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THE HALL EFFECT DUE TO HOPPING
CONDUCTION IN THE LOCALIZED STATES OF AMORPHOUS SEMICONDUCTORS
L. Friedman, M. Pollak
To cite this version:
L. Friedman, M. Pollak. THE HALL EFFECT DUE TO HOPPING CONDUCTION IN THE LO-
CALIZED STATES OF AMORPHOUS SEMICONDUCTORS. Journal de Physique Colloques, 1981,
42 (C4), pp.C4-87-C4-90. �10.1051/jphyscol:1981414�. �jpa-00220760�
THE HALL EFFECT DUE TO HOPPING CONDUCTION I N THE LOCALIZED STATES OF AMORPHOUS SEMICONDUCTORS
L. Friedman
+
and M . ~ o l l a k *Gl'K L a b o r a t o r i e s , WaZtham, MA 02254, U.S.A.
*
U n i v e r s i t y o f C a l i f o r n i a a t R i v e r s i d e , R i v e r s i d e , CA 92502, U. S.A.A b s t r a c t . The H a l l m o b i l i t y
u H
d u e t o h o p p i n g t r a n s p o r t i s c a l c u l a t e d f o r s y s t e m s w i t h b o t h p o s i t i o n a l and s i t e - e n e r g y d i s o r d e r . A p e r c o l a t i o n - t h e o r - e t i c a p p r o a c h i s a d o p t e d and t h e Miller-Abrahams jump r a t e s a r e used. I n t h e low t e m p e r a t u r e , v a r i a b l e - r a n g e - h o p p i n g l i m i t , a n u p p e r l i m i t t ou H
i s found (Iu I < l ~ - ~ c m ~ ~ - ~ ~ - ~ f o r a - S i a t T=300K) and i t s t e m p e r a t u r e dependence i s est!?bfblished.I n t r o d u c t i o n . - I n f i t t i n g t r a n s p o r t and H a l l e f f e c t d a t a i n amorphous s e m i c o n d u c t o r s , a n o u t s t a n d i n g problem h a s b e e n t h e magnitude and t e m p e r a t u r e dependence o f t h e H a l l m o b i l i t y
u
due t o h o p p i n g c o n d u c t i o n s among l o c a l i z e d s t a t e s below t h e m o b i l i t y e d g e . I nh i s
p a p e r , p H i s c a l c u l a t e d i n t h e l o w - t e m p e r a t u r e , v a r i a b l e - r a n g e l i m i t o f t h e Miller-Abrahams I m p u r i t y c o n d u c t i o n r e g i m e , i n c o r p o r a t i n g randomness i n b o t h t h e s i t e p o s i t i o n s and e n e r g i e s .An o u t l i n e o f t h e p r o c e d u r e u s e d i s a s f o l l o w s : A p e r c o l a t i o n - t h e o r e t i c a p p r o a c h i s employed, i n which t h e c u r r e n t i s c a r r i e d by t h e backbone p a r t o f t h e p e r c o l a t i o n c l u s t e r (no dead e n d s ) which c o n s i s t s o f b r a n c h e s c n n n e c t e d a t j u n c t i o n s , m a i n l y t h r e e b r a n c h e s t o a j u n c t i o n . Only a t t h e s e j u n c t i o n s a r e t h e e x t r i n s i c H a l l c u r r e n t s g e n e r a t e d , s i n c e t h e e x i s t e n c e o f a H a l l e f f e c t i n h o p p i n g c o n d u c t
a r i s e s from h i g h e r o r d e r jump p r o c e s s e s i n v o l v i n g a minimum o f t h r e e s i t e s .
tar
However, t h e s e c u r r e n t s need n o t c o n t r i b u t e f u l l y t o t h e measured m a c r o s c o p i c H a l l c u r r e n t , s i n c e p a r t o f t h e e x t r i n s i c c u r r e n t may b e s p e n t on i n t e r n a l f l o w i n o r n e a r t h e j u n c t i o n . T h i s i s t a k e n i n t o a c c o u n t by u s i n g a s i m p l i f i e d r e s i s t a n c e c i r c u i t t o c a l c u l a t e t h e H a l l v o l t a g e V c o n t r i b u t e d by t h e j u n c t i o n u n d e r t r a n s - v e r s e open c i r c u i t c o n d i t i o n s . The l a t e e r i n t u r n depends on t h e s i t e e n e r g i e s and s e p a r a t i o n s o f t h e j u n c t i o n .
The c o n f i g u r a t i o n a l a v e r a g e < V > i s t h e n t a k e n , and i t i s a t t h i s s t a g e t h a t d i f f e r e n c e s among v a r i o u s i n v e s t i g a t ! ? o n s o c c u r . I n o u r c a s e , we t a k e t h e o c c u r r e n c e p r o b a b i l i t i e s o f t h e p o s i t i o n s and e n e r g i e s t o b e i n d e p e n d e n t l y d i s t r i b u t e d , t h e d i s t r i b u t i o n o f t h e s i t e p o s i t i o n s b e i n g t h e u s u a l P o i s s o n d i s t r i b u t i o n f o r a s p a t i a l l y random s y s t e m . The d i s t r i b u t i o n i n e n e r g y c o r r e s p o n d s t o 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 e x p l i c i t l y t a k e s a c c o u n t o f t h e f a c t t h a t s i t e s o f h i g h e n e r g y
( E 5 em) o c c u r w i t h a much s m a l l e r p r o b a b i l i t y t h a n s i t e s w i t h a l a r g e s p a t i a l s e p a r a t i o n , f o r a g i v e n v a l u e of t h e i n t e r - s i t e impedance. We f i n d t h a t we must d i s t i n g u i s h t h e t h r e e s i t e s a t a j u n c t i o n a c c o r d i n g t o w h e t h e r o r n o t t h e i r e n e r g i e s a r e on t h e same s i d e o f t h e Fermi l e v e l ; t h e f o r m e r make t h e dominant c o n t r i b u t i o n t o V
H '
To o b t a i n t h e m a c r o s c o p i c H a l l f i e l d , i t is a l s o n e c e s s a r y t o know t h e d e n s i t y o f j u n c t i o n s . T h i s is o b t a i n e d from t h e d e t e r m i n a t i o n o f t h e p a r t i c u l a r p e r c o l a t i o n ' p a r t s of t h i s work w e r e done a t U n i v e r s i t y o f S t . Andrews, S c o t l a n d , U n i v e r s i t y o f
C a l i f o r n i a a t R i v e r s i d e , s u p p o r t e d i n p a r t b y NSF g r a n t DMR78-21959-01, and Brown U n i v e r s i t y , P r o v i d e n c e , R I , USA, t h e l a t t e r s u p p o r t e d by t h e M a t e r i a l s R e s e a r c h L a b o r a t o r y , funded b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981414
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c l u s t e r which c a r r i e s t h e dominant c u r r e n t . C r i t i c a l e x p o n e n t s a r e u s e d h e r e . ( 2 ) From t h e mean ( c o r r e l a t i o n ) d i s t a n c e between j u n c t i o n s L i n t h e t r a n s v e r s e d i r e c t i o n , we o b t a i n t h e open c i r c u i t H a l l f i e l d E = <V > / L . T h i s i n t u r n g i v e s t h e s h o r t c i r c u i t H a l l c u r r e n t j = oEH and f i n a l a y t h e % a l l a n g l e O = jH/jC and H a l l m o b i l i t y pH, where jC
ys
t h e c o n d u c t i o n c u r r e n t d e n s i t y , a y s o depending on t h e d e n s i t y o f c o n d u c t i n g p a t h s .Having o u t l i n e d o u r p r o c e d u r e i n some d e t a i l , we now proceed t o g i v e some o f t h e k e y e q u a t i o n s and r e s u l t s . The s t a r t i n g p o i n t i s t h e r a t e e q u a t i o n f o r t h e s i t e o c c u p a t i o n p r o b a b i l i t i e s f .
T h i s i s l i n e a r i z e d by w r i t i n g
t h e sum of e q u i l i b r i u m terms p l u s p a r t s l i n e a r i n t h e e l e c t r i c f i e l d (E), t h e mag- n e t i c f i e l d (H), and t h e i r p r o d u c t (EH). The z e r o o r d e r p a r t g i v e s t h e u s u a l d e t a i l e d b a l a n c e i n e q u i l i b r i u m , t h e e q u a t i o n l i n e a r i n E g i v e s t h e c o n d u c t i v i t y e q u a t i o n s f o r ohmic c o n d u c t i o n , t h a t l i n e a r i n H t h e d i a g m a g n e t i c c u r r e n t s , w h i l e t o o r d e r (EH) we have
The m a g n e t i c - f i e l d - p r o p o r t i o n a l p a r t of t h e jump r a t e i n v o l v e s t h e i n t e r m e d i - a r y o f a t h i r d s i t e and i s w r i t t e n a s a n e l e c t r o n p a r t and h o l e p a r t , p r o p o r t i o n a l t o t h e t h i r d s i t e b e i n g empty o r o c c u p i e d , r e s p e c t i v e l y
Here v:;: i s t h e t h r e e s i t e t r a n s i t i o n r a t e f o r e l e c t r o n s , and vijk (h) i s o b t a i n e d from vijk by t h e p r e s c r i p t i o n (e) (E.+S 3 .+-.I.., H+-H). A f t e r c o n s i d e r a b l e m a n i p u l a t i o n ,
I i' i j 1~
t h e l a s t t h r e e ( d r i v i n g ) t e r m s of ( 3 ) , which g i v e t h e e x t r i n s i c c u r r e n t i n t o a g i v e n s i t e (k) become
where @ = l / k ~ ,
J ( 6 )
and t h e t o t a l s i t e p o t e n t i a l i s
w i t h $ . = e g r . t h e e l e c t r o s t a t i c p o t e n t i a l and eVi t h e p e r t u r b a t i o n t o t h e l o c a l c h e m i c i l p o t e n $ i a l d u e t o
E.
For t h e M i l l e r Abrahams c a s e , (3)
where J i j i s t h e t r a n s f e r i n t e g r a l ,
hji
t h e v e c t o r a r e a of t h e t r i a n g l e ( i j k ) , andS u b s t i t u t i n g (8) and ( 9 ) i n t o ( 5 ) , we o b t a i n ( e l
wH.
= vijk(Ei = 5 = ck = O ) f ( r i . s j . c k ) .i j k j
with
1 ) + e x p - B ( ~ . . + ~ c . ) + e x p - B ( e
k
] [
j ik-I 1'] [
jk''ji-"j'd} i l l )where
and
A s explained e a r l i e r , t h e e x t r i n s i c c u r r e n t (5) do n o t c o n s t i t u t e t h e Hall c u r r e n t . To o b t a i n t h i s , t h e H a l l v o l t a g e developed a c r o s s any two of t h e t h r e e s i t e s of t h e t r i a n g l e i s c a l c u l a t e d , given t h e e x t r i n s i c c u r r e n t s a t each s i t e ( i . e . eqn ( 5 ) and c y c l i c permutation). T h i s g i v e s ( l a b e l l i n g t h e s i t e s 1 , 2 , 3 )
where j i s t h e (ohmic) c u r r e n t i n t o t h e j u n c t i o n and
w i t h
The "short-ing" e f f e c t due t o t h e impedances e x t e r n a l t o t h e t r i a n g l e h a s been shown t o produce a small e r r o r i n a p e r c o l a t i o n approach.(4)
The f i n a l t a s k i s t o average (13) o v e r a l l p o s s i b l e c o n f i g u r a t i o n s of s i t e p o s i t i o n s and e n e r g i e s of t h e j u n c t i o n . The p r o b a b i l i t i e s a r e assumed t o be independently d i s t r i b u t e d
P ( R 1 2 ~ R 2 3 , R 3 1 ; ~ 1 , ~ 2 ~ & 3 ) = W ( R l 2 , R 2 3 ~ R 3 1 ) ~ c ( c l ~ ~ 2 , c 3 ) , (15) where W i s taken t o b e t h a t f o r a s p a t i a l l y random system(5) with s i t e c o n c e n t r a t i o n N,
R ) = 8 n 2 ~ R R R
W(R12'R239 31 12 23 31 (16)
and
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for a constant density of states, v(E)=v. The quantity P~(E~)=P (E.IZ ) is the conditional probability that the second smallest impedance emanagin; fyom site i with energy E. is Z or less, and constrains site i to be on the percolation cluster.
hi;
resuTfts in sites of large ener y (E.<E ) being very unlikely,(6) and is responsible for the small magnitude of 111 Hf
obtirnzd.In taking the configurational average of (13), the procedure is to integrate over the R. .Is up to Rij (m)=(~/~o)~~T~ij ) for fixed values of the ci.'s. The maxi- mum contri6ation occurs at Ri .=R
i j In performing the subsequent Anergy inte- grations, we use the fact thaz
BE^>>^
to approximate the integrals by their (exponentially large) maximum multiplied by an appropriate half-width. The final result is of the formwhere a-I is the radius of the localized wavefunction, A is the distance between largest impedances Zm, BE~=(T~/T)'I~, f are incomplete Gamma functions and n=13/4.
For parameters appropriate to a-Si (a-f=101, v=1019 (e~)-lcrn-~, T=300K), we estimate
This is an upper limit, for reasons which we cannot give for lack of space. We note that P has a power law dependepce on (T IT)%, rather than the exponential dependence pound for o exp[-(T
IT)<].
suchOan exponential dependence (with a smaller T ) has been obtained byOother workers. (7) Their procedure agrees with ours up tg eqn (13), but they apparently average R and E separately with theconstraint ij ij
in contrast with our procedure described just before eqn (18).
A detailed account of this work will appear in a future issue of the Philosophical Magazine (B).
References.
(1) FRIEDMAN, L., HOLSTEIN, T., Ann. Phys.
2
(1963) 494.(2) SKAL, A.S., SHKLOVSKII, B.I., Soviet Phys. Semicond.
3
(1975) 1029.(3) MILLER, A., ABRAHAMS, E., Phys. Rev.
120
(1960) 795.(4) FRIEDMAN, L., POLLAK, M., Phil. Mag. 38 (1978) 173.
(5) HOLSTEIN, T., Phys. Rev.
2
(1961) 1 3 9 .(6) POLLAK, M., in 'The Metal Non-Metal Transition in Disordered Systems', edited by L. Friedman and D. Tunstall, Proc. of 19th Scottish Universities Summer School in Physics (1978) 239.
(7) GRUNEWALD, U., MUELLER, H., THOMAS, P., WUERTZ, D., Solid State Comm. (to be published).
