A NON-PERTURBATIVE GREEN FUNCTION TECHNIQUE FOR CALCULATIONS OF NON-LINEAR TRANSPORT PROPERTIES

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A NON-PERTURBATIVE GREEN FUNCTION TECHNIQUE FOR CALCULATIONS OF NON-LINEAR TRANSPORT PROPERTIES

A. Jauho, J. Wilkins, F. Esposito

To cite this version:

A. Jauho, J. Wilkins, F. Esposito. A NON-PERTURBATIVE GREEN FUNCTION TECHNIQUE

FOR CALCULATIONS OF NON-LINEAR TRANSPORT PROPERTIES. Journal de Physique Col-

loques, 1981, 42 (C7), pp.C7-301-C7-306. �10.1051/jphyscol:1981736�. �jpa-00221673�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobr-e 1981 page C7-301

Résumé. - Nous avons développé un formalisme pour obtenir les propriétés de transport d'un champ électrique de force arbitraire à partir de fonctions de Green. Dans le cas spécial où un simple niveau résonnant est accouplé à des électrons de conduction notre formalisme a permis le calcul analytique exact de la fonction de Green pour une particule isolée. Nos résultats sont décrits pour la matrice Tdont la partie imaginaire (dans un champ électrique nul) donne la relation de diffusion. On trouve T (e) =v2/[e-E-ir (e,F)]

où E est l'énergie du niveau résonnant et F le champ électrique.

La fonction de largeur r(e,F) est la mesure de la corrélation des électrons de conduction au niveau résonnant. Dans le cas d'un potentiel gaussien pour l'impureté on trouve analytiquement la transition d'un cas dominé par les collisions jusqu'à l'état où les particules sont libres à mesure que le champ augmente.

A NON-PERTURBATIVE GREEN FUNCTION TECHNIQUE FOR CALCULATIONS OF NON-LINEAR TRANSPORT PROPERTIES

A.P. Jauho, J.W. Wilkins* and F.P. Esposito**

NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

* Cornell University, LASSP, Ithaca, N.Y. 14853, U.S.A.

**University of Cincinnati, Physics Department, Cincinnati OH 45221, U.S.A.

Abstract. - We have developed a Green function formalism to cal- culate transport properties in an arbitrarily strong static electric field- In the special case where an isolated resonant level is coupled to the conduction electrons our formalism has permitted an exact analytical evaluation of the single particle Green function. Our results can be characterized in terms of the T-matrix whose imaginary part (in zero electric field) gives the

scattering rate. We find T(e) = V§/[e-E-ir(e,F)], where E is the energy of the resonant level and F is the electric field. The width function r(e,F) is a measure of the coupling of the conduc- tion electrons to the resonant level. For a gaussian impurity potential we demonstrate analytically the transition from colli- sion dominated to free particle behavior as the electric field is increased.

1. Introduction. Most calculations of non-linear transport properties rely on perturbation theory, i.e. one expands the transport coeffici- ents in powers of the external field. Linear response theory has achieved a high level of sophistication and there exists a number of rigorous theorems and sum rules which impose constraints that all rea- sonable theories must satisfy. However, for non-linear transport theory no such information is available and even the existence of the perturbation series in powers of the external field has not been estab- lished .

We have chosen an alternate route in our calculation. Instead of first solving for the Green function in the presence of the scattering

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981736

C7-302 JOURNAL DE PHYSIQUE

mechanisms and t h e n d o i n g p e r t u r b a t i o n t h e o r y i n t h e e x t e r n a l f i e l d , we b e g i n by ' d r e s s i n g ' t h e f r e e Green f u n c t i o n w i t h t h e e x t e r n a l f i e l d and t h e n u s e t h i s f i e l d - d e p e n d e n t Green f u n c t i o n i n s u b s e q u e n t c a l c u l a - t i o n s .

W e d e r i v e f i r s t a n e x p l i c i t e x p r e s s i o n f o r t h e f i e l d - d e p e n d e n t f r e e Green f u n c t i o n . Our r e s u l t i s v a l i d f o r a n e l e c t r o n i n a n empty band and t h e r e have indeed been s p e c u l a t i o n s 1 t h a t t h i s might b e t h e a p p r o p r i a t e model f o r d e s c r i b i n g t r a n s p o r t i n submicron d e v i c e s . We proceed t o c a r r y o u t a n o n - l i n e a r , n o n - p e r t u r b a t i v e e v a l u a t i o n of t h e Green f u n c t i o n f o r t h e r e s o n a n t l e v e l model 2 (RLM). The RLM i s a model o f a n energy-dependent s c a t t e r i n g c e n t e r and a l l o f i t s e q u i l i b r i u m and l i n e a r t r a n s p o r t p r o p e r t i e s a r e known. Accordingly i t seemed an i d e a l c a n d i d a t e f o r a n o n - l i n e a r c a l c u l a t i o n .

2 . F i e l d dependent Green f u n c t i o n . The Green f u n c t i o n i s d e f i n e d i n t h e customary way:

+ +

+

G ( k , k ' , t - t ' ) = -i < ~ { c g ( t ) c j ; , ( t ' ) ]> ; (1) i t s dynamics a r e governed by t h e Hamiltonian (F + i s a s t a t i c e l e c t r i c f i e l d )

We have chosen a system o f u n i t s i n which h = me = ] e l = 1. The r e s u l t i n g e q u a t i o n o f motion f o r G i s

where we have used a c o o r d i n a t e system where t h e f i e l d d e f i n e s a pa- r a l l e l d i r e c t i o n and t h e o t h e r two s p a t i a l dimensions a r e r e f e r r e d t o a s t h e p e r p e n d i c u l a r d i r e c t i o n s . E q . ( 3 ) c a n b e s o l v e d t o g i v e

t-t' F -+ + -f

G ( k r k l , t - t f ) =

-

i6(ki-$1) 6 ( k , . -k:, - F ( t - t ' ) ) 8 (t-t' ) e x p [ - i I ~ ~ ( r ) d r ] ,

where 0 (4)

i s t h e time-dependent k i n e t i c energy. We may u n d e r s t a n d Eq.(4) a s a n a t u r a l m o d i f i c a t i o n i n an e l e c t r i c f i e l d of t h e z e r o - f i e l d Green f u n c t i o n ,

The two changes a r e ( i ) t h e s h i f t i n t h e k,, -momentum c o n s e r v a t i o n law t o a c c o u n t f o r t h e a c c e l e r a t i o n d u e t o t h e e l e c t r i c f i e l d and (ii) t h e t i m e e v o l u t i o n o f t h e k i n e t i c e n e r g y , E q . ( 5 ) .

3 . Resonant l e v e l model (RLM1. The r e s o n a n t l e v e l model i s c h a r a c t e - r i z e d by an energy l e v e l E which i n t e r a c t s w i t h t h e c o n d u c t i o n e l e c - t r o n s w i t h a m a t r i x e l e m e n t

v(Z)

^:

where t h e c g ' s r e f e r t o t h e c o n d u c t i o n e l e c t r o n s and b i s t h e r e s o n a n t l e v e l d e s t r u c t i o n o p e r a t o r . The RLP.J1s Green f u n c t i o n s a t i s f i e s t h e e q u a t i o n

3 +

GRLM(l'k1"k , w ) = 6 (k-k') G o ( % , w )

+

G~ ( % , w ) ~ ( % ) g0(w) IV*

(if) ~"~(c,P,w)

( 8 )

3

4

I n E q . ( 8 ) go(w) i s t h e f r e e p r o p a g a t o r f o r t h e r e s o n a n t l e v e l , go(w) = ( w - ~ l - ' . The s o l u t i o n o f Eq. ( 8 ) c a n b e w r i t t e n a s

where

Consider now t h e d e f i n i t i o n o f t h e T-matrix, G = Go

+

GOT Go, which l e a d s t o t h e i d e n t i f i c a t i o n

I n t h e RLM one u s u a l l y t a k e s v(;) = Vo and a c o n s t a n t d e n s i t y o f s t a t e s and we do s o i n t h i s p a p e r . Of c o u r s e a more r e a l i s t i c d e n s i t y of s t a t e s c o u l d b e u s e d i n o u r formalism. Within t h e s e a p p r o x i m a t i o n s , Eq.(ll) h a s a s i m p l e i n t e r p r e t a t i o n i n t e r m s o f a l e v e l w i d t h

The imaginary p a r t o f t h e T-matrix ~ ( % , z , w ) i s t h e n e g a t i v e of t h e s c a t t e r i n g r a t e l / ~ ( w ) which h a s t h e expected r e s o n a n t s t r u c t u r e

4 . F i e l d dependence i n t h e RLM. We w i l l now t u r n t o t h e f i e l d depen- d e n t g e n e r a l i z a t i o n s o f Eqs. ( 8 ) and (11). I n p a r t i c u l a r we w i l l

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examine t h e f i e l d dependence of t h e l e v e l w i d t h I'. The e q u a t i o n of motion a n a l y s i s l e a d i n g t o E q . ( 8 ) c a n b e r e p e a t e d e s s e n t i a l l y unchanged l e a d i n g t o

F + + F + -t

~ ( % , % ' , w ) = G ( k , k ' , w ) +

1

^G ( k . q l l w ) ~ ( ~ 1 ) ~ * ( ~ 2 ~ g o ( ~ ) ~ ( ~ 2 ~ ~ ' I ~ ) - ( 1 4 ) + -+

91%

E q . ( 1 4 ) c a n a g a i n b e s o l v e d t o g i v e t h e f i e l d - d e p e n d e n t g e n e r a l i z a t i o n o f Eq. ( 9 ) :

F + + F + + F - t -t F - t +

~ ( g , g ' , w )

⁼ G ( k r k a 1 w ) +

1

^G ( k I q l l w ) T (ql,q2,w)G (q2,k',w), (15) + -+

9142 where

From E q . ( 1 6 ) we can i d e n t i f y t h e f i e l d - d e p e n d e n t g e n e r a l i z a t i o n o f t h e l e v e l w i d t h (from t h i s on we assume t h a t V(q) + i s r e a l ) :

From t h e form of Eq. (17) one c a n s e e t h a t

r

( w ,F) depends o n l y on even powers o f F. By p e r f o r m i n g a s e r i e s e x p a n s i o n i n powers of t h e e x t e r - n a l f i e l d we c a n r e c o v e r r e s u l t s s i m i l a r t o t h o s e which ~ a r k e r ~ ob- t a i n e d w i t h a s u p e r - o p e r a t o r formalism: t h e r i g h t - h a n d s i d e o f t h e t r a n s p o r t e q u a t i o n a q u i r e s a n a d d i t i o n a l f i e l d - d e p e n d e n t t e r m which i s p r o p o r t i o n a l t o t h e e n e r g y d e r i v a t i v e of t h e s c a t t e r i n g p o t e n t i a l .

I n s t e a d o f p u r s u i n g t h i s a n a l o g y f u r t h e r we w i l l e v a l u a t e E q . ( 1 7 ) e x p l i c i t l y f o r a model i n t e r a c t i o n . I n p a r t i c u l a r we have chosen

T h i s f u n c t i o n a l form h a s t h e a p p e a l i n g f e a t u r e s of making t h e i n t e - g r a l s i n Eq.(17) q u i t e s i m p l e and we c a n a l s o examine t h e e f f e c t s o f t h e r a n g e o f t h e i n t e r a c t i o n on t r a n s p o r t p r o p e r t i e s . The e x p r e s s i o n t o b e e v a l u a t e d i s

Using t h e c o n s t a n t d e n s i t y o f s t a t e s a p p r o x i m a t i o n and p e r f o r m i n g t h e t i m e i n t e g r a t i o n we o b t a i n

where

and A i ( x ) i s t h e Airy f u n c t i o n 4

.

^{A s} a check we n o t e t h a t t h e zero-

CO

f i e l d r e s u l t , Eq. ( 1 2 ) , can be r e c o v e r e d by n o t i n g t h a t

-

i d x A i ( x ) = 1.

For e x t r e m e l y h i g h f i e l d s t h e i n t e g r a l i n E q . ( 2 0 ) c a n be eva- l u a t e d by u s i n g an a s y m p t o t i c e x p r e s s i o n f o r t h e i n t e q r a l of an A i r y f u n c t i o n 5 and we g e t t h e f o l l o w i n g r e s u l t :

The s c a t t e r i n g i s t h u s s u p p r e s s e d and t h e t r a n s p o r t p r o c e s s becomes f r e e p a r t i c l e l i k e .

A n u m e r i c a l e v a l u a t i o n o f Eq. (20) w i t h p a r a m e t e r v a l u e s X = 20 A and ~ = 2 = 2 . 4 - 2Tr 1 0 1 2 ~ z y i e l d s t h e r e s u l t shown i n F i g . l a . We d i s t i n g u i s h t h r e e d i f f e r e n t r e g i o n s : 1) low f i e l d regime where t h e l e v e l w i d t h r e - t a i n s a p p r o x i m a t e l y i t s z e r o f i e l d v a l u e , 2) t r a n s i t i o n a l regime where t h e l e v e l w i d t h i s a d e c r e a s i n g f u n c t i o n of t h e e l e c t r i c f i e l d and 3 ) h i g h f i e l d regime where t h e l e v e l w i d t h i s v e r y s m a l l t h u s i n d i c a t -

FIELD ( V I M ) FIELD (VIM)

F i g . 1

( a ) The l e v e l w i d t h a s a f u n c t i o n of e l e c t r i c f i e l d f o r X = 20 A and v =

*

2 Tr ⁼ 2.4-1012 Hz. The f i e l d s t r e n g t h s F12 and F23 mark t h e t r a n s i - t i o n s between low f i e l d , t r a n s i t i o n a l and h i g h f i e l d regimes ( s e e t e x t ) .

(b) The t h r e e regimes f o r g e n e r a l v a l u e s of w . The l i n e a t w = wo d e n o t e s t h e f r e q u e n c y which was used w h i l e c o n s t r u c t i n g F i g . l a .

C7-306 JOURNAL DE PHYSIQUE

i n g d e c o u p l i n g of t h e c o n d u c t i o n e l e c t r o n s from t h e r e s o n a n t l e v e l ; s e e a l s o E q . ( 2 2 )

.

F i g u r e l b shows t h e s e t h r e e regimes f o r g e n e r a l v a l u e s of f r e - quency and e l e c t r i c f i e l d . The s h a p e s o f t h e s e b o u n d a r i e s c a n be u n d e r s t o o d by examining how a depends on t h e v a r i a b l e s w , A and F

(Eq. (21) )

.

For a f i x e d a i n t h e low f i e l d regime we f i n d w % F 2/3

,

which i s a l s o d i s p l a y e d by t h e boundary between r e g i o n s 1 and 2 i n F i g . l b . The v a l u e o f

a

a t which t h e z e r o f i e l d l e v e l w i d t h b e g i n s t o de- c r e a s e i s determined by t h e i n t e g r a l i n E q . ( 2 0 ) ; i t t u r n s o u t t h a t f o r a % 2 t h e i n t e g r a l h a s e s s e n t i a l l y a c h i e v e d i t s ' u l l v a l u e . Hence, f o r a 2 o r F ~

2

/w ~t h e l e v e l w i d t h i s a d e c r e a s i n g f u n c t i o n of t h e e x t e r - n a l f i e l d . T h i s c o n c l u s i o n i s independent of t h e c h o i c e o f t h e s c r e e n - i n g parameter and i n p a r t i c u l a r would remain v a l i d f o r A = 0 , i . e . c o r - r e s p o n d i n g t o a p o t e n t i a l which i s c o m p l e t e l y l o c a l i z e d i n r e a l s p a c e .

For h i g h f i e l d s it f o l l o w s from Eq.(21) t h a t t h e f r e q u e n c y and t h e f i e l d a r e r e l a t e d by U % F ' which i s confirmed by t h e h i g h f i e l d s l o p e of t h e boundary l i n e s . The boundary between r e g i o n s 2 and 3 h a s been chosen t o mark where a changes i t s s i g n . The e x a c t p o s i t i o n of t h i s l i n e depends on t h e c h o i c e of A .

I t i s i n t e r e s t i n g t o n o t e t h a t f o r a f u l l y l o c a l i z e d p o t e n t i a l ( A = O ) t h e r e g i o n 3 d o e s n o t e x i s t a t a l l , i . e . no m a t t e r how s t r o n g t h e f i e l d i s i t i s n o t a b l e t o s u p p r e s s t h e s c a t t e r i n g t o t a l l y . The s m a l l - e s t a t t a i n a b l e l e v e l w i d t h i s one t h i r d of t h e z e r o f i e l d v a l u e . We g i v e t h e f b l l o w i n g q u a l i t a t i v e e x p l a n a t i o n of t h i s e f f e c t . I n E q . ( 1 9 ) t h e r e a r e two f a c t o r s i n t h e exponent which c a u s e r e d u c t i o n from t h e

2 2

z e r o f i e l d v a l u e : ( i ) -i/24 ~~t~ and ( i i ) -A2/4 F t

.

The f i r s t one can b e a s s o c i a t e d t o p r o p a g a t i o n between c o l l i s i o n s whereas t h e second one r e f e r s t o something which happens " i n s i d e " t h e c o l l i s i o n . The r a p i d o s c i l l a t i o n s caused by ( i ) a r e s u f f i c i e n t t o r e d u c e t h e s c a t t e r - i n g s t r e n g t h b u t one needs t o b e a b l e t o g o

"

i n s i d e " t h e c o l l i s i o n b e f o r e a f u l l s u p p r e s s i o n i s a c h i e v e d and t h i s i s , o f c o u r s e , impossib- l e f o r a f u l l y l o c a l i z e d p o t e n t i a l .

T h i s r e s e a r c h was s u p p o r t e d i n p a r t by t h e ONR under c o n t r a c t N0014-80-C-0489.

R e f e r e n c e s :

1) K . Hess and N . Holonyak, J r . , P h y s i c s Today, October 1980, p.40 2) M. Salomaa, Z e i t s c h r i f t f . Physik B25, 49 (1976)

3 ) J . R . B a r k e r , J - P h y s . C5, 2663 ( 1 9 7 3 F

4) M. Abramowitz and I.A. S t e g u n , e d i t o r s , "Handbook of Mathematical Functions!', Washington D.C.: N a t i o n a l Bureau o f S t a n d a r d s (1964) pp. 446-452

5) Ref. 4 . Ch. 1 0 , p . 449, Eq.(4.82)