MONTE CARLO SIMULATION OF VARIABLE RANGE HOPPING AT THE FERMI LEVEL

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MONTE CARLO SIMULATION OF VARIABLE RANGE HOPPING AT THE FERMI LEVEL

G. Schönherr, H. Bässler, M. Silver

To cite this version:

G. Schönherr, H. Bässler, M. Silver. MONTE CARLO SIMULATION OF VARIABLE RANGE

HOPPING AT THE FERMI LEVEL. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-111-C4-

114. �10.1051/jphyscol:1981420�. �jpa-00220827�

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JOURNAL DE PHYSIQUE

Colloque C4, supplement au n°10, Tome 42, ootobre 1981 page C4-111

MONTE CARLO SIMULATION OF VARIABLE RANGE HOPPING AT THE FERMI LEVEL

G. Schonherr, H. Bassler and M. Silver

Fachbeveich Physikalisehe Chemie, Philipp-Universitat, Hans-Meerwein-Stra&e, D-SSSO Marburg, F.R.G.

Department of Physios and Astronomy, University of North Carolina, Chapel Hill, N.C. 27514, U.S.A.

Abstract.- Employing the Monte Carlo technique hopping at the Fermi level in a cubic lattice has been simulated as a function of temperature, electric field, hopping parameter 2a, and time. It has been found that the predictions of the simple theories agree remarkably well with our computer experiments. In our work we find a true d.c. conductivity proportional to exp -(T /T)!/1* at low temperatures. The time dependence shows dispersion at short times due to energy relaxation and-later equilibrium to the average hopping distance.

Introduction.- Perhaps the most simple, yet elegant approach to hopping at the Fermi level was the optimization approach by Mott (1) where the fameous In a T 1'1+ law was found. This law has since been verified (see for examples G.G.Robert et al.(2))and now put on firm theoretical grounds by Movaghar and Schirmacher (3). While experiments (2) have shown the T~^-/^ law the experimental data gave a conductivity which was too large by a factor of more than 10 . Because of this discrepancy we have decided to perform a computer simulation to test these simple theoretical ideas and approxima- tions. We have found excellent agreement between our Monte Carlo simulations and theory regarding temperature, hopping parameter, and electric field. At high temperature an activated conductivity is found and simple arguments are presented for the magnitude of the activation energy.

Computational Methods.- Because of its simplicity the Monte Carlo Method has- fre- quently been applied to treat transport properties of disordered solids (see e.g.

(4,5)). In the present computation a volume of 30x30x69 sites with cubic symmetry (lattice parameter a = 10"' cm) was used employing periodic boundary conditions along y and z rendering the effective sample dimensions in transverse directions practically infinite. To mimic hopping at the Fermi level the distribution of the energy of the hopping sites was approximately by a half-Gaussian, N(e) =

2 exp(-e2/2tr2) /(27r)*/2a3o- forS^O, the Fermi level being located at e = 0. We chose a = 0.3 eV which is » k T at all temperatures studied. Hopping between two sites j and k is governed by

13 -1

To simulate real systems we assumed v = 10 s . For details of the simulation method see refs.(5,6).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981420

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I n o r d e r t o s t u d y t h e energy r e l a x a t i o n we i n i t i a l l y c r e a t e a hopping p a r t i c l e a t random on an a r b i t r a r y s i t e l o c a t e d w i t h i n t h e x = 1 p l a n e . This g i v e s us an i n i - t i a l e n e r g e t i c non e q u i l i b r i u m . However, s i n c e t h e p a r t i c l e t h e r m a l i z e i n only a few jumps, t h i s does n o t d i s t o r t our d . c . c o n d u c t i v i t y r e s u l t s . We s t u d y 20 p a r t i c l e s p e r l a t t i c e c o n f i g u r a t i o n and average over 50 d i f f e r e n t c o n f i g u r a t i o n s . We s t o r e n ( x , t , s ) where

n

is t h e number o f jumping p a r t i c l e s . We t h u s can c a l c u l a t e < x ( t ) >

and < & ( t ) > from which we o b t a i n t h e c o n d u c t i v i t y and t h e average energy. These ave- r a g e v a l u e s a r e s t u d i e s a s a f u n c t i o n of temperature and e l e c t r i c f i e l d f o r 2aa r a n g i n g from 3 t o 10.

R e s u l t s and Discussion.- F i g u r e 1 summarizes o u r r e s u l t s f o r l o g i m v s T - " ~ f o r v a r i o u s v a l u e s of 2aa i n t h e l o n g time l i m i t (thermodynamic e q u i l i b r i u m ) . A t high temperatures t h e c u r r e n t is a c t i v a t e d w h i l e a t low t e m p e r a t u r e l o g i m a ( ~ o / ~ ) 1 / 4 . Our To d a t a a r e b e s t f i t t e d by To

=

( 2 , 3 ) where NF is t h e d e n s i t y of s t a t e s a t t h e Fermi l e v e l , i n t h e p r e s e n t c a s e NF 2 / ( 2 ~ r ) ' / ~ o a ~

.

Using t h e simple r e l a t i o n - s h i p d e r i v e d by Mott ( 1 ) we a l s o can d e r l v e a value o f t h e most probable hopping d i s t a n c e <R>. Data f o r T = 25OK a r e l i s t e d i n t a b l e 1.

F i g . 1: l o g i- vs Tel/'. Open c i r c l e s

P

a r e f o r a f i e l d F

=

105 ~ c m - 1 , c l o s e d c i r c l e s a r e c o r r e c t e d f o r t h e high f i e l d e f f e c t shown i n f i g . 2 . The i n s e r t shows Ea d e r i v e d from t h e high temperature d a t a . The dashed li e i n d i c a t e s Ea = (NFaSP)-l.

Table 1: Simulation r e s u l t s f o r hopping a t t h e Fermi l e v e l . Data f o r

<R>/a r e f e r t o T = 25 K.

The f a c t t h a t <R> c o n s i d e r a b l y exceeds t h e l a t t i c e parameter a , e x p l a i n s why we can r e c o v e r t h e a n a l y t i c r e s u l t s , w h i c h have o r i g i n a l l y been worked o u t f o r a much more complicated s p a t i a l a r r a y of hopping s i t e s ( 3 ) , b y c o n f i n i n g t h e simu-

l a t i o n s t o a s i m p l e c u b i c l a t t i c e . Also shown i n t h e i n s e r t i n f i g u r e 1 i s t h e high temperature a c t i v a t i o n energy Ea vs t h e hopping parameter 2aa.

I t i s e a s i l y v e r i f i e d t h a t it should be given by Ea = ( ~ ~ awhere p is t h e ~ ~ ) - ~ number o f s i t e s which on average a par- t i c l e l o c a t e d a given s i t e can jump t o . S i n c e , & i m , i ( T ) a p exp(-2aa), p i s ob- t a i n e d by e x t r a p o l a t i n g t h e l o g i - = ~ - ' r e g i o n t o T + m . Notice t h a t our d a t a f o r Ea can very w e l l be f i t t e d by u s i n g t h e p v a l u e s from t a b l e 1.

Apsley and Hughes ( 7 ) have p r e d i c t e d t h e c o n d u c t i v i t y behavior 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 i n t h e low temperature regime. The r e s u l t s of o u r s i m u l a t i o n s shown i n f i g u r e 2 demonstrate e x c e l l e n t agreement w i t h t h e i r p r e d i c t i o n s . Concequent- l y , t h e e n t i r e s i m p l i f i e d a n a l y t i c model f o r hopping a t t h e Fermi l e v e l a g r e e s w i t h our computer experiment. Even t h e magnitude of t h e c u r r e n t i s i n accord with t h e o r y when we normalize it t o our e m p i r i c a l jump frequency.

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loL as in6 id

Electrlc Field IVlcm)

-

Pig.2: F i e l d dependence of t h e c u r r e n t f o r a a = 3 and v a r i o u s T . S o l i d l i n e s a r e t h e o r e t i c a l ( 7 ) .

L o g i ( t ) , mean energy

<E ( t ) > and l o g NNS vs l o g t . The simula- t i o n parameters a r e T

=

25 K , F

=

2.105

~ c m - 1 , ya = 3 and a

=

10-7 cm.

One f i n a l p o i n t is of i n t e r e s t . The simula- t i o n s allow us t o examine t h e dynamics of r e l a x a - t i o n t o t h e q u a s i e q u i l i b r i u m hopping a t t h e Fermi l e v e l . I n f i g u r e 3 we show t h e l o g i vs l o g t , l o g < & > vs l o g t and f i n a l l y l o g NNS vs l o g t where

<E> i s t h e average energy of t h e hopping p a r t i -

c l e s , and NNS i s t h e number of new s i t e s a p a r t i c l e h a s v i s i t e d a f t e r a given time t . A s s t a t e d e a r - l i e r , we s t a r t o u r p a r t i c l e s o f f with a p o p u l a t i o n p r o p o r t i o n a l t o t h e d e n s i t y of s t a t e s . One s e e s t h a t t h e energy r e l a x a t i o n t o t h e e q u i l i b r i u m va- l u e i s very f a s t , o c c u r r i n g a f t e r approximately 10 jumps. Within t h i s time regime t h e c u r r e n t de- cays i n a power-law f a s h i o n y i e l d i n g a s t r a i g h t l i n e with s l o p e al-1 when p l o t t e d on a l o g t vs l o g t s c a l e i n d i c a t i n g t h a t t r a n s p o r t i s d i s p e r - s i v e ( 8 ) . However, even a f t e r < ~ ( t ) > h a s become c o n s t a n t , i ( t ) c o n t i n u e s t o d e c r e a s e a s i ( t ) %

ta2-' with a p a l . This s i g n a l s e x i s t e n c e of a s e - cond d i s p e r s i v e regime a t low temperature. I t i s a l s o seen t h e l o g NNS vs l o g t - p l o t which shows two l i n e a r segments with s l o p e s a1<a2<1. The mag- n i t u d e of a 2 d e c r e a s e s with d e c r e a s i n g T . The se- cond d i s p e r s i o n obviously i s n o t due t o t h e energy r e l a x a t i o n term b u t i s a r e s u l t o f t h e i n h e r e n t d i s o r d e r a r i s i n g from t h e i n c l u s i o n of a l l p o s s i b l e hops. A t low temperature more t h a n 100 new s i t e s v i s i t e d a r e needed even i n t h i s simple system be- f o r e t h e t r u e d . ~ . c o n d u c t i v i t y is achieved. For t h e parameters used t o c a l c u l a t e t h e d a t a of f i g . 3 t h i s f i g u r e even exceeds t h e t o t a l number of new s i t e s a p a r t i c l e v i s i t s b e f o r e being c o l l e c t e d a t t h e a b s o r b i n g boundary which t e r m i n a t e s t h e simu- l a t i o n volume. Only f o r T > 50 K t r u e thermodynamic e q u i l i b r i u m i s a t t a i n e d w i t h i n a time l e s s t h a n t h e c a r r i e r t r a n s i t time through t h e sample. This s u g g e s t s t h a t perhaps one should s t u d y t h e a . c . c o n d u c t i v i t y f o r Fermi l e v e l hopping a s w e l l a s t h e d . c . c o n d u c t i v i t y . Perhaps some of t h e d i s - c r e p e n c i e s between t h e e x p e r i m e n t a l r e s u l t s and simple t h e o r y may b e r e v e a l e d .

I n c o n c l u s i o n , we have s i m u l a t e d hopping a t t h e Fermi l e v e l i n a c u b i c l a t t i c e and show r e - markably good agreement with s i m p l e t h e o r i e s . We show t h e high temperature s a t u r a t i o n p r e d i c t e d by Movaghar and Schirmacher ( 3 ) i n c l u d i n g an a c t i - v a t i o n energy which has a p a r t i c u l a r l y s i m p l e i n t e r p r e t a t i o n .

Supported i n p a r t by t h e N a t i o n a l Science Foundation under g r a n t D M R 790023 and NATO g r a n t (SA-2-05B(1953)/870(80)AG).

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