A percolation treatment of dc hopping conduction

JOURNAL OF NON-CRYSTALLINESOLIDS 11 (1972) 1-24 © North-Holland Publishing Co.,

A PERCOLATION TREATMENT OF

dc H O P P I N G C O N D U C T I O N *+

M. POLLAK Department of Physics, University of California, Riverside, California 92502, U.S.A. Received 28 September 1971; revised manuscript received 3 February 1972 It is shown that the current paths considered by Miller and Abrahams in their theory of hopping conduction must be modified for large enough samples, such as are met under most experimental conditions. A possible exception is the case of transverse measurements on thin amorphous films. A percolation model for hopping conduction is set up using Miller and Abrahams' impedance representation of the problem. Approximate solutions for the model yield expressions for dc hopping conductivity. It is found that the density of states functions effects the temperature dependence of the conductivity. When the density of states is constant within a certain region around the Fermi level Mott's T ~ law is obtained. For a density appropriate for impurity conduction at small compensations an activated conductivity is obtained. When the density of states increases as the nth power of the distance from the Fermi energy the logarithm of the resistivity is a linear function of T t,~+l)/(n+4). Some comparison of the theory with experiments on impurity conduction and on amorphous semiconductors is made.

I. Introduction It is the purpose of this note to modify Miller and A b r a h a m ' s 1) (MA) theory for h o p p i n g conductivity in the framework of a percolation theory. A n i m p o r t a n t part of the M A theory is to show that the p r o b l e m of cond u c t i o n t h r o u g h localized states can be replaced by the p r o b l e m of conduction through a n e t w o r k of impedances which interconnect the sites of the localized states. The impedances in the theory are functions of three r a n d o m variables, the two energies of the sites between which an impedance is connected, a n d the distance between them. M A proceed to find an a p p r o x i m a t e solution for the r a n d o m network of impedances by assuming that the ira-

* An extended abstract of this work appears in the Proceedings of the Fourth Intern. Conf. on Amorphous and Liquid Semiconductors. Published in J. Non-Crystalline Solids 8-10 (1972) 486. * Supported in part by an Intramural Grant of the University of California.

2

M. P O L L A K

portant paths of the current are formed by going from site to site via the smallest impedance in the forward direction. It is shown that this assumption must be modified, and more general paths for the current must be considered. A general theory is developed in the second section. Sequences of numbers are constructed which are in a one to one correspondence with all possible paths originating at some site. Each number in the sequence is associated with a site in the path by the order in which they appear. The numbers themselves indicate to which site the path goes next, e.g. a number k directs the path to the kth nearest neighbor from the site with which k is associated. For reasons explained later, only sites which lie in the general direction of the applied field are considered. The average number ~ of paths, originating from some site, whose largest impedance is less than a certain value, can be obtained from the statistical properties of the material. Required are the probabilities Pk(Z) that the kth smallest impedance connected to a site is Z or less. Percolation occurs when is non-zero. This condition gives a simple equation for the largest impedance Zm in a path which characterizes the onset of percolation. When the distribution of impedances is extremely broad, as it is in hopping conduction, the resistivity can be evaluated from Z m. It is shown that it is not necessary to evaluate all pk(Z). TO evaluate Z m it is sufficient to know the average number P(Z) ofimpedences of magnitude Z or less connected to a given site. In the first part of the next section it is assumed that P(Z) is independent of this site. Then P(Z) can be very easily calculated from the statistical properties of the material. The assumption is, however, not justified for many systems of interest. The second section later generalizes the theory to include such systems. The equation for Zm remains similar to that for the simpler case, but the average number P(Z) of impedances less than Z, connected to a site, must now be properly averaged over the various distributions which can occur for different sites. A part of the next section, in small print, explains in some detail how to evaluate _P(Z) [-the averaged P (Z)] from a known density of states function. An approximate formula which relates the resistivity to the impedance Z m is developed. The third section applies the general theory to some specific cases of hopping conduction. In detail are treated the cases of impurity conduction at small compensation, and hopping conduction when the density of states is constant within a region around the Fermi energy. Several other cases are discussed in less detail. The last section makes some comparison between the theoretical predictions of this paper, and some available experimental results.

PERCOLATION TREATMENT OF d c HOPPING CONDUCTION

3

2. General theory 2.1. THE PERCOLATIONCONDITION FOR SITE-INDEPENDENTP (Z) The reason that it is necessary to modify the MA current path is that the path suggested by MA cannot be operative for macroscopic samples, as demonstrated below. Call Pl (Z) the probability that a smallest impedance has the value Z or less. Clearly Pl ( Z ) < 1 for any finite Z, and Pl ( o o ) = 1. In order that a path through some impedances be useful, the highest Z within the path must be, with a non-zero probability, within some finite limits. This must be true even for macroscopic samples where the number of impedances N within a path becomes very large. In macroscopic samples where the resistance of the sample is proportional to the length of the sample the value of the largest Z in a chain must, in fact, be independent of N for large enough N. Hence it must have a finite value in the limit as N--, oe. The probability that the smallest impedance in a MA chain with N elements has the value Z is [p~ (Z)] ~ if no correlation between impedances exists. Clearly, as N ~ o o , this probability is non-zero only for Z = oe. It is advantageous to avoid the places where the smallest outgoing impedance is very large by utilizing occasionally other than smallest impedances, if these happen to be reasonably small. The admission of these impendances gives us a choice of many more possible paths, and it is shown that there is a finite probability that at least some of them consist of only finite impedances in an infinite chain. If Z ' is the largest impedance in a chain, the smallest value of Z', say Z m, which permits the construction of chains penetrating a sample from electrode to electrode is required to obtain an expression for the conductivity. We proceed to evaluate Zm. Sometimes the analogous quantity rm, the smallest separation between states necessary for percolation may be needed. For the sake of brevity, the former is in the following referred to as Z-percolation, the latter as r-percolation. To set up the percolation problem, begin on some site on one surface of the sample. Any one of the possible paths originating on this site can be specified by a sequence of numbers n, each specifying the nth smallest impedance (or, for r-percolation, the nth nearest neighbour) to be taken. Thus the sequence 1, 3, 1, 2, ... means a path which goes from the first site through the smallest impedance to the next site, from there through the third smallest, then the smallest, then the second smallest impedance, etc. We write p, (Z) to be the probability that the nth smallest impedance has the value Z or less (or, for r-percolation, that the nth nearest neighbour is at a distance r or less) and %, to be the number of numbers n in a sequence N impedances long. Then the probability that none of the N impedances in a

4

M.POLLAK

specific sequence is larger than Z is given by

p(N, {vn}, Z) = 1~ [p~(Z)]v~,

(1)

n

and the expectation value o f the n u m b e r o f sequences, with no link larger than Z, out o f all the sequences with the same set o f numbers {v~} is

~(N, {v~},Z) (N!/Hvn!)H Ep~(z)yo, =

n

(2)

n

~v n = N. n

The expectation value o f the n u m b e r o f sequences with all links less that Z, out o f all the possible sequences which begin at some specific site is

F1(X, Z) = E (n!/Hv~!) I-[ [P. (z)] "°. {v.}

n

n

(3)

Clearly percolation can exist only for such Z which makes ri (N, Z ) finite (non-zero) as N becomes very large. Thus we look for the smallest Z which satisfies lim ti (N, Z) > 0. (4) N - + ct~

Following the usual methods of statistical mechanics, we replace the sum in eq. (3) by its largest term. This is given by*

vn = Npn ( z ) / E p~ ( z ) . n

We further approximate the factorial functions by the Stirling formula and obtain lira r~ (N, Z) = ( 2 ~ N ) - ' 1~ [ ~ Pn (Z)/p~ (Z)] ~ [ ~ p~ (Z)] N , l

n

n

which can be shown to imply E P ~ ( Z ) = 1.

(5)

n

Eq. (5) is the condition for the onset o f percolation t. It is now necessary to find the proper expressions for p~(Z). These are derivable f r o m P(Z), the average n u m b e r o f impedances with a value Z or less connected to a site, by the relationship * The following expression is an approximation of Feller's 2) expression for the largest term in a multinominal distribution. t Eq. (5) is actually a lower bound for the onset of percolation. The proper condition for percolation is lim~wo~p (N, Z) > 0, where p (N, Z) is the probability that a site is on an N impedance long path with no impedance larger than Z. Obviously t~ (N, Z) > p (N, Z), so eq. (5) is a lower bound. However, for reasons which follow eq. (6), this point will not be of practical importance here. I am indebted to Dr. S. Kirkpatrick for making me aware of it.

PERCOLATION TREATMENT OF dc HOPPING CONDUCTION

5

P(Z) p,(Z) = [ ( n - l) !]-1

i"

e-P

pn-I

dp = e-P(Z, ~,

d

[p(z)]k/k!

(6)

k=tl

0

2.2. DIRECTIONALCONSTRAINTS Following the method of MA we modify the above expression for p,(Z) by introducing a factor y whose purpose is to restrict the path of the current in a general forward direction. We stress that this condition is extraneous to the percolation problem. Its introduction reflects the fact that the important impedance paths not necessarily identical with the critical percolation paths. These can meander aimlessly through the material, but the current may prefer to chose a straighter path between electrodes even though it may include somewhat larger impedances than the largest impedances in the critical percolation path. It is, essentially, a matter of optimization between a very long path, with somewhat smaller impedances and a short path with somewhat larger impedances. MA coped with a similar problem in their theory. We adopt MA's result of a variational calculation, according to which the current is in most cases restricted to flow in the forward hemisphere (y=½). Only when a difficult hop in the forward direction can be avoided by an easy hop in the backward direction is this easy hop allowed, so for large Z, y = 1. Hence y= 1 forZ>Z',

7=½forZ
where Z' is some relatively large impedance, perhaps sensitive to the exact choice of Z'.

1 Z m.

(7) The result is not

The expression for p. (Z) is now modified as follows

p,, (Z) = exp [ - ½P (Z)] ~ [½P (z)]k/k!

for Z < Z',

k=n

p.(z) = exp [ - P (Z)] i

[P (z)]R/k! --

exp [-- P (Z')] ~., [P (Z')]k/k!

k=n

+exp[-½P(Z')]

k=n

~

[½P(Z')]k/k!

forZ>Z',

(8)

k=n

which makes the condition for percolation to be Z P. (Z) = P (Zm) - ½P (Z') = 1. n

(9)

6

M. P O L L A K

Writing ~ for

[-1--½P(Z')/P(Zm)]-1'

this becomes

(lO) Clearly if ~ were ½ for all Z, we would have ~ = 2. The fact that 7 = 1 for impedances close to Zm brings ~ closer to unity. We estimated ~ = 1.7, but since this result is based on MA's result which cannot be automatically taken over for the percolation theory of hopping conduction, the value of a cannot be considered to be accurately determined at this time. A possibility for an experimental determination of a is discussed in a later section. We make use of an additional constraint, somewhat similar to the above, to show that in this theory the conductance in a macroscopic sample is proportional to its cross sectional area. For this purpose we constrain the current to flow inside channels of identical, macroscopic, but small cross sectional areas. The channeling can be accomplished by assigning the states at the surface of the channels a value of 7 half of that which they would have without channeling. This causes a deflection of the current from the boundaries back into the channel. In the presence of channels the conductance is proportional to the number of channels and therefore to the cross-sectional area of the sample. Thus the proof that the current in macroscopic samples is proportional to the area of the cross section is accomplished by showing that these artificial boundaries have no effect on the results, i.e. that the conductivity is the same with them and without them. This will be so if the condition for percolation does not depend on these boundaries. The value of ~, here is not, as before, a function of Z, but depends on the site. Since different sites have a different 7 the expression for p~,°)(Z) (c here indicates the value with the channeling constraint) must be written

p~¢)(Z) = pop(,°) (Z) + pipti) (Z), where Po and Pi are the probabilities that a site is on the boundary, or inside a channel, respectively, and p~O)(Z) and p~i)(Z) are the corresponding values o f p , ( Z ) . Specifically, p(,])(Z) is given by eq. (8) and p~°)(Z) can be constructed from eq. (8) by replacing each P by ½P and each P ' by ½P'. For sufficiently large channels Po/Pi"~ 1, and Po '~ l. The condition for percolation inside channels then is

[ ~ PoP(°)(Zm)/Pi+ ~ P(1)(Zm)](l-po)= n=l

n=l

= [po P°

(Z,,)/pi + P (Zm)]/o~= 1.

Since P 0 (Zm) = ½P (Z~) we get for the percolation condition P (Zm) = C~which is identical with eq. (10). This proves that the conductance is proportional

PERCOLATION TREATMENT OF dc HOPPING CONDUCTION

7

to the area in macroscopic samples, providing that the cross sectional dimensions are sufficiently large. According to Adkins 3) this means at least 400 A for amorphous carbon. For thinner films Adkins find that the constraints on the percolation paths make the conductance decrease more rapidly than the thickness. So far there is no differer.ce between r-percolation and Z-percolation, Z in any of the above equations can be replaced by r. One difference between the two arises from the different dependence of P on either argument. We begin with the simpler distribution P ( r ) , that is with the case of the r-percolation. For most cases, the radial distribution can be taken to be P ( r ) ---~4~Nsr3 ,

(1 l )

where N s is the density of localized states. The condition for percolation, eq. (10), then gives rm =

1.2 rs,

rs

(12)

~ (34-~N~) - 1 / 3

For Z-percolation we need an expression for P ( Z ) . This is neither as readily available, nor as generally valid as P ( r ) above. To obtain P ( Z ) we may begin with an expression for Z from MA, given in their eqs. (!II-22) to (1II-26). These expressions can be manipulated to give

Z-

posSli49(tCoa'~2(~)2e2r/akTcoth( A ) eaz2 8~ e 2 J A 2kT A x

sinhkT +sinh

El - Ev kT

+sinh

E2 - E v l - 1 k T _] '

(13)

where 1 and 2 label the two sites connected by Z, E v is the Fermi energy, E z and E 2 the energies of the localized states on the two sites respectively,

A = IE~ - E 2 [, r is the separation between the two sites, -~ is the deformation potential, and the rest of the notation is in accordance with MA. The expression can be shown to be symmetric with the interchange of the indices 1 and 2, as well as symmetric for reflection around Ev. In most cases A > kT, so the coth term can be approximated by unity. Of the three sinh terms only one is important for almost any Z. We call E the energy in the argument of the dominating sinh term. If both E1 and E 2 are on the same side of the Fermi energy, E is the larger distance of the two from the Fermi energy. If the two sites are on the opposite sides of the Fermi level, E = A. With this, we get

Z = C (air) 2 e 2"/" ( k T / A ) e E/kT, C =- (97tPosSh'/8e2F- 2) (~Coa/e2) 2 .

(14)

8

M.POLLAK

It must be stressed that the impedances of eq. (14) were derived by MA for conditions prevailing in impurity conduction. Impedances representing hopping conduction in other systems may differ in the pre-exponential, but the exponential functions in eq. (14) are quite general4). In the following we shall use the full eq. (14) unless stated otherwise. The function P(Z) can now be evaluated from the distributions of the random variables E, r, and A which enter the expression for Z, Z

P(Z)=fp(z)dz,

p(z) dz=

0

f

p(E,r,A)dEdrdA

(15)

Z = const.

where the integration in the second equation extends over surfaces of constant Z in the E, r, A space, p(z)dz is the differential probability that at least one impedance connected to a site has the value z+)dz, p(E, r, A) is the probability density that a site has a neighbour at r, E, and A. Instead of using the rather complicated exact surfaces of constant Z, we shall use the approximate ones given by

E/kT + 2r/a = constant.

(16)

With this approximation, the random variable A becomes unimportant and only the distribution p (E, r) is required for eq. (15). It is convenient to change the variables E and r to

E/kT + 2r/a = 4,

E/kT - 2r/a = r/.

(17)

The integration over curves of constant Z is equivalent to the integration over constant 4, i.e. over r/, and dE dr = ¼kTa d~ dr/. Eq. (10) then gives

Z m = Ca2kT m e era,

(18)

where the triangular brackets include some characteristic value or r and A compatible with ~m.* Since the condition ~ = constant does not accurately describe a surface of constant impedance, because of variations in the preexponential, it is sensible to chose for m the median of 1/r2A at

~=~m. The fact that there exists a maximum value of ~ within which the important conduction occurs has the important physical implication that there exists a maximum distance E m from the Fermi level within which conduction occurs, as well as a maximum length r m for hops which contribute to the dc conduc* The necessity for this approximation arises from the fact that an approximate rather than an exact condition for constant Z was used to calculate p (Z) dZ.

PERCOLATION TREATMENT OF d c HOPPING CONDUCTION

9

tion process,

Em

---~

kT~m,

rm --- ½a~m.

(19)

2.3. THE PERCOLATIONCONDmON FOR SITE-DEPENDENTP ( Z ) So far it has been assumed that p,(Z) is uniform for all sites in the material. In most cases this is a good assumption for p,(r) but not very often for p,(Z). It will be seen later that it is a particularly poor assumption when conduction occurs near the Fermi level. The reason that p,(Z) is nonuniform is its dependence on the site energy Ei. This results in a lack of stochastic independence between successive impedences in a chain. For example, impedances with a large E will occur only in pairs because E is at least E i both for an incoming and for an outgoing impedance from a site at El. We stress that this correlation between successive impedances exists even when site energies are entirely uncorrelated. The above correlation does not alter the fact that a sequence of numbers n determines uniquely any path from a given site. The probability that all the impedances in such a path are Z or less is

p,, (Z,)p,2(Z2 I Z I ) . . . p , k ( Z k I Zk_l, ..., Zl) ....

(20)

The indices k mark the kth impedance in the chain. The notation of the conditional probabilities is the usual one. For example, p,3(Z3 ] Z1, Z2) is the probability that the n3th smallest impedance emerging from the third site in the chain is at most Z3 providing that the preceding impedances were not larger than Z1 and Z2, respectively. The following three important observations permit the simplification of eq. (20) into a much simpler form : (1) When the energies of the sites are uncorrelated, only nearest-neighbor correlations between impedances in a chain exist. Thus only the first Z is needed to the right of any bar in eq. (20). (2) In a macroscopically homogeneous material the conditional probabilities are stationary in the sense that they do not depend on the location in the material and hence on the index k. (3) The distribution of impedances at a site, say k, does not depend on how many impedances smaller than Zk_ 1 entered the site k - 1 , i.e. it does not depend on nk-1. The probability given by eq. (20) thus does not depend on the actual sequence of the numbers n but only on the values of v. Hence the left hand side ofeq. (l) can also be used as a left hand side for eq. (20). With the simplifications resulting from the above points an equation similar to eq. (2) can be written

~(N, {v.}, z) -- I-I (s!/F[vk!) [p.(Z]Z)] v", n

k

(21)

I0

M.POLLAK

where pn(Z [Z') is the probability that the nth smallest impedance leaving a site is Z or less provided that the site was entercd through an impcdance Z' or less.The percolation condition derived from eq. (21) is

[Pn (Zm [ Zm)] v" =

(~-

(22)

n

Since the correlation between impedances comes only from the site energies E~ we have P~ (mm[ Zm)

= f Pn (mm I Ei) dp (E i I Zm)'

(23)

where pn(Z[ E~) is the conditional probability that the nth smallest impedance is Z or less if the site energy is E~, and dp (El [ Z ) is the differential conditional probability that the energy of a site is Ei 4- ½dEg if it is entered via an impedance Z ' or less. If ¢' is the exponent ¢ of the impedance Z ' then dp(Ei [ Z') vanishes for all Ei less than - ¢'kT or larger than ¢'kT, so these can be used for the limits of integration. Substituting eq. (23) into eq. (22) and changing the order of summation and integration the equation for Zm becomes Em /~ (Zm)

~ _I" P (Ei' zm) dp (E i [ Zm) =

0~,

(24)

-- Em

where E m~ emkT, and e(ei,Zm)-Y.p.(Zm I E,) is the average number of impedances Z m or less which are connected to a site with energy E v Since S dp(Ei ] Zm) = 1, we recover eq. (10) from eq. (24) w h e n P (E, ] Zm)is independent of E i. The dependence of P (Ei,Z) and of dp (E, I Z) on z is again most conveniently obtained by using { and i/in place of the random variables E and r. 2.4. COMPUTATION OF Z m FROM THE DENSITY OF STATESFUNCTION

We now derive expressions for dp(Ei]Z) and P (Ei,Z) valid for rather general density of stated functions, but with the assumption that the distributions of the random variables E and r are mutually uncorrelated. Fig. 1 shows some general density of states and the Fermi energy which serves here also as the zero of the energy scale. The ordinate gives the energy Ej of a site from which an impedance can enter the site at E v To simplify the procedure the approximation is again adopted in which the variation of the impedance is accounted for by the variation of¢. The probability dp (Et [ Z ) can be viewed as the fractional number of impedances of magnitude Z or less which enter sites at the energy E~4-½dEv It

PERCOLATION TREATMENT OF d c HOPPING CONDUCTION

11

Ei

Ei TT

EF~O

~(Ej)

Fig. 1. A plot of some arbitrary density of localized states. Some particular state at the energy E~ has impedances connected to any other state at Ej.

is convenient to devide the range of Ej into regions I, II, and III for E s > E,, E i > Ej > 0, and Ej < 0, respectively. The division is shown in fig. 1. All the impedances which enter the site at E~ from region I have E = Ej, all those which enter from region II have E = E , and all those which enter from region III have E = E i - Ej. I f E~ is negative, a reflection of the Ej axis through Ej = 0 provides the same result when the reflected co-ordinate is used in the above relations. I f the random variable E is given, the condition that the impedance is Z or less is r ~< (E m E) a/kT. -

-

We thus have

dp(EiIZm)--O,

for

Ei>E m

and for

EI<-Em,

E~

Ei

dp(E,IZm)=K[f v(Es)(Em-EydE+(Em-Ey f v(E)dEj E~

0

0

-F

f - (Era-

(Em-'bEj-Ei)3v(Ej)dEj]v(Ei)dEi E~)

for

E~>E,>0,

(25)

12

M.POLLAK El

0

dp(Ei[Zm)=K[f v(Ej)(Em+Ej)adEj+(Em+Ei)3f v(Ej)dEj -- E m

El

Era+El

-b f

v(Ej)(Em-Ej+EI)adEj]v(Ei)dEi

for

O>EI>-E m.

0

The three terms on the right hand side of the equations come from regions I, II, and III, respectively. The normalization factor K is obtained from Em

f dp(EilZrn)=l. -- Em

We now turn to the evaluation of P(E~,Z), the average number of impedances Z or less connected to a site at energy E~. In analogy with eq. (15) we write Zm

f 0

(26)

z = coflst.

The quantities p(z [E,) and p(E,r ]Ei) are analogous to p(z) and p(E,r) ofeq. (15.), but refer specifically to sites with the energy E i. Since we assumed lack of correlation between the distributions of E and of r, p (E,r [ Ei) can be written as a product p(E[Ei)p(r lEt). The distribution of r is in practice independent of site energy, and is simply 4zcr 2. From the previous discussion and from fig. 1 it is clear that p (El Ei) is

p (E [ e,) = 0,

for E < E,, E~

p(EIEi)=6(E-E,)

f

v ( E j / d E i,

forE=Ei,

El>O,

0

Ei,

p (E [ E,) = v (E) + v (El - E),

for E >

p (~ [ E,) = 0,

for E < IE,I,

(27)

0

P

(E[E,)=6(E+E,)fv(Ej)dEj,

forE=[Ei[,

El
El

p(E[Ei)=v(-E)+v(Ei+E ), for E > IE, I. obtain P(E~, Z), two integrations ofp(EIEi), as indicated

To in eq. (26) are necessary. The procedure is illustarted in fig. 2. Substituting ~ and ~ for r

PERCOLATIONTREATMENTOF dc HOPPINGCONDUCTION

13

a n d E, p (E, r ] E,)drdE is t r a n s f o r m e d into p (¢, r/] E~)d ¢ dr/. The integration over c o n s t a n t Z surfaces is a p p r o x i m a t e d again by the integration over r/. The integration over z then c o r r e s p o n d s to i n t e g r a t i o n over ~. T h e domain o f i n t e g r a t i o n is illustrated in fig. 2 by the shaded triangle. H a v i n g established the p r o c e d u r e it is advantegous, for algebraic simplicity, to revert

t_~ + 2El/kT r Fig. 2. The coordinate systems of the random variables E, r and ~, t/. The two pairs of variables are connected by eq. (17). Only the upper right quarter of the plane has physical significance. Lines parallel to the r/axis are approximately lines of constant impedance. The dark area is the domain of integration of eq. (26). All the impedances from region II of fig. 1 are contained within the heavy line at Ez in this plot. All the impedances of region I and III are contained above this line. No impedances exist below this line. to the variables o f integration r and E. Performing the r integration first, the limits o f r are zero and ( E r a - E)a/2kT. F o r the other integration, over E, the limits then are Eg and E m. The result is Ei

P(Ei, Z ) ~ ~ ( a / 2 k T ) 3 t ( E m - Ei) 3 f v ( E i ) d E i t o Em "['- t" I v ( E ) - ~ v ( E i - E ) ] (E m - E) 3 d E t . J Ei

(28)

~d

The first term comes f r o m the 6 function o f p ( E I Eg) at E = E~, the second term comes f r o m the c o n t i n u o u s p a r t o f p (E [ El) S u b s t i t u t i o n o f eqs. (28) and (25) into eq. (24), and integration over E i, gives the e q u a t i o n for Z m. 2.5. AN APPROXIMATE EXPRESSION FOR THE RESISTIVITY The resistivity 9 is o b t a i n e d f r o m Zm by* * It is implicitly assumed here that the d chains are parallel and disconnected from each other except at the electrodes.

14

M.POLLAK

p = R/d,

(29)

where R is the resistance of a unit lenghth of a percolating path, and d is the density of percolating paths per unit area. The value of R is given by Zm

n

Zm k~l

~

Z=O

J 0

n=l

pn(Zm) L, (30)

where pn(Z)-[.pn(E, [Z)dp(E, [ Z m ) a n d L is the length of the chain of N=~nv~ links. The series converges very rapidly, so only a few terms are needed. Utilizing the fact that the integrand is sharply peaked at the upper limit of the integral we obtain the result R = 0.39 NP(~m ) where

Zm/L --- 1.17p(~m ) Zm/(r),

(31)

/3 (era) - d/~ (¢)/d¢ [ ¢~"

The second equality in eq. (31) comes from L = ½N(r),

(32)

where ( r ) is the length of an impedance averaged over all those with Z < Z mThe factor ½ is for the directional averaging. (Actually this factor should be a little smaller because a few impedances are allowed to go backwards, but we ignore this.) The value of ( r ) can be obtained from E

( r ) = _I" ( r (Ei)) dp ( e i [ Z m )

,

(33)

-E

where (r (Ei)) is the average length of impedances Z m or less connected to a site at the energy El. The value of d is taken to be the density of points through which percolating paths cross in a plane perpendicular to the current. It is calculated approximately as the density of paths which can penetrate a slab of a certain thickness of the material investigated. The paths have the same charachteristics as the percolation paths. The highest impedance permitted in a path is Z m, and the directional condition is also maintained. If the thickness of the slab is much less than rm, many of the paths which can penetrate the slab are not able to propagate for long distances outside the slab. If the thickness of the slab is much larger than rm, many paths will merge inside the slab and their density will lose the desired meaning. We therefore chose r m to be the thickness of the slab. The result of tile calculation is d = 0.33 N~.

(34)

PERCOLATION TREATMENT OF dc HOPPING CONDUCTION

15

The result is also in approximate agreement with percolation theories on various lattices. In any case, d occurs only in the pre-exponential of the resistivity, and the result is in keeping with the accuracy of the other approximations used in the derivation of the pre-exponential. The expression for p becomes p = 3.55 Ns-~ ( r ) -~ /~ ( ~ m ) Zm.

(35)

3. Applications As already stated, the distribution p (r, E) has no generally valid form, but differs from case to case. The differing distributions in various cases can result in different features of the hopping conduction. For example, it will be shown that the difference between the impurity conduction which exhibits an activation energy and Mott's T -~ dependence observed in some amorphous semiconductors 5) arises from different p (r, E). We first investigate a simple case characteristic of impurity conduction at small compensation. Fig. 3 illustrates the model. It shows that the spatial autocorrelation of the (one-particle) potential and the carrier--carrier correlation cause nearly all the states involved in conduction to be at an energy ~e2/xrs . This is even more true in the three-dimensional case. E being nearly constant for the majority of impedances, Z m can be obtained using the simpler r-percolation. The result is

Zm

=

C (a/1.2

rs) 2

(kT/(A)m) exp (2.4 rs/a ) exp (eZ/KrskT).

(36)

(A)m is now the median of A for states separated by r m= 1.2 r~, in the potential of ionized donor-acceptor pairs. The results for the other quantities needed in eq. (35) to obtain the pre-exponential of the resistivity are

N~a,

(37)

( r ) = 0.59 N~-+ .

(38)

/o (~m)

=

19

and Substituting these into eq. (35), p = (160

CrsN~a3kT/(A)m) exp (2.4 r~/a) exp (e2/xr~kT).

(38a)

We turn next to Mott's case where the conduction occurs at the Fermi level, and the density of states can be assumed to be constant. Such a situation may be representative of a number of amorphous semiconductors. The correlation between successive impedances is important here and must be taken into account. To obtain Z m we use eq. (24). We note that no correlation between the distributions of E and of r exists in Mott's model, so we

16

M.POLLAK

=0 ~ ) i ' ~ o - -

\°o

;

/o •

\\

e'l,

e...~e °

oo

o~. x'o

// I I

/ e°'" ~°°\

/ I

I

~

I

/ I

I

I I

I I

+

+

4"-

MINORITY

ION

4-

MAJORITY

ION

•

UNOCCUPIED

/

/ ~

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~ ~ I

I

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+

MAJORITY

Fig. 3. A one-dimensional model for the energies of impurity states for small compensations. The dashed curve neglects carrier-carrier interactions. The solid curve is the potential including carrier-carrier interactions as experienced by a carrier activated from the neighbourhood of a distant minority ion. The latter has relatively minor fluctuations around E = 0. These are even less pronounced for the three-dimensional situation.

can use eqs. (25) and (27) for dp (Eli Z) and P (El, Z). The results are

and

P(Ei, Zm) = 7tvkTa3~4(1 + 2x - 6x 2 + 8x 3 - 5x4)/21,

(39)

dp (E, [ Zm) = 5 (1 -- X) 3 (1 + X) dx/3,

(40)

where x = Eifl/¢m. Substituting eqs. (39) and (40) into eq. (24) and integrating, we obtain for Cm ~m ~--" 1.82 (vkZa3) ¼. (41) We thus get M o t t ' s T - ~ dependence. The factor 1.82 differs slightly from the factor ~ 2 obtained in an independent study by A m b e g o a k a r , Halperin and Langer4), and is close to the upper value (10)~= 1.78 obtained by Jones and Schaich 6). The pre-exponential of Z m is calculated according to eq. (18) and the remarks which follow it. The exact evaluation is difficult, but a g o o d approximate value for ( r - 2A - 1)m is 27/a2kT~ 3 = 4.5 v ~a ~'(kT)- ~, and Z m = 4.5 C (vkTa3) ~ exp I-1.82 (vkTa3)-~:].

(42)

To calculate p f r o m eq. (35) we need P(~m) and ( r ) . F r o m the first o f eqs. (37) we have P (¢rn) = ~-~nvkTa3¢3m = 5.2 (vkTaa) -~ .

PERCOLATION TREATMENT OF d c FLOPPING CONDUCTION

17

The value of ( r > is obtained from eq. (33) ( r ) = l a ~ m = 0.45 a (vkTa3) -¼. The concentration Ns in eq. (34) is replaced here by 2v~mkT, the concentration of localized states between the limiting energies ~mkT and -~mkT. With this, we finally get p = 78 Ca (kTa3v) ¼exp [1.82 (vkTa3)-~]. (43) A model which, in a way, is a hybrid between the two models studied so far is also of physical importance. In this model there is a rather small density of states at the Fermi level, but a much larger density of localized states somewhere above and/or below the Fermi level. This corresponds qualitatively to the case of impurity conduction at large compensations, where the Fermi level lies in states which are either unusually close to a minority ion, or are within a bunched group of minority ions. Either situation is unlikely, so the density of states at the Fermi level must be small. On the other hand, it is much more probable to find a state at the average distance from a minority ion, so the density of states at this higher energy must be much larger. The model may conceivably also represent, in a qualitative way, some amorphous semiconductors. In a very simple version of the model there is a constant density of states v. At and above some positive energy E o the density of states is much larger, say va. This model is illustrated in fig. 4. At sufficiently low temperatures the

Ej

E o-

E F ~0

I

Fig. 4. An idealized density of states used as a model for some hopping systems.

18

M. POLLAK

states above the energy E o do not participate in the conduction process (E~, < Eo) so at very low temperatures the conductivity is that of eq. (41). At sufficiently high temperatures, on the other hand, the states below E o can be ignored because v2>> v. In this region the solution of eqs. (25), (28), and (24) is very simple and gives p oc exp [Eo/kT + 1.5 (v2a3kT)-+].

(44)

If v2 is sufficiently large the second term of the argument of the exponential function can be neglected, so one obtains an activated conductivity with the activation energy E o. The temperature which separates the regimes where p obeys eq. (41) and where it obeys eq. (44) is found by equating the exponents of the two equations. The result is T = 0.28 (Eo/k) a (~rcvEo) ~ Eqs. (25), (28), and (24) can be brought into a form which reveals the temperature dependence of the conductivity for the more general densities of localized states

v(Ej) =

{0E~

for for

Ej>0, Es < 0,

(46)

where c is a constant. Obviously the same temperature dependence is obtained when the density of states is zero for positive energies and increases as [Ej[" for negative energies. It is also obtained for the symmetric case v(Ej)=cIEj[". The derivation for the symmetrical case is somewhat more tedious, but just as straightforward, as the following derivation for the conditions of eq. (46)* Substituting eq. (46) into eq. (25) and writing the normalization coefficient explicitly gives Ei

0 Em

Ern

+ f (~m-E/kT)3EndE]E~dEi/c2(½a)Z f []E~dEi, (47) Ei

0

where the brackets in the denominator are identical with the brackets in the numerator. It is convenient to substitute E/E m and Ei/E m for the two variables of integration. It is then easily seen that all the factors which are constants of the variables of integration cancel in the numerator and the deno* It is obviously impossible, due to the very nature of the Fermi level, that v (E) should vanish on one side of it. However, in some cases most of the states on the one side may be so far removed from the Fermi level that they cannot participate in conduction and therefore can be neglected.

PERCOLATIONTREATMENTOF dc HOPPINGCONDUCTION

19

minator, so the numerator and the denominator are only functions of the limits of integration El~Era, and Em/Em. Since the latter is a pure number, the numerator is only a function of x =-Ei/Em, and the denominator is only a number. Thus dp (El [ Z)=f(x)dx. Substituting now eq. (46) into eq. (28), we obtain P (E,, Zm) = {1j~3ac/(kZ)n+ 1} [(~m -- Ei/kZ) 3 Ei/kr

× f

,~m

(E/kT) n d ( E / k T ) +

0

; ( ~ m - - E / k T ) 3 (E/kT)nd(E/kT)]

•

(48)

Ei/kt

Dividing inside the bracket by ~nm+4and multiplying outside by the same factor, the inside becomes a function of x only, say (p (x), since flEi/~m= = EJEm = x. Eq. (24) for Zm becomes 1

~n+4 (kT)n+ 1K (la)3 c ~ (P ( x ) f ( x ) dx =

(49)

0 Since the integral is a pure number it follows that under the conditions of eq. (46) ~m 0(2 T -(n+ 1~/(n+4) (50) For n =0, i.e. for a constant density of states, this gives again Mott's T -'~ behaviour. For n ~ oo, which corresponds to a sudden very large increase in the density of states at some distance from the Fermi level, the conductivity shows an activated behavior, This is consistent with eq. (44) when v2 is sufficiently large to make the second term in the argument of the exponential negligible. It is worth mentioning that while the symmetric density of states shows here the same temperature dependence as the asymmetric one, the resistivity is not, as may appear at first thought, half that of the asymmetric case but is smaller. The reason for this is that in the symmetric case conduction does not occur independently on the two sides of the Fermi level. Transitions across the Fermi level are important, which means that conducting paths will include impedances connected between sites which are on the opposite sides of the Fermi level. The formal expression of this is the second term in the second integral ofeq. (28). This term does not exist when v (E) on one side of the Fermi level can be neglected.

4. Connection with experiments 4.1. IMPURITYCONDUCTION The most pronounced difference between the present theory, eq. (38a),

20

M.POLLAK

and the MA theory is in the form of the exponential. To compare which is in better agreemnt with experimental results, we use the experiments of Fritzsche and Cuevas 7). The comparison is shown in fig. 5. Fig. 5a reproduces fig. 5 of ref. 7, and fig. 5b replots it in such a way that the present theory should give a straight line. The agreement with the present theory seems to be somewhat better. The value of the radius a of the acceptor state according to this theory is 62 A compared to a value of 90 A obtained from the theory of MA, as reported in ref. 7. The value obtained from ac conductivity by Golin s) is 75 A. As an alternative to the computation of a we could use Golin's value of 75 A to obtain from Fritzsche and Cuevas' data thevalue of ~tin eqs. (10) and (24). The result of this procedure is ~=2.9, a value which seems unreasonably large. The value of a will have to be known with F o r e confidence before this value for a can be accepted. One possible reason why Golin's result for a may not be accurate is the fact that Golin used for his experiments material with a compensation ratio K=0.4, while his theory is expected to work well primarilly in the low compensation regime. In this regime the condition A "~kT, needed for ac hopping conduction, is particularly difficult to fulfill. Using such a theory for cases where the condition A "~kTis not as restrictive would overestimate the value of a and hence of ~.* Recent measurements of conductivity by F. Alien on highly compensated germanium to very low temperatures reveal a transition from a T - ~ behaviour to an activated behaviour 10). This is in agreement with the behaviour predicted here for the simple model used for highly compensated materials. The situation in lightly doped materials is not clear at this time. No evidence for a T - ~ region in these materials exists, and this would be in agreement with this theory. However, the only experimental work reported for lightly compensated materials 11) below 1 °K appears to be in the concentration region where conduction may not be by hopping. 4.2. AMORPHOUSSEMICONDUCTORS The T -* dependence which we obtain is not new and has been observed in a number of cases since Mott 5) first predicted that it should occur. The observation of the behaviour is nicely illustrated in fig. ll of Fritzsche's review paper 12). The additional features in this theory as compared to Mott's is the knowledge of the factor of (vkTa3) -~ in the exponent of eq. (43) and an expression for the pre-exponential. The numerical constant allows us to evaluate the density of states from experiments when some value for a is * An opposing argument could be made that the ac conductivity at small compensation should be relatively large because of correlation effects such as were discussed in ref. 9. The temperature dependence observed by Golin suggests, however, that such an argument does not apply to his experiments.

PERCOLATION TREATMENT OF

dc

21

HOPPING CONDUCTION

o D

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I

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LO

~O

~"

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--

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~

~.~

oo

:-=~

~

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m ~

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0

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22

M, POLLAK

assumed. The densities of states obtained from Fritzsche's plot 12) are as follows. For amorphous germanium v--7.0× 1019 eV -1 cm -3, for amorphous silicon 7.5 x 1019, and for amorphous carbon 2.5 x 1021. A radius a = 3 A has been assumed for the radii of the localized states. The pre-exponential in eq. (43) permits, in principal, an independent determination of a. Unfortunately, it is not possible to utilize this because details of the electron-phonon interaction in amorphous semiconductors are unknown at the present. They are probably different from those prevailing in impurity conduction and used by MA. The material constant C, as well as the factor ( a / r ) 2 (kT/A) in the pre-exponential of Z thus may not be correct for amorphous semiconductors. The pre-exponential in eq. (43) in the present form is therefore probably not useful for the evaluation of a. Although this theory permits the derivation of the pre-exponential, more will have to be learned about the electron-phonon interaction before the preexponential can be derived with some confidence. Even then, it is not clear whether the theory itself is sufficiently accurate to utilize the pre-exponential for the evaluation of a. Some of the specific questions concerning the theory are the implicit assumption of parallel disconnected paths, the implicit assumption that the Bohr radius a is fixed and the neglect of some correlation effects which have been suggested to be important in amorphous semiconductors 9). These questions are currently under investigation. Some work on the correlation effect is reported in ref. 13. Instead of trying to evaluate a from the pre-exponential, it may be somewhat more reasonable to try to evaluate the ratio between the a's in different materials. We note that C in eq. (14) contains the factor a 2 and assume that the rest of C is comparable for the different materials and cancels out. Applying this procedure to the plot in ref. 12, we find that a for silicon and germanium are very similar but for carbon it is about 3.7 times larger. This would bring down the density of states for carbon by a factor of about fifty but the large radius may not be reasonable. It is possible that the mechanism of conduction in carbon is quite different. We note that the entire observed variation in carbon is by less than two orders of magnitude, so the fit to a T -~ dependence may easily be fortuitous. It has also been stressed that an apparent T -+ dependence may occur for other reasons, e.g. a mobility which increases with distance from the Fermi level 1°, 14).

4.3. THIN FILMS An interesting question arises in connection with the discussion of MA's current paths in the beginning of this paper. It was argued that these cannot be the important current carrying paths in large samples. However, short enough samples can be crossed by paths with the largest impedance

PERCOLATION TREATMENT OF

dc

HOPPING

CONDUCTION

23

considerably shorter than Zm. This is so because [PI (Z)] ~ is vanishingly small only when N becomes extremely large. It is desirable to know the length of the samples where the MA current paths cease to be important and the percolation paths discussed in this paper begin to take over. In view of the numerous experiments on conductivity across thin films of amorphous semiconductor materials the question may have some practical importance. It is not a priori obvious that percolation theory should apply here. In the following semiqualitative discussion we assume for the sake of simplicity that the hopping conduction exhibits an activation energy so that r-percolation is adequate for the percolation theory. We require to find the boundary below which the current is carried primarily in the MA paths, and above which it is carried primarily in the percolating paths. The desired boundary is therefore defined by an equal current density in both types of paths. The current density in either case is proportional to the cross-sectional density of current paths divided by the largest impedance in a path. For the paths discussed in this paper this is 0.33N)/Zm, for the MA paths with a largest impedance Z it is roughly {[Pl (Z)] N Ns}~/Z. If the thickness of the sample is t then

N = 3t/ ~ 3tl[rsF($, (,'/r,)3)]. Inserting p for firs, exp(2rs/a)(pm-p) for Zm/Z, and e x p ( - ½ p 3) for Pl (z), we get for the thickness which separates the MA and the percolation regimes t ~ F (5,/93) rs [ln 3 + (2 rs/a ) (,om - p)].

(51)

The value of t is largest f o r / 9 ~ 1, in which case eq. (50) gives t ~- 0.24 r2/a. For impurity conduction the thickness of samples is always very much larger, so the percolation theory always holds here. While the above duscussion does not strictly apply to the thin films where hopping conduction follows a T - + law, we use it as a qualitative criterion. Using the values for v and for a cited previously for amorphous silicon and germanium, and putting 4 rs=(TrcCmkT ) - - ±~, we get t-~ 1000 ~ at liquid nitrogen temperatures. It is thus conceivable that for films below this thickness a transition to a different mode of conduction across the film should occur. If so, experiments should reveal that the transverse conductivity increases with decreasing thickness of the film. In conclusion, we stress that some further work on this theory is necessary before its general applicability. This must primarily include the introduction of the Bohr radius as a random variable, the clarification of the parameter of eq. (2), a detailed justification of the implicit assumption of parallel dis-

24

M. POLLAK

connected paths, a n d the inclusion o f correlation effects which have been suggested 9) to be o f i m p o r t a n c e in a m o r p h o u s semiconductors.

Acknowledgements I a m i n d e b t e d to Dr. S. K i r k p a t r i c k a n d to Dr. M. K n o t e k for very helpful discussions a n d comments. Professor T. D. Holstein was the first to express to me d o u b t a b o u t the M A paths. I wish to t h a n k Professor H. Fritzsche, Dr. B. I. Halperin, and Drs. R. Jones a n d W. Schaich for preprints o f their p a p e r s sent p r i o r to publication. T h a n k s are also due to Dr. E. A. Davis for pointing o u t to me some c o m p u t a t i o n a l mistakes.

References 1) A. Miller and E. Abrahams, Phys. Rev. 120 (1960) 745. 2) W. Feller, An Introduction to Probability Theory and its Application, Vol. 1, 3rd ed. (Wiley, New York, 1968) p. 171. 3) C. J. Adkins and E. M. Hamilton, in: Proc. Intern. Conf. on Conduction in Low Mobility Materials (Taylor and Francis, London, 1971) pp. 229 If. 4) V. Ambegoakar, B.I. Halperin and J. S. Langer, J. Non-Crystalline Solids 8-10 (1972) 492; V. Ambegoakar, B. I. Halperin and J. S. Langer, Phys. Rev. B 4 (1971) 2612. 5) N. F. Mott, Phil. Mag. 19 (1969) 835. 6) R. Jones and W. Schaich, J. Phys. C 5 (1972) 43. 7) H. Fritzsche and M. Cuevas, Phys. Rev. 119 (1960) 1238. 8) S. Golin, Phys. Rev. 132 (1963) 178. 9) M. Pollak, Discussions Faraday Soc. 52 (1970) 13; Phil. Mag. 23 (1971) 519. 10) N. F. Mott, J. Non-Crystalline Solids 8-10 (1972) 1. 11) G. Sadasiv, Phys. Rev. 128 (1962) 113. 12) H. Fritzsche, in press; M. Morgan and P. A. Walley, Phil. Mag. 23 (1971) 661. 13) M. Knotek and M. Pollak, J. Non-Crystalline Solids 8-10 (1972) 505. 14) M. H. Cohen, private communication.