THE HALL MOBILITY OF HOPPING CARRIERS

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THE HALL MOBILITY OF HOPPING CARRIERS

P. Butcher, J. Mcinnes

To cite this version:

P. Butcher, J. Mcinnes. THE HALL MOBILITY OF HOPPING CARRIERS. Journal de Physique

Colloques, 1981, 42 (C4), pp.C4-91-C4-94. �10.1051/jphyscol:1981415�. �jpa-00220772�

JOURNAL DE PHYSIQUE

CoZloque C4, suppldment au nO1O, Tome 42, octobre

1981

page

C4-91

T H E H A L L M O B I L I T Y O F H O P P I N G C A R R I E R S

P.N. Butcher and J . A . McInnes

Department o f Physics, U n i v e r s i t y o f Warwick, Cooentry CV4

7AL,

England

A b s t r a c t . - We r e p o r t numerical c a l c u l a t i o n s of dc Hall m o b i l i t y f o r a H o l s t e i n 3 - s i t e hopping p r o c e s s . I n t h e r - p e r c o l a t i o n c a s e 7 p o i n t s a r e a v a i l a b l e with ons-1/3 i n t h e range 2 t o 6. They a r e used t o study t h e s u c c e s s of r e c e n t a n a l y t i c a l t r e a t m e n t s . An a n a l y t i c a l c a l c u l a t i o n i s made i n t h e low-tempera- t u r e regime.

I n t r o d u c t i o n . - We d i s c u s s t h e Hall e f f e c t i n a random 3D system i n which a l l t h e e l e c t r o n s t a t e s a r e l o c a l i z e d and t h e e l e c t r o n s move by hopping. A t t e n t i o n has been focused r e c e n t l y on c a l c u l a t i o n s of t h e dc Hall m o b i l i t y which i n v o l v e p e r c o l a t i o n t h e o r y ( 1 , 2 , 3 ) . A Green f u n c t i o n approach has a l s o been developed by Movaghar e t a l . ( 4 ) . We p r e s e n t h e r e v a l u e s f o r t h e Hall m o b i l i t y c a l c u l a t e d numerically and compare them with t h e a n a l y t i c a l p r e d i c t i o n s .

The t r e a t m e n t s r e f e r r e d t o above a r e f o r high temperatures. We g i v e t h e gene- r a l t h e o r y i n s e c t i o n 2 and p r e s e n t t h e numerical r e s u l t s d e r i v e d f r o n i t . I n sec- t i o n 3 we improve t h e a n s a t z used by Butcher and Kumar ( 3 ) a t low s i t e d e n s i t i e s and d i s c u s s t h e a n a l y t i c a l r e s u l t which it y i e l d s t o g e t h e r w i t h t h o s e o b t a i n e d by o t h e r a u t h o r s . I n s e c t i o n 4 we employ t h e improved a n s a t z t o make an a n a l y t i c a l c a l c u l a t i o n o f t h e H a l l m o b i l i t y a t low-temperatures.

General formula and numerical r e s u l t s f o r t h e r - p e r c o l a t i o n c a s e . - F u l l d e t a i l s a r e g i v e n i n r e f e r e n c e s ( 3 ) and ( 5 ) . A f t e r l i n e a r i z a t i o n t h e r a t e e q u a t i o n s may be w r i t t e n a s c i r c u i t e q u a t i o n s modified t o allow f o r t h e presence of t h e magnetic f i e l d . The e q u a t i o n s a r e

0 . a t frequenc:~ 11. Here: C i s a d i a g o n a l m a t r i x of s i t e c a p a c i t i e s , g 1s t h e symmetrical m a t r i x of Miller-Abrahams conductances and gH i s t h e antisymmetric p e r t u r b a t i o n i n t r o d u c e d by t h e magnetic f i e l d . The v e c t o r s

x

and V_ a r e column m a t r i c e s o f s i t e c o o r d i n a t e s and s i t e v o l t a g e s . We suppose t h a t E and B a r e i n t h e x and z d i r e c t i o n s and f o r m a l l y s o l v e e q u a t i o n ( 1 ) f o r V_ up t o terms l i n e a r i n E , B and EB. The Ohmic and Hall c o n d u c t i v i t i e s

cXx

and oxy may then b e d e r i v e d by i n - s p e c t i o n of t h e time d e r i v a t i v e of t h e p o l a r i z a t i o n . For a n i s o t r o p i c system with volume

fi

t h e y a r e , when w + 0 ,

Here: gmn i s t h e conductance between m and n, gmpn i s t h e c o n t r i b u t i o n t o t h e (mn)th Hz m a t r i x element of gH i n v o l v i n g t h e i n t e r m e d i a t e s i t e p, V: is t h e p o t e n t i a l d i f f e - r e n c e between m and n when E i s t h e x - d i r e c t i o n and V =

(vtP. ^vY , v Z

) .

+P mp mp

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981415

JOURNAL DE PHYSIQUE

The H a l l m o b i l i t y p = Oxy/OxxB has been c a l c u l a t e d numerically from t h e s e e q u a t i o n s f o r 2025 randomly l o c a t e d s i t e s w i t h v a r i o u s d e n s i t i e s n when

gHZ _mnp = ga exp H (-olr _mpn ) eB A' _mpn

+I-'

where r i s t h e d i s t a n c e between m and n and r and A~ a r e t h e p e r i m e t e r and z-

mn mnP mPn

component of 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 mpn. The p o t e n t i a l d i f f e r e n c e s a r e - 1

c a l c u l a t e d by d i r e c t numerical s o l u t i o n of e q u a t i o n ( 1 ) w i t h w = B = 0 ,

a

= 1 - 3 5 nm and ga/ga H = 0.145. The r e s u l t s o b t a i n e d a r e t h e p o i n t s i n f i g u r e 1.

P a r t i c u l a r formula f o r t h e r - p e r c o l a t i o n case.- We may c o n f i g u r a t i o n average equa- t i o n s (2) and (3) t o o b t a i n

-

⁰⁰

where t h e primed a v e r a g e s a r e t o b e e v a l u a t e d w i t h s i t e s 1, 0 and 2 f i x e d a t r 0 and r

.

We have used t h e i s o t r o p y of t h e system t o r e p l a c e v e c t o r components-& - i n v a ~ z a n t s c a l a r s .

The e v a l u a t i o n of formula (6) i s c o n v e n i e n t l y completed by u s i n g t h e a n s a t z 2 / 2 2 f 2

= E r

1 1 (8)

where f = 1 when g < gp and f l = g /g when gO1, > gp. Here: gP = ga exp(-s )

is t h e c r i t i c a l p e r c o l a t i o n conductanze 0 1 ( 6 ) . The l n t e g r a l is dominated by c o n t r i b u t i o n s from n e a r t h e c r i t i c a l p e r c o l a t i o n d i s t a n c e rp = sp/2a % nS-lI3 and we o b t a i n t h e f a m i l i a r exponent exp (-2arp) i n

<a

>. To complete t h e e v a l u a t i o n

XX

/

of formula (7) we need a corresponding ansacz f o r

<Il0

x

q2>.

For macroscopic t r i a n g l e s t h i s q u a n t i t y reduces t o 2 EL times t h e v e c t o r a r e a A However, high

-102'.

conductances have e x p o n e n t i a l l y s m a l l v o l t a g e d r o p s a c r o s s them. S i n c e only

50

and 3 2 a r e involved, Butcher and K u m a r suggested t h a t one should simply m u l t i p l y ,,

2 E~~~~~ by f l f 2 . Then <O > c o n t a i n s t h e same e x p o n e n t i a l f a c t o r a s

<axx>

and XY

which i s t h e dash-dot curve i n f i g u r e 1 ( 3 ) . However, t h e above a n s a t z does n o t p r e s e r v e t h e symmetry o f <V x V+,2f w i t h r e s p e c t t o t h e s i t e i n d i c e s . To do t h a t

-1 0

we r e p l a c e f l f 2 by f o f l f 2 a s t h e m u l t i p l y i n g f a c t o r , where f o = 1, g12<gp and f o = gp/g12, gI2>gp. Then

<a

> i s dominated by c o n t r i b u t i o n s from e q u i l a t e r a l

xy

t r i a n g l e s w i t h p e r i m e t e r s 3 r and c o n t a i n s t h e e x p o n e n t i a l exp (-3ar 1. Consequent-

P P

l y 3nnS a-3 is r e p l a c e d by 4 exp (-Clr ) i n e q u a t i o n (9) which y i e l d s t h e f u l l s t r a i g n t l i n e i n f i g u r e 1 ( 5 ) . P

The Green f u n c t i o n formalism of Movaghar e t a l . ( 4 ) y i e l d s t h e dashed curve i n f i g u r e 1 which i s e x c e l l e n t agreement w i t h a l l o u r computed p o i n t s . B o t t g e r and Bryksin (1) and Friedman and P o l l a k (2) c a l c u l a t e

%

by c o n s i d e r i n g t h e c o n t r i b u - t i o n t o t h e H a l l c u r r e n t ( o r t h e H a l l v o l t a g e ) from a small t r i a n g l e of s i t e s 102 l o c a t e d a t a branching p o i n t of t h e p e r c o l a t i o n c l u s t e r which c a r r i e s most o f t h e Ohmic c u r r e n t . The r e s u l t s f o r pH a r e r e s p e c t i v e l y two ( B o t t g e r and Bryksin) and

Fig.1: I i a l l m o b i l i t y a s a f u n c t i o n of ans -1/3

one (Friedman and P o l l a k ) o r d e r o f magnitude below our c a l c u l a t e d v a l u e s (31.

The low temperature case.- Much o f t h e g e n e r a l a n a l y s i s goes through unchanged a t low temperatures. B o t t g e r and Byrksin develop t h e r a t e e q u a t i o n s f o r t h i s c a s e . The approach of Butcher and K u m a r a g a i n y i e l d s e q u a t i o n s ( 2 ) and ( 3 ) . I n t h e s i m p l e s t model ( 7 , 8 ) t h e s i t e e n e r g i e s appear because gmn and g&, i n e q u a t i o n s ( 4 ) and (5) i n v o l v e a d d i t i o n a l f a c t o r s

't,,

= exp ( - 0 ~ ~ ~ ) and Qmpn = [ ~ f n ~ c $ ~ e x p ( B l €

1

) + c . P - 1/3

- 1 P

where 13 = (kT)

,

t h e energy of s i t e p i s w r i t t e n E and

€,,=(I

e,l+l cnI+I em-€,\ ) / 2 . P

Assuming a c o n s t a n t d e n s i t y of s t a t e s

+

we f i n d t h a t t h e a d d i t i o n a l energy averaging which i s now r e q u i r e d is o b t a i n e d by r e p l a c i n g ns by pF ir. e q u a t i o n s ( 6 ) and ( 7 ) and i n t e g r a t i n g over t h e e n e r g i e s of t h e s i t e s involved.

The a n s a t z ( 8 ) c o n t i n u e s t o p r o v i d e a good r o u t e t o t h e f i n a l e v a l u a t i o n of

<axx> which now i n v o l v e s t h e e x p o n e n t i a l e x p ( - s ) w i t h s e q u a l t o ( 8 4 e 3 / p F k ~ ) * (6).

P P

The s u c c e s s of t h e symmetrized a n s a t z i n t h e r - p e r c o l a t i o n c a s e s u g g e s t s t h a t we proceed s i m i l a r l y i n t h e low temperature c a s e . Thus we w r i t e

where f i i s d e f i n e d a s b e f o r e . For f i x e d e n e r g i e s t h e r - i n t e g r a l i n ( 7 ) i s dominated by c o n t r i b u t i o n s from t r i a n g l e s of c r i t i c a l conductances. We s e e from e q u a t i o n ( 5 ) t h a t t h e e x p o n e n t i a l i n t h e r e s u l t i s c o n t a i n e d i n

QlO2 exp [3s - ~ E ~ ~ - B E ~ ~ - G E ~ ~ ) /2

1 .

This f u n c t i o n a c h i e v e s a n e g a t i v e exponent P

C4-94 JOURNAL DE PHYSIQUE

with a maximum magnitude of s a l o n g c e r t a i n l i n e s i n energy space. Hence >

i n v o l v e s t h e same exponentialpas

<o

> and we f i n d simply: XY

XX

For p = l 0 I 9 ev-' cm -3

,

T = 300 K and t h e o t h e r parameter v a l u e s a s b e f o r e : 2 -1 -1

u H s 0 . 5 c m V s

.

Conclusion.- We s e e from f i g u r e 1 t h a t t h e symmetrized a n s a t z and t h e Green f u n c t i o n formalism a r e b o t h i n good agreement w i t h our computer r e s u l t s i n t h e r - p e r c o l a t i o n case. On t h e o t h e r hand, t h e low-temperature t h e o r y p r e s e n t e d i n t h e p r e v i o u s sec- t i o n must b e regarded a s c o n t r o v e r s i a l . Friedma? and-Pollak ( 7 ) and Grunewald e t a l . (8) have t a k e n up t h e same problem and f i n d uH<10-4 cm2 V-I s - l w i t h a

power law dependence on T i n t h e f i r s t c a s e and on e x p o n e n t i a l f a c t o r e x p ( - 0 . 3 7 5 ~ ~ ) i n t h e second. Computer c a l c u l a t i o n s a r e i n p r o g r e s s t o e l u c i d a t e t h e s i t u a t i o n . References.

1 . B o t t g e r H. and Bryksin V . V . , Phys.Stat.So1. (b)

81

(1977) 433.

2. Friedman L. and P o l l a k M . , Phil.Mag. B

38

(1978) 173.

3. Butcher P.N. and Kumar A.A., Phil-Mag. B

42

(1980) 201.

4. Movaghar B., Pohlmann B. and Wuertz D., t o b e p u b l i s h e d i n Journ-Phys. C . 5. Butcher P.N. and M. Innes J . A . t o be p u b l i s h e d i n Phil.Mag. B.

6. Butcher P.N., Phil-Mag. B

42

(1980) 799.

7. Friedman L. and P o l l a k M . , t o b e p u b l i s h e d i n Phil.Mag. B.

8 . Grunewald M., Mueller H . , Thomas P. and Wuertz D . , t o be p u b l i s h e d i n S o l . S t a t e Comm.

Acknowledgements.- We a r e indebted t o t h e a u t h o r s of r e f e r e n c e s 4 , 7 and 8 f o r p r e - p u b l i c a t i o n c o p i e s of t h e i r p a p e r s . PNB would a l s o l i k e t o thank P r o f . G. Landwehr f o r h i s h o s p i t a l i t y a t t h e Max-Planck-Institut i n Grenoble.