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QUANTUM PHYSICS OF RETARDED TRANSPORT
J. Barker
To cite this version:
J. Barker. QUANTUM PHYSICS OF RETARDED TRANSPORT. Journal de Physique Colloques,
1981, 42 (C7), pp.C7-245-C7-252. �10.1051/jphyscol:1981729�. �jpa-00221666�
QUANTUM PHYSICS OF RETARDED TRANSPORT
J.R. Barker
Warwick University, Coventry CV4 7AL, United Kingdom
Abstract: - Many problems in hot electron physics, in particular the transport pro- perties of sub micron devices, involve space and time scales such that the semi- classical Boltzmann transport theory is invalid. Here we examine the temporal and spatial retardation effects associated with the evolution of the microscopic Wigner distributions for carriers using many-body Green function techniques applicable to inhomogeneous high field quasi-particle transport.
1. Introduction: - Boltzmann transport theory is widely deployed in hot electron physics, yet there are a number of circumstances where its validity is questionable
[1,2,3] and we have to make recourse to a deeper theory. In the present paper we attempt to focus on a particular problem: the spatio-temporal response of the carrier and phonon distribution functions. It is already well appreciated that the macro- scopic integrals of motion e.g. average velocity, temperature, calculated within the Boltzmann theory, show retardation effects in space and time when the conditions for a steady state are violated, or when the driving fields have high frequency components [1,4]. Thus overshoot and other transient relaxation effects occur on time scales for which the considered transit time is less than or comparable to the classical momentum and energy relaxation times within the system. In these
circumstances the time evolution of the average response is not Markovian: some memory of the initial transients and state preparation persists. An extreme case is the idealised situation of true ballistic transport where it is supposed that no collisions occur so that the carrier dynamics are strictly time-reversible [4,5].
On the other hand, overshoot, ballistic transport etc cannot be rigoriously handled by Boltzmann transport theory because the time/space scales on which these effects occur, are generally too short for the Boltzmann equation itself to be valid, although the required modification may only be perturbative [6-8]. In previous JOURNAL DE PHYSIQUE
Colloque C7, supplément au n°10, Tome 42, octobre 1981 Pa9e C7-245
Résumé: - Beaucoup de problèmes dans la physique des électrons chauds, en particulier les propriétés de transport dans les dispositifs sub-microniques, font intervenir des échelles de temps et d'espace de telle sorte que la théorie de transport semi-classi- que de Boltzmann n'est plus valable. Nous examinons ici les effets de retard spaciaux et temporels associés à l'évolution de la distribution microscopique de Wigner pour les porteurs en utilisant des techniques de fonction de Green pour des systèmes à plusieurs corps, applicable au transport de quasi-particules dans de forts champs in- homogènes.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981729
JOURNAL DE PHYSIQUE
s t u d i e s [l-81, t h e emphasis h a s been on t h e r e t a r d e d quantum t r a n s p o r t p h y s i c s o f b a r e e l e c t r o n s and phonons w i t h i n a homogeneous r e g i o n of semiconductor m a t e r i a l . But i n sub-micron c o n f i g u r e d h i g h f i e l d d e v i c e s , where t r a n s i e n t e f f e c t s a r e t h e norm, it i s n o t s u f f i c i e n t t o c o n s i d e r t h e i n t r a - d e v i c e volume a l o n e . For example, c a r r i e r s w i t h i n t h e d e v i c e volume a r e g e n e r a l l y swept i n t o t h e channel from s t a t e s w i t h i n t h e c o n t a c t r e g i o n s where t h e l a n d s t r u c t u r e , s c a t t e r i n g environment and e l e c t r i c f i e l d s a r e v e r y d i f f e r e n t from t h e channel i t s e l f . This i s a l s o t r u e f o r h o t e l e c t r o n i n j e c t i o n from semiconductors a t i n s u l a t o r s and vice-versa. Now w i t h i n any homogeneous r e g i o n it i s u s u a l t o d e s c r i b e t h e c a r r i e r j p h o n o n s t a t e s a s r e - n o r m a l ~ s e d s t a t e s , p a r t i c u l a r l y i n p o l a r media where t h e d r e s s i n g e f f e c t s of t h e s c r e e n i n g , and electron-phonon i n t e r a c t i o n s may be s u b s t a n t i a l i n s o f a r a s t h e y e f f e c t t h e q u a s i - p a r t i c l e masses and l i f e t i m e s , and indeed t h e e f f e c t i v e s t r e n g t h of t h e q u a s i - p a r t i c l e s c a t t e r i n g i n t e r a c t i o n s . The q u e s t i o n which we wish t o a d d r e s s concerns how t h e "memory" o f t h e dynamical s t a t e s o f c a r r i e r s / p h o n o n s w i t h i n a source r e g i o n (e.g. a c o n t a c t ) i s l o s t o r p r e s e r v e d when t h e c a r r i e r s / p h o n o n s a r e swept i n t o a d i f f e r e n t channel r e g i o n . T h i s and r e l a t e d q u e s t i o n s cannot be posed w i t h i n Boltzmann t r a n s p o r t t h e o r y , n o r indeed w i t h i n t h e p r e v i o u s l y deployed quantum k i n e t i c e q u a t i o n s C21 based on s i n g l e time Green f u n c t i o n s (Wigner f u n c t i o n s ) . I n s t e a d we s h a l l use t h e language o f non-equilibrium double-time thermodynamic Green f u n c t i o n s f i r s t s t u d i e d f o r l i n e a r t r a n s p o r t problems by Kadanoff and Baym C91.
The s t r u c t u r e o f t h e g e n e r a l non-linear t h e o r y i s o u t l i n e d i n an accompanying paper
[lo].
Here we f o c u s on t h e n a t u r e and o r i g i n of t h e r e t a r d a t i o n o f response t o inhomogeneous systems of coupled e l e c t r o n s and phonons.2. The e x a c t quantum t r a n s p o r t e q u a t i o n s
Following our p r e v i o u s s t u d i e s [21, we work e n t i r e l y w i t h t h e s i n g l e c a r r i e r Wigner d i s t r i b u t i o n f ( p , R , t ) , which i s r e l a t e d t o t h e more fundamental double-time r e t a r d e d Green f u n c t i o n G (1,l') R whexe ( 1 , l ' ) E ( r l , t l ; r 2 , t 2 ) o r i n terms of Wigner c o o r d i n a t e s : R = ( r
+
r ) / 2 , r = rl - r 2 , T = ( t l+
t ) / 2 , t = tl - t2-
R 1 2 2
G ( r , t ; R , t ) . 1
where $ J ~ , $ a r e e l e c t r o n f i e l d o p e r a t o r s d e f i n e d w i t h i n t h e Heisenberg r e p r e s e n t a - t i o n such t h a t t h e ensemble average < > i n v o l v e s a many body t r a c e o v e r t h e i n i t i a l system d e n s i t y m a t r i x a t time T = 0. I t i s supposed h e r e f o r s i m i p l i c i t y t h a t t h e e x t e r n a l f i e l d s a r e i n i t i a t e d a t T > 0 , s o t h a t $, $ t evolve under a Hamiltonian which c o n t a i n s t h e coupling t o an e x t e r n a l a p p l i e d p o t e n t i a l ba(R,T). Note t h a t G R (pwRT) i s t h e space-time F o u r i e r t r a n s f o m o v e r t h e r e l a t i v e c o o r d i n a t e s n , t of
R R A
G ( r , t ; R T ) . We a l s o observe t h a t G and t h e advanced Green f u n c t i o n G a r e r e l a t e d t o t h e f u l l Green f u n c t i o n G d e f i n e d by
R A
where T i s t h e fermion time-ordering o p e r a t o r , G, G
,
G maybe most c o n v e n i e n t l y a n a l y s e d by c o n s t r u c t i n g e q u a t i o n o f motion f o r an analogous imaginary time Green f u n c t i o n G ( l , l ' , S ) d e f i n e d by:-G ( I . , ~ ' , s ) = -i <Ts+(L)$ t (1') >/<$s;
where to-it3
S = expC-i d 2 $ a ( 2 ) n ( 2 ) 1
n ( 2 ) i s t h e number d e n s i t y o p e r a t o r , f3 = 1 /kT ( i n i t i a l t e m p e r a t u r e ) , and t h e t i m e s a r e d e f i n e d i n t h e imaginary time domain: 0 < i(t-to) < B (to r e a l ) . G ( l , l ' , s ) i n v o l v e s a system evolving from an inhomogeneous e q u i l i b r i u m d e n s i t y m a t r i x
1t maybe shown [9,11,121, t h a t G ( 1 , l ' ) i s j u s t t h e a n a l y t i c c o n t i n u a t i o n of G ( l , l ' , S ) t o t h e r e a l time domain provided t -t
--.
The e q u a t i o n of motion fox G ( l , l r , S ) may be f o r m a l l y d e r i v e d by f u n c t i o n a l d e r i v a t i v e t e c h n i q u e s t o have an "apparently" c l o s e d form by i n t r o d u c i n g t h e s e l f - energy C f o r t h e c a r r i e r s : -
where 9 i s t h e e f f e c t i v e t o t a l d r i v i n g p o t e n t i a l comprising t h e induced f i e l d p o t e n t i a l s p l u s t h e a p p l i e d p o t e n t i a l . I f we perform t h e a n a l y t i c c o n t i n u a t i o n , t h e e x a c t e q u a t i o n of motion f o r G (pw;RT) B i s o b t a i n e d from:-
2 -m
dl" zR(l",l') ( G ~ - G ~ ) C i , 1 " 1
JOURNAL DE PHYSIQUE
3. The Boltzmann l i m i t
The Boltzmann t r a n s p o r t theory may be recovered from ( 7 ) by: ( a ) assuming slowly varying disturbances; (b) taking t h e asymptotic l i m i t T +
-
r21. Let usf i r s i l examine t h e appearance of t h e l o c a l d r i v i n g f o r c e , i.e. of t h e normal form
[aT
+ v.aR + eE.3 I ~ ( ~ , R ; T )P ( 8 )
We note f i r s t t h a t the LHS of ( 7 ) has t h e Wigner coordinate form:-
In thermal equilibrium, t h e undressed version of G R i s diagonal, and has a s p a t i a l range of t h e order of a thermal wavelength X
*
(3%6/2m)lL, and a temporal decay time of t h e order46.
The l o c a l homogeneity approximation (LHA C l ] ) , t h i s supposes t h a t i f C$ v a r i e s slowly on t h e s e s c a l e s , we may expand $ i n a Taylor s e r i e s t o lowest order:
The d r i v i n g term f o r f(p,R,T) i s then e a s i l y obtained a s expression ( 8 ) . More generally, one should use t h e exact form:-
where H I p 2 /2m
+ +
(R,T) ; A-
3p.a R- a
R E .a+ aTaw - awaT.
i s t h e generalised Poisson b r a c e l e t o p e r a t o r C101. Expression (10) i sexact
f o r constant, uniform f i e l d s .I f we use the LHA on t h e d r i v i n g terms, and n e g l e c t a l l q u a n t i t i e s of order t , r compared t o R, T t h e RHS of
( a ) ,
we find:-Now G~ i s r e l a t e d t o ($ v i a t h e s p e c i a l function A(pwRT) A = GR
-
GAR A
From t h e equation of motion f o r G G we t h u s o b t a i n , within t h e same approximation:-
l a
+
vaR- a
$a+ a + a
] A = oT R p T w (14)
I t may a l s o be shown ( e x a c t l y ) t h a t
The l i n e a r t r a n s p o r t study by Kadanoff and B a p C61 takes:
a s t h e s o l u t i o n t o (14). By taking t h e Born approximation t o t h e s e l f energy it i s
t h e n v e r y e a s y t o show t h a t e q u a t i o n (12) reduces e x a c t l y t o t h e B o l t Z m a ~ equation w i t h t h e a i d of ( 1 3 ) , (151, ( 1 6 ) . This r e s u l t i s a weak coupling r e s u l t and n e g l e c t a l l q u a s i - p a r t i c l e e f f e c t s . Moreover, t h e d e l i b e r a t e
local
approximation made t o-
t h e time and space dependence of t h e " c o l l i s i o n i n t e g r a l s " o r t h e RHS o f (171, p l u s t h e l o c a l s o l u t i o n ( 1 6 ) , a r e p r i m a r i l y r e s p o n s i b l e f o r t h e r e s u l t i n g Markovian ( i . e . non-retarded) form of t h e Boltzmann equation. No i n t r a - c o l l i s i o n a l f i e l d e f f e c t s a r e v i s i b l e d e s p i t e t h e appearance of t h e f i e l d i n (16).
4. Retarded t r a n s p o r t
We have a l r e a d y s e e n from e x p r e s s i o n (9) t h a t t h e d r i v i n g f o r c e s a x e g e n e r a l l y non-local i n space and time. Let u s now observe t h a t t h e " c o l l i s i o n " terms a r e r a t i o r e t a r d e d . The appearance of r e t a r d a t i o n i s i n t i m a t e l y r e l a t e d t o t h e degree of d i s s i p a t i v i t y w i t h i n t h e s y s t a . Bolt-nn t r a n s p o r t i s a d i s s i p a t i v e theory:
c o l l i s i o n s occur a s p o i n t e v e n t s which a r e completed i r r e v e r s i b l y .
To s e e how t h e c o l l i s i o n i n t e g r a l c o n t a i n s r e t a r d a t i o n , l e t u s go back t o e q u a t i o n ( 7 ) , and look a t t h e l i m i t t = tl-tl, + 0, which we need t o o b t a i n t h e Wigner d i s t r i b u t i o n f . Let u s keep t h e same approximation f o r t h e s p a t i a l
dependence of X , G b u t t r e a t t h e time-dependence e x a c t l y . The e q u i v a l e n t of t h e RHS of e q u a t i o n (12) i s t h e n found t o be
T
dr{expLZi(w'-w") (T-r) I
+
c.c} x-03
The asymptotic l i m i t i s now e a s i l y seen: i f we l e t T +
-,
t h e e x p o n e n t i a l f a c t o r s c o l l a p s e t o a r e p r e s e n t a t i o n of a & - f u n c t i o n , d e s c r i b i n g t h e c o n s e r v a t i o n of energy:-Neglect of t h e t i m e v a r i a t i o n i n t h e e a r l i e r approximation i s t h u s e q u i v a l e n t t o a time-coarse q r a i n i n g o f t h e system.
We n e x t observe t h a t t h e asymptotic l i m i t cannot be t a k e n without examining t h e asymptotic behaviour of Z and G. Indeed s i n c e b o t h C and G can be expressed e n t i r e l y i n terms of t h e s p e c t r a l f u n c t i o n A(pwRT) t h e Wigner f u n c t i o n f(poRT)
(and t h e phonon d i s t r i b u t i o n i n g e n e r a l c a s e s ) , t o g e t an e x p l i c i t form f o r t h e c o l l i s i o n i n t e g r a l we s h a v e an approximation f o r A. I f we go back t o e q u a t i o n
(14) f o r example, it i s obvious t h a t Kadanoff and Baym's s o l u t i o n i s
not
c o r r e c t e x c e p t f o r t h e r m a l e q u i l i b r i u m . To s e e t h i s we t r a n s f o r m e q u a t i o n (14) t o p a t h v a r i a b l e form C2 1 :C7-250 JOURNAL DE PHYSIQUE
where d ~ / d r = 1; d ~ / d r = p ( T ) /m; d p / d ~ = -aR$ ( R ( T ) , T ( T ) ; d w / d ~ = aT$ (R,T)
.
Theg e n e r a l s o l u t i o n t o (19) i s i n f a c t
ACT]
= c o n s t a n t . Thus i f A h a s t h e boundary c o n d i t i o n ( i n i t i a l ) v a l u e A = 2a6 ( w - ~ [ p l ),
ELPI= p /2m we f i n d : - 2A(p,w,R,T) = 2s6[w
- ~ ( g ) ]
(20)where t h e t i l d a s r e f e r t o r e t a r d e d q u a n t i t i e s : -
I f eqn (20) i s i n s e r t e d i n t o t h e Born approximation f o r C, and t h e n i n t o e q u a t i o n (17) we r e c o v e r e x a c t l y t h e g e n e r a l i s e d k i n e t i c e q u a t i o n o f Barker and F e r r y C31, showing t h e temporal r e t a r d a t i o n o f t h e p o s i t i o n , momentum and time dependence of t h e d i f f e r e n t i a l s c a t t e r i n g r a t e s plus t h e i n t r a - c o l l i s i o n a l f i e l d e f f e c t [I]. For electron-phonon s c a t t e r i n g bhe e q u a t i o n f o r f r e a d s (homogeneous c a s e ) : -
r
T[aT
+ e ~ a ~f = JOd' $,is ($,$*,T) ~ ( $ - . T - T )I n t h i s c a s e , t a k i n g t h e asymptotic l i m i t T + m, does not r e c o v e r t h e s h a r p energy c o n s e r v a t i o n o f Boltzmann t r a n s p o r t t h e o r y even f o r c o n s t a n t uniform f i e l d s , because t h e s c a t t e r i n g i s d e s c r i b e d between a c c e l e r a t i n g s t a t e s ; t h e s t a t e s p , p' a r e t h u s s h i f t e d and smeared o u t by l e v e l s h i f t and broadening e f f e c t s due t o t h e t r a n s i t i o n induced by t h e f i e l d w i t h i n t h e c o l l i s i o n d u r a t i o n E l l .
5. Q u a s i - p a r t i c l e e f f e c t s
Let u s now r e l a x t h e s p a t i a l approximations t o C and G made i n s e c t i o n s 3,4.
We f i r s t e v a l u a t e t h e s e q u a n t i t i e s a t t h e same l e v e l of approximation a s t h e d r i v i n g terms i . e . use t h e LHA. Thus terms such a s G ( r p , R-(r-r')/2) which appear i n (7) a r e approximated by
The second term i n (25) c o n t r i b u t e s two a d d i t i o n a l terms t o equation ( 1 7 ) . One h a s an asymptotic l i m i t i n t h e form o f a Poisson b r a c k e t : -
-{ReC (pwRT)
,
G R (BURT)1
where
Consequently, t h e terms i n
a
G,a
G,a
G can be r e t u r n e d t o t h e RHS of e q u a t i o n (17)T R p
s o a s t o renormalise t h e d r i v i n g terms. Only one of t h e s e terms w i l l be d i s c u s s e d h e r e : t h e term i n
a
ReC aRG R.
This term e s s e n t i a l l y r e n o r m a l i s e s t h e base v e l o c i t yP
v = pRm t o t h e q u a s i - p a r t i c l e v e l o c i t y v + p/m
+ a
ReC. A s i m i l a r m o d i f i c a t i o n Pf o l l o w s through i n t h e e q u a t i o n of motion f o r t h e s p e c t r a l f u n c t i o n . I f t h e asymp- t o t i c l i m i t i s n o t taken we f i n d t h a t t h e q u a s i - p a r t i c l e v e l o c i t y h a s aretarded form with t h e s t r u c t u r e
where M i s a memory f u n c t i o n a l . I f t h e d r i v i n g f i e l d s a r e s t r o n g o r v a r y r a p i d l y i n time v i s found t o reduce t o p/m. Only when t h e f i e l d s a r e slowly p e r t u r b a t i v e
( a d i a b a t i c ) does t h e e q u i l i b r i u m q u a s i - p a r t i c l e behaviour appear.
I n a d d i t i o n , a f u r t h e r term i s c o n t r i b u t e d by (25) which i n i t s asymptotic form adds t o (25) a term:-
This asymptotic form d e s c r i b e s t h e decay of i n i t i a l c o r r e l a t i o n s i n t h e i n i t i a l s t a t e a t T = 0. It does n o t appear t o be s i g n i f i c a n t f o r l o w frequency t r a n s p o r t , b u t i s dominant f o r f r e q u e n c i e s i n e x c e s s o f t h e c o l l i s i o n a l r e l a x a t i o n f r e q u e n c i e s
(WT >> 1 ) o r f o r high magnetic f i e l d s ( W T >> 1 ) . A d i s c u s s i o n o f t h i s c o n t r i b u t i o n h a s been given from a d i f f e r e n t p o i n t of view i n r e f . : l l .
A r i g o r o u s t r e a t m e n t of t h e e x a c t e q u a t i o n ( 7 ) h a s been made r e c e n t l y 1121. I t r e q u i r e s t h a t t h e b a s i c modelling i n v o l v e a dynamical approximation t o t h e s e l f - energy X. The s p e c t r a l f u n c t i o n s and i n i t i a l s t a t e a r e t h e n e v a l u a t e d s e l f c o n s i s t e n t l y with t h i s approximation s o a s t o v a l i f y t h e microscopic c o n s e r v a t i o n laws (Ward i d e n t i t i e s [ I ] ) . I n a d d i t i o n t h e l e v e l of time-space c o a r s e g r a i n i n g
(e.g. t h e LHA) i s decided by examining w i t h i n t h e model which time-space s c a l e v a r i a t i o n s i n t h e f i e l d o r t h e h e t e r o j u n c t i o n s t r u c t u r e a r e comparable w i t h t h e i n t r i n s i c time-space s c a l e s f o r q u a s i - p a r t i c l e d r e s s i n g . A s an example we q u o t e t h e r e s u l t s f o r t h e c a s e where t h e f i e l d s a r e s u f f i c i e n t l y t o modify energy c o n s e r v a t i o n w i t h i n c o l l i s i o n s , b u t do n o t d e s t r o y t h e q u a s i - p a r t i c l e s t r u c t u r e : t h i s c a s e assumes a homogeneous medium, b u t n o t n e c e s s a r i l y homogeneous t r a n s p o r t . We f i n d t h a t t h e dominant r e t a r d a t i o n p r o c e s s i s f u l l y d e s c r i b e d by t h e Barker-Ferry eqns.
C31, b u t t h e momentum s t a t e s appearing on t h e d r i v i n g terms (group v e l o c i t y ) and c o l l i s i o n i n t e g r a l s a r e modified t o t h e field-dependent q u a s i - p a r t i c l e v e l o c i t i e s . I n a d d i t i o n t h e r e t a r d e d c o l l i s i o n i n t e g r a i s a r e modified by a q u a s i p a r t i c l e e x p o t e n t i a l decoy term. The l a t t e r i s e a s i l y e v a l u a t e d by s o l v i n g f o r t h e s p e c t r a l f u n c t i o n from t h e r i g o r o u s equation o f motion
JOURNAL DE PHYSIQUE
R .
where
r
= CA-
C 1s t h e imaginary p a r t o f t h e s e l f energy.The r e s u l t i n g e q u a t i o n s show f o r example, t h a t t h e p o l a r o n i c c a r r i e r s i n weak coupling media, a r e p r o g r e s s i v e l y s t u f f e d t o t h e b a r e l a n d mass s t a t e s ( a c c e l e r a t i n g s t a t e s ) a s h i g h f i e l d s a c c e l e r a t e c a r r i e r s beyond t h e o f t e n phonon t h r e s h o l d .
On t h e q u e s t i o n of s e l f - c o n s i s t a n c y , we f i n d t h a t t h e Born approximation t o Z i s never v a l i d ( s e e a l s o E l l ) , b u t one can use t h e g e n e r a l i s e d Born approximations.
The l a t t e r u s e s a s e l f c o n s i s t a n t form f o r t h e s p e c t r a l f u n c t i o n i n t h e e x p r e s s i o n f o r t h e s e l f energy, u n l i k e t h e Born approximation which use t h e f r e e c a r r i e r form.
The e f f e c t i s t o g i v e c o l l i s i o n a l l e v e l s h i f t s i n t h e energy c o n s e r v a t i o n and t o broaden t h e s h a r p energy c o n s e r v a t i o n due t o q u a s i - p a r t i c l e l i f e t i m e e f f e c t s .
6. Refeaences
1. Barker, J.R., i n Handbook of Semiconductors, Vol. 1, W. P a u l , Ed. (North Holland 19811, Ch. 13.
2. Barker, J . R . , i n P h y s i c s o f non-linear t r a n s p o r t i n Semiconductors, e d by Ferry,D.
Barker, J . , and Jacoboni, C. (Plenum, New York, 1980) p.126.
3. Barker, J.R., and F e r r y , D . K . , Phys. Rev. L e t t e r s ,
42
(1979) 1776.4. Grubin, H.L., F e r r y , D.K., L a f r a t e , G . J . , and Barker, J.R. i n b ~ c r o s t r u c t u r e s Science and ~ e c h n o l o g y / V L ~ ~ , Vol. 3 , Ed. by Einspruch, N. (Academic, New York, i n p r e s s )
.
5. Barker, J.R., F e r r y , D.K., and Grubin, H.L., IEEE E l . Dev. L e t t e r s EDL-1 (1980) 209.
6. Barker, J.R., and F e r r y , D.K., S o l i d S t a t e E l e c t r o n i c s , 23 (1980) 519.
7. Barker, J.R., and F e r r y , D.K., S o l i d S t a t e E l e c t r o n i c s , 23 (1980) 531.
8. F e r r y , D.K., and Barker, J.R., J. Phys. Chem. S o l i d s , 41 (1980) 1083.
9. Kadanoff, L.P. and Baym, G . , Quantum S t a t i s t i c a l Mechanics (Benjamin, 1962).
10. Imre, K . , Dzizmir, E . , Rosenbaum, M., Zweifel, P. F., J. Math. Phys. 8 (1967), 1097.
11. Barker, J.R., and Lowe, D . , t h e s e proceedings.
12. Barker, J.R., and Lowe, D . , t o be published.
