THE THEORY OF TRANSPORT IN AMORPHOUS SEMICONDUCTORS

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THE THEORY OF TRANSPORT IN AMORPHOUS SEMICONDUCTORS

B. Movaghar

To cite this version:

B. Movaghar. THE THEORY OF TRANSPORT IN AMORPHOUS SEMICONDUCTORS. Journal

de Physique Colloques, 1981, 42 (C4), pp.C4-73-C4-82. �10.1051/jphyscol:1981412�. �jpa-00220737�

JOURNAL D E PHYSIQUE

CoZZoque C4, suppZ6ment au nO1O, Tome 42, octobre 1981

THE THEORY OF TRANSPORT I N AMORPHOUS SEMICONDUCTORS

B. Movaghar

Fachbereich Physik, Universitat Marburg, F. R. G.

Abstract.- The theory of transport in disordered systems is approached using the Master equation point of view. Emphasis is laid on a micros- copic interpretation. An approximate selfconsistent theory is presented with which it is possible to evaluate hopping d.c and a.c conductivity and Hall mobility. The theory is compared with computer simulations and experiments on amorphous semiconductors. The method is extended to treat diffusion limited relaxation and recombination and applied to evaluate the spin-lattice relaxation time of electronic and nuclear spins in amor- phous semiconductors.

Introduction.- Electronic hopping transport, nuclear and electronic spin diffusion and relaxation, recombination and trapping in the region of localized states in amor- phous systems are all examples of phase incoherent diffusion problems. The particle or excitation density ni(t) generally obeys a Master equation involving diffusion and relaxation terms in a disordered network of localized states

dni (t)

-

= -

1 ^w..

n.(t)(l-n.(t))+l

w..

n.(t)(l-ni(t))-6i ni(t) (1)

dt 1 1 1 3

j ^{3 1 3}

where the W.. represent transition frequencies from a site "i" to a site "j", and 1 1

the loss rates 6i in general will differ from site to site. Depending on the pro- blem under consideration, the €ii can for example represent spin relaxation or re- combination rates. In the absence of loss terms this equation describes ordinary diffusion in a disordered system. This type of description has wide range of appli- cations. Here I want to concentrate on three particularly interesting applications in the field of amorphous semiconductors.

In the first place there is the hopping conductivity problem. In a large class of evaporated and sputtered materials one experimentally finds a frequency dependent hopping conductivity1 of the form

where uxx(0) is the d.c contribution which usually obeys the Mott law uxx(0) = .a e - (T~/T)

'I4

The exponent s is a function of temperature with s C a - b T . A typical and particu- larly complete set of data taken by Balkan and ~orrg' on a-Ge are shown in fig. (1).

In the

~iller- bra hams^

type of approach one transforms the microscopic dif- fusion problem illustrated by ( 1 ) into a macroscopic conductance network. The d.c conductivity is evaluated using percolation techniques4, and yields in 3 dimensions

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981412

C4-74 JOURNAL DE PHYSIQUE

I I I I and for a constant density of states

(crff') LOP the Mott law with To = 2.7 To(Mott)

.

^The

frequency dependence however, is usual- ly attributed to pair processes with

- 4 - electrons hopping forward and backward

between sites producing fluctuating electric dipole moments with a distri- bution of relaxation times. The Austin-

-

~ o t t ~ , ~ o l l a k - ~ e b a l l e ~ pair theories do indeed yield an wS law, the exponent s is however not dependent on temperature as found experimentally. To remedy this

-

discrepancy in the framework of the pair model ~lliott', for example has used the barrier model of pike9 in

0.6

.

place of the usual phonon assisted hop-

- ping rates. We shall see that this dif- ficulty is simply due to the artificial separation between a.c and d.c effect.

I 1 In fact the macroscopic network inter-

I 2 3 'Lo9:r,,,r) pretation, though extremely useful for

accurate computation of the d.c conduc-

~ i 1: ~h~ total conductance of an amor- ~ . tivity, has obscured somewhat the phous germanium sample at the physics and the generality inherent in perature indicated. Inset: the gra- the

dient of the high frequency part of

the conductance characteristics (s) A second example is the spin re- against temperature. From ref. (2) laxation problem in a-Si and a-Ge. Ex-

perimentally Connell and pawlikll,

10 ^.

-

and Voget-Grote and Stuke1° observe

- - -

- - -

- - - -

-

-

a ^- - -

- - -

that the temperature dependent line- AHppiOl----;

width can be written as

n A$,(T) = AHpp(0) + C ( u ~ ~ ( O ) ) ~ ,

I

10 0.5<)1< 1 (3)

01 The temperature dependent linewidth

is therefore proportional to the d.c conductivity to the power of an ex- ponent I.! which is usually less than 1 (fig. 2)

.

^Since

AH^^

^(T) - AH (0) is caused by lifetime broadening or PP spin lattice relaxation, the depend- ence on uxx is a clear indication that hopping is the basic mechanism responsible for the relaxation pro- cess12, transporting the spin to particular fast relaxing centres and bonds. A microscopic random walk

U 10 .q of view is inescapable in this

A third example is spin-lat- a

. 7 7

' 021 022 023 0 2 4 025 026 027 tice relaxation of Hydrogen nuclear

(IIT)~M-~) spins in a-Si:H. The data of Carlos

and ~ a y l o r l ~ are shown in ficr. (3).

Fig. 2: (a) Temperature dependent part of the These a in peak to peak linewidth which the spin excitation diffuses (b) conductivity of evaporated a-Si via dipolar spin-diffusion to par- at four annealing temperatures ticular relaxing centres: hydrogen plotted against T - ~ / ~ . From clusters coupled to structural re- ref. (10) laxation modes. By deuterising the

samples, Reimer et al. l4 have recent- ly supplied direct experimental

evidence for spin-diffusion.

Fig. 3: Nuclear spin-lattice relaxation time of Hydrogen in a-Si:H as a function of tem- perature; vo is the resonance frequency.

From ref. (13)

The motivation of the present work is there- fore to present a microscopic theory of diffuse tran*

port capable of reproducing the well established per- colation results in the d.c limit but valid for all densities,frequencies, and temperatures. The theory is then generalized to include random relaxation or recombination processes and applied to ESR and NMR.

The realisation that a microscopic random walk theory of transport is more powerful and more general is not new. This has been the object of the Continuous Time Random Walk CTRW description due to Montroll, Scher and ~ a x l ~ , 16. In the CTRW formalism, one maps the true random walk onto a regular lattice and assumes that all the disorder can be included in a single site waiting time distribution function $(t). This assumption has now been given rigorous justification by Klafter and silbey17. The approximation for $(t) used by Scher and ~ a x l ~ however yield a d.c. conduc- tivity which is the Miller-Abrahams value rather than the correct percolation result. As far as we know, the theory has not been generalized to treat asym- metric hopping frequencies characterising for example hopping in amorphous semiconductors.

The conductivity.- Let us begin by considering the conductivity. Several authors3. ' ' 1 have shown how to linearise the Master equation in the presence of an electric and a magnetic field. The general expres- sions for the frequency dependent longitudinal

(ox, ( iw) ) and transverse uxy (iw) (Hall) conductivity can be written a s 2 0 r 2 1 r 2 2

respectively, where e is the electronic charge, V is the volume, xi (yi) are the x - ~ j ) ~ , the pointed bracket denotes a (y) coordinates of the "i" th site, x. . = (x.

configurational average, and F i = fi(1

-

fi) where fi is the Fermi function in the limit of infinite Hubbard repulsion energy. The quantity

~ f i

^(iw) can be interpreted as a Green function or matrix element of the operator (iw - K)-', where K is analo- gous to a tight binding Hamiltonian21 with diagonal element K ~ ( ~ ' = -

1

^{K ~ P ) and off-}

diagonal terms K~ j(H) = -rij (H) / F ~ with 1

and the index "H" denotes a magnetic field H. A G ~ $ ~ ) ( ~ U ) is the linear term in an expansion of G. !H)(iw) in powers of H. The GijH)(iw) represents the Fourier-Laplace

1 7

transform of the time dependent Green function GijH) (t) which is the solution of the linearized Master equation,

and can be given a microscopic interpretation as describing the probability of find- int the particle at site "j" at time t if it was created at site "i" at time t = O .

C4-76 JOURNAL DE PHYSIQUE

The problem that one faces in evaluating avv(x) is twofold. For arbitrary disordered networks an exact solution is not possible so that one must 1) find a reasonable approximation to the probability function Gi,(t), in other words to the sum over all paths a particle can take in going form "i" to "j", 2) carry out the configurational average over space and energy. To deal with the path summation we expand Gij (iw) = < i

1

^(iw

-

^K)

-' 1

^j > using the renormalized perturbation expansion

(RPE)21. This has the great advantage of eliminating all the repeated indices leaving only self-avoiding walks from "in to "j". The basis of our approximation which is discussed in detail in ref. (21.22) is now to develop systematic two-site, three-site etc. selfconsistent approximations. In the two-site theory, we treat pair memory effects exactly and approximate the 3-site and higher order memory effects

(i.e. closed loops). Similarly in the 3-site theory we keep up to three site memo- ries and approximate higher order effects. This means that the "particle" always remembers the previous or the previous two sites it has visited on its paths and randomises over the rest. Selfconsistency is achieved by using in each case the cor- responding Effective Medium Approximation (EMA) 5 1 21. The higher order correlation memory effects lead, in this approximation, to an effective reduction of the site density after the first jump. In the present language the Scher-Lax theory is equi- valent to keeping no memory of previous jumps at all i.e. rerandomising after every step. The exact treatment of 3-site processes is essential for the Hall-conductivity for UXx(w) however, it turns out that the two-site selfconsistent theory is per- fectly adequate. It is also interesting to note that our path summation approxima- tions would be exact if the system were an infinitely coordinated a) Cayley tree b) triangular tree lattice. The two-site selfconsistent theory is summarized by the following set of equations

e2 n2 ^p (E) p (El) R2 d8 dE dE'

( 8 ) (T ( E , E ~ ,E) 1-1

+

^{(iw F(E*)} + a1 (E* ,iw)

where ol(EV,w) satisfies the integral equation

and n is the site density, p(E) the normalized energy distribution function (density of states), ap is the density reduction factor due to closed loops21 ap = e-l. We have assumed a random spatial distribution of sites. The thermopower can be evalu- ated21 using the energy dependent conductivity o(E,O)

.

Considerable simplification is achieved if the jump rates W.. are symmetric, here we find

1 I

where nc is the densityof carriers and D(iw) the single particle diffusion coeffi- cient given by

where

P (E) a (iw) = nap

1 (w(E,R) 1-1

+

^(iw + a (iw) )-I 1

Equation (11) can also be written in the more elegant form D(iw) = l*a (iw) which then defines an effective mean squared hopping distance trivially obtainable using ( 11) and (12)

.

^A similar result has been derived by Gochanour et a1. 23 in con- nection with the exciton diffusion problem using a diagrammatic technique. Their re- sult is identical to (11) and (12) if ap is taken as ap = 1/2.

The frequency dependent Hall-conductivity in the 3-site selfconsistent de- scription becomes22

where

; : : * A

represents the first order modification of

~ f r )

in an interference pro- cess over an intermediate site "0" in a magnetic fieldz4. Again oxy (iw) simplifies considerably for symmetric hopping frequencies. One then replaces ol(E,iw) + o (iu), F(E) 3-site hopping junctions (Hall triangles) dominate the Hall conductivity in the hop- + 1 and T

=

³ . +Wi,. In deriving oXy (iu)

,

we have made the usual assumptionls that ping regime. The above equations for uxx(iw) and uXy(iw) are completely general, they apply to any phase incoherent transport processes describable by the Master equation with no constraints on the hopping lengths. All one has to do is to sub- stitute the corresponding "jump frequencies".

The frequency dependence of oxx(iw) has a physically more transparent inter- pretation in time space. For simplicity we shall consider only the symmetric case, here we have

The time dependent diffusion coefficient D(t) defined by (14) is the inverseFourier- Laplace transform

e-l)

^{of D (iw)} /iu. The frequency (time) dependence of D (iw) there- fore reflects the fact that the derivative of the mean-squared distance < ~ ~ ( t ) >

moved by a particle is not a constant in a disordered system.

Fig. 4: The conductivity ux,(0) (left curve) and Hall mobility

uH

(right curve) plotted against ~ t n - 1 1 ~ in the R-hopping problem (Wij = vo ee2~Rij). Solid lines: present theory using eq. (10) and (13). The points are the numerical simulations due to McInnes and ButcherJ1, and the dotted lines are theories due to the same authors. The curves marked BB and FP refer to the work in refs. (18) and (33) respectively, QCA is the diffusion equation approxima- tion (ref. 22).

JOURNAL DE PHYSIQIJE

a,,-"3 : 5

Butcher eC al

-

R W

---- _SL

0n-l/3 = g Butcher et.al

-

R W

- - - - _SL

Defining the degree of dispersion d as d = R e D(w= -)/Re D(w=O), then d,l in general and thus D(t) decreases with time eventually reaching a constant d.c value. At short times the diffusion is do- minated by the well conducting clusters,

in the long time limit, however, the par- ticle must eventually see the slow conduc- ting channels. The time dependence of D(t) is believed to be the cause of the so- called dispersive transport phenome- nonl5, 251 26r 27 which can also be produced by diagonal disorder i.e. trapping and release of particles2*. The degree of dis- persion is dependent on the model con-

sidered. In some models D(t) can in fact t w

vanish, a situation referred to a quasi- locali~ation~~. One should note that our conductivity equations are exact in the high frequency (a+-) and high density limits (n + m) , in each case we obtain the diffusion equation limit, for example

1 ^-

Application to hopping transport.- We have

-6 -6 .j .i -5 .i .j .i .i 6 i

W1v7

analysed the validity of our approximat~on for oxx(a) by comparing our results with

Adlwl the computer simulations of the R-hopping

-5- 90" problem by McInnes and B tcher30. Here we

-7 simply have Wij = vo

1 ,

^{the rates}

a,i"3 = 16 are symmetric and the appropriate equa-

/+

^{et 01 tions are ( 1 1 ) and ( 1 2 )}

^.

The simulation

-9. results are compared with csxx(0) in fig.

S L ( 4 ) . The agreement is excellent for all

-'@I

values of the effective density parameter

The frequency dependence31 Re ax, ( i w ) is shown in fig. 5a-c. At hlgher densities a11-~/~>5, both the Scher-Lax and our theory are ingood agreement, for low densities the Scher-Lax low frequency results begin

-12 -10 -0 -6 -4 -2 -0 to deviate quite strongly as expected. The a.c conductivity follows an wS law with s decreasing with increasing "density"

Fig- 5: The conductivity Re ₍₀₎ for ,1/3 a-i. The reason is that with increas- the R-hopping problemXrs plot-

'L ing n the spatial disorder decreases, even-

ted against frequency (w/VO) tually\ the system becomes a "homogeneous for three different values of

liquid".

Solid line present

theory; points are numerical OXy (0)

The d.c Hall mobility

uH=-

simulations due to McInnes and Ox, (0) ^H-

Butcher, dotted line is the is shown in fig. (4). The theoretical curve Scher-Lax the0 ~ 3 1 . W.. = v using (13) with the symmetric R-hopping

( a .

.

^/

- I .

^I' frequencies is in good agreement with the

1 3 computer points due to McInnes and

~utcher~'. At low densities our results agree with the percolation approach of McInnes and Butcher, Bottger and Bryksin (except for the prefactor) but deviate from those of Pollak and ~ r i e d m a n ~ ~ . In the percolation approach one has to limit the size of the contributing "Hall triangles" and this has been the subject of some con- t r o ~ e r s y ~ ~ ~ ~ ~ . Friedman and Pollak cut off the lengths by using the first nearest neighbour distribution function. The selfconsistent equation (13) however tells us without further assumption that the cut-off occurs at the critical hopping distance

(as originally proposed by Bottger

1u2 and Bryksin18). This is the reason

for the discrepancy. Unfortunately no computer points are available

1u4 in this region of density to check

these predictions and conclude the

1u6 discussion. The Hall mobility in-

-

cluding spatial and energetic dis-

- so

2 W4 order has recently been evaluated

6- by Gr"newa1d et a1. 34 using perco-

8 lation theory.

Consider now hopping at the Fermi level. The rates rij are here the Ambegaokar-Halperin-Langer 35 rates

Fig. 6: The Re ol(o,T) from eq. (12) is plotted

.

^exp[-(\El

⁺

^IE'

I +

^{IE-E' I)/2kB~1}

against frequency (w/Vo) at several tern-

oeratures T- = 5 .

lo7

K. and o(E,iw) has to be solved using

U the integral equation (9). Numeri-

cal solutions show that it is quite permissible to work with the simpler symmetric theory and to interpret the energy dependence as due to symmetric barriers. Writing therefore W . . 1 3 - - v 0 e-2al~ijl

-

IE[/kBT, and substituting this in (9) for a constant density of states p , we

10" obtain at low temperatures

' . T I

x 3 d x

d

_{exa (0) 1}

+

^{1 (16)}

which for T < < T o gives us Mott's law ol (0) =voe- (TO/T)

'I4

^with

To=eTo (~,tt)

,

and is roughly the percolation value. Our treatment Fig. 7: The Re D (o) (solid curve) using (1 1 ) is Of loops effec-

compared with the data of fig. 1 (open tively reproduces the percola- circles) at two temperatures. The 77 K tion number vc in 3 dimensions.

d.c conductivity value is fitted. Mott's optimization procedure is equivalent to treating the lattice as an infinitely coordinated Cayley tree, where a p = l (vc= 1). The path summation approximation is less good in 2 dimensions, but excellent results for the d.c and a.c conductivity are obtained if one identifies ap-' with the corresponding percolation number vc 3 6 .

Turning to the frequency dependence, the numerical solution37 of Re 01 (w) is shown in fig. ( 6 ) . The temperature dependence of the exponent s in the wS region is simply a consequence of the fact that the dispersion d decreases with temperature as it obviously must: with increasing T the energetic disorder disappears. For a given w

,

s decreases roughly linearly37 with T when p (E) = constant. In fact when T - f m and n + m , s + o i.e. there is no frequency dependence in the diffusion equation limit. In time space D(t) behaves as t-S initially, and reaches the d.c value when t l t c where tc = vo-I e(To/T)1/4, the critical hopping ttme. Whether transport is

JOURNAL DE PHYSlQUE

dispersive or not therefore depends in the hopping model on the time scale of the experiment in relation to tc. For sufficiently short times D(t) will always be time dependent in a disordered system.

Application of the theory to experiments on amorphous materials poses some very serious problems. Fig. ( 7 ) is a comparison of Re D(w) with a set of data by Long and ~ a l k a n ~ . The fit to the frequency dependence is perfect however two pro- blems arise: the theoretical d . ~ value is two to three orders of magnitude too small.

Even if one discards this discrepancy, the required value of vo is unreasonably large (vo % 1 0 ~ ~ s - l ) in asimple phonon assisted hopping modeL This is just another example of the problem encountered in all attempts to explain the conductivity of amorphous semiconductors: one needs to explain even the d.c conductivity in the framework of the hopping theory, very large values of vo in the range [1016 - s-l] and large variations in this number because annealing and preparation cause oxx(0) to change by many orders of magnitude whereas the value of To hardly changes at all. The above observation is therefore an example of a more fundamental difficulty not connected with any theoretical approximation, but rather to the fact that our basic under- standing of the fundamental hopping process in amorphous semiconductors is still in- complete. The dilemma is that on the one hand so many qualitative features of the hopping model seem to fit with observation (i.e. the T ~ law, the wS dependence and / ~ the temperature dependence of s) and on the other hand that the absolute values and numbers cannot be explained on the basis of the simple Miller-Abrahams hopping rates.

Diffusion and relaxation.- Let us now extend the diffusion theory to treat relaxa- tion problems. In the electronic and nuclear spin-relaxation problem, one is inter- ested in the time decay of the longitudinal magnetisation. We consider therefore symmetric jump frequencies and let pi(t) = [ni4(t) -niS(t)l be the local magnetisa- tion at the site "i". This obeys the linear Master equation38

dpi (t)

-= -

dt

1

^{pi(t) W . . +}

1

^{W.. p. (t)} - Gi(pi(t) -po) -2Api(t)

3

¹⁷

,

^{3 1 7}

where po is the equilibrium value and 2A the spin-flip rate due to an external r.f field. The decay of the average magnetisation p(t) per site or more generally, the number of excitations surviving at time t, can be evaluated using (po= 2 A = 0 ) ,

p(t)

=<I ,

^{G . . 17 (t) >}

where Gij(t) is the probability function in the presence of loss terms. Before at- tempting to evaluate Gi,(t), we have to distinguish two physically different situa- tions: a) the loss rates "Ai" are local i.e. the effective range of a relaxation centre is less than the average intersite spacing. Here we can assume that the rates 6i are distributed independently p(6i, 6j

. . .

^{6N) =} ~ ( 6 ~ ) p(6. )

. . .

^{p (6N) , remain}

in the localized or site representation and use standard scatte?ing theory tech- niques for diagonal disorder in particular the well established Coherent Potential Approximation (CPA) 39r40, b) In the other limit, when the relaxation rate associated with a given centre is long ranged like for example in the model of nuclear spin- relaxation of paramagnetic impurities treated by de ~ e n n e s ~ l , it is advantageous to work in the continuum representation. In both the ESR and NMR problems of inter- est we can assume that we are in case a). For a more general discussion of diffu- sion and relaxation when the centres have long ranges see ref. (42). Applying the CPA to (18), we obtain

where s(iw) is a single-site frequency dependent relaxation rate. For an ordered network this is given by

where Goo(iw) is the local Green function of the ordered lattice and

and Z(iw) = (~ib(i.w) - iw), in the EMA, the disordered network is obtained by re- placing Z(iw) with oilol (iw). The EMA does not correctly treat relatively strongly isolated sites and clusters and therefore overestimates the long time decay of p(t).

The statistical weight of such configurations is small, it is therefore also a reasonable approximation in the present case and we find

It is evident that in general the decay of p(t) is non-exponential. In the long time limit it is, however, with a time constant

Applying this formula to hopping at the Fermi level at low temperature and identi- fying T - ~ with the spin-lattice relaxation rate ~ 1 - l we obtain

The relationship to (3) is apparent, although we have to remember that (24) is still a complicated selfconsistency equation because of the relation between x(o) and 01 (A). A particularly simple and instructive case is the "deep sink" model with

1-x(0)

6 j = " and 6 i = 0 with probability c and 1-c respectively. Here x(o) :c and the rela-

tion ~ ~(axx) - p ~fcllows with a p = (1-c) A more realistic model12 is to take 6 =

1

yij2wij where yij2 is the matrix element of the hopping induced spin-flip pro-

:

cess! The quantit.ative evaluation of T ~ - ' is considerably more difficult because of the correlation between 6i and Wij. Furthermore one should include spin-diffusion as well as hopping motion. The qualitative conclusion is that in the hopping regime, the spin-relaxation takes place in the fast conductivity paths, in general the exci- tation docs not need to take the critical hopping paths which determine the d.c con- ductivjty12, and the experimental relation ( 3 ) expresses just that.

Application to NMR in a-Si:H.- We can apply this formalism directly to NMR in a-Si:H.

The diffusion rates W . . are now the spin diffusion rates43

1 1

R

w.

_{1 1} . = vo

(s)

^{6 , vo~06 =}

¹

_{20 Y~} ⁴ _fi ² _T2 ⁽²⁵⁾

1 3

Where yN and T2 are the nuclear gyromagnetic ratio and spin-spin relaxation times respectively. The model of Carlos and ~ a ~ l o r l ~ is specified by writing

where ho2 is the mean-squared fluctuations field caused by the modulation of the dipolar interaction in a local mode activation energy Ei and wo is the resonance frequency. For a constant distribution of Ei in the range we can analy- tically evaluate T 1 (N) and find

where A = (4yi -yihz/ol) 2 > 0 , and o 1 = - 20 9e2 T2 n2. Here n is the density of

JOURNAL DE PHYSIQUE

hydrogen sites per cm3 and x is the concentration of relaxation centres. Taking thc experimental. parameters for n and vo and the ones assumed by Carlos and Taylor

(%ax =, 80 K, = 25 K, T, = IO-'s) , one can reproduce the minimum found experimen- tally In fig. (3), but to obtain Tl(min) %0.25 s requires x 2 0 . 1 or in, other words, that more than 1 0 E of these Hydrogen participate ln local mode relaxation clusters.

One s::oul.d expect to see such clusters in the NMR spcctrum, the ones observed how- ever do not correspond to fast relaxing centres14. There 1.5 obviously a problem here and the magnitude of x casts some doubt on the relaxation mechanism for the nuclear spin proposed by Carlos and Taylor. It seems that more experimental and theoretical work is needed to clarify the situation.

References. -

1. M O W N.F., DAVIS E.A., Electronic Processes in Non-Crystalline Materials, Oxford, Clarendon Press (1979)

2. LONG A.R., BALKAN N., Proc. Int. Conf. on Amorphous and Liquid Semiconductors, Cambridqe, Mass. (1979)

3. MILLER A., ABRAIiAMS E., Phys.Rev.

120

(1960) 745 4. POLLAK M., J. Non-Cryst.Solids

11

(1972) 1 5. KIRKPATRICK S., Rev. Mod. PhyS.

5

(1973) 574 6. AUSTIN I.G., MOTT N.F., Adv. Phys.

18

^{(1969) 41}

7. POLLAK M., GEBALLE T.H., Phys. Rev.

122

(1961) 1742 8. ELLIOTT S.R., Phil.Mag. 36 (1977) 1291

9. PIKE G.E., Phys. Rev. B c(1972) 1572

10. VOGET-GROTE U., STUKE J., WAGNER H., ~illiamsburg 1976, AIP, New York (1976) 91 11. CONNELL G.A.N., PAWLIK J.R., Phys. Rev. B

13

(1976) 787

12. MOVAGHAR B., SC3WIWEITZER L., OVERHOF H., Phil. Mag. B

11

(1978) 603 13. CARLOS W.E., TAYLOR P.C., Phys. Rev. Lett.

45

(1980) 358

14. REIMER J.A., VAUGHAN R.W., KNIGHTS J.C., Phys. Rev. Lett.

44

(1980) 193 and Phys. Rev. B (in press)

15. SCHER H., MONTROLL E.W., PhyS. REV. B

12

(1975) 2455 16. SCHER H., I X M., Phys. Rev. B

10

(1973) 4491 and 4502 17. KLAFl'ER J . , SILI3EY R., Phys. Rev. Lett. 44 (1980) 55

18. B(STTGER J., BRYKSIN V., phys.stat.so1.

(5

(1977) 569 and

81

(1977) 433 19. BUTCHER P.N., in "Linear and Nonlinear Electronic Transport in Solids", Ed.

Devreese J.T. and van Doren V.E., Plenum Press, New York (1976) 341 20. BUTCHER P.N., KUMAR A.A., Phil. Mag. B

42

(1980) 201

21. MOVAGHAR B., SCkIIRMACHER W., J. Phys. C

14

(1981) 859 22. MOVAGHAR B., PO~ILMANN B., WURTZ D., J. phys.

c

(in press)

23. GOCHANOUR C.R., ANDERSF3l H.C., FAYER M.D., J. Chem. Phys.

70

^{(1979) 9}

26. HOLSTEIN T., Phys. Rev.

124

(1961) 1329 25. PFISTPR G., SCBER Y., ABv. Phys.

21

(1978) 747

26. SCHONHERR G., BASSLER H., SILVER M., Phil. Mag. B (in press) and this conferr?ncc 27. SCHIRMACHER W., Proc. Int. Conf. on Amorphous and Liquid Semiconductors,

Grenoble ( 1981 )

28. SCHMIDLIN F.W., Phil. Mag. B

41

(1980) 535

29. ALEXANDER S., BERNASCONI J., SCHNEIDER W.R., ORBACH R., Rev. Mod. Phys.

53

(April 1981)

30. McINNES I.A., BUTCHER P.N., Phil. Mag. B

39

(1979) 1 31. McINNES I.A., BUTCHER P.N., Phil. Mag. B

41

(1980) 1 32. McINNES I.A., BUTCHER P.N., Phil. Mag. (in press) 33. FRIEDMAN L., PO1,LAK M., Phil. Mag. B

38

(1978) 173

34. GRUNEWALD M., M ~ ~ L L E R H., THOMAS P.,

WRTZ

D., Solid State Comm. (in press) 35. AMBEGAOKAR V., HALPERIN B.F., LANGER J.S., Phys. Rev. B

4

(1971) 2612 36. IMGRUND H., OVERHOF H., this conference

37. MOVAGHAR B., P O H L W N B., SAUER G., phys.stat.so1. (b)

2

(1980) 533

38. MOVAGHAR B., SCHWGITZER L., in "Tetrahedrally Bonded Amorphous Semiconductors", Carefree Arizona, AIP (1981) in press

39. SCHWARTZ L., EHRENREICH H., Ann.Phys. NY

64

(1971) 100 40. MOVAGHAR B., J. Phys. C

13

(1980) 4915

41. DE GENNES P.G., J. Phys. Chem. Solids

7

(1958) 345 42. HUBER D.L., Phys. Rev. B

2

(1979) 2307

43. BLOEMBERGEN N., Physica

15

(1949) 386