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Submitted on 1 Jan 1981
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A SCALING THEORY FOR HOPPING CONDUCTION
M. Pollak
To cite this version:
M. Pollak. A SCALING THEORY FOR HOPPING CONDUCTION. Journal de Physique Colloques,
1981, 42 (C4), pp.C4-141-C4-142. �10.1051/jphyscol:1981427�. �jpa-00220883�
JOURNAL DE PHYSIQUE
Colloque C4, supplement au n°10, Tome 42, ootobve 1981 page Q4-141
A SCALING THEORY FOR HOPPING CONDUCTION*
M. P o l l a k
University of California, Riverside, CA 92502, U.S.A.
Extended Abstract. A scaling theory has been developed for hopping transport in disordered systems. It provides an alternative to other existing theories for hopping transport, notably percolation theory.(I* The scaling theory is consid- erably easier to use, but provides somewhat less information than percolation theory.
The following features underlie both the scaling and percolation theories.
The Miller-Abrahams random impedance network is utilized, in which any two sites are connected by a resistance Z = Z0 exp £. £ can depend on several random variables, usually energies E and distances x. The exponential dependence on £ gives Z a very wide statistical distribution. This has two important consequences.
1) The resistance of a current path is dominated by the largest resistances in the path, say with value Z = Z exp£.
2) Where a number of current paths exist in parallel, the path with the smallest 5 carries most of the current.
For the various hopping processes usually investigated, 5 is a sum of two homogeneous functions,
The symbol "v< denotes that E,x can be arrays, say of m and n random variables, respectively.
2 2 2 1/2 For example, for single-phonon assisted hopping in 3D, £x = 2a(x +y +2 ) ,
£E = gE, so i=j=m=l, and n=3. Here a is the localization radius; x,y,z the components of the intersite separation, B=l/kT, and E a random energy, explained in detail in ref. 1.
We consider an ensemble of all possible realizations S of a given hopping system, at some temperature T'. In any specimen 6 of the ensemble there will be a dominant current path, consisting of a cluster of resistances {Z^}. According to 1) above, the largest exponent B.I in the cluster determines the resistivity of specimen S at T'. If the hopping system considered has a well defined resistivity, then gi must be the same (say £') for almost all realizations 6.
We can scale any of the clusters {Z^}^ by scaling x by, say, a., and E by say, b_. A certain combination of _a,l) generates a "conservative"scaling which is such that the scaled clusters, say {Z^} have the same probability of occurance in the ensemble as the original clusters iZ^}r. We denote the largest Z in 'CZ^}* by Z5 = Z0exp5fi. According to 2) above, one should be able to obtain the resistivity at any T by finding {Zi>g with the minimum possible g, at T, i.e. optimizing 55
at T with respect to the scaling parameters a., b^. Such a procedure amounts to Mott's original procedure for variable range hopping (2). The procedure will be meaningful if such scaling will result in the same f. for all S, and if, in
* Supported in part by NSF
t A detailed version of this work has been submitted to J. Phys. C.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981427
JOURNAL DE PHYSIQUE
a d d i t i o n , t h e r e e x i g t s n o c o n f i g u r a t i o n , n o t o b t a i n e d f r o m t h e s c a l i n g , w h i c h h a s a s m a l l e r v a l u e o f 5 a t T t h a n
c6.
I t is shown i n t h e f u l l p a p e r t h a t c o n s e r v a t i v e s c a l i n g f u l f i l l s b o t h c o n d i t i o n s , p r o v i d e d t h a t a l l t h e random v a r i a b l e s a r e u n c o r r e l a t e d .T h e p o s s i b i l i t y t o o b t a i n t h e v a l u e o f
5
a t any t e m p e r a t u r e T f r o m a n assumed v a l u e o f 5 ' a t some t e m p e r a t u r e T' i m p l i e s t h a t t h e t e m p e r a t u r e d e p e n d e n c e o f-
5 = Rnp i s o b t a i n e d . (p is t h e r e s i s t i v i t y ) .
As shown i n t h e f u l l p a p e r , t h e d e p e n d e n c e of on t h e l o c a l i z a t i o n p a r a m e t e r a , a n d o n a d e n s i t y o f s t a t e s 14 is a l s o o b t a i n e d f r o m t h e s c a l i n g p r o c e d u r e . But t h e s c a l i n g p r o c e d u r e g i v e s <(T,ci,N) o n l y up t o a m u l t i p l i c a t i v e c o n s t a n t , w h i c h must b e o b t a i n e d f r o m o t h e r c o n s i d e r a t i o n s ( s e e f u l l p a p e r f o r d e t a i l s ) .
T h e most g e n e r a l r e s u l t f o r t h e r e s i s t i v i t y , o b t a i n e d s o f a r f r o m t h e s c a l i n g t h e o r y , js f o r power law distributions o f t h e random v a r i a b l e s ,
w h e r e Gt, K, are some c o n s t a n t s . The r e s i s t i v i t y i s g i v e n by
i j j v + i i j v / ( j v + i v ) N - i j / ( j v + i P ) ], ( 1 )
P ~ ~ e x p [ ~ ~ k ~ ) -
n m n m
w h e r e 5 i s a c o n s t a n t , N =
n
Kt q l ~ t ,
v = s z l ( ~ s + l ) and P = 1 ( 9 + I )s = l s t = l t
E q . ( l ) a p p l i e s t o a v e r y w i d e r a n g e of h o p p i n g s y s t e m s a n d h o p p i n g p r o c e s s e s , f o r e x a m p l e t o G a u s s i a n - l o c a l i z e d e l e c t r o n s ( i = 2 ) , t o p r o c e s s e s w h e r e b a r r i e r e n e r g i e s a n d s i t e e n e r g i e s a r e i m p o r t a n t random v a r i a b l e s (m=2), t o n o n - u n i f o r m d i s t r i b u t i o n s o f x , w h i c h may o c c u r i n g r a n u l a r m a t e r i a l s , w h e r e x may r e p r e s e n t t h e t h i c k n e s s o f a n o x i d e l a y e r (qt#O). E q . ( 1 ) a l s o c o n t a i n s many of t h e r e s u l t s known f r o m p e r c o l a t i o n t h e o r y , s u c h a s i m p u r i t y c o n d u c t i o n (v=3, p=O, i = l , j = l ) , v a r i a b l e r a n g e h o p p i n g ( p = l , i = 1 , j = l ) i n two d i m e n s i o n s (v=2) a n d i n t h r e e d i m e n s i o n s ( v = 3 ) , a n d h o p p i n g i n a n o n - u n i f o r m d e n s i t y of s t a t e s , N(E) Eq.
R e f e r e n c e s
(1) F o r l a t e s t r e v i e w s s e e BUTCHER, P. N., P h i l . Mag. 0% ( 1 9 8 0 ) ; ZWAGIN, I . P . , P h y s . S t a t . S o l . (B)
101
( 1 9 8 0 ) 9 .( 2 ) MOTT, N . F . , P h i l . Mag.
