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A COMPARISON OF COHERENT POTENTIAL AND RANDOM WALK METHODS FOR CALCULATING THE TRANSPORT PROPERTIES OF AMORPHOUS
SEMICONDUCTORS
V. Halpern
To cite this version:
V. Halpern. A COMPARISON OF COHERENT POTENTIAL AND RANDOM WALK METH- ODS FOR CALCULATING THE TRANSPORT PROPERTIES OF AMORPHOUS SEMICONDUC- TORS. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-119-C4-122. �10.1051/jphyscol:1981422�.
�jpa-00220849�
JOURNAL UE PHYSIQUE
Colloque C4, supple'menl: au nOIO, Tome 4 2 , oetobre 1981 page c4- I 19
A COMPARISON OF COHERENT POTENTIAL AND RANDOM WALK METHODS FOR CALCU- L A T I N G THE TRANSPORT PROPERTIES OF AMORPHOUS SEPIICONDUCTORS
V. H a l p e r n
Department of Physics, Bar-%Zan University, Hamat-Can, Israel
A b s t r a c t - The motion o f p a r t i c l e s between d i s c r e t e l o c a l i s e d s t a t e s i s s t u d i e d i n t e r m s o f two t y p e s o f a p p r o x i m a t i o n t o t h e s o l u t i o n o f t h e a p p r o p r i a t e r a t e e q u a t i o n s . The first o f t h e s e i n v o l v e s an e x t e n s i o n o f t h e d e c o u p l i n g o f c o n f i g u r a t i o n a l a v e r a g e s from which t h e u s u a l con- t i n u o u s t i m e random walk (CTRW) a p p r o x i m a t i o n c a n b e d e r i v e d t o d e f i n e a CTKW(n) a p p r o x i m a t i o n . The o t h e r a p p r o a c h , which i s p r e s e n t e d i n some d e t a i l , i s based on a m o d i f i c a t i o n o f t h e homomorphic c l u s t e r e x t e n s i o n o f t h e c o h e r e n t p o t e n t i a l a p p r o x i m a t i o n (HCCPA). A comparison o f t h e r e s u l t s o f t h e s e methods f o r t h e d i f f u s i o n c o e f f i c i e n t i n some model systems i n d i c a t e s t h a t a l l t h e s e methods g i v e t h e same r e s u l t s a t h i g h f r e q u e n c i e s . A t i n t e r m e d i a t e f r e q u e n c i e s t h e r e s u l t s o f t h e CTRW(2) method a r e much c l o s e r t h a n t h o s e o f t h e CTRW(1) method t o t h o s e of t h e HCCPA, b u t a t low f r e q u e n c i e s t h e r e s u l t s o f b o t h t h e s e CTRW methods d i f f e r a p p r e c i a b l y from t h o s e o f t h e HCCPA method.
I n t r o d u c t i o n . - The problem o f c a l c u l a t i n g t h e t r a n s p o r t p r o p e r t i e s o f d i s o r d e r e d systems i n which p a r t i c l e s hop between d i s c r e t e l o c a l i s e d s t a t e s i s s o complex t h a t numerous a p p r o x i m a t i o n s have been p r o p o s e d f o r t r e a t i n g i t . I f t h e t r a n s p o r t p r o - p e r t i e s a r e d e r i v e d from a s e t o f r a t e e q u a t i o n s , t h e y can g e n e r a l l y b e e x p r e s s e d i n t e r m s o f t h e c o n f i g u r a t i o n a l a v e r a g e <G(u)> o f t h e G r e e n ' s f u n c t i o n G(u) a s s o c i a t e d w i t h t h e Laplace t r a n s f o r m o f t h e s e e q u a t i o n s . A t l e a s t t h r e e d i s t i n c t a p p r o a c h e s e x i s t t o t h e c a l c u l a t i o n o f <G(u)>. One p o s s i b i l i t y i s t o use diagram t e c h n i q u e s
[ I ] o r o t h e r p e r t u r b a t i o n t h e o r y t e c h n i q u e s [ 2 ] t o f i n d a s e l f - e n e r g y from which
<G(u)> c a n be c a l c u l a t e d . Another a p p r o a c h , which c o r r e s p o n d s t o t h e well-known c o n t i n u o u s t i m e random walk (CTRW) a p p r o x i m a t i o n o f M o n t r o l l , S c h e r and Lax [ 3 ] , i s t o d e c o u p l e t e r n s on t a k i n g t h e c o n f i g u r a t i o n a l a v e r a g e o f a Dyson e q u a t i o n f o r G(u) [ 4 ] . A t h i r d a p p r o a c h , on which we c o n c e n t r a t e i n t h i s p a p e r , i s t o u s e a c o h e r e n t p o t e n t i a l a p p r o x i m a t i o n (CPA) t o f i n d an e f f e c t i v e o p e r a t o r whose G r e e n ' s f u n c t i o n a p p r o x i m a t e l y e q u a l s <G(u)> by r e q u i r i n g t h a t t h e s c a t t e r i n g produced by t h e d i f f e r - e n c e between t h i s o p e r a t o r and t h a t o f t h e a c t u a l s y s t e m , a s d e s c r i b e d by t h e T- m a t r i x , i s on a v e r a g e a p p r o x i m a t e l y z e r o . For t h i s p u r p o s e , a s l i g h t m o d i f i c a t i o n o f t h e homomorphic c l u s t e r CPA [S] i s e x t r e m e l y s u i t a b l e .
G r e e n ' s F u n c t i o n s f o r t h e R a t e E q u a t i o n s , and t h e CTRW Approximations.- I t i s f r e - q u e n t l y p o s s i b l e t o map t h e l o c a l i s e d s t a t e s o f t h e d i s o r d e r e d s y s t e m o n t o a p e r i o d - i c l a t t i c e , w i t h ~ ( l _ , t l ~ ~ ) , t h e p r o b a b i l i t y t h a t a p a r t i c l e s t a r t i n g a t s i t e a t t i m e t = 0 w i l l bc a t s i t e
1
a t t i m e t , s a t i s f y i n g t h e r a t e e q u a t i o n s 161Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981422
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where W(L1,1), t h e e f f e c t i v e hopping r a t e from s i t e
1'
t o s i t e1,
i s a random v a r i - a b l e . I t i s convenient t o c o n s i d e r t h e Laplace transform o f equations ( I ) , and t o e x p r e s s t h i s i n m a t r i x form by d e f i n i n g t h e m a t r i c e s G(u), V , and A whose ( l , ~ ) elements a r e , r e s p e c t i v e l y , t h e Laplace t r a n s f o r m of P ( l , tIn),
W(n,l) andx
W(1,p)61,n-
We r e a d i l y f i n d [4] t h a t t h e s e m a t r i c e s s a t i s f y t h e e q u a t i o nP -
- - - -
[ u I - (V - A)]G(u) E (uI - H)G(u) = I (2)
s o t h a t G(u) i s t h e Green's f u n c t i o n of t h e o p e r a t o r H = V - A. The average t r a n s - p o r t p r o p e r t i e s o f t h e system, such a s t h e d i f f u s i o n c o e f f i c i e n t a t frequency w , D(w), can then r e a d i l y be expressed i n terms o f t h e c o n f i g u r a t i o n a l average
<G(iw)> [ 6 ] .
One p o s s i b l e development of e q u a t i o n ( 2 ) i s t o w r i t e i t i n t h e form
where Go(u) i s t h e random diagonal m a t r i x ( u I + A)-'. On t a k i n g t h e c o n f i g u r a t i o n a l average o f both s i d e s o f e q u a t i o n ( 3 ) , and decoupling G(u) i n t h e l a s t term, we o b t a i n
<G (u) > = <Go (u) > + <G (u) ><VGo (u) >, (4) which i s e x a c t l y e q u i v a l e n t t o t h e u s u a l CTRW approximation of [ 3 ] , a s we have shown elsewhere [ 4 ] . An obvious extension of t h i s approach, which we c a l l t h e CTRW(n) approximation [41, i s t o w r i t e
The Modified Homomorphic C l u s t e r Coherent P o t e n t i a l ~ ~ ~ r o x i m a t i o n (HCCPA)
.
- Another p o s s i b l e development o f e q u a t i o n ( 2 ) i s t o look f o r an e f f e c t i v e Hamiltonian He(u) such t h a tFor a system such a s o u r s , i n which H h a s d i s o r d e r i n i t s o f f - d i a g o n a l elements a s well a s i n i t s diagonal ones, t h e HCCPA method i s very s u i t a b l e . I n t h i s method, which h a s been shown t o be an a n a l y t i c e x t e n s i o n of t h e u s u a l CPA method [ S ] , t h e d i s o r d e r e d system i s p a r t i t i o n e d i n t o homomorphic c l u s t e r s ( i . e . , c l u s t e r s a l l t h e same s h a p e ) , C , each c o n t a i n i n g a s e t of r b a s i s f u n c t i o n s t h a t we denote by I c ) . This p a r t i t i o n i n g must be such t h a t both t h e o p e r a t o r H o f equation (2) and t h e e f f e c t i v e o p e r a t o r H o f equation (6) can be expressed a s sums over t h e c l u s t e r s . We t h e n w r i t e
where h C and s ( u ) a r e r x r m a t r i c e s and s ( u ) i s i n d e p e n d e n t o f C. I n t h e s p i r i t o f t h e u s u a l CPA, we d e t e r m i n e s ( u ) by r e q u i r i n g t h a t t h e mean o f t h e T - m a t r i x a s s o c - i a t e d w i t h t h e o p e r a t o r H-H between c l u s t e r s C be z e r o . Thus, i f g e ( u ) d e n o t e s t h e m a t r i x o f Ge(u) between t h e s t a t e s o f a c l u s t e r , we r e q u i r e t h a t t h e c o n f i g u r a t i o n a l a v e r a g e
For c o n v e n i e n c e , we c o n s i d e r h e r e s y s t e m s i n which p a r t i c l e s c a n hop o n l y from one s i t e t o any of t h e n a d j a c e n t s i t e s , and f o r which V i s a symmetrix m a t r i x , a l t h o u g h t h e method can be a p p l i e d t o more g e n e r a l s y s t e m s . For a s y s t e m l i k e t h i s , w i t h hopping o n l y between n e a r e s t n e i g h b o u r s , i t i s n a t u r a l t o choose C t o b e a p a i r o f a d j a c e n t s t a t e s . I n t h a t c a s e , i f we d e n o t e by - s t and s" t h e d i a g o n a l and o f f - d i a g o n a l e l e m e n t s o f s ( u ) , r e s p e c t i v e l y , t h o s e o f He w i l l be - n s l and s" r e s p e c t i v e l y . I n t h e HCCPA method a s o r i g i n a l l y p r o p o s e d [S], t h e d i a g o n a l e l e m e n t s of hc were chosen s i m i l a r l y t o be r ; - t i m e t h o s e o f H. 1 However, f o r o u r problem, a d i f f e r e n t c h o i c e i s more a p p r o p r i a t e . S i n c e t h e d i a g o n a l e l e m e n t s o f H, -A1, a r e j u s t minus t h e sum o f a l l t h e t r a n s i t i o n r a t e s from L t o o t h e r s t a t e s , it i s - n a t u r a l t o a s s o c - i a t e w i t h t h e c l u s t e r C c o n t a i n i n g s t a t e s
-
1 and - 1 ' t h o s e p a r t s o f - A 1 and -A1, t h a t c o r r e s p o n d t o t r a n s i t i o n s from1
t o1'
and v i c e v e r s a . Thus, f o r t h F s c l u s t ~ r we c a n w r i t eSimple c a l c u l a t i o n s t h e n show t h a t s" = s f , a c o n d i t i o n which a l s o f o l l o w s from t h e c o n s e r v a t i o n o f p a r t i c l e s f o r t h e e f f e c t i v e medium, and t h a t s ' i s determined by
H e r e , g l l and g12 d e n o t e t h e d i a g o n a l and o f f - d i a g o n a l e l e m e n t s o f g e ( u ) , r e s p e c t - i v e l y , and a r e known f u n c t i o n s o f s ' , s i n c e s ' d e t e r m i n e s He. Thus, f o r any g i v e n d i s t r i b u t i o n o f t h e t r a n s i t i o n r a t e s V e q u a t i o n (10) can b e s o l v e d t o f i n d s f , and hence Ge(u) = < G ( u ) > . Moreover, i t can r e a d i l y be shown t h a t t h e d i f f u s i o n co- e f f i c i e n t o f t h e system a t f r e q u e n c y w , D(w), i s e q u a l t o a r e a l c o n s t a n t t i m e s t h e v a l u e o f s ' f o r u = i w .
R e s u l t s o f C a l c u l a t i o n s on Model Systems.- I t f o l l o w s from e q u a t i o n (10) t h a t , f o r a l l s y s t e m s , a t s u f f i c i e n t l y h i g h f r e q u e n c i e s s ' = C V ~ , ~ ' . T h i s l e a d s t o t h e same f o r m u l a f o r D ( w ) a s t h e p a i r a p p r o x i m a t i o n , which i s Known [ 6 ] t o be e x a c t i n t h e h i g h f r e q u e n c y l i m i t . As t h e f r e q u e n c y w + 0 , t h e l i m i t i n g v a l u e o f s ' depends on t h e c o n n e c t i v i t y o f t h e system; i n p a r t i c u l a r , it c a n be shown t h a t f o r one-dimen- s i o n a l s y s t e m s , l / s l + < l / V 1 A t i n t e r m e d i a t e f r e q u e n c i e s , t h e r e s u l t s depend on
-> -
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-Figure 1: The r a t i o o f D,!,(w), t h e v a l u e o f t h e r e a l p a r t of t h e d i f f u s i o n c o e f f i c i e n t as given by t h e CTRW(n) method, t o D&(w), i t s value from t h e HCCPA method, a s a fun- c t i o n of t h e reduced frequency @/A, f o r ey =
lo3.
- n = 2; - - - n = 1.t h e d e t a i l s of t h e system. I n o r d e r t o compare t h e r e s u l t s of t h e HCCPA method with t h o s e of t h e CTRW methods, we c o n s i d e r t h e same one-dimensional system a s i n 141, with V1,ll a symmetric random v a r i a b l e having p r o b a b i l i t y d e n s i t y
- -
$(V) = 1/2yV f o r ~ e - ' < V < ~e~ and z e r o elsewhere. (11) The r e s u l t s o f t h e c a l c u l a t i o n s f o r d i f f e r e n t v a l u e s o f y a r e very s i m i l a r , and q u a l i t a t i v e l y resemble t h o s e of t h e CTRW methods [ 4 ] . Thus, f o r i n s t a n c e , D'(w), t h e r e a l p a r t o f D(w), s a t u r a t e s a t h i g h f r e q u e n c i e s , and a s t h e frequency i s lowered has an e x t e n s i v e r e g i o n i n which D ' (w) i s approximately p r o p o r t i o n a l t o wa with O<a<l, t h e e x t e n t of which i n c r e a s e s with i n c r e a s i n g y . However, a d e t a i l e d compar- i s o n o f D1(w) a s given by t h e HCCPA, CTRW(1) and CTRW(2) methods r e v e a l s s i g n i f i c a n t q u a n t i t a t i v e d i f f e r e n c e s , and a s e t of t y p i c a l r e s u l t s i s shown i n f i g . 1.
Discussion.- A t s u f f i c i e n t l y hi-gh f r e q u e n c i e s , where t h e CTRW and HCCPA methods a r e a l l e x a c t [ 4 , 6 ] , t h e y obviously a l l g i v e t h e same v a l u e of Df(w). However, a s can be seen from f i g . 1, when t h e frequency d e c r e a s e s t h e r e s u l t s o f t h e CTRW(1) method s t a r t t o d i v e r g e from t h o s e o f t h e HCCPA method a t a h i g h e r frequency, and i n i t i a l l y t o a much g r e a t e r e x t e n t , t h a n do t h o s e o f t h e CTRW(2) method. The reason f o r t h i s i s t h a t , a t such f r e q u e n c i e s , p a t h s i n v o l v i n g two s t e p s s t a r t becoming o f importance i n determining D1(w), and t h e c o r r e l a t i o n s between t h e s e s t e p s a r e t a k e n i n t o account by t h e CTRW(2) method b u t n o t by t h e u s u a l CTRW(1) method. A t s t i l l lower f r e q u e n c i e s , when p a t h s i n v o l v i n g s e v e r a l c o r r e l a t e d s t e p s can become i m p o r t a n t , n e i t h e r t h e CTRW(1) n o r t h e CTRW(2) method i s r e l i a b l e , and a s we s e e from t h e f i g - u r e t h e i r r e s u l t s begin t o d i v e r g e r a p i d l y from t h o s e o f t h e HCCPA method. Only t h i s l a t t e r method, which t r e a t s t h e p r o p e r t i e s of an average p a i r o f s t a t e s , can t a k e account o f r e p e a t e d hops between a p a i r o f s i t e s , a form of c o r r e l a t i o n which i s of s p e c i a l importance i n one-dimensional systems because o f t h e i r low c o n n e c t i v i t y . References
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