MODELLING OF THE BEHAVIOUR OF HOPPING CONDUCTIVITY IN A SOLID EXHIBITING A SEMICONDUCTOR-INSULATOR TRANSITION, AS A FUNCTION OF FREQUENCY

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MODELLING OF THE BEHAVIOUR OF HOPPING CONDUCTIVITY IN A SOLID EXHIBITING A SEMICONDUCTOR-INSULATOR TRANSITION, AS

A FUNCTION OF FREQUENCY

J. Giuntini, J. Jacquemin, J. Zanchetta, G. Bordure

To cite this version:

J. Giuntini, J. Jacquemin, J. Zanchetta, G. Bordure. MODELLING OF THE BEHAVIOUR OF HOP-

PING CONDUCTIVITY IN A SOLID EXHIBITING A SEMICONDUCTOR-INSULATOR TRAN-

SITION, AS A FUNCTION OF FREQUENCY. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-

95-C4-98. �10.1051/jphyscol:1981416�. �jpa-00220783�

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JOURNAL DE PHYSIQUE

CoZZoque C4, supple'ment au nO1O, Tome 42, octobre

1981

page

c4-95

MODELLING O F T H E BEHAVIOUR O F HOPPING CONDUCTIVITY IN A SOLID EXHIBITING A SEMICONDUCTOR-INSULATOR TRANSITION, A S A FUNCTION O F FREQUENCY

* *

J.C. ~ i u n t i n i * , J.L. Jacquemin

,

J.V. ~ a n c h e t t a * and G .

ordure**

Universite' des Sciences e t Techniques du Languedoc, PZace

E .

BataiZZon, 34060 Montpe ZZier Cedex, France

* ~ a b o r a t o i r e de Chimie Physique

** Centre d1Etudes drEZectronique des SoZides

A b s t r a c t . We propose a model which e x p l a i n s t h e e v o l u t i o n of t h e frequency dependent c o n d u c t i v i t y w h i l e a n organized s t r u c t u r e appears i n an i n i t i a l l y a m o r - phous system. We assume t h a t t h e c r i s t a l l i n e o r g a n i z a t i o n i s connected w i t h t h e e x i s t e n c e of an a n i s o t r o p y . An a n i s o t r o p y parameter i s c a l c u l a t e d . An"apriori!' c a l c u l a t i o n of t h e c o n d u c t i v i t y i s developped, which c o r r e c t l y accounts f o r t h e experimental r e s u l t s o b t a i n e d on "low temperature" carbons.

I n t r o d u c t i o n . - The t h e o r e t i c a l s t u d y of amorphous s o l i d s has widely used t h e con- c e p t of l o c a l i z e d s t a t e s bound t o t h e e x i s t e n c e of a random p o t e n t i a l i n t h e s e ma- t e r i a l s . The numbering of s i t e s (donors and a c c e p t o r s ) among which t h e e l e c t r o n s mo- v e i n a n e l e c t r i c f i e l d can be o b t a i n e d by observing t h e behaviour of c o n d u c t i v i t y a s a f u n c t i o n of frequency ( 1 , 2 ) . The i n t e r p r e t a t i o n of r e s u l t s t h u s o b t a i n e d requi- r e d an implement of a s e t of models ( 3 , 4 ) based, f o r most of them, on t h e assumption t h a t t h e s o l i d can be r e p r e s e n t e d by a p o p u l a t i o n of l o c a l i z e d s t a t e s ( 5 ) . Some au- t h o r s (6,7) have d e s c r i b e d t h e conduction mechanism i n t h e s e m a t e r i a l s by modelling t h e random path of e l e c t r o n s : t h u s t h e h i n t t o c r e a t e s i m u l a t i o n s (8) u s i n g perco- l a t i o n s methods ( 9 ) . A l l t h e s e models c o n t r i b u t e t o p o i n t o u t a c o n d u c t i v i t y law a s a f u n c t i o n of frequency i d e n t i c a l t o t h e experimental law o(w) 2. us.

Some works have r e p o r t e d m e t a l - n o n m e t a l t r a n s i t i o n s (2) due t o a g r a d u a l r e o r g a n i z a t i o n of t h e c r y s t a l s t r u c t u r e of t h e compounds. I t seems i n t e r e s t i n g t o c o n s i d e r a d e s c r i p t i o n of t h e s e t r a n s i t i o n s by t r y i n g t o e s t a b l i s h a c o r r e l a t i o n between t h e s exponent, t h a t c h a r a c t e r i z e s t h e behaviour of conduction a s a f u n c t i o n of frequency, and t h e d e g r e e of s t r u c t u r e o r g a n i z a t i o n , measured with t h e h e l p of a n o r d e r parameter a s d e s c r i b e d by Landau ( 1 0 ) . By doing s o , we s t a r t f--om a model, d e s c r i b i n g t h e p r o p e r t i e s of amorphous compounds, which c o n t a i n s parameters c a p a b l e of i n t r o d u c i n g g r a d u a l l y t h e concept of o r d e r .

I

-

Proposed model.- Generally i t i s considered t h a t t h e p r o p e r t i e s of amorphous s o l i d s a r e explained w i t h t h e h e l p of i s o t r o p i c models. Thus t h e way t o c o n s i d e r t h e appearance of an o r d e r c o n s i s t s i n i n t r o d u c i n g an a n i s o t r o p y f a c t o r i n t o t h e c r y s t a l s t r u c t u r e . The d i e l e c t r i c c o n s t a n t becomes a t e n s o r whose t h r e e components E

, EY

E, can be determined. Consequently t h e p o t e n t i a l around an atom must a l s o bexanlso- t r o p i c , which i m p l i e s t h a t , f o r a given normal mode t h e phonons frequency changes according t o t h e considered d i r e c t i o n ( l l ) , t h a t i s v,, vy, v z . For t h e s a k e of s i n - p l i c i t y , we have supposed t h a t E, = c y ( t h u s vS = v y ) . Thus t h e d i e l e c t r i c c o n s t a n t a n i s o t r o p y induces a d i s t o r s i o n of e q u i p o t e n t l a l s u r f a c e s around a donor s i t e i n t h i s model. The s u r f a c e becomes e l l i p s o i d which l a r g e a x i s , i . e . p a r a l l e l t o Oz, h a s a l e n g t h 2 8 , a n d which s h o r t a x i s h a s a l e n g t h 2 b , f o r a given v a l u e W, of t h e energy, W , r e p r e s e n t s t h e h e i g h t of t h e p o t e n t i a l b a r r i e r t h a t an e l e c t r o n s e t t l e d i n a do- nor s i t e a t t h e c e n t e r of t h e e l l i p s o i d must overcome t o r e a c h a s i t e a t a d i s t a n c e R on t h e e l l i p s o i d . It i s due t o t h e coulombian i n t e r a c t i o n f o r c e s (12,13). I f we c a l l WM t h e energy r e q u i r e d t o e x t r a c t a n e l e c t r o n from a donor s i t e ( 3 ) we o b t a i n : WM-Wx = e2(bcX)-1 ; WM-W, = e 2 ( a c z ) - 1 ; and a l o n g a d i r e c t i o n making an a n g l e €I w i t h

t h e Ox a x i s , Wu-We = e 2 ( ~ ~ e ~ ) - l , Re and E

e

being t h e v a l u e s of t h e r a d i u s v e c t o r and of t h e permittivity along t h e g i v e n d i r e c t i o n . Obviously, on an e q u i p o t e n t i a l s u r f a - c e , we have : W =

y (

0 = Wx = W o r E ~ / E= b / a . When an e l e c t r i c f i e l d ~

3

of c i r c u - l a r frequency w i s a p p l i e d , which d i r e c t i o n makes an a n g l e Y $ w i t h t h e Ox a x i s , each p a i r of s i t e s is l o c a t e d by y ( a n g l e w i t h t h e f i e l d d i r e c t i o n ) and R ( d i s t a n c e b e t - Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981416

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C4-96 JOURNAL DE PHYSIQUE

ween both sites). Under these conditions, the pairs of sites thus characterized par- ticipate to the conductivity. The real part is (14) :

i(w) represents the current density, n(y,R) is the number of identical pairs of sites and T is the relaxation time corresponding to electron hop.

Let dp(R, y) be the number of pairs in the elementary volume dV ; general- lizing relation (1) leads to the expression of the real elementary conductivity,

(assuming that ch2(w/2k~)

2

1 (5,14)) :

2 2 2

du'(w) = dp (y ,R)e R cos y(4k~)-1w2~ (I+w~T~)-' (2) Let NA and ND be the number of donors and acceptors sites perunitvolume, we find the distribution dp(y,R) of the number of pairs in the blementary volume

dV : dp(y,R) = NAND~V. The quantity dV is the volume determined by the variations dB and d$(J, is the angle of rotation of the ellipsoid around the Z axis), and between two equipotential ellipsoidal surfaces, both centered on the donor site and characterized by the semi-axes a and a + da. Using the parametric form of the el- lipsoyd in cylindric coordinates we obtain : dV = a2<~z/~x)2da sin6 dBd$, where

B

is related to 8 by the equation a cos = R cos 8. The appearance of an ordered structure in an initially amorphous solid generally begins by short distance order around localized sites (15). Gradually these sites can be changed inao conducting microdomains in which the electronic energy levels can be considered as delocalized (16).

However, at the beginning of this evolution, i.e. before the appearance of a long distance order, these organized microdomains are still randomly oriented in the material (2). This phenomenon is taken into account by considering the mean value of do'(,), in the whole space. Thus, we obtain the mean contribution to the conductivity of dipoles which acceptor site is between two homothetic ellipsoidalsurfaces centered on the donor site :

2 2

do' (w) = NANDe (36kT)-' ( E ~ / E ~ )

(I

+ 2 (Ezirx)2)u2r(l + u2~')-la4da (3) The calculated conductivity du1(w), is the mean contribution to the conductivity of all dipoles characterized by a ho ping time between T and T + d ~ . The relaxation time in the z direction is : r = uiYexp (Zaa), where a-' represents the Bohr radius of the localized sites. The value da = (2ar)-'d~, introduced in equation (3) gives

the conductivity ul(w) : rw

The integral can be calculated (3,4,5,11) - : lo

2 2 4

0' (u) = iTNAND(144 k~)-'e2(EZ/Ex) (1 + 2(EZ/sX) ),aw,

where a, is the longer semi-axis of the ellipsord such that the relaxation time corresponding to the hop from the donor site (at the center of the ellipsoid) to the acceptor site located on the equipotential surface is T,, with along the z direction UT, = 1. The relaxation time previously defined leads to : w = T-1 = uzexp(-2aa,).

The value of % can thus be evaluated : a, = (2a)-lln (vz/w). ~h$! investigated expression is written :

o' (w) = ~ ~ ~ ~ ~ ( 7 2 k T ) - l (2a)-5e2(~z/~x) (1 2 + ~(E~/E~)~)w'(~~(v~/w)! 4

- - ..

The random orientation of microdomains gives rise to any orientation of the axis of the various

ellipsoids

that characterize the short distance organization of each do- nor site. Consequently, the only parameter bound to phonons, and experimentally ac- ceptable, is the mean frequency vph(17) :

v2

- L(v2 + v2

+

vz)=

4 .

^(v:

⁺

2v2), avec vx = v,,. Let us call

n

= u2/v2, which allog: ;03esfabli~h a measurement of k e anisotropy. It is equal to 1 fGr zn isotropic structure which, in our model, represents the amorphous state, and it takes very high values for a very anisotropic structure, i.e. an ordered structure (1 6

n < mil.

The expression of vph gives :

vz = vph (I(l+2n))-'I2. Calling d = (-(l+~g))-"~ one obtains v = dvph. The quantity d, is equa? to 1 when 11 = 1 , that is when the structure is totzlly 3 disordered and tends to 0 when the system is ordered. Thus this parameter can be related to an anisotropy parameter (10). In fact, the latter must be equal to zero for the most symetric phase and different from zero in the least symetric phase. With p = 1

-

^{d2, we}

(4)

define an anisotropy parameter that meets this criterion capable to describe a transition due to the appearance of an ordered structure in an amorphous material :

P = (v:

-

vi)

lv: ^(I

⁺⁽^v^:^/²^v^:⁾^f¹ ^{with vx}^)I^vz ⁽⁴⁾

Taking into account vZ = dvDh, the equation (2) leads to an equation containing d : 0' (a) = ITN~N,(~~~T)-~ (2~t)-~e*(e~/~~) 2

(I

+ 2(~~/~~)~)w(ln(dv ph /u)14

Case 1. Isotropic : q = 1 or d = 1.

The solid correspondsto a totally disordered material, i.e. amorphous material. Using the approximation suggested by MOTT (4,5), NA = ND = kTN(EF), we find an expression identical (if we except the constant value) to the equation generally proposed for amorphous compounds :

0' (a) = (r/3)kT(N(EF)) 2a-5e2&n(v h/W))4

Under these conditions it has been shown (14) that the reaf part of the conductivity

~'(w) varies as wS, that is :

o(ln(dv lw)) = CIA',

where C is a constant. studyingPkhis function leads to a value of s of the form (4):

dvph

s = 1

-

^4ibn^(-) ⁽⁵⁾

One can see that in the case where dvph/wiE large (with d = I), which is always ex- perinlentally verified,~ + I .

Case 2. q large, d small.

For a given phonon mean frequency vpb,.if d decreases, dvDh/W is such that according to equation (5), s + 0. The conductlvlty becomes independent of the frequency. Between these two limiting cases, s is a decreasing uniform monotonous function of d in the interval ]0,1].

I1 - Experimental verification. We have used the frequency dependent conductivity of

"low temperature carbons" around a non-metal-metal transition (2). In fig. 1 we have drawn the conductivity as a function of frequency (logarithmic scale, T = 186 K) for 3 carbons of heat treatment temperature (HTT) 600, 650 and 7 3 0 " ~ . The values of s are 0.77, 0.61 and 0, respectively (2). We observe that s + o when HTT increases.

Theses carbons are gradually organized when HTT increases, i.e. the dimension of p e r fectly organized crystallites increases regularly (18). This result can be considered as a rather satisfactory experimental picture of the behaviour foreseen by the model proposed.

HTT 730°C

t

t-kC++++++t+t+klt

(Q-lm-l'

^itti.

Frange limit 0,5

10-I 2 4 6 8 ~ ~

Fig.1- Frequency depen- Fig.2- Simulated frequen- s 3 -Effect of the dence of the conductivi- cy dependent conductivity anisotropy factor FA,

-

ty of 3 carbons (T=186K) as a function of the ani- on the exponent s.

with increasing HTT

.

sotropy factor FA.

111 - Calculation of the conductivity in an anisotropic model. To simulate numerical- 17 the proposed model we used a method of ~ercolation. & A The lattice used here is der- wed from previous works (19,20) and is a square two dimensional 30x30 node lattice.

Each node is bound to its neighbours by orthogonal bounds. The bound is considered as a good or a poor conductor according to the value of r : if r is greater than a

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CS-98 JOURNAL DE PHYSIQIJE

g i v e n t h r e s h o l d r , f o r m a l l y d e f i n e d , t h e b o u n d i s c o n s i d e r e d a s a g o o d c o n d u c t o r a n d i s t h e n r e p r e s e n t e d b y a r e s i s t o r & . I f n o t , i t i s c o n s i d e r e d a s a p o o r c o n d u c t o r a n d r e p r e s e l r t e d b y a r e s i s t o r R 1 a n d a c a p a c i t o r C 1 i n p a r a l l e l . I n t h i s w o r k , t h e r a t i o R 1 / R o i s t a k e n e q u a l t o 1 0 4 . ~ e h a v e s i m u l a t e d th e a n i s o t r o p y o f t h e m a t e r i a l by i n t r o d u c i n g a n a n i s o - t r o p i c f a c t o r FA, a s f o l l o w s : e v e r y bound of t h e l a t t i c e i s s c a n n e d o n e a f t e r t h e o t h e r . E a c h t i m e we m e e t a c o n d u c t i n g bound B, i t i s e x t e n d e d by FA-] c o n d u c t i n g bounds a l o n g t h e d i r e c t i o n p e r p e n d i c u l a r t o t h e e l e c t r o d e s , i n d e p e n d a n t l y o f t h e p r e - v i o u s v a l u e s o f t h e s e b o u n d s . Thus we i n t r o d u c e i n t o t h e l a t t i c e s e q u e n c e s o f FAcon- d u c t i n g bounds a l o n g t h e d i r e c t i o n p e r p e n d i c u l a r t o t h e e l e c t r o d e s . O b v i o u s l y FA d e - p e n d s o n t h e e x t e n t o f o r g a n i z a t i o n o f t h e l a t t i c e , s o t h a t FA = 1 r e p r e s e n t s a n i s o t r o p i c s t r u c t u r e . We o b t a i n a new c o m p l e x c o r r e l a t e d l a t t i c e i n w h i c h t h e p o t e n - t i a l i s c a l c u l a t e d a t e a c h mode by t h e G a u s s - S e i d e l i t e r a t i o n p r o c e d u r e improved by a r e l a x a t i o n p r o c e s s d e s c r i b e d e l s e w h e r e ( 2 1 ) . I n o r d e r t o r e d u c e t h e c o m p u t a t i o n t i m e , t h e r a t e o f c o n d u c t i n g bounds i s s e t t o 1 5 . 3 X. Then t h e c o n d u c t i v i t y o o f t h e s i m u l a t e d m a t e r i a l i s known f o r a g i v e n v o l t a g e a n d f r e q u e n c y . Drawing l o g o a s a f u n c t i o n o f l o g w f o r v a r i o u s v a l u e s o f t h e a n i s o t r o p i c f a c t o r , we o b t a i n t h e r e s u l t s shown i n f i g . 2 . We o b s e r v e a n o t e w o r t h y l i k e n e s s b e t w e e n t h e f a m i l y o f e x p e r i m e n t a l c u r v e s ( f i g . 1 ) and t h e c u r v e s o b t a i n e d by s i m u l a t i o n ( f i g . 2 ) . Moreover we f i n d t h e e x p e r i m e n t a l e v o l u t i o n o f s ( f i g . 3 ) . The h i g h v a l u e s f o u n d f o r s , i n a n i s o t r o p i c s o l i d c a n b e e x p l a i n e d by t h e s i m p l i c i t y o f t h e t w o - d i m e n s i o n a l model d e s i g n e d t o i n - t e r p r e t t h e b e h a v i o u r o f a t h r e e - d i m e n s i o n a l s o l i d .

I11

-

C o n c l u s i o n . C o n s i d e r i n g t h e e q u a t i o n (5), we f i n d : v ( I - p ) 1 / 2

s = I

-

4 / t n ( p h 1

w h e r e p i s t h e a n i s o t r o p y p a r a m e t e r f o r m a l l y d e f i n e d ( e q u a t i o n 4 ) . The p r o p o s e d mo- d e l l e a d s t o a r e l a t i o n t h a t b i n d s e x p l i c i t e l y t h e s e x p o n a n t t o a s t r u c t u r a l p a r a - m e t e r p, o n a l a r g e r a n g e o f f r e q u e n c i e s . p c h a r a c t e r i z e s some i n s u l a t o r - s e m i c o n d u c - t o r t r a n s i t i o n s a s s o c i a t e d t o t h e g r a d u a l a p p e a r a n c e o f a n o r d e r i n i n i t i a l l y amorphous s o l i d s . The s p a r a m e t e r , t h a t c a n b e d i r e c t l y o b t a i n e d e x p e r i m e n t a l l y i s a c o n - v e n i e n t way t o e v a l u a t e t h e o r d e r i n some c a s e s .

R e f e r e n c e s .

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( 1 9 5 5 ) 8 5 . ( 1 8 ) F . CARMONA a n d P . DELHAES - J . A p p l . P h y s . 4 9 (1978) 6 1 8 .

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