TRANSPORT IN SUBMICRON DEVICES

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TRANSPORT IN SUBMICRON DEVICES

D. Ferry

To cite this version:

D. Ferry. TRANSPORT IN SUBMICRON DEVICES. Journal de Physique Colloques, 1981, 42 (C7),

pp.C7-253-C7-261. �10.1051/jphyscol:1981730�. �jpa-00221667�

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JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, ootobve 1981 page C7-253

Résumé - Alors que la taille des composants semiconducteurs devient plus petite, on peut s'attendre à ce que les échelles de temps et de distance dans ces composants soient suffisamment faibles pour que l'approche semiclassique de la théorie du transport incorporée dans l'Equation de Transport de Boltzmann (BTE) ne soit plus valable. Les réponses transitoires spatiale et temporelle durent en fait pendant une fraction appréciable du temps de transit des porteurs dans le composant. Il faut donc développer des équations cinétiques de transport

correctement retardées incluant les effets de mémoire et de durée non nulle des collisions. Dans cet article, celles-ci sont obtenues à partir d'une équation fondamentale du transport quantique. Nous en déduisons les équations cinétiques de variables dynamiques comme le moment et l'énergie.

TRANSPORT IN SUBMICRON DEVICES x

D.K. Ferry

Colorado State University, Fort Collins, Colorado 80523, U.S.A.

Abstract - As semiconductor devices become smaller, it is expected that the relevant temporal and spatial time scales in these devices become sufficiently small that the semi-classical approach to transport theory, as embodied in the Boltzmann transport equation (BTE), is no longer valid. The temporal and spatial transient response in fact lasts for a sizable fraction of the transit time of the carriers through the device. Thus, kinetic transport equations must be developed that are properly retarded to include memory functional and non-zero collision duration effects. In this paper, properly retarded transport equations are derived from a basic quantum transport equation and kinetic equations for dynamic observables such as momentum and energy are developed.

1. Introduction.- Essentially all theoretical investigations of hot electron transport in semiconductors are based on a one-electron transport equation, usually the Boltzmann transport equation (BTE) [1]. Indeed, the overriding theoretical concern in such high-field transport is primarily the solution of the transport equation to ascertain the form which the distribution function takes in the presence of the electric field. However, for transport purposes, this is not an end product, since integrals must be carried out over the distribution function in order to evaluate the transport coefficients. In applications to semiconductor devices, however, the full solution of the BTE is usually too complicated to determine at each spatial point within the device, and transport equations for relevant observables such as energy and momentum are preferred [2,3]. Such transport equations are obtained by taking moments of the BTE and these directly relate to the normal hydrodynamic semiconductor equations. In general, the complicated nature of the BTE precludes solv- ing it analytically, and the existence of the various moment equations is based upon

KT h i s work supported in part by the U.S. Office of Naval Research

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981730

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JOURNAL DE PHYSIQUE

a number of assumptions, t h e most common of which is t h a t t h e d i s t r i b u t i o n f u n c t i o n can be r e p r e s e n t e d a s a d i s p l a c e d Maxwellian [4].

I n small semiconductor d e v i c e s , however, t h e time s c a l e s a r e such t h a t t h e u s e of t h e BTE must b e questioned [5]. T r a d i t i o n a l s e m i - c l a s s i c a l approaches, such a s t h a t of t h e BTE, assume t h a t t h e response of t h e c a r r i e r s t o any a p p l i e d f o r c e o c c u r s simultaneously w i t h t h e a p p l i e d f o r c e , even though t h e system may undergo subsequent r e l a x a t i o n t o a nonequilibrium s t e a d y - s t a t e . On t h e short-time s c a l e though, a t r u l y c a u s a l t h e o r y i n t r o d u c e s memory e f f e c t s which l e a d t o c o n v o l u t i o n i n t e g r a l s i n t h e t r a n s p o r t c o e f f i c i e n t s 16-10], s o t h a t t h e r e s u l t a n t k i n e t i c equa- t i o n i s n o t of t h e Markovian type. For t h e s t e a d y - s t a t e , t h i s r e s u l t s i n a c o l l i - s i o n o p e r a t o r t h a t depends upon t h e frequency of t h e d r i v i n g f i e l d . The problem i s complicated a s w e l l by t h e f a c t t h a t t h e time s c a l e s of i n t e r e s t a r e u s u a l l y s u f f i - c i e n t l y s h o r t t h a t t h e c o l l i s i o n d u r a t i o n T i s no l o n g e r n e g l i g i b l e i n comparison, t h u s l e a d i n g t o f u r t h e r frequency dependences of t h e c o l l i s i o n o p e r a t o r s . I n t h e BTE, t h e c o l l i s i o n terms assume p o i n t c o l l i s i o n s of z e r o d u r a t i o n , which i s f i n e when t h e mean t i m e between c o l l i s i o n s i s l a r g e , but a t high f i e l d s i n s m a l l semiconductor d e v i c e s , t h i s i s no l o n g e r t h e c a s e , and c o r r e c t i o n terms must be gener- a t e d t o account f o r t h e s e "memory" e f f e c t s .

P r e v i o u s l y , we have d i s c u s s e d t h e development of b a l a n c e e q u a t i o n s from a modi- f i e d BTE [ l l ] , based upon a f u l l quantum k i n e t i c equation. D i r e c t and g e n e r a l methods f o r t h e d e r i v a t i o n of such quantum k i n e t i c e q u a t i o n s , f o r t h e one-electron d e n s i t y m a t r i x f o r e l e c t r o n s i n phonon and o t h e r a r b i t r a r y f i e l d s of f o r c e , have e x i s t e d f o r some time 1121. Such t r e a t m e n t s a v o i d t h e random phase approximation and y i e l d k i n e t i c e q u a t i o n s which indeed a r e non-Markovian. I n a d d i t i o n , e f f e c t s due t o i n t e r f e r e n c e between t h e d r i v i n g f i e l d s and t h e s c a t t e r e r s a r e o b t a i n e d , l e a d i n g t o c o l l i s i o n a l s h i e l d i n g of t h e d r i v i n g f i e l d as w e l l a s t o an 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 [12-141, which i s a d i s t u r b a n c e of t h e s c a t t e r i n g by t h e d r i v i n g f i e l d . Using such a quantum k i n e t i c e q u a t i o n f o r t h e d i s t r i b u t i o n f u n c t i o n , t h e moment e q u a t i o n s c a n be o b t a i n e d i n a s t r a i g h t f o r w a r d manner and a r e modified from t h e i r more normal BTE-derived c o u n t e r p a r t s . These e q u a t i o n s show memory e f f e c t s a s w e l l a s a d d i t i o n a l r e t a r d a t i o n due t o t h e non-zero c o l l i s i o n d u r a t i o n . However, t h e b a s i s of t h e s e e q u a t i o n s is s t i l l s u b j e c t t o r a t h e r s t r i n g e n t approximations.

Indeed, a r a t h e r unique form must be assumed f o r t h e d i s t r i b u t i o n f u n c t i o n [ I l l ; i t is s p e c i f i e d a s a r e t a r d e d , d i s p l a c e d Maxwellian of r a t h e r s p e c i a l form. It remains u n c l e a r t o what e x t e n t t h e form assumed f o r t h e d i s t r i b u t i o n f u n c t i o n a f f e c t s t h e r e s u l t i n g moment e q u a t i o n s and whether t h e a c t u a l form assumed i s a proper a n s a t z .

The above concerns o v e r t h e d e t a i l e d form of t h e moment e q u a t i o n s c a n b e removed by d e r i v i n g t h e s e e q u a t i o n s d i r e c t l y from t h e quantum t r a n s p o r t e q u a t i o n s . The e x a c t s o l u t i o n s of t h e L i o u v i l l e e q u a t i o n s d e s c r i b e t h e time e v o l u t i o n of a s t a t i s t i c a l ensemble a t any time i n t e r v a l a f t e r a p p l y i n g an e x t e r n a l f i e l d . I f t h e r a t e of i n t e r - c a r r i e r s c a t t e r i n g i s h i g h , t h e n a f t e r a s h o r t time i n t e r v a l T

1'

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s m a l l e r t h a n any a p p r o p r i a t e time-scale of i n t e r e s t , t h e e v o l u t i o n of t h e nonequi- l i b r i u m d e n s i t y m a t r i x must b e independent of t h e i n i t i a l d i s t r i b u t i o n , and t h e r e should be a r e d u c t i o n i n t h e number of parameters n e c e s s a r y t o d e s c r i b e t h e non- e q u i l i b r i u m response of t h e system [15]. It i s t h e r e f o r e p o s s i b l e t o assume a non- e q u i l i b r i m s t a t i s t i c a l o p e r a t o r which i s smoothed i n i t s m i c r o - f l u c t u a t i o n s and from t h e very beginning d e s c r i b e t h e slow e v o l u t i o n of t h e system f o r time i n t e r v a l s which a r e l a r g e r t h a n rl 116,171. T h i s approach h a s p r e v i o u s l y been a p p l i e d t o h o t e l e c t r o n problems by Kalashnikov [18,19]. By u t i l i z i n g such a n approach, both t h e r e l e v a n t moment e q u a t i o n s and t h e form of t h e d i s t r i b u t i o n f u n c t i o n i t s e l f a r e ob- t a i n e d d i r e c t l y p r i o r t o t h e e x t e n s i o n t o t h e s e m i - c l a s s i c a l t r a n s p o r t p r o p e r t i e s . I n t h i s paper, we review t h i s approach. The nonequilibrium d e n s i t y m a t r i x and t h e moment e q u a t i o n s a r e developed. It i s shown t h a t t h e e q u a t i o n s o b t a i n e d i n [ l l ] a r e c o r r e c t under c e r t a i n c o n d i t i o n s , f o r which t h e assumed form of t h e d i s t r i b u t i o n f u n c t i o n i s a l s o c o r r e c t . I n t h i s t r e a t m e n t , a homogeneous f i e l d i s assumed, although t h e t r e a t m e n t i s r e a d i l y e x t e n d a b l e t o t a k e i n t o account t h e s p a t i a l in- homogeneities of t h e system and t h e d e r i v a t i o n of t h e s c a t t e r e r s from t h e r m a l equi- l i b r i u m [19], such a s i n i n t e n s e picosecond l a s e r e x c i t a t i o n and l i g h t s c a t t e r i n g

[201

2. The Nonequilibrium D i s t r i b u t i o n . - It i s assumed, based upon t h e work of Bugoluibov [15], t h a t f o r time t > rl, c o r r e l a t i o n s with l i f e t i m e s l e s s t h a n

T1 Can be ignored and t h e s t a t e of t h e system may b e d e s c r i b e d by a reduced s e t of macro- s c o p i c o b s e r v a b l e s Q,(t) which a r e t h e average v a l u e s , over t h e nonequilibrium ensemble, of a s e t of dynamical v a r i a b l e s P These dynamical v a r i a b l e s , and t h e i r

m '

conjugate f o r c e s f ( t ) , may t h e n b e used t o d e f i n e a nonequilibrium s t a t i s t i c a l m ensemble. The approach i s based upon ~ o i n c a r g ' s theorem on i n t e g r a l i n v a r i a n t s g e n e r a l i z e d t o quantum systems [19,21]. Thus, t h e system d e n s i t y m a t r i x should b e c o n s t r u c t a b l e from t h e p r i n c i p l e i n v a r i a n t s of t h e system. A d d i t i o n a l l y , a n aux- i l i a r y ( l o c a l ) e q u i l i b r i u m d e n s i t y m a t r i x p i s d e f i n e d a s a q u a s i - e q u i l i b r i u m q u a n t i t y . This l a t t e r f u n c t i o n s e r v e s a s an i d e a l i z e d i n i t i a l c o n d i t i o n f o r t h e system a f t e r t h e randomization, c h a r a c t e r i z e d by rl, h a s o c c u r r e d , and from which t h e system evolves under dynamical laws governed by i t s Hamiltonian. This means t h a t a t a l l times t > t h e nonequilibrium d e n s i t y m a t r i x , which must s a t i s f y t h e Liouville-Von Neumann e q u a t i o n , i s a f u n c t i o n a l of p

0'

Thus, on a time s c a l e g r e a t e r t h a n rl, t h e system may b e c h a r a c t e r i z e d by t h e average v a l u e s of t h e s e t of o p e r a t o r s Pm, d e f i n e d a s

where P i s t h e o p e r a t o r f o r t h e t o t a l momentum of t h e c a r r i e r s and N i s t h e c a r r i e r number o p e r a t o r . The term H c o n t a i n s t h e c a r r i e r Hamiltonian, t h e c o h e r e n t p a r t of any i n t e r a c t i o n w i t h t h e environment, and t h e c a r r i e r - c a r r i e r i n t e r a c t i o n . However,

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C7-256 JOURNAL DE PHYSIQUE

it is assumed that carrier eigenfunctions can be chosen such that this term is diagonal. The set P in (1) is a minimal basis set, in that it is assumed here that

m

the lattice remains in equilibrium. The P satisfy equations of motion m

where the dot over P indicates the time derivative. Thus, for m

where H contains the incoherent interaction with the lattice (and environment), CP

where P = H P (the carat over H in (2) and (4) indicates a commutator-genera- m(L) cp m

ting super-operator). The next step is to introduce a set of time-dependent parameters f (t) (which are c-numbers) constructed to be thermodynamic conjugates of the

m

%

^as

where B (t) multiplies each term in the brackets and is the inverse carrier temperature (= l/k T ), vd(t) is the drift velocity, and ~ ( t ) is the chemical potential.

B c

The asymptotic, time-smoothed quantities B may be defined from the Pm,fm(t) m

as

50

B,,,(t) = s

1

dt'e-st'fm(t-t')~m(t')

,

0

where

+iit

'

/h P (t') = e

m Pm(o)

.

The quantities B (t) satisfy the Liouville equation in the limit s -t 0

+

and are the m

quasi-invariant portions of the products P f (t) with respect to evolution under m m

the Hamiltonian H

.

Thus, they are integrals of motion in this limit and any statistical operator constructed from these quantities will also be an integral of Liouville's equation. The operation of taking the quasi-invariant part is also used in scattering theory to assure that retarded, causal solutions to Schrodinger's equation are used [ 2 2 ] . In order to assure that f (t) and Qm = <P >t (the t super-

m m

script denotes a time-varying average) are thermodynamically oonjugate, we impose the condition

(6)

where <. >t denotes an average over the nonequilibrium density matrix p(t) and <- >t indicates an average over the quasi-equilibrium p (t).

We can now write the nonequilibrium density operator in the form

where

The choice of (9) for p(t) assures that this quantity reduces to the generalized Gibbs distribution in thermal equilibrium and results in a positive entropy produc- tion in the system. Indeed, by defining the entropy operator as

with

Q0 = In Tr exp (-

1

^{Pmf, (t}⁾1

,

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m

we may combine (6) and (9) so that

m

p(t> = expf- ~(t,o)

- ,/

d t ~ e - ~ " ~ ( t - t ~ , t ~ ) l

,

0

after integration by parts, where

Clearly then p (t) = exp[- S(t,o)] is the quasi-equilibrium operator. Here, $m can be found from ( 4 ) , and

where the 6 indicates a functional derivative.

It is obvious that terms of first and second order in H appear in (14).

CP

Therefore, to obtain the balance equations up to second-order in the electron- phonon interaction (the first non-vanishing order), we use the following iterative expansion of the density matrix:

m -TS(~,O)

~s

(t 9 0 )

p(t)

'

^(I+

(

^dtle-st'

,/

^{dr e} i(t-tl,t') }p0(t) , (16)

0 0

where P (t) is the quasi-equilibrium density matrix corresponding to the isolated carrier distribution at given values of the average energy and momentum.

3. The Balance Equations.- With the non-equilibrium density matrix of (16), we can now find explicit transport equations for the generalized operators. We suppose, as in (4), that the equations of motion of the operators P have the form

m

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C7-258 JOURNAL DE PHYSIQUE

where t h e a and

G

a r e defined i n ( 4 ) . Then, using (16) t o c o n s t r u c t t h e

w m(L)

average, we f i n d t h a t t

Gm>

=

1

am%>

+

( d t ' G m ( L ) ; ( ~ ) ( t ' ) > t f L ( t - t ' ) ⁺

n ₀

where t h e l a s t te.rm has been i n t e g r a t e d by p a r t s and t h e l i m i t s - t o has been

+

taken throughout. The upper l i m i t on t h e second term on t h e right-hand s i d e follows from t h e assumption t h a t f L ( t ) = 0 f o r t < 0. The quantum c o r r e l a t i o n functions a r e defined by

0

and B ( t ' ) i s given by (7). The energy and momentum balance equations a r e then given by

where 3' = Be

-

B, and

Over a broad i n t e r v a l of t h e e l e c t r i c f i e l d s t r e n g t h , t h e k i n e t i c energy a s s o c i a t e d with t h e e l e c t r o n d r i f t i s much smaller than t h e mean energy of a c a r r i e r ; i . e . ,

mvd 2 < < E [I]. Therefore, i n t h e expression f o r t h e quasi-equilibrium d e n s i t y matrix

p o ( t ) we may n e g l e c t t h e d r i f t v e l o c i t y v Then t h e c o r r e l a t i o n f u n c t i o n s which

contain t h e products P and

k 8'

c ( L ) ~ C (L) c(L) c(L) vanish a s i n t h e c a s e of thermal equilibrium [18j. Then, t h e c o r r e l a t i o n f u n c t i o n s s i m p l i f y and (20-21) become

- = dcE>

eFv ( t ) [l- <PC; Pc(o)> ke(t)/m]

-

d t d

t

- j

B:(t-t')<fic(L); k ( L ) ( t v ) >

.

0

and dv d

i

^t

dt

= eF

-

d t ' f? ( t - t ' ) ~ ~ ( t - t ' ) < P ~ ( ~ ) ; P c ( , ) ( t l ) > ⁹

(8)

where we have used <P > = mNvd. The remaining correlation functions are the collision-relaxation memory functions [6], and the forms of (221, (23) are that obtained in the drifted Maxwellian approximation [Ill, although the details of the scattering function are somewhat different. Indeed, ~ ( t ) is almost identical to that taken in

[ll] for this case. The quantity multiplying the force term in (22) has an additional term not found from the normal semi-classical equations, and represents an additional delay on the increase of energy. The second factor in the square brackets usually produces only a negligible change in the velocity response. How- ever, it is significant during the transient phase for the energy response and appears even in classical Monte Carlo calculations [23]. That this memory effect

3 -1 should appear in the energy equation follows from the fact that E

-

_{2 B e}^{k T}^{= - 6}₂_{e '}

The temperature is in effect a measure of the fluctuations and cannot be changed until the correlation of these fluctuations is broken up by the field.

The relaxation functions have a gain-loss structure as is well known. For example, the energy-loss rate can be written as [18]

where [241

0

from time-dependent perturbation theory, and

The limit of t >> T gives g(x,t) as the collision-duration broadened joint-spectral- density function [14]. But in high electric fields, where the duration of g(x,t) can be long, it is possible to have field acceleration during the collision due to the time variation of energy in (24) [12-14,241. This intra-collisional field effect can be introduced into (22), (23) by an additional retardation due to the effective collision duration T- as [ll]

U

where

r

under the integral is evaluated with the long-time joint-spectral density function. Here T can be evaluated directly from the principle duration of (25).

4. Discussion.- It is clear from the present approach that the normal semi- classical belance equations can be obtained from a more accurate quantum transport approach. However, several points must be made. First, as is evident in (20) and (21), the individual balance equations are a fully coupled pair unless (mv 2

I<€>)

d

<< 1. It is clear that a difference arises here from the semi-classical approach

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C7-260 JOURNAL DE PHYSIQUE

a s s e v e r a l t r e a t m e n t s of semiconductor t r a n s p o r t have found t h i s term t o b e compar- a b l e t o I i n c e r t a i n o p e r a t i n g r e g i o n s of semiconductor d e v i c e s [25,26]. The e x t r a terms l e a d i n g t o t h e d i r e c t c o u p l i n g between t h e e q u a t i o n s a p p e a r t o b e r e s i d u a l terms from t h e f i e l d - s c a t t e r e r i n t e r f e r e n c e e f f e c t s 112,141. I n t h i s r o l e , t h e y l e a d t o a memory e f f e c t which m o d i f i e s t h e e f f e c t i v e mass e q u a t i o n a t t = 0 -I-

.

Indeed, t h e e x t r a c o r r e l a t i o n f u n c t i o n i n t h e d r i v i n g t e r m of (22) r e p r e s e n t s t h e e f f e c t s on t h e motion i n t h e s i n g l e - p a r t i c l e s p a c e due t o i n i t i a l c o r r e l a t i o n s w i t h o t h e r p a r t i c l e s [27]. While t h i s term i s common i n quantum t r a n s p o r t [27,28], i t h a s g e n e r a l l y been r e g a r d e d a s s m a l l and n e g l i g i b l e i n c l a s s i c a l t r a n s p o r t

[29,30]. Such i s n o t t h e c a s e i n t h e f a s t , t r a n s i e n t r e s p o n s e regime. The i n d i r e c t c o u p l i n g a p p e a r i n g i n t h e r e l a x a t i o n a l c o r r e l a t i o n f u n c t i o n s d i r e c t l y p o i n t s o u t , t h a t a l t h o u g h t h e two c o r r e l a t i o n f u n c t i o n s d i f f e r s l i g h t l y , one cannot r e a d i l y i d e n t i f y a s o l e l y momentum r e l a x a t i o n p r o c e s s and a s o l e l y energy r e l a x a t i o n p r o c e s s .

The second c r i t i c a l p o i n t i s t h a t usage of b a l a n c e e q u a t i o n s i n which t h e coxrvolution i n t e g r a l s a r e o m i t t e d i s r e s t r i c t e d t o c a s e s of t >> T e, T ~ .Indeed, i n t h i s l i m i t t h e r e l a x a t i o n terms can b e s i m p l i f i e d . However, t h e u s e of t h e s e un- r e t a r d e d s i m p l i f i c a t i o n s t o t r e a t t r a n s i e n t response,where t 5 r e , T ~ ,i s c l e a r l y i n e r r o r .

R e f e r e n c e s

1. F e r r y D. K., i n P h y s i c s of N o n l i n e a r T r a n s p o r t i n Semiconductors, Ed. by P e r r y D. K . , Barker J . R . , and J a c o b o n i C. (Plenum, New York, 1980) p. 117.

2. Bosch R. and Thim H., IEEE Trans. E l e c t r o n Dev. ED-21 (1974) 16.

3. Grubin H. L., F e r r y D. K., I a f r a t e G. J., and Barker J . R., i n M i c r o s t r u c t u r e S c i e n c e and T e c h n o l o g y / V ~ S I , Vol. 3, Ed. by Einspruch N. (Academic, New York, i n p r e s s ) .

4. ~ r o h l i c h H., Proc. Roy. Soc. A188 (1974) 521.

5. Barker J. R. and F e r r y D. K . , S o l . - S t a t e E l e c t r o n . 23 (1980) 519.

6. Zwanzig R., J. Chem. Phys. 40 (1964) 2527.

7. Green M. S., J. Chem. Phys. 20 (1952) 1281; 22 (1954) 398.

8. Kubo R., J. Phys. Soc. Jpn. 1 2 (1957) 570.

9 . Mori H., Phys. Rev. 112 (1958) 1829.

10. Zwanzig R., Phys. Rev. 124 (1961) 983.

11. F e r r y D.K. and Barker J . R., J. Phys. Chem. S o l i d s 4 1 (1980) 1083.

12. Argyres P. N., Phys. Rev. 132 (1963) 1527.

13. Thornber K. K., Phys. Rev. B3 (1971) 1929.

14. Barker J. R., J. Phy. C: S o l . S t a t e Phys. 6 (1973) 2663.

15. Bogoliubov N. N., Problemi dinam. t e o r i i u s t a t . F i z . , Moscow 1946, a l s o i n S t u d i e s i n S t a t i s t i c a l Mechanics, Vol. 1, d e Boer J. and Uhlenbeck G. E . , Eds.

(North-Holland, Amsterdam, 1962) p. 11.

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Zubarev D. N. Dokl. Akad. Nauk. SSSR 140 (1961) 92 [Sov. Phys. Doklady 6 (1972) 7761.

McLennan J. A. Jr., Phys. of Fluids 4 (1961) 1319.

Kalaghnikov V. P., Physica 48 (1970) 93.

The nonequilibrium statistical operator is reviewed by Zubarev D. N.,

Nan-

equilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974).

See also the discussion on this approach in Ferry D. K. and Barker J. R., Sol.-State Electron. 23 (1980) 531.

Luzzi R. amd Vasconcellos A. R., J. Stat. Phys. 23 (1980) 539.

Poinear6 H. Mgthodes Nouvelles de Mgcanique CCleste, Vol. 1 (Gauthier-Villars, Paris, 1892); reprinted by (Dover, New York, 1957). See also the discussion

in Callen H. B., Thermodynamics (Wiley, New York, 1960).

Gell-Mann M. and Goldberger M. L., Phys. Rev. 91 (1953) 398.

Lugli P., Zimmermann J. and Ferry D. K., these proceedings.

Fujita S., Introduction to Nonequilibrium Quantum Statistical Mechanics (Sanders, Philadelphia, L966).

Butcher P. N., Repts. Progr. Phys. 30 (1967) 97.

Grubin H. C., Ferry K. D. and Barker J. R., in Proc. 1979 International Electron Device Mtg. (IEEE Press, New York, 1979).

Boercker D. 3. and Dufty J. W., Phys. Rev. A 23 (1981) 1952.

Barker J. R., in Physics of Nonlinear Transport in Semiconductors, Ed. by Ferry D. K., Barker J. R. and Jacoboni C. (Plenum, New York, 1980) p. 126.

Zwanzig R. W., in Lectures in Theoretical Physics, Vol. 3, Ed. by Britton W. E., Downs B. W., and Downs J. (Interscience, New York, 1961) p. 106.

Balescu R. and Prigogine I., Physica 25 (1959) 281, 302.