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Submitted on 1 Jan 1981
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A MICROSCOPIC TRANSPORT THEORY OF
ELECTRON-PHONON SYSTEMS
A. Wagh
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, supple'ment au n o 12, Tome
4 2 ,de'cembre 1981
page
c6-471A
MICROSCOPIC TRANSPORT THEORY OF ELECTRON-PHONON SYSTEMSA.S. Wagh
Physics Department, University of West Indies, Mona, Kingston, Jamaica, W.I.
p b s t r a c t - The two-time Green f u n c t i o n s a r e used t o d e r i v e k i n e t i c e q u a t i o n s of a system of e l e c t r o n s and phonons i n t e r a c t i n g w i t h F r o h l i c h hamiltonian. A s u i t -
a b l e connected diagram a n a l y s i s i s used t o d e r i v e simul- taneous e q u a t i o n s s i m i l a r t o Kadanoff-Baym e q u a t i o n s . The i n i t i a l c o r r e l a t i o n s a r e shown t o decay i n time com- p a r a b l e t o t h e macroscopic time s c a l e . The g e n e r a l t r a n s p o r t e q u a t i o n s d e r i v e d i n t h e s t e a d y s t a t e a r e a p p l i c a b l e t o v a r i o u s t r a n s p o r t p r o c e s s e s . I t i s shown t h a t t h e p r e v i o u s assumption of c o n s i d e r i n g t h e phonons t o b e i n thermal e q u i l i b r i u m i s j u s t i f i e d o n l y f o r d i - l u t e systems and phonon drag i s a p p r e c i a b l e i n d e n s e r systems.
1. I n t r o d u c t i o n . The two-time Green f u n c t i o n s have been e x t e n s i v e l y used by Kadanoff and F3aym1 t o s t u d y t h e e v o l u t i o n and t r a n s p o r t of a quantum mechanical i m p e r f e c t gas. We extend t h i s f o r m u l a t i o n t o t h e e l e c t r o n phonon system. A connec-
2
t e d diagram a n a l y s i s i s used t o d e r i v e t h e e v o l u t i o n e q u a t i o n s from t h e theorem by
3
Bloch and Domenicis
.
We o b t a i n f o u r t r a n s p o r t e q u a t i o n s two f o r e l e c t r o n s and two f o r phonons, t h e l a t t e r d e s c r i b i n g phonon drag. Using t h e s e e q u a t i o n s it i s p o s s i b l e t o show t h a t t h e phonon d r a g i s n e g l i g i b l e f o r d i l u t e systems such a s e l e c t r o n s i n semiconductors b u t can b e a p p r e c i a b l e i n metals.2. The E v o l u t i o n Equations. The F r o h l i c h hamiltonian f o r e l e c t r o n p h o n o n i n t e r a c - t i o n i s g i v e n by
Here a ( p ) - and b ( p ) - s t a n d f o r a n n i h i l a t i o n o p e r a t o r s f o r e l e c t r o n s and phonons,
4
i s t h e e x t e r n a l f i e l d and o t h e r symbols have t h e i r u s u a l meaning. The p a r t i a l Green> >
f u n c t i o n s g< f o r e l e c t r o n s and n' f o r phonons a r e d e f i n e d by
C6-472 JOURNAL DE PHYSIQUE
where t h e synbols 1 and 2 s t a n d f o r ( ~ ~ , t ~ ) and ( p , t ) . The e v o l u t i o n o f g' (1.2) -2 2
may b e s t u d i e d by d i f f e r e n t i a t i n g each of t h e s e w i t h r e s p e c t t o t h e t i m e s tl and t2. One o b t a i n s f o r example
-iT.r{pa+(2) u+(t1) t a ( g l ) , H I U(tl) }
=
I
+I
-
i ~ r ( ~ X J d p y [af ( 2 ) a ( q - E 3 , t 1 ) b ( e t )-3 P3 3' 1
This e q u a t i o n i s n o t i n c l o s e d form. One needs t o analyze i t by means o f p e r t u r b a - t i o n and connected diagram t e c h n i q u e , t h e e s s e n t i a l f e a t u r e s being s i m i l a r t o t h e one o b t a i n e d from Wick's theorem f o r an i m p e r f e c t gas. Here however, we s t a r t from Bloch and ~ o m i n i c i s 3 theorem t h a t t h e average<ABC..> f o r o p e r a t o r s A,B,C, e t c . may be decomposed i n t o averages of t h e p a i r - p r o d u c t s , v i z .
where
+
s t a n d s f o r phonons and-
f o r e l e c t r o n s . The d e s c r i p t i o n s of t h e s e l f - e n e r g y p a r t s i s g i v e n i n t h e t a b l e . One o b t a i n s f o r g2
2
t2 t 2i.a4
= ~ h ~ ( ~ ~ ) + ~ $ ( ~ ~ ) l g ~ +!
d t 3 1 d 5 ~ ~ ( l , 3 ) g : ( 3 , 2 )-
t l j d t 3 ~ d p 3 Z Z ( 1 , 3 )g1(3,2)-
d t 3 j d 3 ~ ' ( l , 3 ) g3(3,2)+
i ~gS(%.i8,,2) ~ ) (5)>
'
+
where C< = E<
+
E-. The l a s t term i n eq. ( 5 ) i s due t o i n i t i a l c o r r e l a t i o n s , which a r e due t o o u r c h o i c e o f i n i t i a l time a s t = 0 . I n a s i m i l a r manner one w r i t e s1
e q u a t i o n s f o r
a2
and f o r n2 a l l having same form of Kadanoff and Baym e g u a t i o n s . at23 . Decay of I n i t i a l C o r r e l a t i o n s . Though t h e electron-phonon i n t e r a c t i o n i s a l o n g range p o t e n t i a l , we show h e r e t h a t t h e i n i t i a l c o r r e l a t i o n s d i e o u t by t h e time s t e a d y r a t e i s reached.
I f t h e system i s i n i t i a l l y i n thermal e q u i l i b r i u m , t h e i n i t i a l Green f u n c t i o n s f o r e l e c t r o n s obeying c l a s s i c a l s t a t i s t i c s a r e
where
p
=t(p
-p ) , € and a r e p a r t i c l e energy and f u g a c i t y . T h e i n i t i a l c o r r e l a - 1 -2t i o n term involved i n eq. ( 5 ) f o r a l i n e a r l a t t i c e w i t h one atom p e r l a t t i c e i n bina- r y c o i l i s i o n approximation i s
if3
1 =
{
d(i~l)fdg3~<(1,P3,iBl)g'(-E3r iB1,2)§ ( p ) = § , s i n f p ' a = f § , p ' a
where 5, i s a c o n s t a n t o f t h e c r y s t a l and a i s a l a t t i c e c o n s t a n t , we o b t a i n
where we have used energy c o n s e r v a t i o n r e l a t i o n e ( p ) + § ( p l ) =
$(p
Q ' ) .1 1
I decays a t t h e most a s - o r -
.
Thus a s tl, t2- t h e whole term v a n i s h e s . tl t2Thus i n i t i a l c o r r e l a t i o n s a r e completely removed from t h e d e s c r i p t i o n o f t h e s t e a d y s t a t e which i s s t u d i e d only a f t e r a l o n g time.
4 . The Steady S t a t e T r a n s p o r t Equations. Dropping t h e i n i t i a l c o r r e l a t i o n s and f o l l o w i n g t h e u s u a l procedure1, t h e s t e a d y s t a t e t r a n s p o r t e q u a t i o n s c a n e a s i l y b e o b t a i n e d a s
3
where G' and S s t a n d f o r e l e c t r o n and phonon s t e a d y s t a t e Green f u n c t i o n s each a l o n g w i t h t h e s e l f energy p a r t s b e i n g f u n c t i o n s of macroscopic momentum p, p a r t i c l e energy
w,
p o s i t i o nR
and time T. The square b r a c k e t s a r e g e n e r a l i z e d Poisson brac- k e t s . These e q u a t i o n s a r e Markoffian and homogeneous. They can b e used f o r t h e s t u d y of v a r i o u s t r a n s p o r t s . Here we s h a l l use them t o s t u d y t h e r o l e o f phonon d r a g i n e l e c t r i c a l conduction.5 . Phonon Drag. The e q u a t i o n s (10) can b e l i n e a r i z e d i n t h e e l e c t r i c f i e l d . To s e e t h e e f f e c t of t h e phonon d r a g , t h e l i n e a r p a r t s o f t h e Green f u n c t i o n s G: and
P
S- can be expanded f o r a homogeneous system a s
S u b s t i t u t i o n of t h e s e i n t o t h e l i n e a r e q u a t i o n s and comparing t h e terms of t h e same o r d e r of A y i e l d s i n t h e l o w e s t o r d e r
2
+
s i m i l a r terms i n GLl+
6 ( p g ' + p * ? ~ ( u - u ' + u ' I ~ ~ ~ ( ~ ' W ' ) S : ~ ( P ' ' w ' '10' (pro) e qP
+
s i m i l a r terms i n G 1 L (12)where V i s t h e frequency of t h e f i e l d and G P SP a r e e q u i l i b r i u m Green f u n c t i o n s . eq' eq
C6-4 74 JOURNAL DE PHYSIQUE
TABLE
I.
Simplest self -energy diagrams and the corresponding expressions.
Symbol
Simplest Diagram
Expression in the second order
I 2
Z-
\z4
Ides
&,
s5
g<(pl-
e4
t,, ,?,-p, t 31
x
n((_p4
t,,_Ps 1 3 - - --\dp,\dp5&4
&5g<(p1-p4 t1,p3-pst3)
X "< ( p 4 t,,P5 t 3 ) . - . . - - - - -- 112-$
\
d
\ d ~ ~ & ~
~
~
2 p 5 g > ( p 1 - ~ 4
t I , ~ f h 13)(-P4
t,.F,
1,) References1. L. Kadanoff & G. Baym, The Quantum S t a t i s t i c a l Mechanics (Benjamin, N.Y. 1 9 6 2 ) . 2 . S . F u j i t a , Phys. Rev. A4, 1114 (1971).
