QUANTUM THEORY OF HOT ELECTRON-PHONON TRANSPORT IN INHOMOGENEOUS SEMICONDUCTORS

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QUANTUM THEORY OF HOT

ELECTRON-PHONON TRANSPORT IN INHOMOGENEOUS SEMICONDUCTORS

J. Barker, D. Lowe

To cite this version:

J. Barker, D. Lowe. QUANTUM THEORY OF HOT ELECTRON-PHONON TRANSPORT IN

INHOMOGENEOUS SEMICONDUCTORS. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-

293-C7-300. �10.1051/jphyscol:1981735�. �jpa-00221672�

QUANTUM THEORY OF HOT ELECTRON-PHONON TRANSPORT IN INHOMOGENEOUS SEMICONDUCTORS

J.R. Barker and D. Lowe

Warwick University, Coventry CV4 7AL, United Kingdom

Abstract - Functional derivative techniques are used to devise systems of coupled transport equations for the electron and'phonon Wigner distributions evolving in the presence of inhomogeneous time-dependent strong electric fields.

The formalism is designed to handle the quasi-particle dressing and undressing effects anticipated in different domains of evolution and includes a description of the dynamic field-dependent self-consistent screening of the interactions.

1. Introduction.- The recent concentration of interest on electron injection and transport in small device structures, layered media and transient dynamics of hot electrons has exposed many weaknesses in the traditional Boltzmann-Bloch. approach to transport physics [1,2]. In previous studies we have developed a quantum

mechanical framework for incorporating intra-collisional field effects [3], field- dependent dynamical screening [4], retarded spatio-temporal response [5] and non- local effects of the driving electron fields [6,7], However, in all these studies the basic models have assured an already renormalised Hamiltonian for electrons and phonons, with the exception of [3,4] which included the influence of conduction electrons in screening the electron-phonon interaction. In general, this procedure is suspect, as first pointed out for linear response theory by Martin in 1967 [8]:

for non-equilibrium problems it is not possible to define a renormalised Hamiltonian which can be used to construct transport equations, unless the space-time scales of interest are such that the renormalisation of phonon frequencies, polaronic masses, coupling constants etc. takes place on scales short-fast compared to the variation of the driving fields (and, indeed, is insensitive to the detailed struct- JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10

Tome 42, octobre 1981 page C7-293

Résumé - Des techniques de dérivation de fonctionnelles sont utilisées pour décrire les systèmes couplés d'équation de transport des distributions de Wigner d'électrons et de phonons évoluant en présence de forts champs électriques inhomogènes et dépen- dant du temps. Le formalisme est conçu pour traiter les effets d'habillement et de déshabillement des quasi-particules prévu dans les différents domaines d'évolution et inclut une description de l'écrantage self-consistant et dépendant du champ dyna- mique des interactions.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981735

C7-294 JOURNAL DE PHYSIQUE

u r e of t h e carrier/phonon d i s t r i b u t i o n s ) . In. p r i n c i p l e , one should i n s t e a d s o l v e f o r t h e d r e s s i n g of t h e electron/phonon s t a t e s simultaneously with t h e d e r i v a t i o n of t r a n s p o r t equations. The problem would be academic, were it n o t f o r t h e i n t e r e s t i n t h e i n j e c t i o n of c a r r i e r s a c r o s s i n t e r f a c e s i n t o channel r e g i o n s where t h e r e exist s t r o n g d i s c o n t i n u i t i e s i n c a r r i e r d e n s i t y , electron-phonon coupling c o n s t a n t s e t c . Thus, i n c o n s i d e r i n g s h o r t channel h i g h f i e l d t r a n s p o r t , t h e f o l l o w i n g q u e s t i o n s a r i s e : - t o what e x t e n t can t h e c a r r i e r s w i t h i n t h e channel b e d e s c r i b e d by t h e e q u i v a l e n t b u l k channel e l e c t r o n s t a t e s ; how much of t h e pre- i n j e c t i o n quasi-electron/phonon s t r u c t u r e s u r v i v e s i n j e c t i o n ; how i s t h e dynamical t r a n s i e n t c u r r e n t f l o w determined by t h e s c a t t e r i n g / d r i v i n g f o r c e s ; what i s t h e n a t u r e of t h e t r a n s i e n t coupling t o s c a t t e r i n g mechanisms and s o on. I n a t t e m p t i n g t o a d d r e s s t h e s e q u e s t i o n s we have been f o r c e d back t o a more fundamental approach t o non-equilibrium t r a n s p o r t based on double time Green f u n c t i o n techniques a s f i r s t developed f o r l i n e a r r e s p o n s e by KadalnoK and Baym

The non-linear t h e o r y d i f f e r s s u b s t a n t i a l l y from Kadwoff and Baym, p a r t i c u l a r l y w i t h r e s p e c t t o t h e i n f l u e n c e of t h e f i e l d s and t h e non-asymptotic n a t u r e of t h e s c a t t e r i n g processes.

The p r e s e n t paper s k e t c h e s an o u t l i n e of t h e k i n e t i c t h e o r y , a d e t a i l e d a n a l y s i s w i l l b e g i v e n elsewhere [lo].

2. F u n c t i o n a l t r a n s p o r t equations. -

p r i n c i p a l concern i s t o e s t a b l i s h k i n e t i c e q u a t i o n s f o r t h e d r e s s e d e l e c t r o n and phonon Wigner d i s t r i b u t i o n s , where t h e former i s d e f i n e d a s i n t h e accompanying paper [Ill, i n terms of t h e r e t a r d e d r e a l time Green f u n c t i o n

The b a s i c approach i s s i m i l a r t o t h a t of Kadanoff and Baym [2], e x c e p t t h a t ( a ) we i n c l u d e non-linear inhomogeneous time dependent a p p l i e d f i e l d s which can couple t o t h e e l e c t r o n s o r phonons; ( b ) we c o n s i d e r electron-phonon s c a t t e r i n g , and non-equilibrium s c r e e n i n g of p o l a r and non-polar phonon i n t e r a c t i o n s ; ( c ) we u s e non-equilibrium phonon d i s t r i b u t i o n s ; ( d ) t h e a n a l y s i s of t h e r e t a r d e d response i s performed d i f f e r e n t l y t o i n c l u d 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 s and quasi- p a r t i c l e s t r i p p i n g a s i n [ll]. The e q u a t i o n s of motion f o r

and t h e e q u i v a l e n t phonon d i s t r i b u t i o n s a r e obtained from a n a l y t i c c o n t i n u a t i o n s of t h e imaginary t i m e Green f u n c t i o n s f o l l o w i n g e s s e n t i a l l y t h e same p r e s c r i p t i o n a s i n [11,9].

The i n i t i a l s t a t e of t h e system i s supposed t o b e a thermal e q u i l i b r i u m s t a t e w i t h no f i e l d s p r e s e n t . The d r i v i n g f i e l d s we i n i t i a t e d a t time T=O and g i v e s r i s e t o a d d i t i o n a l terms i n t h e Hamiltonian

where &,

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 , q i s t h e normal phonon displacement

o p e r a t o r , and J i s a c l a s s i c a l f i e l d coupled t o t h e phonons ( t h i s f i e l d i s s e t t o

z e r o a t t h e end of t h e c a l c u l a t i o n s

u l t r a sound d r i v i n g f o r c e s a r e n e g l e c t e d ) .

The a p p l i e d e l e c t r i c p o t e n t i a l

( r , t ) may a l s o b e coupled t o t h e p o l a r phonon modes b u t we s h a l l l e a v e d e t a i l s of t h i s case t o a subsequent paper.

The imaginary time Green f u n c t i o n s ( l a b e l l e d by S ) a r e d e f i n e d b y

t o

S =

exp { - i p d2 [PIa(2)&+(2)+(2) - ~ ( 2 ) ~ ( 2 ) ] ] to-ia

T

t h e time o r d e r i n g o p e r a t o r , and t h e averages < > a r e o v e r t h e i n i t i a l thermal e q u i l i b r i u m d e n s i t y matrix. ( 1 , l f )

( r l t l , r i t l t ) .

The Heisenberg e q u a t i o n s of motion f o r

e t c . a r e coupled via. t h e two p a r t i c l e Green f u n c t i o n s

e t c . t o t h e complete many-body h i e r a r c h y . Decoupling can f o r m a l l y b e achieved by u s i n g f u n c t i o n a l d e r i v a t i v e s s i n c e we can show

The e l e c t r o n e q u a t i o n of motion i s t h u s e x a c t l y (imaginary t i m e s )

where cp i s t h e b a r e coulomb p o t e n t i a l ,

i s t h e b a r e electron-phonon coupling, and 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 i n c l u d e s t h e induced mean f i e l d s : -

To c o n s i d e r s c r e e n i n g we a c t u a l l y need t h e non-equilibrium screened i n t e r -

a c t i o n s . These a r e d e f i n e d f o r t h e coulomb s c a t t e r i n g and electron-phonon

s c a t t e r i n g by t h e non l o c a l forms:-

C7-296 JOURNAL DE PHYSIQUE

From ( 7 ) - (10) and (11) we s e e t h a t G i s coupled t o t h e phonon Green f u n c t i o n s . The phonon e q u a t i o n of motion

s i m i l a r l y ( b u t i n momentum coordinates):-

Both t h e e l e c t r o n and phonon e q u a t i o n may b e s i m p l i f i e d by i n t r o d u c i n g t h e e q u i v a l e n t s e l f e n e r g i e s

and n

Here 2 i s t h e b a r e phonon frequency. Now T i s d e f i n e d by t h e f u n c t i o n a l e q u a t i o n

Z

Jd2 63 V(1-2)C(lr3) c1 ( 3 , 1 1 ) + Coulomk term (18)

6 J ( 2 )

where we i n c l u d e only t h e electron-phonon c o n t r i b u t i o n . Since

we can i t e r a t e equ (16) t o o b t a i n s e l f - c o n s i s t e n t dynamical approximation t o

Thus we s t o p a t t h e g e n e r a l i s e d Born approximation [12], t o g e t

Using t h e f u n c t i o n a l i d e n t i t i e s ( 7 ) - ( l o ) , (13)-(14) we f i n d Y i n terms of s e l f c o n s i s t e n t l y screened p o t e n t i a l s

S i m i l a r l y t h e phonon self-energy i s

3. Screening:- To proceed we must e v a l u a t e t h e screened p o t e n t i a l s . For c o n s i s t e n c y we u s e t h e same l e v e l of i t e r a t i o n on t h e d e f i n i n g e q u a t i o n s f 1 2 ) ( 1 3 )

(14) a s f o r t h e s e l f e n e r g i e s . Thus from (13)

This e x p r e s s i o n may b e Wigner transformed under t h e l o c a l homogeneity approximation [ 3 , l l ] ( w h i c h assumes t h e mean f i e l d s a r e slowly varying on t h e s c a l e o f t h e c o l l i s - i o n d u r a t i o n s and c o l l i s i o n w i d t h s ) t o o b t a i n :

where L i s t h e Wigner transform of

(34) ~ ( 4 3 + ) . A s i m i l a r procedure f o l l o w s f o r

s o t h a t f i n a l l y we o b t a i n f o r t h e sum of t h e screened p o t e n t i a l s : -

We observe t h a t t h e q u a n t i t y (1-0 L) i s e s s e n t i a l l y t h e non-local imaginary time non-equilibrium d i e l e c t r i c f u n c t i o n e (k,w,

.

It remains t o e v a l u a t e t h e renormalised phonon f r e q u e n c i e s which a r i s e a s a consequence of b o t h t h e electron-phonon i n t e r a c t i o n and t h e s c r e e n i n g o f t h a t i n t e r a c t i o n . The renormalised phonon f r e q u e n c i e s rn a r e determined i n t h e q u a s i - p a r t i c l e approximation [12] by i d e n t i f y i n g t h e p o l e s i n t h e phonon Green f u n c t i o n . A s t a n d a r d argument g i v e s t h e r e n o r m a l i s a t i o n e q u a t i o n

lbut w i t h i n our approximation scheme

i s

s o t h a t a f t e r some a l g e b r a we r e c o v e r

2 2

w

/(l-GcL) where wo i s g i v e n by:-

wo2

a 2 -

n w ( w o )

lwhere

i s t h e b a r e electron-phonon c o n t r i b u t i o n t o

. Discussion of phonon lsoftening e f f e c t s e t c . due t o h i g h f i e l d s and t r a n s i e n t i n j e c t i o n i s d e f e r r e d t o a /subsequent p u b l i c a t i o n (10).

14. Transport equations:- Having determined a p p r o p r i a t e approximations f o r X,

@SC KJ

we n e x t follow t h e g e n e r a l procedure d i s c u s s e d i n rll]. The Equations (16) and

C7-298 JOURNAL DE PHYSIQUE

(17) a r e a n a l y t i c a l l y continued i n t o t h e r e a l time domain, which g e n e r a t e s two

R A

coupled e q u a t i o n s f o r G , G t h e r e t a r d e d and advanced Green f u n c t i o n s , and two

A R

coupled e q u a t i o n s f o r

and D . The e q u a t i o n f o r G , f o r example, i s

Here we have used t h e l o c a l homogeneity approximation on t h e L.H.S. ( s e e [ 3 1 ) and taken t h e s p a t i a l l y l o c a l l i m i t on t h e

( t h e r e l a x a t i o n of t h i s approxim- a t i o n i s d i s c u s s e d i n [ l ] ) : t h e time dependence i s exact.

s i m i l a r e q u a t i o n h o l d s f o r t h e phonons. These e q u a t i o n s may b e converted i n t o e q u a t i o n s f o r t h e

R .

e l e c t r o n and phonon d i s t r i b u t i o n s alone. We f i r s t n o t e t h a t

1s r e l a t e d t o

via. t h e s p e c t r a l f u n c t i o n A(p,w;R,T) [9,11] o r i n terms of t h e Wigner f u n c t i o n

S i m i l a r l y t h e phonon d i s t r i b u t i o n n(pRT)

c o n s t r u c t e d from t h e phonon s p e c t r a l f u n c t i o n B(pwRT)

D(w + i o + ) -D(w-io+)

Using t h e s e d e f i n i t i o n s and i n t e g r a t i n g over w we a r r i v e a t two e q u a t i o n s f o r

and n, and two e q u a t i o n s f o r

and

The e q u a t i o n f o r

i s

which h a s t h e s o l u t i o n [ll] :- A

- e(p)]

where p -

p - _J I' d-reE [ R ( T ) ,

I n s e r t i n g t h e e x p r e s s i o n f o r

and a s i m i l a r expres- s i o n f o r

i n t o t h e s e l f - e n e r g y terms we o b t a i n t h e r e q u i r e d t r a n s p o r t e q u a t i o n s . o '.?he e q u a t i o n f o r f i s s i m i l a r t o t h e quantum k i n e t i c e q u a t i o n s d e r i v e d i n [3,5]

e x c e p t

(a) t h e space dependence i s i n c l u d e d

(b) t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n and e l e c t r o n phonon i n t e r a c t i o n terms i n v o l v e t h e s c r e e n e d p o t e n t i a l s .

The k i n e t i c e q u a t i o n s a r e rbT + M R - ^b$ bp] f (pRT)

-

b t c o l l i s i o n s CbT + v b 1 n (p,R, T)

- On

P R -1 b t c o l l i s i o n s

where v i s t h e e l e c t r o n group v e l o c i t y ( a c t u a l l y a q u a s i - p a r t i c l e group v e l o c i t y : s e e [ I l l ) , V i s t h e phonon q u a s i - p a r t i c l e group v e l o c i t y . We s h a l l d i s c u s s t h e

P

g e n e r a l s t r u c t u r e s of t h e c o l l i s i o n i n t e g r a l s i n a subsequent paper; h e r e it s u f f i c e s t o g i v e t h e s i m p l i f i e d forms ( l o c a l space dependent s t r u c t u r e s and no s e l f energy e f f e c t s )

This term d e s c r i b e s Coulomb s c a t t e r i n g ( i f we l e t

+ 6 (m - ^{e ( k )} 1, and

t h i s term i s t h e normal screened Born approximation c o l l i s i o n i n t e g r a l ) . The momentum c o n s e r v a t i o n term combined w i t h t h e r e t a r d e d forms f o r t h e s p e c t r a l f u n c t i o n s h a s t h e e f f e c t of inducing t h e momentum and t i m e r e t a r d a t i o n i n t h e t e r m s i n

( f i e l d dependent), and a l s o g i v e s r i s e t o t h e g e n e r a l form f o r 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 . It should b e note2 t h a t t h e w , C2, i n t e g r a l s induce a d i r e c t f i e l d dependence i n t o t h e screened Coulomb p o t e n t i a l via. t h e g e n e r a l i s e d d i e l e c t r i c f u n c t i o n , t h e e l e c t r o n i c component of which is:-

This Lindhard l i k e form was d e r i v e d by a d i f f e r e n t approach i n [4]. The p r e s e n t t h e o r y i s more g e n e r a l : i t i n c l u d e s t h e phonon c o n t r i b u t i o n s t o e, and indeed can handle t h e l a t t i c e p o l a r i s a t i o n , and i s designed t o handle f u l l q u a s i - p a r t i c l e e f f e c t s . E x p r e s s i o n

a new r e s u l t . S i m i l a r forms e x i s t f o r bf/bt/e-ph b u t s i n c e t h e y a r e analogous t o s t r u c t u r e s d i s c u s s e d i n [3 1 we omit. them h e r e , e x c e p t t o n o t e t h a t t h e e l e c t r o n phonon coupling i s screened i n t h e form:-

and t h e phonon e n e r g i e s a r e renormalised ( i n a f i e l d dependent form). The phonon

s c a t t e r i n g r a t e ( b n / b t ) c o l l i s i o n s i s a l s o of t h e above s t r u c t u r e ; no e s p e c i a l l y

new f e a t u r e s emerge s o i t s d e t a i l e d form i s omitted. We n o t e t h a t a l l t h e forms

reduce t o electron/phonon Boltzmann e q u a t i o n s i f we t a k e t h e asymptotic l i m i t

T+m, n e g l e c t 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 , and t h e s p a t i a l n o n - l o c a l i t y of t h e

s e l f e n e r g i e s and Green f u n c t i o n s . F i n a l l y we n o t e t h a t i f t h e phonon d r i v i n g

f i e l d J i s n o t s e t t o zero, it appears i n a renormalised form on t h e LHS of

JOmNAL DE PHYSIQUE

e q u a t i o n

5. Conclusion:- I n t h i s paper we have very b r i e f l y o u t l i n e d a r i g o r o u s approach t o inhomogeneous time dependent coupled electron-phonon t r a n s p o r t which i s intended t o b e a b a s e f o r model c a l c u l a t i o n s of e l e c t r o n i n j e c t i o n

i n t e r f a c e s , d l e l e c - t r i c response i n h e t e r o s t r u c t u r e s and sub micron d e v i c e physics. Space does n o t allow u s t o e l a b o r a t e on 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 s , although some d i s c u s s i o n i s given i n

We n o t e f i n a l l y t h a t t h i s approach does n o t presuppose a screened o r renormalised Hamiltonian , and a s such g i v e s a uniform, non-phenomenological model f o r quantum t r a n s p o r t .

6. References :-

1. Barker,

and F e r r y

Sol-State E l e c t r o n i c s 23 (1980) 519.

2. Barker,

and F e r r y ,

Sol-State E l e c t r o n i c s 23 (1980) 531.

3. Barker,

i n Non-linear t r a n s p o r t i n Semiconductors, ed F e r r y

Barker, J., and Jacoloni, C. (Plenum, New York, 1980) 127.

4.

Barker,

Sol-State Communications 32 (1979) 1013.

5. Barker,

and Ferry,

Phys Rev Letters 42 (1979) 1776.

6. Barker,

J, Luminescence, 1981, i n t h e p r e s s . 7. Barker,

Sol-State E l e c t r o n i c s 21 (1978) 267.

8. Martin, P.C., Phys. Rev. 161 (1967) 143.

9. Kadanoff,

and Baym, G., Quantum S t a t i s t i c a l Mechanics (Benjamin, 1962).

10. Barker,

and Lowe,

t o b e published.

11. Barker,

t h e s e proceedings.

12. Barker,

Handbook of Semiconductors vol.1, ed Paul W. (North Holland, 1981)

13.