169 1.9064 1.1219 2.0270 1.1459 2.1505 1.5804 2.1581 2.1670
Ec (kV/cm)
ni
n2
A
52.4242 103.4550 63.4305 148.9098 93.8151 207.5922 151.8870 220.8936 365.5529 245.5794 455.4437 304.5541 428.1290 386.2440 447.0339
3.8501 4.2379 4.1330 4.0635 4.8807 4.7193 6.0373 7.2044 5.3193 7.8138 5.0264 9.4438 7.8166 12.5795 17.3681
0.6078 1.1227 0.6725 1.0849 0.7395 1.0239 0.7670 0.7857 1.0396 0.7897 1.0016 0.8080 1.0196 0.8324 0.8554
2.2623 3.0295 2.7321 3.0052 3.7387 3.6204 5.1797 6.1973 3.2332 6.9502 2.6055 8.0022 2.4359 8.6037 8.7253
cjoo o>—
v_
2.5
/o
o •o
/° /°
2
-
1.5
1 •n Q
New High-Field-Mobility Model Monte Carlo
0
J
-
0.5 W
rx 1
'
100
,
1
'
»
200
300
400
500
600
Applied Electric Field [kV/cm]
Figure 13. Least squares fitting of low-field mobility model to Monte Carlo calculations in wurtzite phase GaN.
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559
As observed from Figure 13, the electron velocity initially increases with applied field, but after a critical field, it starts to decrease. This is the negative differential resistance effect and is typical of multivalley semiconductors. This effect occurs as a result of electron intervalley scattering from a valley with low effective mass to an upper valley with a higher effective mass.
oft
0
1
1
1
1
1
1
100
200
300
400
500
600
Applied Electric Field [kV/cm]
Figure 14. Calculated electron drift velocity versus applied electric field for GaN, Al2Ga8N, Al5Ga5N, Al8Ga2N, and A1N. For this calculation, the random alloy potential was set equal to conduction band offsets. Lattice temperature is at 300 K, and electron concentration is equal to 1017 cm'3.
3.3. Velocity-field calculations Figure 14 shows the calculated electron steady-state drift velocity versus applied electric field, for GaN, Alo.2Gao.8N, Alo.5Gao.5N, Alo.8Gao.2N, and A1N materials. The calculation is performed at a lattice temperature of 300 K and an ionized doping concentration equal to 1017 cm"3. Also, the random alloy potential is assumed equal to the conduction band offsets. Selecting the random alloy potential in this way, results in a strong alloy scattering rate, which we believe, sets an upper limit on its effect on the transport dynamics. In this way, we can bracket the effect of alloy scattering by comparing the calculations to the case where alloy scattering is omitted. The results of the calculation with no alloy scattering are shown in Figure 15. It is interesting to compare the results of Figure 14 and Figure 15. It is observed that alloy scattering becomes the dominant scattering mechanism in these materials under the present simulation conditions, when the random alloy potential is set to the conduction band offset between GaN and A1N. A significantly lower drift velocity is observed when
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alloy scattering is present which is simply a result of the higher total scattering rate. Moreover in the presence of alloy scattering the peak velocity occurs at a higher applied field compared to the case when alloy scattering is neglected. This is also due to the fact that in the presence of alloy scattering, the total scattering rate is higher; thus a higher field is required to heat the carriers prior to the onset of intervalley transfer. It should be mentioned that if the bowing parameters are small, the Phillip's theory of electronegativity differences is consistent with a very small alloy scattering potential. Experimental evidence and several calculations16 indicate that the bowing parameters may indeed be small. Under these conditions, the alloy scattering becomes negligible and the electron drift velocity as a function of the applied field approaches that shown in Figure 15. Figure 16 shows the calculated electron drift velocity versus applied electric field for the GaN, Ino.2Gao8N, Ino.5Gao.5N, In0.8Gao.2N, and InN materials. The calculation is again performed at a lattice temperature of 300 K, and an ionized dopant concentration equal to 1017 cm'3. The random alloy potential is set equal to the conduction band offsets. Similar to the case of the AljGa^N system, a second set of simulations, with zero alloy scattering is made in order to bracket the effect of alloy scattering. The results of the latter calculation for the electron drift velocity are shown in Figure 17. Similar comparisons and arguments applied to the case of Figure 14 versus Figure 15 also apply here. However it should be mentioned that for the InxGa!. XN compound the random alloy scattering potential calculated from Phillips' theory of electronegativity differences results in values on the same order as the conduction band offsets. Therefore if these values are used to calculate the alloy scattering, the results would be closer to that shown in Figure 16.
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561
GaN
o
•c
Q a p W
200
300
400
Applied Electric Field [kV/cm]
Figure 15 Calculated electron drift velocity versus applied electric field for GaN, Al2Ga8N, Al5Ga5N, Al8Ga2N, and AIN. For this alculation, the random alloy potential was set to zero. Lattice temperature is at 300 K, and electron concentration is equal to 1017 cm"3.
200
300
400
Applied Electric Field [kV/cm]
Figure 16. Calculated electron drift velocity versus applied electric field for GaN, In2GagN, In5Ga5N, InsGa2N, and InN. For this calculation, the random alloy potential was set equal to conduction band offsets. Lattice temperature is at 300 K, and electron concentration is equal to 10" cm"3.
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-e—
0
100
200
300
400
GaN
500
600
Applied Electric Field [kV/cm]
Figure 17. Calculated electron drift velocity versus applied electric field for GaN, Ino.2Ga0.gN, Ino.5Gao.5N, Ino.8Gao.2N, and InN. For this calculation, the random alloy potential was set to zero. Lattice temperature is at 300 K, and electron concentration is equal to 1017 cm"3.
It might be thought that since the electron peak velocity of InN is higher than that of GaN, it should also be higher than that of In0 8Gao.2N. However, Figure 17 shows a higher peak velocity in In0.8Gao.2N and Inn.5Gao.5N, than InN. This arises from the fact that the peak velocity depends nonlinearly on the electron mobility in the lowest valley and the energy separation between this valley and the secondary valleys. By changing the In mole fraction in IrixGa^N from 1 to 0, the mobility decreases from that of InN to that of GaN, however the energy separation between the lowest valley and secondary valleys increases. Due to the nonlinear dependence of the peak velocity on these two factors, an initial increase followed by a decrease in the peak velocity with increasing Ga composition is observed, as shown in Figure 17.
3.4.
High applied field strength
In this section two examples of high field transport study are presented. The study of high field transport in wurtzite phase InN employs an energy dependent scattering rates model both for the interaction with phonons and impact ionization. For the study of high field hole transport in 4H-SiC the hole-phonon interaction has to be included using a wavevector dependent formulation because of the extreme complexity of the valence band structure even at very low energy.
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3.4.1. High field electron transport in InN The principal input for the ensemble Monte Carlo model used in the transport calculations, is the band structure description of hexagonal InN. The present study employs a band structure obtained from the empirical pseudo-potential method (EPM)5. The electronic structure of InN is still under investigation. Many uncertainties remain concerning the position of the satellite valleys, and a more accurate determination of the energy gap has to be performed. The InN band structure has been studied by means of ab-initio75,16 and empirical pseudo-potential, EPM, methods77'78. The choice of EPM in this study has been made to provide a reasonably accurate description of the electronic structure without sacrificing computational efficiency in calculating the transport parameters. The computational scheme is based on an expansion of the pseudowavefunction in 135 cell periodic plane waves. This choice enables the use of symmetry transformations79 for the determination of the pseudo-wavefunction outside the irreducible wedge. The form factors are determined by fitting the band structure to the available experimental data. When no experimental data were available information from an ab-initio calculation was used. The InN band structure is shown in Figure 18.
T
MK
T A
L H
A M L K H
Figure 18. Calculated pseudopotential band structure of InN.
The first five conduction bands are included in the model. Particularly important for the low field calculation is the correct fitting of the effective masses at the conduction band edge. The density of states effective masses are calculated from the pseudopotential band structure. It is found that the mass in the direction parallel to the c axis is slightly higher (0.12) than the mass in the basal plan (0.1). Furthermore, the
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position of the satellite valleys is not very well known. Different band structure calculations place the lowest secondary valley (or saddle point) at the A point or at theT-point (r 3 symmetry)75'76. In our band structure and in an another recent EPM calculation78 the lowest secondary valley is found to be at the T point (IYsymmetry). The next valley is on the line between the L and M point, similar to what occurs in GaN. Although the relative valley position is the same, the energy separation is different. The energy gap between the first and second conduction band at the r point is lower in our band-structure than for the EPM calculations of Ref. 78 and higher than the one in Ref.77. The relative energies of the secondary valley at point A and the one on the L-M axis are reasonably consistent for all the different EPM calculations, even though the absolute energies change. Polar-optical, acoustic phonon, ionized impurity and piezoelectric scattering are included for electron energies below 0.9eV. Polar optical, deformation potential phonon and impact ionization are the scattering mechanisms used for energies greater than 0.9 eV. This choice has been made to avoid using inter-valley scattering explicitly. Since the inter-valley coupling constants are unknown, it is useful to reduce the parameterization by incorporating all of the inter-valley and band-to-band mechanisms into one general mechanism. For this reason, our model uses only one isotropic deformation potential scattering based on the realistic final density of states at energies in excess of 0.9 eV. The realistic band structure takes into account the effect of the higher lying valleys as well as each of the higher energy bands. The coupling constant for this deformation potential scattering is determined by matching the high-energy rate to the one at low energy. The scattering rate is computed by using the realistic density of states from the EPM band structure. It is very well known that Ill-nitrides exhibit a strong piezoelectric behavior. It is then relevant to investigate the effect of piezoelectric scattering on the transport parameters. The rate for this scattering mechanism has been computed following the formalism of reference 80. The relevant coupling constant K2av 81 has been found to be comparable to the one for ZnO. At 300K the scattering rate for the piezoelectric interaction is comparable to the acoustic scattering rate at low energy. Some of the coupling constants for the other scattering mechanisms have neither been experimentally determined nor computed in other ways. This makes the selection of these parameters difficult. In this work, these coupling constants are estimated from values taken from other similar compounds82. The transport parameters used in the model have been reported elsewhere83. Band to band impact ionization is also considered at high energy. The transition rate for this mechanism has been computed numerically from the realistic band structure using the wave-vector dependent dielectric function. The energy-dependent impact ionization transition rate is determined by integrating Fermi's golden rule for a two-body, screened Coulomb interaction over the possible final states using a numerically generated dielectric function and pseudowavefunctions. The wave-vector-dependent rate is first computed on a 924 point grid for the first five conduction bands within the irreducible wedge.
202
Simulation 10'
:
I
of Carrier Transport in Wide Band Gap Semiconductors
1
.
565
!
r 4Jr
|io13 .2 "55 c
:
c
110" t3
nj
^1010
Numerical Analytical fitting \ , ,
/ 10°
1
3
4
5 Energy (eV)
6
7
Figure 19. Calculated impact ionization transition rate for wurtzite InN.
The total transition rate, as shown in Figure 19, has been obtained by averaging the wave-vector-dependent rate over constant energy surfaces. The energy threshold of 2.1 eV for the onset of impact ionization is slightly higher than the direct energy gap of InN of 1.87 eV. The transition rate has two distinct behaviors as shown by the two curves in Figure 19. The first region corresponds to energies between 2 and 4 eV (measured from the conduction band edge). In this region the rate increases in magnitude relatively slowly saturating at a value ~1012 sec"1. The second region lies above 4eV. The rate increases much faster in magnitude in this region saturating at a value of ~ 6 x 1014 sec"1. The ionization rate in each of these regions can be expressed using a Keldysh formula-like50'84 fit as,
T(E)
=
(r
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f E-E'
\2
(43)
V
where E is the electron energy, E1^ = 1.95eV , E2^ = 4.2 eV, P(1) = 0.85 x 1012 sec"1, P 0 ' = 6.75 x 1014 sec"1 and 0 is the step function. Similar behavior of the ionization transition rate has been observed experimentally in silicon85. A possible explanation for the form of the transition rate can be traced back to the shape of the InN band structure. Only those states lying at relatively high energies for which both energy and momentum conservation are satisfied contribute to the ionization process. In InN, states exist within the first two conduction bands from which ionization events can proceed. Ionization events originating from these states
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within the first two bands are responsible for the first region of the transition rate plot. The slope of the transition rate decreases with increasing energy since the first two bands ultimately end around 3-4 eV. The second region of the transition rate, which lies above 4 eV, arises from ionization events occuring from states within the third and higher conduction bands. The shape of the transition rate of InN is in sharp contrast to that observed previously in GaN65 and ZnS55,86. In these materials the transition rate smoothly increases throughout the full range of energy. GaN and ZnS have substantially larger band gaps and as such, few ionization events originate from states within the first two conduction bands. Therefore, the third and higher energy bands dominate the ionization process at all energies and as such, the transition rate has only one region. The calculation of the ionization coefficients requires knowledge of the high energy phonon scattering rates in addition to the ionization transition rate. It is well known that the phonon scattering rate greatly influences the ionization rate coefficients. Unfortunately, little information about the high energy phonon scattering rates is known for even the most widely studied semiconductors. The high energy electron-phonon scattering rate is dominated by the optical deformation potential mechanism87. This mechanism is, to lowest order, isotropic and proportional to the final density of states through a single deformation potential constant. The density of states is calculated directly from the pseudopotential band structure. Typically, the deformation potential is assumed constant and its value is ascertained indirectly from comparison of the Monte Carlo generated ionization coefficients to experimental data87,55. Although the high-energy deformation potential may have some angular dependence, it is taken as a scalar constant for simplicity and due to our lack of any information regarding it for InN. No experimental data are presently available. Therefore, the deformation potential cannot be ascertained through comparison to experimental measurements. Instead, we select a baseline value of the deformation potential constant such that the high and low energy scattering rates match at 0.9 eV, as described in Section 2. Due to the inherent uncertainties in the selection of the deformation potential constant, the sensitivity of the ionization coefficient to variations in the deformation potential is analyzed. The model also includes a treatment of band mixing and crossing points54,60. Previous work has shown that the inclusion of this treatment for electron transport can be important but it is mostly of concern for valence band transport of holes66'60. Nevertheless, for completeness, it is included here.
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Figure 20. Calculated impact ionization coefficients for wurtzite InN.
The Monte Carlo calculated electron ionization coefficient for fields applied in T-K and T-A directions in bulk InN is shown in Figure 20 for field strengths between 500kV/cm and 2MV/cm. Calculations were also made for the field applied in the T-M direction. The coefficients for the T-M and the T-K directions are about the same. For clarity only those along the T-K direction are shown in the figure. The solid curves shown in Figure 20 connect the calculated ionization coefficients using the baseline phonon scattering rate. As can be seen from the figure, a substantial anisotropy in the coefficients is present throughout the full range of applied fields examined. Note that the coefficients are substantially higher for the case when the field is applied along the T-K direction. The primary reason for this behavior can be attributed to the band structure since the high energy scattering rate is assumed to be isotropic. Inspection of the band structure shows that the first two conduction bands are well separated from the third and higher bands except near the K point. When the field is applied along the T-K direction, the electrons can readily drift to states from which they can be scattered into the third and higher conduction bands. As discussed above, the ionization transition rate is substantially larger within the third and higher conduction bands and, as a result, the ionization coefficient is greater. Conversely, when the field is applied along the T-A direction, the electrons cannot directly drift to states from which they can transfer into the third and higher conduction bands. Given the large separation of the energy bands along this direction far fewer electrons, on average, then will transfer into the third and higher bands. As a result, the ionization coefficients are substantially lower for an applied field along the T-A direction.
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The top end of the error bars shown in Figure 20 represent the variation of the ionization coefficient for a 5% decrease in the high energy scattering rate. The bottom end of the error bars represents the variation of the ionization coefficient for a 5% increase in the high energy scattering rate. The total variation of the ionization coefficient is of the order of 25% in the T-K direction and 45% in the T-A direction. As is generally observed, the ionization coefficient is highly sensitive to variations in the scattering rate.
4 5 Energy (eV) Figure 21. Percentage of electron ionization events as a function of the initiating electron energy for three applied field strengths. The topmost plot is computed for a field of 500 kV/cm, the middle plot for a field of 1 MV/cm and the bottom plot for a field of 2 MV/cm.
To further illustrate the physical features of the impact ionization process in InN, it is instructive to examine the percentage of ionization events as a function of the initiating carrier energy. This is shown in Figure 21 for different values of the field applied along the T-M direction. Inspection of the three curves shows a clear progression in the origin of the initiating carriers. The uppermost plot in Figure 21 shows that, for a field of 500kV/cm, there is a sharp peak at an energy of 2.7 eV above
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Simulation of Carrier Transport in Wide Band Gap Semiconductors
569
the conduction band minimum. In addition, virtually all of the events originate from electrons with energies less than ~ 4 eV. Since there are few states within the third and higher conduction bands at this energy, the vast majority of the ionization events must originate from the first two conduction bands at a field of 500 kV/cm. This assertion is further substantiated by Figure 22. As can be seen from Figure 22, at an applied field of 500 kV/cm, virtually all of the ionization events originate from electrons in bands 1 or 2. As the field is increased to 1 MV/cm, two peaks appear in the ionization events plot (center plot of Fig. 21). The first peak is again at 2.7 eV while a second peak occurs at 4.8 eV. The first peak is again due to ionization events originating from the first two conduction bands. The higher applied field acts to heat the electrons such that some of the carriers move to the third and higher energy bands. This is the origin of the second peak. As can be seen from Fig. 22, at lMV/cm there is a substantial increase in the number of ionization events originating from the third band. The bottom plot of Figure 21 shows that for a field of 2 MV/cm a single peak appears at an energy of 5 eV. In addition, the majority of carriers now ionize with energies greater than 4 eV, implying that most of the ionization events originate from the third and higher conduction bands. This is confirmed from inspection of Figure 22, where it is seen that most of the ionization events originate from either the third or fourth bands. The dip between the peaks at 2.7 and 4.8 eV in the ionization events plot (center plot of Figure 21) arises from the following. As discussed above, the impact ionization transition rate shows two different behaviors, fit by the different functions given by equation 51. Inspection of Figure 19 clearly shows that at 3-4 eV, the first rate dominates the process. Not until the carriers achieve a higher energy does the second and much higher rate become important. In addition, the density of states decreases at this energy resulting in fewer available states from which ionization can occur. Therefore, fewer ionization events at this energy, 3-4 eV, occur. As can be seen from Figure 21, a substantial amount of carriers survive up to an energy of 5.5-6 eV, nearly three times the energy gap. This is an indication of a soft impact ionization threshold.
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Electric Field (KV/cm) Figure 22. Percentage of electron ionization events originating from each of the first five conduction bands as a function of the applied electric field strength.
3.4.2. High Field Hole Transport in 4H-SiC As discussed above, the complex electronic structure of 4H-SiC complicates its transport physics. It was found that the numerous band intersections and crossings within the band structure, particularly within the valence band, necessitates the inclusion of band to band tunneling within the model. Besides the electronic structure, the Monte Carlo model also requires as inputs the deformation potentials for acoustic and non-polar interaction. Very little is known about these parameters in 4H-SiC. It is also very difficult to find an efficient technique to compute them from first principles. Because of the strong crystal anisotropy it is reasonable to expect an angular dependence of the deformation potentials. However, the application of techniques such as the pseudo-rigid-ion method8,88, to compute this dependence is presently not available. For simplicity, we have thus chosen to employ constant isotropic values both for the acoustic and non-polar optical deformation potentials. The values have been determined by fitting the calculations to experimental values of the low field temperature dependent mobility89. More details about the low field results can be found elsewhere90, and will be extensively discussed in a future work. The transport parameters have been subsequently determined for a range of electric field strengths varying from lOkV/cm to 4MV/cm.
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Figure 23 presents the calculated hole drift velocity as a function of the applied electric field strength. The dashed line is for a field applied perpendicular to the c-axis (basal plane) and the continuous one corresponds to a field applied parallel to the c-axis. The plot shows an evident anisotropy in the velocity field characteristic in the two directions. For transport in the basal plane the velocity increases steadily up to a field strength of 2.25MV/cm to reach a value of 10.5 x 106 cm/s. As the field strength is increased to 4.0MV/cm the velocity decreases down to 9.5 x 106 cm/s. The calculated negative differential resistance (NDR) is very small for transport in the basal plane. The velocity field curve for an electric field applied parallel to the c-axis shows a very pronounced NDR. The velocity peaks at a field strength around 750 kV/cm to a value of 6.4 x 106 cm/s and decreases down to 4.2 x 106 cm/s at a field strength of 4MV/cm. Along the c-axis, the bands are relatively flat, and, as a consequence of the small vertical size of the Brillouin zone, the zone edge is reached at lower electric field strengths than for that in the basal plane. As the particles' energies increase, only a fraction of the whole ensemble transfers to the bands where the group velocity is higher. Up to an applied field of lMV/cm, the majority of the particles remain located in the lowest four valence bands. As the field strength is increased the particles start to enter the higher valence bands, in particular the seventh, eighth and ninth. Since these bands are very flat the ensemble velocity is reduced. If the field is increased to 4MV/cm, a substantial amount of carriers reach the highest valence bands,
209
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where they almost immediately undergo impact ionization. The amount of NDR shows little dependence on the selection of the final state for the impact ionizing holes, but is essentially controlled by the band population and consequently by the average velocity in each band. The results discussed above have been obtained including the treatment of the mixing and crossing points using the overlap integral test. Within this framework particles are able to tunnel between bands in the proximity of these points. This introduces an additional inter-band transfer mechanism. It is important to understand what happens when this is no longer allowed. The dash-dotted line in Figure 23 represents the calculated velocity field characteristic when inter-band tunneling is not allowed. From a careful analysis of the band population, it can be seen that almost the entire ensemble is located within the four lowest valence bands. Very few particles can move to high energy states, and the ensemble velocity decreases. As opposed to the dramatic change in the velocity field characteristic for transport along the c-axis direction, the drift velocity changes very little in case of transport in the basal plane. The dotted line in Figure 23 shows the calculated drift velocity for field strengths applied perpendicular to the c-axis when inter-band tunneling is not allowed. The structure of the bands in the basal plane is such that the effect of the mixing and crossing points is not as critical as in the c-axis direction. In the basal plane the presence of mixing and crossing points is accompanied by many alternative ways for the particles to move to higher energy states. The same behavior can be seen by analyzing the calculated average hole energy as a function of the electric field strength. The average energy is almost unchanged for transport parallel to the basal plane, the absence of the inter-band tunneling transfer leads to a collapse of the average hole energy for transport parallel to the c-axis. Clearly, the absence of interband tunneling through the overlap integral test at crossing and mixing points results in an extremely cool hole energy distribution function. To test which picture is physically correct, the inclusion of tunneling via the overlap integral test or its omission at mixing and crossing points, comparison of the calculated ionization coefficients with and without tunneling to experimental measurement is made. The ionization coefficients strongly reflect the temperature of the carrier distribution. Figure 24 presents the calculated and experimental values of the hole ionization coefficients. The open squares and diamonds represent the calculated values respectively for transport perpendicular and parallel to the c-axis. The ionization coefficients for transport perpendicular to the c-axis are from five to ten times greater than the values for transport for fields applied parallel to the c-axis. Further inspection of Figure 24 shows that the calculated values including interband tunneling for a field applied in the <0001> direction are in very good agreement with the experimental measurement of Raghunathan and Baliga70. The calculated ionization coefficients for transport in <0001> without interband tunneling in the proximity of crossing and mixing points, shown as open circles in Figure 24, are very much lower than the experimental measurements. The calculated values are at least two orders of magnitude lower than the lowest experimental values. This clearly shows how the inclusion of the
210
Simulation
of Carrier Transport in Wide Band Gap Semiconductors
573
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Inverse Electric Field (1CT7 cm/V) Figure 24. Measured and calculated hole impact ionization coefficients for 4H-SiC. The open square and the + symbols represent, respectively, the calculated values for an applied electric field perpendicular to the caxis when the band to band tunneling is allowed and when it is not allowed. The open diamond and the open circle symbols represent, respectively, the calculated values for an applied electric field parallel to the c-axis when the band to band tunneling is allowed and when it is not allowed. The pentagon symbols represent the experimental values measured by Raghunathan and Baliga for a field applied parallel to the c-axis. The asteriks represent the experimental values measured by Konstantinov et al. for a field applied parallel to the caxis.
Konstantinov91 and coworkers have measured the ionization coefficient for transport in <0001> as well, and their results are reported in Figure 24 as asterisks. Their measured values are higher than the ones obtained by Raghunathan and Baliga70. The disagreement seems to be related to a different quality of the samples used. Raghunathan and Baliga have pointed out that the material used by Konstantinov and coworkers may not be completely defect free, and consequently the higher ionization coefficient could be due to defect assisted ionization. In either case, the measured ionization coefficients are very much larger than those predicted using the Monte Carlo model without the use of the overlap formulation for the inter-band tunneling. Hence, we conclude that the inclusion of inter-band tunneling at crossing and mixing points is essential to properly determine the transport properties in highly complex materials.
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4. Conclusions In this chapter the study of transport properties of technologically relevant wide band gap semiconductor material systems has been presented. The focus of this study has been on non-cubic semiconductors, in particular indium nitride, and the 4H silicon carbide polytype. The transport properties have been computed using a model based on semiclassical transport theory, and the solution of the Boltzmann Transport Equation has been determined by means of the Monte Carlo method. Several important key problems have been addressed and a solution proposed. First of all the electronic structures to be employed in the full band transport model have been determined. The empirical pseudopotential method has been extended to handle materials with 4H crystal structure. In particular, a set of effective atomic potentials has been determined for use in band structure calculations for different SiC polytypes. A careful comparison between the band structure obtained from the EPM and ab-initio methods has been performed to evaluate the quality of the fitted atomic potentials. Effective masses and energy gaps at high symmetry points have been chosen as comparison parameters. A new set of symmetric transformations has been developed for the 4H crystal structure. This allows only pseudo wave functions for the k-points in the irreducible wedge to be stored. The pseudo wave functions outside the irreducible wedge can than be determined and used for the scattering and impact ionization rate calculation. To perform the high field transport study the impact ionization transition rates have been computed for all the material systems of interest. This is an essential feature of the model since it is well known that it is not possible to determine particle-phonon scattering and impact ionization rates by fitting only the impact ionization coefficients. Moreover, for new materials, experimental information on the transport properties is very scarce and it becomes crucial to be able to compute as much data as possible from first principles. The impact ionization transition rates have been determined by numerically integrating Fermi's golden rule using the full band structure. A wave vector or energy dependent formulation for the impact ionization transition rate has been included in the transport models according to the relative importance of the anisotropy of the crystal structure. The electron-phonon and hole-phonon scattering rates have been determined by using different techniques. For materials, such has zinc sulfide and indium nitride, because of the symmetry of the lowest conduction band, T valley, an energy dependent model has been used. It has been shown that this is a very good approximation for hexagonal indium nitride because of the small anisotropy of the effective mass at T. The high-energy scattering rate has been modeled as a single energy dependent deformation potential mechanism and the total rate has been computed using the realistic final density of states. When necessary, collision broadening has been included using a formulation based on the self-energy. As opposed to electron transport, it is not possible to use an energy dependent scattering rate for holes. The pronounced warping of the valence band structure requires the use of numerically calculated k-dependent scattering rates. For cubic and 4H silicon carbide, the hole phonon scattering rates have
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been computed on a grid throughout the irreducible wedge, and a linear-logarithmic interpolation is used to determine the rates on k points outside the grid. The transport simulation has been performed taking into account the symmetry of the Brillouin zones of different polytypes. Figure 25 presents a summary of the velocity field curves for electrons calculated for several different wide band gap material systems. It is important to appreciate how both the peak and high field drift velocities are much higher than the values in silicon. This is important since the knowledge of these parameters allows the design of faster devices compared to the present silicon-based approach. It is noticeable that the operational field value for devices made from these materials is in excess of five times the maximum field allowed by silicon devices. Figure 26 presents a summary of the velocity field curves for holes in cubic and 4H silicon carbide. For the cubic phase the predicted drift velocity is much higher than what has been measured for silicon. It is important to appreciate the directional dependence of the drift velocity as a function of the field strength. The calculated value presents an appreciable anisotropy. Similar results are obtained for silicon. The calculated drift velocity for holes in 4H polytype presents a strong anisotropy as well. The intrinsic anisotropy of the crystal structure is such that the directional dependence of the drift velocity is greatly enhanced by the lower dimensionality of the Brillouin zone for this polytype. Of particular importance to the description of the high field transport properties are the ionization coefficients. Figure 27 collects the calculated values of the ionization coefficients for electrons for cubic and hexagonal zinc sulfide, hexagonal indium nitride, and cubic and hexagonal gallium nitride. The plot also reports experimental values measured for zinc sulfide. Similarly, Figure 28 presents the calculated ionization coefficients for holes for cubic silicon carbide, and cubic and hexagonal gallium nitride. The plot also reports experimental values measured for 6HSiC and Gao.52Ino.48P- The energy gap of each material is reported beside the corresponding curve. Acknowledgements The work at Georgia Tech was supported in part by the National Science Foundation through Grant ECS-9811366, by the Office of Naval Research through Contract E21K19, by the Office of Naval Research through Subcontract E21-K69 made to Georgia Tech through the UCSB MURI program, and by the Yamacraw Initiative. Work at the University of Minnesota was supported by the National Science Foundation through Contract ECS-9811366, by the Office of Naval Research, and by the Minnesota Supercomputer Institute. The work at Politecnico di Torino was partially supported by CNR (Italian National Research Council) through the MADESS II project.
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100
150
200 250 300 Electric field (kV/cm)
350
400
450
Figure 25. Calculated electron drift velocity for cubic and hexagonal zinc sulfide, hexagonal indium nitride, and cubic and hexagonal gallium nitride. 2.5 3C-SiC 3C-SiC 3C-SiC 4H-SiC 4H-SiC
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200
400
600
800 1000 1200 Electric field (kV/cm)
1400
1600
1800
2000
Figure 26. Calculated hole drift velocity for cubic and 4H polytype silicon carbide.
214
Simulation
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\2.0eV
WZ-lnN[1000] WZ-InN [0001] ZnS Exp.1 ZnS Exp.2 ZB-ZnS[100] WZ-ZnS[1000] ZB-GaN[100] WZ-GaN[1000
10 Inverse electric field (10~7 cm/V)
15
5
577
Figure 27. Calculated impact ionization coefficients for electrons for cubic and hexagonal zinc sulfide, hexagonal indium nitride, and cubic and hexagonal gallium nitride. The plot also reports experimental values measured for zinc sulfide. The energy gap of each material is reported beside the corresponding curve.
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Figure 28. Calculated impact ionization coefficients for holes for cubic silicon carbide, and cubic and hexagonal gallium nitride. The plot also reports experimental values measured for 6H-SiC and Gao.52Ino.48P. The energy gap of each material is reported beside the corresponding curve.
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578 E. Bellotti et al. References
1. K. F. Brennan, E. Bellotti, M. Farahmand, J. Haralson II, P. P. Ruden, J. D. Albrecht, and A. Sutandi, "Materials theory based modeling of wide band gap semiconductors: from basic properties to devices," Solid-State Electron. 44 (2000) 195-204. 2. K. F. Brennan, J. Kolnik, I. H. Oguzman, E. Bellotti, M. Farahmand, P. P. Ruden, R. Wang, and J. D. Albrecht, "Materials theory based modeling of GaN devices", in GaN and Related Materials n, Gordon and Breach, Amsterdam, Vol. 7, edited by S. J. Pearton, 2000, 305-359. 3. H. Shichijo and K. Hess, "Band-structure-dependent transport and impact ionization in GaAs," Phys. Rev. B 23 (1981) 4197-4207. 4. M. V. Fischetti and S. E. Laux, "Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects," Phys. Rev. B 38 (1988) 9721-9745. 5. M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, (Springer, Berlin, 1988) 6. K. Kunch, "Dynamique de resau de composes ANB8"N presentant la structure de la blende", Annales the Physique, (1973-1974), p.319 7. Ab-initio calculation of phonon spectra, edited by J.T. Devreese, V.E. Van Doren, and P.E. Van Camp, Plenum Press, New York, 1983 8. S. Zollner, S. Gopalan, and M. Cardona,"Microscopic theory of intervalley scattering in GaAs: k dependence of deformation potentials and scattering rates" J.App.Phys. 68 (1990) 1682-1693. 9. M.V. Fischetti, and J. Hingman, in Monte Carlo Device Simulation: Full Band and Beyond, edited by K. Hess, Kluwer, Dordrecht, 1991, p 123. 10. S. Zollner, Private Communication 11. M. L. Cohen and T. K. Bergstresser,"Band structure and pseudopotential form factors for fourteen semiconductures of the diamond and zincblende structures", Phys. Rev. 141 (1966) 789796. 12. J.R. Chelikowski, and M.L. Cohen,"Non-local pseudopotential calculation for the electronic structure of eleven diamond and zinc-blende semiconductors",Phys. Rev. B14 (1976) 556-582. 13. D.J. Singh, Planewaves, Pseudopotentials, and the LAPW Method, (Kluwer Academic Publisher, Norwell, 1994). 14. R.M. Dreizler, and E.K.U. Gross, Density Functional Theory: an approach to the quantum many-body problem, (Springer-Verlag, Berlin, 1990)
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15. M.L. Cohen, and V. Heine, "The fitting of pseudopotentials to experimental data and their subsequent application", in Solid State Physics, vol.24, edited by F. Seitz, and D. Thurnbull, Academic Press, New York, 1970. 16. M. Goano, E. Bellotti, E. Ghillino, C. Garetto, G. Ghione, and K. F. Brennan, "Band structure nonlocal pseudopotential calculation of the Hi-nitride wurtzite phase material system. Part II: ternary alloys AlxGal-xN, InxGal-xN, and InxAll-xN," J. Appl. Phys. 88 (2000) 6476-6482. 17. M.V. Fischetti and S. Laux, "Band structure, deformation potentials, and carrier mobility in strained Si, Ge, and SiGe", J. App. Phys. 80 (1996) 2234-2252. 18. A.P. Dimitriev, N.V. Evalkov, A.S. Furman, "Pseudopotential calculation of the band structure of the solid solution SiC-AIN", Semiconductors 30 (1996) 60-67 19. C. Persson, and U. Lindefelt, "Relativistic band structure calculation of cubic and hexagonal SiC polytypes", J. App. Phys. 82 (1997) 5496-5508. 20. B. Wenzien, P. Kackell, F. Bechstedt, and G. Cappellini, "Quasiparticle band structure of silicon carbide polytytpes", Phys. Rev. B52, (1995) 10897-10905. 21. D.K. Ferry, Semiconductors (McMillan, New York, 1988). 22. C. Kittel, Quantun Theory of Solids (Wiley, New York, 1987), p. 103 23. J.P. Walter, and M.L. Cohen, "Frequency and wave-vector dependent dielectric function for Silicon", Phys. Rev. B5, (1972) 3101-3110. 24 R. Wang, and P.P. Ruden, J. Kolnik, I.H. Oguzman, K.F. Brennan, "Dielectric properties of wurtzite and zincblened struture gallium nitride", J. Phys. Chem. Sol. 58 (1997) 913-918. 25. K F. Brennan, The Physics of Semiconductors with Applications to Optoelectronic Devices, (Cambridge University Press, Cambridge, 1999). 26. J.M. Hammersley and D.C Handscomb, The Monte Carlo Method (Metheun, London, 1964) 27. R.W. Hockney and J.W. Eastwood, Computer simulation using particles, ( Mc Graw Hill, New York, 1981) 28. C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation ( Kluwer Academic Publisher, Norwell, 1990). 29. C. Moglestue, Monte Carlo simulation of semiconductor devices (Chapman & Hill, London, 1993) 30. I Lux and L. Koblinger, Monte Carlo particle transport method: neurtron and photon calculation (CRC Press , Boca Raton, 1991). 31. J. Bude, in Monte Carlo Device Simulation: Full Band and Beyond, edited by K. Hess (Kluwer, Dordrecht, 1991), p 27.
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580 E. Bellotti et al. 32. P. Vogl, "The electron-phonon interaction in semiconductors", in Physics of non linear transport in semiconductors, edited by D.K. Ferry, J.R. Barker, and C. Jacoboni. (Plenum Press, New York, 1980). 33. Y. C. Chang, D.Z.-T. Ning, J.W. Tang, and K. Hess, "Monte Carlo simulation of impact ionization in GaAs including quantum effects", Appl. Phys. Lett. 42 (1983) 26-28.. 34. J.G. Ruch, and W. Fawcett, "Temperature dependence of the transport properties of Gallium Arsenide determined by a Monte Carlo method", J. Appl. Phys. 41 (1970) 3843-3849. 35. M. A. Littlejohn, J. R. Hauser, T. H. Glisson, D. K. Ferry, and J. W. Harrison, "Alloy scattering and high field transport in ternary and quaternary IJI-V semiconductors (FET model)," Solid-State Electronics 21 (1978) 107-14. 36. V. W. L. Chin, B. Zhou, T. L. Tansley, X. Li, "Alloy-scattering dependence of electron mobility in the ternary gallium, indium, and aluminum nitrides," J. Appl. Phys. 77 (1995) 60646066. 37. J. C. Phillips, "tonicity of the chemical bond in crystals," Rev. Mod. Phys. 42 (1970) 317354. 38. J. D. Albrecht, R. P. Wang, P. P. Ruden, M. Farahmand, E. Bellotti, and K. F. Brennan, "Monte Carlo calculation of high- and low-field AlxGai_xN electron transport characteristics," Proceedings of the Nitride Semiconductors Symposium (1998) 815-820. 39. S. -H. Wei, A. Zunger, "Valence band splittings and band offsets of A1N, GaN, and InN," Appl. Phys. Lett. 69 (1996) 2719-2721. 40. W. Quade, E. Sholl, and M. Rudan, "Impact ionization within the hydrodynamin approach to semiconductors transport", Solid State Electronics 36 (1993) 1493-1505. 41. E.O. Kane, "Electron scattering by pair production in silicon", Phys. Rev. 159 (1967) 624631. 42. J. Bude, K. Hess, and G.J. Iafrate, "Impact ionization in semiconductors: Effects of high electric fields and high scattering rates", Phys. Rev B 45 (1992) 10958-10964. 43. Y. Kamakura, H. Mizuno, M. Yamaji ,M. Morifuji, K. Taniguchi, M. Hamaguchi, T. Kunikiyo, M. Takenaka, "Impact ionization model for full band Monte Carlo simulation". J. App. Phys. 75 (1994) 3500-3506. 44. T. Kunikiyo, M. Takenaka, Y. Kakamura, M. Yamaji, H. Mizuno, M. Morifuji, K. Taniguchi, M. Hamaguchi, "A Monte Carlo simulation of anisotropic electron transport in silicon including full band structure and anisotropic impact ionization model". J. App. Phys. 75 (1994) 297-312. 45. M. Stobbe, R. Redmer, and W. Schattke, "Impact ionization rate in GaAs", Phys. Rev. B49 (1994)4494-4500.
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46. Z.H. Levine and S.G. Loui, "New model dielectric function and exchange correlation potential for semiconductors and insulators", Phys. Rev. B25, p.6310,1982 47. N. Sano, M. Tomizawa, and A. Yoshii, "Impact ionization in submicron and sub-0.1 micron MOSFETs", in Hot carriers in semiconductors, edited by K. Hess, J.P. Leburton, and U. Ravaioli (Plenum Press, New York, 1996) p. 337. 48. N. Sano and A Yoshii, "Impact ionization theory consistent with the realistic band structure of silicon", Phys. Rev. B45 (1992) 4171-4180. 49. P.T. Landsberg, Recombination in semiconductors (Cambridge University Press, Cambridge, 1991) 50. L.V. Keldysh, "Concerning the theory of impact ionization in semiconductors", Sov. Phys. JEPT21(1965) 1135-1144. 51. B. K. Ridley, "Lucky-drift mechanism for impact ionization in semiconductors", J. Phys. C 16(1983)3373-3388. 52. M. V. Fischetti, "Monte Carlo simulation of transport in technologically significant semiconductors of the diamond and zincblende structure. Part I: homogeneous transport", IEEE Trans. Electron Dev. 38 (1991) 634 -649. 53.1. H. Oguzman, Y. Wang, J. Kolnik, and K. F. Brennan, "Theoretical study of hole initiated impact ionization in bulk silicon and GaAs using a wave-vector-dependent numerical transition rate formulation within an ensemble Monte Carlo calculation", J. Appl. Phys. 77 (1995) 225-232. 54. J. Kolnik, I. H. Oguzmann, K.F. Brennan, R. Wang, and P.P. Ruden, "Calculation of the wave vector dependent impact ionization transition rates in wurtzite and zincblende phases of bulk GaN", J. App. Phys. 81 (1996) 8836-8840. 55. E. Bellotti, K.F. Brennan, R. Wang, and P.P. Ruden, "Monte Carlo study of electron initiated impact ionization in bulk zincblende and wurtzite ZnS." J. Appl. Phys. 83 (1998) 4765-4772. 56. H-E. Nilsson, M. Hjelm, C. Frojdh, C. Persoon, U. Sannemo, and C.S. Petersson, "Investigation of electron transport in 6H-SiC using a full band Monte Carlo model", J. Appl. Phys. 86(1999)965-973. 57. E. Bellotti, "Advanced modeling of wide band gap semiconductor materials and devices," Ph.D. Thesis, Georgia Tech, Atlanta, GA, June 1999. 58. M.V. Fischetti, N. Sano, S.E. Laux, and K. Natori, "Full-band-structure theory of high-field transport and impact ionization of electrons and holes in Ge, Si, and GaAs", Journal pf Technology Computer Aided Design, February 1997. Available at: http://www.ieee.org/products/online/journal/tcad/accepted/fischetti-feb97/.
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59. W.R.L. Lambrecht, and B. Segall, "Band structure analysis of the conduction band anisotropy in 6H and 4H SiC", Phys. Rev. B52 (1995) R2249-R2252. 60. I. H. Oguzman, E. Bellotti, K. F. Brennan, J. Kolnik, R. Wang, and P. P. Ruden, "Theory of hole initiated impact ionization in bulk zincblende and wurtzite GaN", J. Appl. Phys. 81 (1997) 7827-7834. 61. J.M. Hinckley, and J. Singh, "Hole transport in pseudomorphic Sii_xGex alloys grown on Si(001) substrates", Phys. Rev. B41, 2912 (1990)Phys. Rev. B41 (1990) 2912-2926. 62. E. Bellotti. H-E. Nilsson, K.F. Brennan, and P.P Ruden, "Ensemble Monte Carlo calculation of hole transport in bulk 3C-SiC", J. App. Phys. 85 (1999) 3211-3217. 63. M. Hofman, A. Zywietz, K. Karch, and F. Bechstedt, "Lattice dynamics of SiC polytypes within the bond charge model", Phys. Rev. B50 (1994) 13401-13411. 64. For a general review see: Monte Carlo Device Simulation: Full Band and Beyond, edited by K. Hess (Kluwer Academic Publishers, Boston, 1991). 65. J. Kolnik, I.H. Oguzman, K.F. Brennan, R. Wang, and P.P. Ruden, "Electronic transport properties of bulk zincblende and wurtzite phases of GaN based on an ensemble Monte Carlo calculation including a full zone band structure", J. Appl. Phys. 78 (1996) 1033-1038. 66. I. Oguzman, J. Kolnik, K.F. Brennan, R. Wang, T.-N. Fang, and P.P. Ruden, "Hole transport properties of zincblende and wurtzite phases GaN based on an ensemble Monte Carlo calculation including a full zone band structure", J. Appl. Phys. 80 (1996) 4429-4436. 67. R. Redmer, J. R. Madureira, N. Fitzer, S. M. Goodnick, W. Schattke, and E. Scholl, "Field effect on the impact ionization rate in semiconductors," J. Appl. Phys. 87 (2000) 781-788. 68. E. Bellotti, H.-E. Nilsson, K.F. Brennan, P.P. Ruden, and R.J. Trew, "Monte Carlo calculation of hole initiated impact ionization in 4H phase SiC," J. Appl. Phys., 87, (2000) pp. 3864-3871. 69 P. P. Ruden, E. Bellotti, H.-E. Nilsson, and K. F. Brennan, "Modeling of band-to-band tunneling transitions during drift in Monte Carlo transport simulations," J. Appl. Phys., 88, (2000) pp. 1488-1493. 70. R. Raghunathan and B. J. Baliga, 'Temperature dependence of hole impact ionization coefficients in 4H and 6H-SiC", Solid-State Electron., 43 (1999) 199-211. 71. S. Selberherr, Analysis and simulation of semiconductor devices, (Springer-Verlag, New York, 1984). 72. Z. Yu and R. W. Dutton, "The Manual of SEDAN in - A generalized electronic material device analysis program", Stanford University, 1985.
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73. A. W. Smith, "Light confinement and hydrodynamic modeling of semiconductor structures by volumetric methods", Ph.D. Thesis, Georgia Institute of Technology, 1992. 74. A. Bjorck, Numerical methods for least squares problems, (SLAM, Philadelphia, 1996). 75. W.R.L. Lambrecht and B. Segall, Properties of the Group HI Nitrides, edited by J.E. Edgar (LNSPEC, LEE, London, 1994). 76. M.H. Tsai, D.W. Jenkins, J.D. Dow, and R.V. Kasowski, "Pseudo function theory of the electronic structure of InN", Phys. Rev. B 38 (1988) 1541-1543. 77. C.P. Foley, and T.L. Tansley,, "Pseudopotential band structure of indium nitride", Phys. Rev. B 33 (1986) 1430-1433. 78. Y.C. Yeo, T.C. Chong, and M.F. Li, "Electronic band structures and effective mass parameters of wurtzite InN and GaN", J. Appl. Phys. 83 (1998) 1429-1436. 79. K. F. Brennan, E. Bellotti, M. Farahmand, H.-E. Nilsson, P. P. Ruden, and Y. Zhang, "Monte Carlo simulation of noncubic symmetry semiconducting materials and devices", LEEE Trans. Electron Dev., 47, (2000) 1882-1890. 80. B.K. Ridley Quantum Processes in Semiconductors, (Oxford, Oxford Univ. Press, 1982) p. 123. 81. Following Ref.79 the coupling constant for piezoelectric scattering can be defined as: K2
_\
L
6Ci
/ ,\
T
/
, where ( e L ) , ( e L / > c L » °r . and
£
are respectively the averaged
tCn
longitudinal and transverse piezoelectric constants, the longitudinal and transverse elastic constant, and the optical dielectric constant. 82. V.W. Chin, T.L. Tansley, and T. Osotchan, "Electron mobilities in gallium, indium and aluminum nitrides", J. Appl. Phys. 75 (1994) 7365-7372. 83 E. Bellotti, B.K. Doshi, K.F. Brennan, P.P Ruden, "Ensemble monte carlo study of electron transport in wurtzite InN", J. App. Phys., 85 (1999) 916-923. 84. K. F. Brennan and N. S. Mansour, "Monte Carlo calculation of electron impact ionization in bulk InAs and HgCdTe," J. Appl. Phys., 69 (1991) 7844-7847. 85. E. Cartier, M. V. Fischetti, E. A. Eklund, and F. R. McFeely, "Impact ionization in silicon", Appl. Phys. Lett. 62 (1993) 3339-3341. 86. K. F. Brennan, ""Theory of high field electronic transport in bulk ZnS and ZnSe," J. Appl. Phys. 64 (1988) 4024-4030.
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584 E. Bellotti et al. 87. M. Dur, S. M. Goodnick, S. S. Pennathur, J. F. Wager, M. Reigrotzki and R. Redmer, "High field transport and electroluminescence in ZnS phophor layers", J. Appl. Phys. 83 (1998) 31763185. 88. S. Zollner, S. Gopalan, and M. Cardona, ,Intervalley deformation potentials and scattering rates in zinc blende semiconductors", Appl.Phys. Lett. 54 (1989) 614-616. 89. W.J. Schaffer, G.H. Nagley, K. Irvine, and J.W. Palmour, "Conductivity anisotropy in 6H and 4H-SiC", Mater. Res. Soc. Symp. Proc, 339 (1994) 595-601. 90. H-E Nilsson, E. Bellotti, M.Hjelm, K.F. Brennan, and C.S. Petersson, Proc. of the MACS Conference, Sofia, Bulgaria, June 1999. 91 A. O. Konstantinov, Q. Wahab, N. Nordell, and U. Lindefelt, "Study of avalanche breakdown and impact ionization in 4H silicon carbide," J. Electron. Mat., 27 (1998) 335-341.
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International Journal of High Speed Electronics and Systems, Vol. 11, No. 2 (2001) 585-615 © World Scientific Publishing Company
ELECTRICAL TRANSPORT IN ORGANIC SEMICONDUCTORS
I. H. CAMPBELL and D. L. SMITH Los Alamos National Laboratory, Los Alamos, NM 87545, USA Organic semiconductors have processing and performance advantages for low cost and/or large area applications that have led to their rapid commercialization. Organic semiconductors are rt conjugated materials, either small molecules or polymers. Their electrical transport properties are fundamentally distinct from those of inorganic semiconductors. Organic semiconductor thin films are amorphous or polycrystalline and their electronic structures consist of a distribution of localized electronic states with different energies. The localized sites are either individual molecules or isolated conjugated segments of a polymer chain. Electrical transport results from carrier hopping between neighboring sites. At room temperature, equilibration between neighboring sites of different energy is fast enough that carrier transport can be described using a mobility picture. Hopping transport in these disordered systems leads to a mobility that can depend strongly on both the electric field and carrier density. This article presents experimental measurements and theoretical analysis of the electrical transport properties of representative organic semiconductors.
1. Introduction This article discusses the electrical transport properties of organic semiconductors, a new class of electronic materials that are now widely used in organic light-emitting diodes (LEDs) and field-effect transistors (FETs). These organic electronic devices are attracting considerable interest because they have processing and performance advantages for low cost and/or large area applications. Organic electronic devices use undoped, semiconducting organic materials as the light-emitting and charge transporting layers. The charge carriers in the devices are injected from the contacts. Electronic devices based on doped organic materials have not been developed in a manner analogous to doped inorganic semiconductor devices. This article considers electrical transport in thin films of undoped, conjugated organic materials. These organic semiconductors are the focus of both current scientific research and commercial development. References [1-4] review the electronic properties and Refs. [5-9] review the device applications of organic semiconductors. Organic semiconductors are n conjugated materials, either small molecules or polymers. They have energy gaps ranging from about 1.5 eV to 3.5 eV,2,3 are undoped, and therefore have essentially no free carriers at room temperature. Figure 1 shows the chemical structure of three representative materials: the small molecule tris-(8hydroxyquinolate)-aluminum [Alq], the polymer poly(p-phenyelene vinylene) [PPV], and the small molecule pentacene. Tris-(8-hydroxyquinolate)-aluminurn was used in the first 223
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organic light-emitting diodes. 10 " Poly (p-phenylene vinylene) was the active material used in the first polymer light-emitting diodes.12 It is an insoluble polymer, i.e. it does not dissolve in organic solvents, and is prepared by thermal conversion of a precursor polymer. Soluble derivatives of PPV, such as poly [2-methoxy, 5-(2-ethyl-hexyloxy)1,4-phenylene vinylene] (MEH-PPV),13 which can be spin cast from organic solutions, are now more widely used. Pentacene is widely used in organic thin film transistors.14'15 These materials all have large conjugated units, i.e. regions with resonant single and double bonds, which determine their conduction and valence energy levels and thus their energy gaps.
Fig. 1. Chemical structures of Alq (left), PPV (center) and pentacene (right).
The main difference between the small molecule and polymer materials is their processing method. Thin films of small molecules, such as Alq and pentacene, are usually prepared by vacuum evaporation whereas thin polymer films are formed by solution processing methods such as spin casting. In both cases, the resulting films are amorphous or small grain polycrystalline (5-20 nm crystallites) and are highly disordered.2'1016 The electrical and optical properties of thin films of small molecules and polymers are generally similar. From an electronic structure point of view, the molecular films can be considered as a collection of distinct molecular sites. For the polymer films, the extended polymer chain is broken into independent sites by a combination of structural and chemical defects. In these organic films, electronic conduction occurs by hopping from one localized site to another. Hopping conduction is an essential aspect of the electrical transport in these materials. In order to better understand the electronic properties of the organic thin films, an elementary molecular orbital (MO) picture of the electronic structure of Tt-conjugated materials is first discussed. Because many of the conjugated materials used in organic
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electronic devices contain phenylene (benzene) rings, benzene and biphenyl, a molecule consisting of two benzene rings bonded together, are used as examples. Figure 2 is a 2
3 H
r
26 zp
H
-rr-
y
-26
Fig. 2. Chemical structure of benzene and its 7t-electron molecular orbital energies.
schematic of the molecular structure of benzene. The solid lines represent the a bonds formed primarily from the sp2 hybridized orbitals of carbon and the s orbitals of hydrogen. Neutral excitations and charged states involving these orbitals lie outside of the energy range of interest. The pz orbitals of carbon, which are orthogonal to the plane of the page, form the rc bonds indicated by the circle. The low energy excitations and charged states of device interest are formed from these orbitals. Six spatial states (twelve including spin) can be formed from the p z orbitals. In a simple molecular orbital picture with nearest neighbor interactions, the 6 MOs (not normalized) are: (1,1,1,1,1,1), (0,1,1,0,-1,-1), (2,1,-1,-2,-1,1), (-2,1,1,-2,1,1), (0,1-1,0,1,-1) and (1,-1,1,-1,1,-1) with corresponding MO energies -2(3, - p, - p, P, P and 2p. Here the site ordering for the MOs is (1,2,3,4,3',2') where the site numbers are indicated in the figure and P is the magnitude (a positive number) of the hopping integral. The MO energy levels are schematically illustrated in Fig. 2. MOs 2 and 3, and 4 and 5 are degenerate. In the electronic ground state of the molecule, MOs 1,2 and 3 are each occupied with two electrons. MOs number 2 and 5 have no amplitude on sites 1 and 4. The energy gap between occupied and empty MOs is 2P and the sum of the occupied MO energies is -8p. Figure 3 shows a schematic of the molecular structure of biphenyl, which consists of two benzene molecules bound together. A simple perturbation theory description of the MO energies of biphenyl is shown next to the molecular structure schematic. The lowest energy biphenyl MOs are made up from the first benzene MO (LI for the first MO on the left benzene and Rl for the first MO on the right benzene): (Ll+Rl) and (Ll-Rl) with energies -2P-C/6 and -2P+C/6, respectively. Here C is the magnitude of the hopping integral between carbon-4 on the left benzene and carbon-1 on the right benzene. In the same way symmetric and anti-symmetric biphenyl MOs can also be made from benzene MOs 3, 4 and 6: (LN+RN) and (LN-RN), where N=3, 4 or 6 with energies -P+C/3, -PC/3 for N=3; P-C/3, p+C/3 for N=4; and 2P+C/6, 2P-C/6 for N=6. Because benzene MOs 2 and 5 have no amplitude on carbons 1 and 4, they do not mix when forming
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biphenyl in the simple nearest neighbor interaction model. The inter-benzene hopping matrix element C depends on the angle between the planes of the two benzene molecules, C=C0 |cos0|, where 9 Dis the angle between planes of the two molecules and C0 is the 2P + C/6
2P-C/6 P+C/3
P P - C/3
e
4f-
"P + C/3
4-4- * -ff-
-P~c/3
-ff-
-2P + C/6
_ff-
-2P-C/6
Fig. 3. Chemical structure of biphenyl and its jr-electron molecular orbital energies.
magnitude of the matrix element when the benzene molecules are coplanar. The 6 lowest energy MOs are each occupied by 2 electrons in the neutral ground state. The energy gap between occupied and empty MOs is 2(P-C/3) and the sum of the occupied MO energies is -16(3. The energy gap of biphenyl is lowered compared with benzene by -2C/3. This lowering of the energy gap is a consequence of the more spatially extended nature of the states in biphenyl. The extent of this gap lowering depends on the angle between the benzene molecules, but the sum of the MO energies does not depend on the angle at this level of approximation. The lowest energy state with an extra electron added to the biphenyl molecule has the extra electron in the (L4+R4) MO with energy P-C/3. The sum of the occupied MO energies for this negatively charged state is -15P-C/3. The lowest energy state with an electron removed from the biphenyl molecule has a hole in the (L3+R3) MO with energy -P+C/3. The sum of the occupied MO energies for this positively charged state is also -15P-C/3. The sum of the occupied MO energies depends on the angle between the benzene molecules for both the positively and negatively
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charged states, but not for the neutral state. This leads to a coupling between the angular orientation and electronic charges. This molecular orbital picture is highly idealized but its qualitative results are preserved in more complete calculations. Figure 4 shows the result17 of a quantum chemical calculation (AMI level18) of the angular dependence of the energy of the biphenyl molecule (arbitrary energy zero) in the ground electronic state for the neutral, positive and negative ions. For the neutral molecule, there is a weak dependence of the energy on angle (in the simple description above the energy is independent of this angle). The lowest energy angle is approximately 40 degrees. The angular dependence of the 0.6
0.3
3§
0-0
©
c
LU
-0.3
anion neutral cation
-0.6 0
90
180
270
360
Angle (degree) Fig. 4. AMI results for the total energy as a function of torsion angle for biphenyl. Solid, dashed, and dot-dashed lines are for anion, neutral, and cation states, respectively (fromRef. 17).
energy for the two ions are very similar, they are much stronger than for the neutral molecule and have a minimum at 0 and a maximum at 90 degrees. The shape of the angular dependence curves are similar to |cos8| expected from the simple MO model. The minimum energy angle of the neutral molecule does not coincide with that for the ions so that a rotation will occur when an electron or hole is transferred to a neutral molecule to form an ion. In the amorphous or polycrystalline organic films used in electronic devices, there can be considerable variations in the electronic energy levels due to these variations in molecular geometry. The electronic structure of the conjugated organic thin films consists of a distribution of localized electronic states with different energies. The site energy distributions are believed to be approximately Gaussian with standard deviations
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Smith
typically between 0.1 eV and 0.2 eV.4 The localized sites are either individual molecules or isolated conjugation segments of a polymer chain. Electrical transport results from carrier hopping between neighboring sites. At room temperature, equilibration between neighboring sites of different energy is generally fast enough that carrier transport can be described using a mobility picture.4 However, at lower temperatures or for systems with very large disorder the mobility description may not be valid.4 Hopping transport in a disordered system leads to a mobility that can depend strongly on both electric field and carrier density.4 The mobility increases with increasing electric field and with higher carrier densities. As the electric field increases, more states become available for energetically favorable hopping transitions, thus increasing the mobility. At high carrier densities, the mobility is increased because charge transport occurs predominantly in a region with a higher density of states and therefore increased number of energetically favorable hopping sites. Although organic materials can be electronically doped, the dopant ions significantly modify the intrinsic properties of the organic material making carrier mobilities determined in doped materials largely irrelevant to the undoped films used in devices.19 Therefore, measurements of the mobility must be performed on undoped, insulating films. Conventional Hall effect measurements have not been useful for determining carrier mobilities in these insulating organic materials and photo-Hall measurements are complicated by short carrier lifetimes and relatively large exciton binding energies. Two general approaches are used to measure the carrier mobility: 1) measuring the transit time of optically injected carriers across thin films4,20"22 and 2) fitting the current-voltage characteristics of devices.9'23"33 Organic diodes and organic field effect transistors depend on transport normal to and parallel to the plane of the film, respectively. Because the molecular packing in organic thin films is often asymmetrical, particularly for polycrystalline and polymer films, they are likely to have different mobilities normal to and parallel to the plane of the film.34 Organic diodes and FETs also operate in distinct electric field and charge density regimes. Organic diodes typically operate at electric fields of several times 105 V/cm and at carrier densities of up to a few 1017 cm"3.35 In contrast, field effect transistors operate at lateral electric fields of a few 104 V/cm and at carrier densities above 1018 cm"3.36 For carrier densities below about 1017 cm"3, typical of diode operation, the mobility does not depend strongly on the carrier density.17 In this low density limit, interactions between carriers and state filling effects are not significant In contrast, for carrier densities above about 1018 cm"3, typical of FET operation, occupation of the lower energy hopping sites begins to enhance the carrier mobility.17 Because of the anisotropy in molecular structure
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and differences in the operating regimes, the mobilities determined from LED and FET measurements are not necessarily equivalent. Three techniques have been used to determine carrier mobilities in organic electronic materials: time-of-flight (TOF) current transient measurements,4 fitting of single carrier space charge limited (SCL) diode current-voltage (I-V) characteristics,9,23"33 and analysis of field effect transistor current-voltage characteristics.9 These three mobility measurement techniques sample different electric field and charge density regimes. The time-of-flight technique can measure the carrier mobility for electric fields from about 5xl0 4 V/cm to lxlO 6 V/cm but is restricted to very low volume averaged carrier densities, about 1013 cm"3 or less. Because the TOF technique is restricted to very low carrier densities, it is difficult to measure the carrier mobility in the presence of extrinsic trap states. Fitting single carrier, space charge limited diode current-voltage characteristics can probe the mobility for electric fields from about 105 to 106 V/cm and for carrier densities from about 1016 to 1018 cm"3. These measurements are much less sensitive to trapping effects than TOF measurements because the comparatively large density of injected carriers can fill moderate densities of traps without significantly perturbing the mobility measurement. In SCL diode measurements, the electric field and carrier density are functions of position within the device structure. Therefore, fitting the measured I-V characteristics requires assumptions about the carrier density and electric field dependences of the mobility. If both TOF, which requires low trap densities, and SCL diode mobility measurements can be performed, they usually yield consistent results. In FET mobility measurements, the lateral electric fields that are responsible for current flow are typically in the 104 V/cm range and the charge is confined to a thin region of the organic film adjacent to the gate insulator. This leads to very high charge carrier densities of about 1019 cm"3. These carrier densities are high enough to significandy modify the mobility due to changes in the occupation of hopping sites. In addition, the results are sensitive to the local molecular structure near the interface that may differ significantly from that typical of bulk films.37jU The FET mobilities are often significantly larger than TOF and SCL diode results on the same material. The higher mobility inferred from FET measurements may be due to the different carrier density regimes in which the measurements are made or to variations in the local molecular structure in the two kinds of devices. The carrier mobilities of different organic materials, as measured by TOF and SCL diode techniques, vary over a large range from, 10"8 cmVVs to 10"2 cm2/Vs.4 They are orders of magnitude smaller than those in typical inorganic semiconductors.36 The low mobility of the organic materials plays a critical role in determining the characteristics of organic devices. Measured carrier mobilities for films made from a given organic
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material can differ both because of different processing conditions for the films and because of different synthesis conditions used to make the organic material.35 It is difficult to compare detailed mobility results between different groups who may use different processing conditions and materials made under different synthetic conditions. It is possible to standardize processing conditions so that consistent mobility results can be achieved for films fabricated from material made in a given synthetic run. However, especially for conjugated polymers, films fabricated from materials made in different synthetic runs, even when using the same equipment and nominally the same approach for the synthesis, often give somewhat different mobility results. The electrical properties of the polymer film are sensitive to the film morphology, which can be affected by small changes in the polymer molecular weight distribution that varies slightly from batch to batch. This article focuses on mobility measurements of three representative conjugated organic materials: MEH-PPV, Alq, and pentacene. Carrier mobilities determined by TOF and SCL techniques are compared. The electric field dependence of the mobilities is discussed and theoretical models used to interpret the measurements are described.
2. Time-of-Flight Mobility Measurements Time-of-flight is an established technique to measure carrier mobilities in insulating materials. In this technique, a semitransparent blocking contact/insulating film/blocking contact structure is used. An optical pulse incident on the material through the semitransparent contact creates a thin sheet of electron-hole pairs next to that contact and, depending upon the sign of the applied bias, electrons or holes are driven across the sample. The absorption depth of the optical excitation must be small compared to the film thickness and the optical pulse duration must be short compared to the transit time of the charged carriers across the sample. Low intensity optical pulses are used so that the photogenerated charge carrier density does not significantly perturb the spatially uniform electric field in the structure. The carrier mobility, (X, is determined from the measured carrier transient time, T, by H = -^T
(1)
xV where d is the film thickness and V is the applied voltage. The structures used for TOF measurements consisted of a thin, semitransparent Al contact on a glass substrate, an organic film a few \im thick, and a top, thick Al contact. A Nitrogen laser pumped dye laser producing 500 ps pulses was tuned to the peak of the
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Electrical Transport in Organic Semiconductors
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absorption coefficient for the organic material so that the absorption depth of the optical pulse was much smaller than the film thickness. The bandwidth of the current preamplifier was two orders of magnitude greater than the reciprocal of the transit time. The product of the structure capacitance and amplifier input impedance was at least two orders of magnitude smaller than the transit time. The total charge injected into the film was about 0.01 CV, where C is the capacitance of the structure and V the applied voltage.
10'4 ill!
.;—
10"
10" •
i
-I
i
i 11Ti
r
—i
i
,5
40 V .
< >. tSi
e r>
io-;
\
C 3
U
1 III
10"
1
1
10"
10"
100V • 10"
10"
10"5 Time (s)
10'4
Fig. 5. Time-of-flight hole current transients for MEH-PPV at three applied voltages. The structure was semitransparent Al (10 nm)/MEH-PPV (1.8 u.m)/Al (100 nm) (from Ref. 20).
Figure 5 is a log-log plot of the TOF hole current density in MEH-PPV as a function of time after optical excitation for applied biases of 10 V, 40 V and 100 V at room temperature.20 The transit time was determined by the intersection of the asymptotes to
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the plateau and declining slope of the current transient. The MEH-PPV film was 1.8 Jim thick. Figure 6 shows the hole mobility as a function of electric field determined from the TOF measurements (markers) and a least squares fit (solid line) to the Poole-Frenkel form H = HoexP
(2)
where E is the electric field and (io and E0 are parameters describing the mobility. The Poole-Frenkel form for the electric field dependent mobility is frequently observed in organic molecular solids and polymers.4 Figure 6 shows that the Poole-Frenkel form
3
4
5
6
Electric Field (10 s V/cm) Fig. 6. The hole mobility of MEH-PPV as a function of electric field derived from TOF measurements. The markers are TOF results and the solid line is a least squares fit of Eq. IV.2 to the TOF results (from Ref. 20).
describes the measured TOF results reasonably well. The fit to the TOF data yielded the parameters |io = 2.1xl0"7 cm2/Vs and E0 = 8.7xl04 V/cm. The TOF current transients are close to the dispersive regime at the largest electric fields.4 In linear-linear current transient plots it becomes difficult to distinguish the current plateau at electric fields above about 4xl05 V/cm. The error bars on the mobility shown in Fig. 6 are estimated from TOF measurements on several different devices. It was not possible to measure hole TOF transients at significantly lower temperatures. At 250K the TOF plateau and falling edge could no longer be clearly distinguished over a significant range of electric fields. Neither was it possible to measure electron TOF transients. Space charge limited diode I-V measurements, discussed below, show that the electron mobility of MEH-PPV is much smaller than the hole mobility and trapping effects may be significant for electrons.
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Electrical Transport in Organic Semiconductors
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Time (s) Fig. 7. Time-of-flight electron current transients for Alq at three applied voltages. The structure was semitransparent Al (10 nmVAlq (2 nm)/Al (100 nm) (from Ref. 42).
Figure 7 is a log-log plot of the TOF electron current in Alq as a function of time after optical excitation for applied biases of 20 V, 50 V and 100 V at room temperature.42 The transit time was determined by the intersection of the asymptotes to the plateau and declining slope of the current transient. Figure 8 shows the electron mobility as a function of electric field determined from the TOF measurements (markers) and a least squares fit (solid line) to the mobility assuming the Poole-Frenkel form. Figure 8 shows that the Poole-Frenkel form describes the measured TOF results reasonably well. The fit
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1
2
3
4
Electric Field (105V/cm) Fig. 8. The electron mobility of Alq as a function of electric field derived from TOF measurements. The markers are TOF results and the solid line is a least squares fit of Eq. IV.2 to the TOF results (from Ref. 42).
to the TOF data yielded Ho = 1.5xl0*8 cm2/Vs and E0 = 1.5xl04 V/cm. It was not possible to measure the hole mobility in Alq using the TOF technique. Significant hole trapping that obscures the current transients is widely observed in Alq.43
O
2
1
2
3
Electric Field (105V/cm) Fig. 9. The hole mobility of pentacene as a function of electric field derived from TOF measurements.
Figure 9 shows the measured TOF hole mobility of pentacene at room temperature as a function of electric field (markers). The mobility is essentially independent of field over the range of electric fields considered. The mean value of the TOF data yielded |x = 1.2 x 10"3 cm2/Vs. It was not possible to measure the electron mobility in pentacene using the TOF technique in these films.
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Electrical Transport in Organic Semiconductors
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These TOF mobility results are typical of organic electronic materials with a low density of charge carriers in the film. Strong field dependence and low mobility, typical of hopping conductivity in a broad density of states, is observed for both holes in MEHPPV and electrons in Alq. For more ordered systems, the mobility is higher and less strongly field dependent. For example, the pentacene hole mobility is significantly larger and has much weaker field dependence than the MEH-PPV hole and the Alq electron mobility.
3. Mobility from Single Carrier SCL Diode I-V Characteristics The sensitivity of the time-of-flight technique to small densities of extrinsic traps frequently prevents its use to measure carrier mobilities. This problem can be overcome using single carrier diode current-voltage characteristics in the space charge limited current flow regime. To demonstrate the validity of this technique, we compare measured and calculated current-voltage characteristics of MEH-PPV and pentacene hole only devices using the TOF hole mobility measurements presented above. The independently measured hole mobilities were used, without adjustable parameters, to calculate the current-voltage characteristics of device structures with space charge limited hole current. The I-V characteristics were described using the device model of Ref. [44]. For the SCL contacts, the model reduces to a numerical evaluation of space charge limited current with a field dependent mobility that includes carrier drift and diffusion (the diffusion component is small). Figure 10 shows measured (solid) and calculated (dashed) current-voltage characteristics for a Pt/MEH-PPV/Al structure.20 The TOF mobility was used and there were no adjustable parameters in the calculation. Positive bias corresponds to space charge limited hole injection from the Pt contact and negligible electron injection from the Al contact that has a large electron Schottky barrier. There is good agreement between the measured and calculated I-V characteristics over five orders of magnitude in current. The current-voltage measurements were made using devices with 2 \im thick MEH-PPV films prepared identically to those used for the TOF measurements to ensure that the microstructures of the organic films were equivalent.34 The average charge density in the TOF measurement is about two orders of magnitude smaller than that in the space charge limited I-V characteristic. At an applied bias of 50V the average calculated hole density for the diode was about 1015 cm'3, whereas the average hole density in the TOF measurement was about 1013 cm"3. Since the mobility derived from the TOF measurements at low density accurately describes the I-V characteristics at much higher density, trapping effects are not important for holes in this material. The carrier density
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40
60 Bias(V)
Fig. 10. Measured (solid) and calculated (dashed) current density-voltage characteristics for a Pt (10 nm)/MEH-PPV (2 u,m)/ Al (100 nm) structure. Positive bias corresponds to space charge limited hole injection from Pt. The calculation used the fit to the TOF mobility shown in Fig. 6 without adjustable parameters (from Ref. 20).
in these 2 |xm thick devices under space charge limited current flow is about two orders of magnitude smaller than in typical polymer LEDs that have organic layers about 100 nm thick. 0.25
2
3 Bias (V)
Fig. 11. Measured (solid) and calculated (dashed) current density-voltage characteristics for two Pt/pentacene/Ca structures. Positive bias corresponds to space charge limited hole injection from Pt. The calculation used the TOF mobility shown in Fig. 9 without adjustable parameters.
Figure 11 shows measured (solid) and calculated (dashed) current-voltage characteristics for 20 nm and 300 nm thick Pt/pentacene/Ca structures. The TOF mobility was used and there were no adjustable parameters in the calculation. Positive
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Electrical Transport in Organic Semiconductors
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bias corresponds to space charge limited hole injection from the Pt contact and negligible electron injection from the Ca contact that has a large electron Schottky barrier. There is good agreement between the measured and calculated I-V characteristics. The highest charge density in the TOF measurement is about four orders of magnitude smaller than that in the space charge limited I-V characteristic. Since the mobility derived from the TOF measurements at low density accurately describes the I-V characteristics at much higher density, trapping effects are not important for holes in this material.
10
•••' 1
1'
Ca/Ca
/'
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1
II 1 1 1
ft
-
25nm / ' ft It
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j !
if
60nm / i
4)
11
u 4
Ji it
t/ if
io-
lOOnm
If ll > A
t
i
. . . .
10 Bias (V) Fig. 12. Measured (solid line) and calculated (dashed line) current density-voltage characteristics for 25, 60, and 100 run thick Ca/MEH-PPV/Ca electron only devices on linear (upper panel) and log-log (lower panel) scales (from Ref. 35).
The results for hole dominated diodes made from MEH-PPV and pentacene show that if a mobility is known from TOF measurements it can be used to accurately describe the current-voltage characteristics of single carrier SCL diodes. Single carrier SCL diode current-voltage characteristics are next used to determine the mobility for cases in which trapping interfered with the TOF measurements. To determine the electron mobility of MEH-PPV, for which it was not possible to measure the electron mobility using the timeof-flight technique, a series of electron only structures with space charge limited current
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flow were measured and fit using an electric field dependent and charge density independent electron mobility. Current-voltage characteristics were measured for a series of Ca/MEH-PPV/Ca electron only devices in which the polymer thickness was varied. The current is space charge limited because the energy barrier to injection of electrons from Ca into MEH-PPV is small. Figure 12 shows current density versus bias voltage for a thickness series of Ca/Ca electron only devices.35,45 The experimental results are shown as solid lines and the model results as dashed lines. The same electron mobility parameters Ho=5xl0~12 cm2/Vs, and E o =1.0xl0 4 V/cm were used for all the structures. The model describes current-voltage characteristics for Ca/Ca devices over a range of thicknesses, and over several orders of magnitude of current density. The thickness
0u
0
1 50 Position (nm)
u 100
Fig.13. Calculated electron density (upper panel) and electric field (lower panel) profiles for a 110 nm Ca/MEH-PPV/Ca device biased to provide 5xl0' 2 A/cm"2 device current density. The electron injecting contact is on the left (from Ref. 35).
scaling is not V2/L3 as expected form the analytic result for space charge limited current that does not include the field dependence of the mobility.33 Figure 13 shows the calculated electron density and electric field profiles for the 100 nm Ca/Ca device at a bias such that the current density is 5xl0"2 A/cm2. Electrons are injected from the left contact and collected at the right contact. These profiles, for a single bias point, demonstrate the range over which the electric field and charge density vary in SCL conditions.
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Electrical Transport in Organic Semiconductors
601
For single carrier SCL diodes, the carrier mobility is the main physical quantity that determines the current-voltage characteristics. Of course, the device geometry and the dielectric constant must also be known, but they can be independently determined. Therefore, the carrier mobility can be determined from the I-V curves of these devices. If SCL contacts cannot be made, as is the case for both electrons and holes in Alq, it is also necessary to know the injection properties of the contact.
4. Mobility Models The Poole-Frenkel form for the field dependence of mobility was first observed in TOF measurements on molecularly doped polymers.3,4,46,47 These materials consist of isolated molecular dopants in an inert polymer host. The dopant molecules do not introduce charged carriers, i.e they do not dope the material in the usual semiconductor sense of the term, but they provide low energy sites that the injected carriers occupy. In the molecularly doped polymers, transport results from carrier hopping between the dopant molecules, the host polymer provides a matrix for the active dopant sites. Because similar field dependence is observed in molecularly doped polymers and the solid state films used for electronic devices, it is plausible that the physical mechanisms controlling the electrical transport in these two classes of materials are similar. Bassler and coworkers extensively studied mobility in these types of materials using Monte Carlo simulations of the Gaussian disorder model (GDM).48'49 In the GDM, electrical transport results from carrier hopping between localized sites with the site energies randomly distributed according to a Gaussian distribution. The GDM describes some features of the observed mobility. However, the magnitude of the electric fields over which strong field dependence was found in the simulations did not correspond well with experiment. The magnitude of the fields at which the GDM showed strong field dependence, qualitatively similar to the experimental observations, was significantly larger than the field regimes at which that behavior was observed experimentally. Gartstein and Conwell50 showed that a spatially correlated site energy distribution for the carriers can explain the observed field dependence. Physically, strong field dependence occurs when the potential drop (eE £) across a relevant length scale ( t ) for the system becomes comparable to kT. With uncorrected site energies, the only relevant length scale is the distance between hopping sites. This distance is relatively small and strong field dependence only occurs at high fields. Spatial correlations introduce a new longer length scale, the length over which the site energies are correlated. Thus spatial correlations can lead to strong field dependence of the mobility at lower fields than would occur for uncorrelated site energies.
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Many of the dopant molecules used in molecularly doped polymers have large permanent electric dipole moments. Dunlap and coworkers51"55 proposed a model for the mobility of molecularly doped polymers based on the long range nature of the interaction between charged carriers and the dipole moments of the molecular dopants. In this model the energetic site disorder is the result of different electrostatic potentials at the various sites due to the random distribution in orientation of the dipole moments of the nearby dopant molecules. Because the charge-dipole interaction is long range, sites that are spatially close also have nearly the same energy so that there is a correlation between site position and energy. This model has been successful in describing many aspects of the mobility of molecularly doped polymers. Some of the materials used for organic electronic devices, such as Alq, also have large permanent dipole moments and the charge-dipole interaction model probably applies to them as well as to the molecularly doped polymers for which it was originally designed.42 However, other materials used for organic devices, such as PPV, do not have permanent dipole moments and therefore this model does not appear to apply to them in detail. A second physical mechanism, fluctuations in molecular geometry such as the phenylene ring-torsion in PPV, also leads to spatially correlated site energies and applies to conjugated materials without permanent dipoles. The spatial energy correlation is the result of strong inter-molecular restoring forces for ring-torsion fluctuations in dense films of closely packed molecules. By contrast, the intra-molecular restoring force for a ring-torsion is small as seen, for example, in the AMI calculations for biphenyl shown in Fig. 4. For neutral biphenyl, the energy is almost independent of the torsion angle. Because the restoring force is primarily inter-molecular, ring-torsions on neighboring molecules tend to move together. If an extra electron or hole is added to the system, the energy of the charged state depends strongly on the torsion angle, as also seen in the AMI calculations of biphenyl in Fig. 4. Thus there is a strong coupling between the energy of a carrier at a site and the ring orientation at that site. Different sites will have different ring orientations and therefore this coupling leads to disorder in the site energies. Because the rings on near neighbors move together, there is a spatial correlation in the site energies. Details of the site energy density of states and spatial correlation functions for this model are worked out in Ref. [17]. The results are found to be similar to those for the charge-dipole interaction model. Both models give a site energy density of states that is nearly Gaussian and a two site energy correlation function that falls off as the reciprocal of distance between sites. As a result the field dependence of mobility predicted by the two models is very similar although the physical origin of the energy disorder is different. There is a difference in expected temperature dependence in the two models because in the charge-dipole interaction model the
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Electrical Transport in Organic Semiconductors
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disorder is independent of temperature whereas in the molecular geometry fluctuation model, the disorder increases with increasing temperature. A ID Master equation with nearest neighbor hopping that can be used to describe field-dependent mobility has been exactly solved by Derrida.56 In the continuum limit, the mobility can be calculated using this solution to the Master equation.51 The result is
H=
? ° «, ,«. • „ , ,\
(3)
YJo~dye-™(e-0E<°> + e E W)
where y = PeE, E is the electric field, e is the magnitude of the electron charge, e(y) is the site energy at position y, ^ is the mobility if all site energies were the same, and P = 1/kT. Using a Gaussian approximation to calculate the correlation function and the (1/y) dependence for the correlation function(e(y) e(0)) given by both the charge-dipole interaction and the molecular geometry fluctuation models gives R2„2
(4)
(2PoVPeEa) KL (2PaVPeEa )
where a is essentially the standard deviation of the density of site energies, (1/a) is a momentum cutoff, and Ki(z) is the first-order modified Bessel function of the third kind. (Slightly different prescriptions were used for cutting off momentum integrals when going to the continuum limit in Refs. [51] and [17] which leads to a small difference in numerical factors for the forms stated in those papers.) In the charge-dipole disorder model, the standard deviation of the site energy distribution is a —
| e 2 P 2 n r2 r- - where P V 127ie 2 a
is the dipole moment and n 0 is the density of dipoles. In the molecular geometry
I v 2 kT fluctuation model a = J where v is the linear coupling constant between site \ 4rcKa energies and the ring-torsion and K is the angular restoring force constant for the ringtorsion. In the charge-dipole interaction model, v is temperature independent whereas in the molecular geometry fluctuation model, v is proportional to VkT . asymptotic expansion for Ki(z), the high field mobility becomes
n_e-e2°2e(2(WP^).
241
Using an
51
(5)
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J. H. Campbell & D. L.
Smith
In this ID solution both models give the same result for the field-dependence for the mobility, In \i ~
; that is, the Poole-Frenkel form for the field dependence. Because
a has different temperature dependences in the two models, the mobilities have different temperature dependences. The ID results are not generally valid for dense three-dimensional (3D) systems. In 3D, the carriers can take optimal paths to avoid high energy barriers whereas in the ID model there is only one path the carriers can take. The steady state Master equation describing the carrier transport for this system is 0 = ShjPj(l-Pi)-(OjiPi(l-Pj)J.
(6)
j
Here Pj is the probability for the polaron to be on site i and coy is the polaron hopping rate from site j to site i. Double occupation at a site is excluded. After finding the solution for Pj, the average carrier velocity is found from IcOjjPjd-P^Rjj v=-y
(?) J
where R j: is the position difference between sites j and i, and the mobility is found from v = \x E. The 3D Master equation is in general too complex for analytic solution and two numerical approaches are commonly used, Monte Carlo simulation and direct solution of the Master equation using sparse matrix techniques.
Compared with Monte Carlo
simulations, the sparse matrix approach has some advantages: it guarantees the steady state solution; it is more convenient for considering density-dependent effects; and it is often numerically more efficient. For cases in which both approaches can be used they give the same result. The field-dependent mobility in the dilute limit can be found by linearizing the Master equation. Figure 14 shows a calculation of mobility as a function of electric field for the molecular geometry fluctuation model with three values for the coupling parameter, v, between the ring-torsion and the site energy. The ring-torsion restoring force, K, was chosen to describe the measured hole mobility of MEH-PPV using v=0.3 eV. The values for these parameters are consistent with quantum chemical estimates using model systems17. The curves are reasonably close to linear showing that the model gives approximately the Poole-Frenkel form. Figure 15 shows the density of states for the site energies with the same values of the coupling parameters as in Fig. 14. For a
242
Electrical Transport in Organic Semiconductors
400
600
, - 1 / 2 , . .1/2
E
800
605
1000
-1/2.
(V cm
)
Fig. 14. Calculated logarithm of mobility as a function of E"2 with different polarontorsion couplings. Solid, dashed, and dot-dashed lines correspond to v=0.1, 0.2, and 0.3 eV, respectively (from Ref. 17).
system with a stronger coupling and therefore a broader density of site energies, the mobility is low and has strong field dependence, whereas for a system with weak coupling and therefore a narrower density of site energies, the mobility is higher and has weaker field dependence. 16.0
12.0
>
^
A
8.0
A
/-—>.V
4.0
//
Yv
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41
-"•
0.0 -0.4
V=0.2 V=0.3
„ -?,''^/
/
\
-0.2
0
\ * "•
0J2
0.4
C(eV) Fig. 15. Calculated site density of states as a function of energy for different polaron torsion couplings. Solid, dashed, and dot-dashed lines correspond to v=0.1,0.2, and 0.3 eV, respectively (from Ref. 17).
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The results of Fig. 14 suggest that there is correlation between the magnitude of the mobility and the strength of the field dependence because stronger energetic disorder leads to both lower magnitude of the mobility and stronger field dependence. Molecular geometry fluctuations, such as ring-torsions, are a source of energetic disorder. If the molecular structure is constrained in such a way that the restoring force for ring-torsions are increased or their coupling to the site energies are reduced, energetic disorder is reduced. As a result, the magnitude of the mobility is enhanced and its field dependence is weakened. Comparing TOF hole mobility measurements in MEH-PPV and poly(9,9dioctylfluorene) (PFO) show this correlation. The phenylene rings in an isolated neutral MEH-PPV chain can rotate easily. In PFO, two rings are fixed together by bridging bonds so they can only rotate together. As a result, both the inter-molecular restoring force to ring-torsion is increased, because two rings rather than one collide with a neighboring molecule, and the coupling of the ring-torsion to the site energy is reduce, because the charge can more easily delocalize on two rings than on a single ring. Figure 16 compares mobility measurements of MEH-PPV20 and PFO21 with calculations based on the molecular geometry fluctuation model. The observed hole mobility in PFO is about two orders higher than that for MEH-PPV and the field dependence is much weaker. This is the expected qualitative behavior. In the calculations, the hole mobility data of MEH-PPV was fit by adjusting the parameter K describing the ring-torsion intermolecular restoring force and the parameter v describing the strength of the coupling 10- 5
E 11
10
IDT"
200
400
600
— 1 / 2 , . ,18
E (V cm
800
200
-1/2.
400
600
800
.-1/2 .,,1/2 _ - 1 / 2 ,
)
E (V cm j
Fig. 16. Logarithm of hole mobility as a function of E"2. The left panel shows experimental(dots) and calculated (solid line) results for MEH-PPV. The right panel shows experimental (dots) and calculated (solid line) results for PFO (from Ref. 17).
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Electrical Transport in Organic Semiconductors
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between the ring-torsion and the site energy around the values estimated from AMI levelquantum chemistry calculations of model systems. The value of K was then increased to fit the PFO data. In principle, both an increase in K and a decrease in v should occur when going from MEH-PPV to PFO, but such changes in either of the two parameters gives very similar results. Good fits to the mobilities of both materials are achieved for physically reasonable values of the parameters. The mobility can depend on carrier density, because when some carriers fill deep potential sites from which hopping is difficult, the other carriers become more mobile. The density dependence of mobility can be studied by solving the nonlinear Master equation. Figure 17 illustrates the carrier density effects on the mobility in the molecular geometry fluctuation model with parameters appropriate for holes in MEH-PPV. The mobility is enhanced by almost one order of magnitude with increase of the carrier density to n=6.9X 1018 cm'3 at E - 4X 104 V/cm. In the low-field regime, where the field-assisted carrier hopping is less efficient than in the high-field regime, the carrier density effect on mobility is more pronounced. Diode measurements show that at electric fields of a few times 106 V/cm there is not a strong carrier density dependence of the mobility in MEH-PPV for densities up to about 1018 cm"3. By contrast, field-effect transistor measurements have suggested that the mobility increases strongly with increasing carrier density at low fields for carrier densities above about 1018 cm"3. The calculated results in Fig. 17 are consistent with these device measurements and explain why this behavior is expected when the different field/density regimes are sampled.
icr E o 1(T
200
400 . „
, - 1 / 2 . . .1/2
E
600 ,„
-1/S.
(V cm
800
1000
)
Fig. 17. Calculated logarithm of mobility as a function of E"2 with different carrier densities. Dotted, short-dashed, long-dashed, and dot-dashed lines correspond to carrier density n=0.08,0.5, 2, and 6.9 x 1018 cm"3, respectively. The solid line shows the results of solving the linearized Master equation (from Ref. 17).
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Figure 18 shows the effect of deep-traps on the mobility. Traps are randomly distributed with a concentration of 2 X 1017 cm'3 and a trap energy level 0.5 eV below the center of the Gaussian site energy density of states. The molecular geometry fluctuation model with parameters appropriate for holes in MEH-PPV is used. Because of the traps, the mobility is small in the low-field regime for small carrier densities. When the carrier density is sufficiently large to saturate the traps, the mobility is enhanced dramatically. These results show how a small density of traps can have a very large effect on TOF measurements, in which the carrier densities are very small, but do not significantly effect single carrier SCL diode measurements, in which the carrier densities are much larger. 10"'
;>10E o
10" 200
400
600
r-1/2 .. .1/2
E
-1/2.
(V cm
800
1000
)
Fig. 18. Calculated logarithm of mobility as a function of E"2 with different carrier densities for a system with randomly distributed traps. The trap concentration is 2 x 1017 cm'3 and the trap level is -0.5 eV. Short-dashed, long-dashed, dot-dashed, and dotted lines correspond to carrier density n=4.7, 2.4, 1.2, 0.3 x 1017 cm'3, respectively. The solid line shows the results of solving the linearized Master equation without traps (from Ref. 17).
In 3D systems, a carrier can optimize its path to avoid high energy barriers and achieve a higher mobility. Figure 19 illustrates current patterns in the low-field and the high-field regions.
The molecular geometry fluctuation model with parameters
appropriate for holes in MEH-PPV is used. The figure shows a projection of the 3D lattice onto the x-y plane (the field is in the x direction) by summing over the currents in the z direction. The width of each line in the figure is proportional to the current across the bond. Darker lines indicate that the current is opposite to the standard directions
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Electrical Transport in Organic Semiconductors
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Fig. 19. Current patterns for different applied field. The width of a line is proportional to the current across the bond. Upper and lower panels are for E=0.5xl0 s and 2xl0 6 V/cm, respectively (from Ref. 17).
(from left to right and from down to up). In the low-field regime, the carriers take complex paths involving many chains. When such irregular paths occur, a ID model, where the path is always along the field, is not appropriate. In the high-field regime, where the field is strong enough to overcome the energy barriers, the carrier paths are essentially one-dimensional. Because the energetic disorder is electrostatic in the charge-dipole interaction model, it should be the same, except for a sign reversal, for electrons and holes. Thus the charge-dipole interaction model predicts that the mobility for electrons and holes should have similar electric field dependence. This model should be applicable if the energetic disorder is dominated by the random orientation of the molecular dipoles. For Alq, the molecular dipole moment is large, 5.3 Debye,42 and the intermolecular spacing is about 1 nm which leads to disorder with a standard deviation of about 0.1 eV, comparable to the total energetic disorder. Thus, it is likely that dipolar disorder is the major source of energetic disorder in Alq films.
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In MEH-PPV, holes have a higher mobility and a weaker electric field dependence than electrons, suggesting that the energetic disorder is greater for electrons than for holes. Dipolar disorder is not expected to be a major source of the energetic disorder in MEH-PPV films. MEH-PPV has a flexible molecular structure and molecular geometry fluctuations such as ring-torsion should be a significant source of energetic disorder. For the simple example of the biphenyl molecule discussed above, there is an electron-hole symmetry and the coupling of electron and hole states to ring-torsion is the same. This electron-hole symmetry is broken in more complex molecules such as MEH-PPV and it is not expected that electrons and holes have the same field dependence. The hole mobility in pentacene is comparatively high and essentially independent of electric field. (The electron mobility in pentacene is not well studied.) This behavior is expected from a material that does not have spatially correlated energetic disorder. Pentacene does not have a permanent dipole moment and it is structurally rigid. As a result neither charge-dipole interactions nor molecular geometry fluctuations are expected to make major contributions to energetic disorder. Therefore, the energetic disorder in pentacene is expected to be comparatively weak and not spatially correlated.
5. Conclusion This article presented experimental measurements and theoretical analysis of the electrical transport properties of some representative organic semiconductors. Organic semiconductors have processing and performance advantages for low cost and/or large area applications that have led to their rapid commercialization. Their electrical transport properties are fundamentally distinct from those of inorganic semiconductors. Organic semiconductor thin films are amorphous or polycrystalline and their electronic structures consist of a distribution of localized electronic states with different energies. The localized sites are either individual molecules or isolated conjugated segments of a polymer chain. Electrical transport results from carrier hopping between neighboring sites. At room temperature, equilibration between neighboring sites of different energy is fast enough that carrier transport can be described using a mobility picture. Hopping transport in these disordered systems leads to a mobility that can depend strongly on both the electric field and carrier density. The commercial success of organic materials and devices has led to wide interest in designing new organic semiconductors with properties suited for specific applications. The objective is to be able to design an organic molecule or polymer so that thin films made from it have specific electrical and optical properties. This goal requires a thorough understanding of the properties of the molecule or polymer and how those
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properties determine the thin film behavior. One important objective is to increase the carrier mobility of the organic thin films. For example, increasing the carrier mobility would dramatically improve the power efficiency of organic LEDs and make organic diode lasers technologically appealing. The next decade is likely to see continuing rapid progress and exciting new developments in this rich are of science and technology.
Acknowledgements The authors are grateful to many collaborators who have made major contributions to this work including: Alan Bishop, David Brown, Brian Crone, Paul Davids, Joel Kress, Richard Martin, Avadh Saxena and Zhi Gang Yu at Los Alamos, and Nikolai Barashkov, John Ferraris and Charles Neef at the University of Texas at Dallas. This research was supported by the Los Alamos Directed Research and Development Program, the Defense Advanced Research Projects Agency and the Department of Energy.
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References 1. A. J. Heeger, S. Kivelson, J. R. Schrieffer and W. P. Su, "Solitons in conducting polymers", Reviews of Modern Physics (1988) Vol. 60,781--850. 2. N. C. Greenham and R. H. Friend, "Semiconductor device physics in conjugated polymers", in Solid State Physics, H. Ehrenreich and F. Spaepen, eds., Academic Press, New York, 1995, pp. 1149. 3. C. E. Swenberg and M. Pope, Electronic Processes of Organic Crystals and Polymers, Oxford University Press, Oxford, 1999. 4. P. M. Borsenberger and D. S. Weiss, Organic Photoreceptors for Xerography, Marcel Dekker, New York, 1998. 5. L. J. Rothberg and A. J. Lovinger, "Status and prospects for organic electroluminescence", J. Materials Research (1996) Vol. 11, 3174-3187. 6. P. E. Burrows, G. Gu, V. Bulovic, Z. Shen, S. R. Forrest and M. E. Thompson, "Achieving fullcolor organic light-emitting devices for lightweight, flat-panel displays", IEEE Trans, on Electron Devices (1997) Vol. 44,1188-1203. 7. C. H. Chen, J. Shi and C. W. Tang, "Recent developments in molecular organic electroluminescent materials", Macromolecular Symposia (1998) Vol. 125, 1--48. 8. G. Hadziioannou and P. Van Hutten, eds., Semiconducting Polymers: Chemistry, Physics and Engineering, John Wiley & Sons, New York, 2000. 9. G. Horowitz, "Organic field-effect transistors", Advanced Materials (1998) Vol. 10, 365—377. 10. C. W. Tang and S. A. Van Slyke, "Organic electroluminescent diodes", Appl. Phys. Lett. (1987) Vol. 51,913-915. 11. C. W. Tang, S. A. Van Slyke and C. H. Chen, "Electroluminescence of doped organic thin films", J. Appl. Phys. (1989) Vol. 65, 3610-3616. 12. J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, A. B. Holmes, "Light-emitting diodes based on conjugated polymers", Nature (1990) Vol. 347,539-541. 13. D. Braun and A. J. Heeger, "Visible light-emission from semiconducting polymer diodes", Appl. Phys. Lett (1991) Vol. 58, 1982-1984. 14. S. F. Nelson, Y.-Y. Lin, D. J. Gundlach and T. N. Jackson, "Temperature independent transport in high mobility pentacene transistors", Appl. Phys. Lett. (1998) Vol. 72,1854-1856. 15. C. D. Dimitrakopoulos, S. Purushothaman, J. Kymissis, A. Callegari and J. M. Shaw, "Lowvoltage organic transistors on plastic comprising high-dielectric constant gate insulators", Science (1999) Vol. 283, 822-824.
250
Electrical Transport in Organic Semiconductors
613
16. D. J. Gundlach, Y. Y. Lin, T. N. Jackson, S. F. Nelson and D. G. Schlom, "Pentacene organic thin-film transistors : Molecular ordering and mobility", IEEE Electron Device Letters (1997) Vol. 18, 87-89. 17. Z. G. Yu, D. L. Smith, A. Saxena, R. L. Martin and A. R. Bishop, "Molecular geometry fluctuation model for the mobility of conjugated polymers", Phys. Rev. Lett.(2000) Vol. 84, 7 2 1 724. 18. H. J. S. Dewar, E. G. Zoebisch, and E. F. Healy, "The development and use of quantummechanical molecular-models .76. Ami: a new general-purpose quantum-mechanical molecularmodel", J. Am. Chem. See. (1985) Vol. 107, 3902-3909. 19. D.Chen, M.J. Winokur, M.A. Masse and F.E. Karasz, "Structural phases of sodium-doped polyparaphenylene vinylene", Phys. Rev. (1990) Vol. B41,6759-6767. 20.1.H. Campbell, D.L. Smith, C.J. Neef, and J.P. Ferraris, "Consistent time-of-flight mobility measurements and polymer light-emitting diode current-voltage characteristics", Appl. Phys. Lett, (1999) Vol. 74, 2809-2810. 21. M. Redecker, D. D. C. Bradley, M. Inbasekaran and E. P. Woo, "Nondispersive hole transport in an electroluminescent polyfluorene", Appl. Phys. Lett. (1998) Vol. 73,1565-1567. 22. M. Redecker, D. D. C. Bradley, M. Inbasekaran and E. P. Woo, "Mobility enhancement through homogeneous nematic alignment of a liquid-crystalline polyfluorene", Appl. Phys. Lett. (1999) Vol. 74, 1400-1402. 23. H. Sirringhaus, P. J. Brown, R. H. Friend, M. M. Nielsen, K. Bechgaard, B. M. W. LangeveldVoss, A. J. H. Spiering, R. A. J. Jannssen and E. W. Meijer, "Two-dimensional charge transport in self-organized, high-mobility conjugated polymers". Nature (1999) Vol. 401, 685-688. 24. L. Torsi, A. Dodabalapur, L. J. Romberg, A. W. P. Fung and H. E. Katz, "Charge transport in oligothiophene field-effect transistors", Phys. Rev. (1998) Vol. B57, 2271-2275. 25. G. G. Malliaras, J. R. Salem, P. J. Brock, and C. Scott, "Electrical characteristics and efficiency of single-layer organic light-emitting diodes", Phys. Rev. Vol. B58, R13411-R13414. 26. P. E. Burrows, Z. Shen, V. Bulovic, D. M. McCarthy, S. R. Forrest, J. A. Cronin and M. E. Thompson, "Relationship between electroluminescence and current transport in organic heterojunction light-emitting devices", J. Appl. Phys. (1996) Vol. 79,7991-8006. 27. A. J. Campbell, D. D. C. Bradley, and D. G. Lidzey, "Space-charge limited conduction with traps in poly(phenylene vinylene) light emitting diodes", J. Appl. Phys.(1997) Vol. 82,6326— 6342. 28. A. J. Campbell, M. S. Weaver, D. G. Lidzey, and D. D. C. Bradley, "Bulk limited conduction in electroluminescent polymer devices", J. Appl. Phys.(1998) Vol. 84, 6737—6746.
251
614
/. H. Campbell & D. L.
Smith
29. P. W. M. Blom, M. J. M. DeJong and J. M. Vleggaar, "Electron and hole transport in poly(pphenylene vinylene) devices", Appl. Phys. Lett. (1996) Vol. 68,3308--3310. 30. P. W. M. Blom, M. J. M. DeJong and M. G. Van Munster, "Electric-field and temperature dependence of the hole mobility in poly(p-phenylene vinylene)", Phys. Rev. (1997) Vol. B55, R656--R659. 31. H. C. F. Martens, H. B. Brom and P. W. M. Blom, "Frequency-dependent electrical response of holes in poly(p-phenylene vinylene)", Phys. Rev.(1999) Vol. B60, R8489-R8492. 32. L. Bozano, S. A. Carter, J. C. Scott, G. G. Malliaras and P. J. Brock, "Temperature- and fielddependent electron and hole mobilities in polymer light-emitting diodes", Appl. Phys. Lett. (1999) Vol. 74, 1132-1134. 33. M.A. Lampert and P. Mark, Current Injection in Solids, Academic, New York, 1970. 34. C. Y. Yang, F. Hide, M. A. DiazGarcia, A. J. Heeger and Y. Cao, "Microstructure of thin films of photoluminescent semiconducting polymers", Polymer (1998) Vol. 39, 2299-2304. 35. B. K. Crone, I. H. Campbell, P. S. Davids, D. L. Smith, C. J. Neef and J. P. Ferraris, "Device physics of single layer organic light-emitting diodes", J. Appl. Phys. (1999) Vol. 86, 5767-5774. 36. S. M. Sze, Physics of Semiconductor Devices, John Wiley & Sons, New York, 1981. 37. D. D. C. Bradley, M. Grell, A. Grice, A. R. Tajbakhsh, D. F. OBrien and A. Bleyer, "Polymer light emission: control of properties through chemical structure and morphology", Optical Materials (1998) Vol. 9, 1-11. 38. S. Guha, W. Graupner, R. Resel, M. Chandrasekhar, H. R. Chandrasekhar, R. Glaser and G. Leising, "Planarity of para hexaphenyl", Phys. Rev .Lett. (1999) Vol. 82, 3625-3628. 39. L. Athouel, R. Resel, N. Koch, F. Meghdadi, G. Froyer and G. Leising, "Orientation of molecules in phenylene oligomer thin films: influence of the substrate temperature", Synthetic Metals (1999) Vol. 101,627-628. 40. P. A. Lane, M. Liess, X. Weis, J. Partee, J. Shinar, A. J. Frank and Z. V. Vardeny, "Effects of film order on the optical properties of alpha-sexithiophene", Chemical Physics (1998) Vol. 227,57-64. 41. H. J. Schon, C. Kloc, R. A. Laudise and B. Batlogg, "Surface and bulk mobilities of oligothiophene single crystals", Appl. Phys. Lett. (1998) Vol. 73, 3574-3576. 42. R. L. Martin, J. D. Kress, I. H. Campbell and D. L. Smith, "Molecular and solid-state properties of tris-(8-hydroxyquinolate)-aluminum", Phys. Rev. (2000) Vol. B61,15804-15811.
252
Electrical Transport in Organic Semiconductors
615
43. R. G. Kepler, P. M. Beeson, S. J. Jacobs, R. A. Anderson, M. B. Sinclair, V. S. Valencia and P. A. Cahill, "Electron and hole mobility in tris(8-hydroxyquinalinolato-nl,o8) aluminum", Appl. Phys. Lett. (1995) Vol. 66,3618-3620. 44. P.S. Davids, I.H. Campbell, and D.L. Smith, "Device model for single carrier organic diodes", J. Appl. Phys. (1997) Vol. 82,6319-6325. 45. B. K. Crone, P. S. Davids, I. H. Campbell, and D. L. Smith, "Device model investigation of single layer organic light emitting diodes", J. Appl. Phys. (1998) Vol. 84, 833-842. 46. D. M. Pai,"Transient photoconductivity in Poly(N-vinylcarbazole)", .J. Chem. Phys. (1970) Vol. 52, 2285-2291. 47. W. D. Gill, "Drift mobilities in amorphous charge-transfer complexes of trinitrofluorenone and poly-n-vinylcarbazole", J. Appl. Phys. (1972) Vol. 43, 5033-5040. 48. H. Bassler, "Charge transport in disordered organic photoconductors : a monte-carlo simulation study", Phys. Stat. Sol. (b) (1993) Vol. 175, 15-56. 49. D. Hertel, H. Bassler, U. Scherf and H. H. Horhold, "Charge carrier transport in conjugated polymers", J. Chem. Phys. (1999) Vol. 110,9214-9222. 50. Y. N. Gartstein and E. M. Conwell, "High-field hopping mobility of polarons in disordered molecular-solids: a monte-carlo study", Chem. Phys. Lett. (1994) Vol. 217, 41-47. 51. D. H. Dunlap, P. E. Parris and V. M. Kenkre, "Charge-dipole model for the universal field dependence of mobilities in molecularly doped polymers", Phys. Rev. Lett. (1996) Vol. 77, 542— 545. 52. S. V. Novikov, D. H. Dunlap, V. M. Kenkre, P. E. Parris and A. V. Vannikov, "Essential role of correlations in governing charge transport in disordered organic materials", Phys. Rev. Lett. (1998) Vol. 81,4472-4475. 53. P. E. Parris, M. Kus, D. H. Dunlap and V. M. Kenkre, "Nonlinear response theory: Transport coefficients for driving fields of arbitrary magnitude", Phys. Rev. (1997) Vol. E56, 5295-5305. 54. P. E. Parris, D. H. Dunlap and V. M. Kenkre, "Dispersive aspects of the high-field hopping mobility of molecularly doped solids with dipolar disorder", J. Polymer Science (1997) Vol. B35, 2803-2809. 55. V. M. Kenkre, M. Kus, D. H. Dunlap and P. E. Parris, "Nonlinear field dependence of the mobility of a charge subjected to a superposition of dichotomous stochastic potentials", Phys. Rev. (1998) Vol. E58, 99-106. 56. B. Derrida, "Velocity and diffusion constant of a periodic one-dimensional hopping model", J. Statistical Phys. (1983) Vol. 31,433-450.
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Series on Applied Mathematics - Voi. 13
Inverse Problemsfor Electrical Networks by E d w a r d B C u r t i s & J a m e s A M o r r O W (University of Washington, Seattle)
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his book is a very timely exposition of part of an important subject which goes under the general name of Inverse problems*. The analogous problem for continuous media has been very much
studied, with a great deal of difficult mathematics involved, especially partial
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differential equations. Some of the researchers working on the inverse conductivity problem for continuous media (the problem of recovering the conductivity inside from measurements on the outside) have taken an interest in the authors' analysis of this similar problem for resistor networks.
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The authors' treatment of inverse problems for electrical networks is at a fairly elementary level. It is accessible to advanced undergraduates, and mathematics students at the graduate level. The topics are of interest to mathematicians working on inverse problems, and possibly to electrical engineers. A few techniques from other areas of mathematics have been brought together in the treatment. It is this amalgamation of such topics as graph theory medial graphs and matrix algebra, as well as the analogy to inverse problems for partial differential equations, that makes the book both original and interesting. Contents: Circular Planar Graphs; Resistor Networks; Harmonic Functions; Characterization I; Adjoining Edges; Characterization II; Medial Graphs; Recovering a Graph; Layered Networks. Readership: Graduate students and researchers in applied mathematics and electrical and electronic engineering. 196pp
Pub. date: Mar 2000
981-02-4174-7
US$55
£37
For more information, please contact your nearest World Scientific office:
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Uncertain! and UNCERTAINTY
AND FEEDBACK
Hoc Loop-Shaping and the v-Gap Metric
" ^ iuup-sriapiii" ano :hu v -gap i-ieine
by
Glenn Vinnicombe University of Cambridge 1
T:
he principal reason for using feedback is to reduce the effect of uncertainties in the description of a system which is to be controlled. Hoo loop-shaping is emerging as a powerful but straightforward method of designing robust feedback controllers for complex systems. However, in order to use this, or other modern design techniques, it is first necessary to generate an accurate model of the system (thus appearing to remove the reason for needing feedback in the first place). The V-gap metric is an attempt to resolve this paradox — by indicating in what sense a model should be accurate if it is to be useful for feedback design. This book develops in detail the Hoo loop-shaping design method, the V-gap metric and the relationship between the two, showing how they can be used together for successful feedback design. Contents: An IntroductiontoHoo Control; Hoo Loop-Shaping; The V-Gap Metric; More Hoo Loop-Shaping; Complexity and Robustness; Design Examples; Topologies, Metrics and Operator Theory; Approximation in the Graph Topology; The Best Possible Hoo Robustness Results; State-Space Formulae and Proofs; Singular Value Inequalities. Readership; Researchers in control engineering, electrical & electronic engineering, and systems & knowledge engineering. 340pp 1-86094-163-X
Pub. date; Nov 2000 US$78 £53
Published by imperial College Press and distributed by World Scientific Publishing Co.
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