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The ac conductivity of disordered and dielectric solids
Physica 79 B (1975) 323-335 © North-Holland Publishing Company
T H E AC C O N D U C T I V I T Y O F D I S O R D E R E D A N D D I E L E C T R I C SOLIDS I. CLUSTER APPROXIMATIONS
V. HALPERN* Department of Physics, Chelsea College, Pulton Place, London SW6 5PR, England and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel Received 7 February 1975
The ac conductivity is considered of systems in which the current is due to the motion of charge carriers between localized states. It is shown first that for any given frequency co, a number N(w) can be defined such that carrier paths containing more than N(co) steps make a negligible contribution to the conductivity a(w) at that frequency. For sufficiently high frequencies, N(w) equals umty, while as 09 decreases N(co) increases. An analysis of the possible paths in terms of transitions that are slow or fast relative to the frequency 60 leads, in many cases, to a cluster approximation that is valid for frequencies appreciably greater than a critical percolation frequency. The number of states in each of the dusters that are relevant to the calculation of 0(09) decreases as co increases.
1. I n t r o d u c t i o n In m a n y solids, such as a m o r p h o u s s e m i c o n d u c t o r s 1) and suitably d o p e d s e m i c o n d u c t o r s at low temperatures2), the dc electrical c o n d u c t i v i t y is low, and the ac c o n d u c t i v i t y o(co) increases rapidly with increasing f r e q u e n c y co. It is generally believed t h a t the ac c o n d u c t i v i t y in m a n y o f these s y s t e m s is d u e t o t h e h o p p i n g o f charge carders, usually e l e c t r o n s o r holes, b e t w e e n localized states s i t u a t e d at r a n d o m p o s i t i o n s and having r a n d o m energies. This h o p p i n g is a c c o m p a n i e d b y t h e a b s o r p t i o n o r emission o f e i t h e r a single p h o n o n 2, 3) or o f several p h o n o n s 4' 5). S u c h h o p p i n g m a y also be responsible f o r the dc c o n d u c t i v i t y at finite t e m p e r atures 6, 7 ), b u t a larger c o n t r i b u t i o n t o this dc c o n d u c t i v i t y can s o m e t i m e s c o m e f r o m carriers e x c i t e d i n t o band states, as in the e x p e r i m e n t s o f Pollak a n d Geballe 2 ). T h e dc c o n d u c t i v i t y has b e e n studied b o t h in *Present address: Department of Physics, Bar-Ilan University, Ramat-Gan, Israel. 323
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terms of the most probable hop between pairs of states 8) and in terms of an infinite network of resistancesT'9'l°), but most studies of the ac conductivity have concentrated on the hopping between pairs of states a'~x), which is known as the pair approximation. This approximation has the advantages of being exact in the limits of high frequencies and of low densities at any given frequency12), and of being readily visualised physically. Approaches to the ac conductivity which do not make use of this approximation, such as those of Scher and Lax 13) and Moore14), generally involve decoupling procedures, or the partial summation of infinite sets of diagrams, the physical significance and validity of which is not so easy to understand. In this paper, we show that the pair approximation is only a special case of a more general cluster approximation that can be defined for frequencies greater than a critical percolation frequency Pc. A related problem is the ac dielectric loss e" (w), in solid dielectric media such as alkali halide crystals and polymers. Here, too, the loss is usually attributed to the motion of charge carriers, such as vacancies or dipolar groups of ions, between discrete localized states. However, such transitions may involve a consequent rearrangement of the system, and may sometimes be better described as changes in the state of the system rather than as the motion of individual charge carrierslS). The dielectric loss in these systems can be treated by the same formal analysis as that used for the ac conductivity of amorphous semiconductors, with a suitable interpretation of terms such as charge carrier, localized state, etc. It is only for the sake of simplicity that we use in this paper the model of single charge carriers moving between discrete localized states. In order to derive our cluster approximations, we find it convenient to consider the possible paths of a charge carrier between different states. While this approach is not as convenient for purposes of calculation as the rate equation approach used by Pollak and Geballe 2) and by Butcher11), inter alia, it provides a much simpler and more easily understood method of deriving formal results about cluster approximations. In an accompanying paper, we describe and present examples of a convenient general technique for calculating a(60) when the cluster approximation is valid. Specifically, in this paper we consider the contribution of different possible hopping paths to the current produced in the system at time t in response to a step voltage applied at time t = 0, and hence their contribution to the ac conductivity a(60). In section 2, we show that, for any given infinite frequency 6o, it is possible to define a number N(60) such that the contribution to o(60) from all paths of more than N(60) steps is negligible. This number N(60) increases as 60 decreases, and becomes infinite as 60 tends to zero. Such a result can be used as the basis for a cluster approximation in the analysis of the conductivity of general systems.
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However, the size of cluster indicated by this analysis is often much greater than necessary, and in section 3 we show how to define a more useful cluster approximation for frequencies greater than a critical percolation frequency vc. The size of the cluster relevant to the calculation of o(co) depends on w, and in section 4 we show that our approach leads to a much more general form of Pollak's conditions for cluster approximations16). In the latter part of this section we consider some general results that follow from the dependence on frequency and temperature of the size of the relevant clusters in various systems, while a summary of our main conclusions is presented in section 5.
2. General theory It is convenient to consider a system which is initially in equilibrium with no net current flowing in it, in the absence of an external electric field. At time t = 0, a step voltage is applied to the system, so that at all subsequent times there is a small constant applied field Eo. Let j (t) be the current density produced in response to the application of this field. Then the conductivity a(¢o) is related to the Fourier transforms, J(co) and E(¢o) respectively, of the current density and field by the equation J(~) = a(~)E(o~) = o(~)g0/i~.
(1)
We now define a number of properties of the states of the system. Let the contribution to the dipole moment of the system in the direction of the applied field from a particle in state p be/ap, and let ~pa = ~a - #p
(2)
be the change in the dipole moment of the system in this direction when a particle moves from state p to state q. We define the transition rate for such a move, Vpq, by requiring that the probability that a particle in state p at time t moves to state q by time t + 8t be VpqSt; such a definition avoids the need for different treatments for particles obeying different statistics, a problem that is considered in the accompanying paper. We denote by vp the total transition rate out of state p,
Vp = ~qVpq and define a dipole m o m e n t M such that, for all statesp in the system,
(3)
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V.
q
Halpern/Electric conductivity of disordered dielectrics. I
Vpq tlapq [ ~< Mvp.
(4)
Such an upper b o u n d M exists since for large values of [tZpqI the transition probability ~pq decreases very rapidly with I/apql. For tunnelling, for instance, where I#pql is proportional to the distance tunnelled, vpq decreases exponentially with increasing IlapqI. Finally, let qJpq(s)ds be the probability that a particle in state p at time t = 0 remains there until time t = s and then makes a transition to state q within the time interval (s, s + ds): thus
qJpq(t) = Vpq exp ( - u p t), and its Fourier transform,
(5)
~pq (w) is given by
qJpa(~o) = Vpq/(Vp + ico).
(6)
We now consider the contribution,/p (t), to the current density at time t in the direction of the applied field from a particle that was initially in state p at time t = 0. The contribution t o / p (t) from all paths involving only a single transition or step is just
jO)(t) = ~ Cpq(t)lapq,
(7)
q
and these contribute to ,/(co) a term J(~) (¢o), the modulus of which satisfies the inequality IJ~l)(¢o)l < ~ q
II~pqIvpq/Ivp 4- i6ol ~Mvp/lup 4- i6ol.
(8)
It is convenient to define
cp(¢o) = up/Ivp + i¢ol.
(9a)
and c ( ~ ) = max P
Cp(~).
(9b)
Thus, for all states p,
Ij~l)(co)l~
(I0)
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The contribution t o / ( t ) from all paths of t w o steps starting from p is t
/(2)(t) = ~ ~ f ~pq(S)~qr(t-s)dslzqr , q
(11)
r o
and its Fourier transform is j(2)(6o)
= ~ ~ q r
idqrVqrPpq (Vp q- i¢o)-1 (Vq + i6o)-1.
(12)
Hence IJ[2)(6o)1 < M Y
q
<~M Y
vpqvq
Ivp ÷ i6ol-~ Ivq + i6ol -!
c(6o)Vpq/lup+ icbl
(13)
q
Similarly, the contribution o f all paths of exactly n steps to J(6o) satisfies
IJptn)(6o)l <
Mc(6O) n .
(14)
If 6o is non-zero, c(6o) < 1 since up is always finite, and so the contribution from all paths o f more than N steps to J(6o) tends to zero as N tends to infinity. Thus, in calculation of J(6o) and v(6o) it is only necessary to consider paths involving no more than N(6o) steps. Since paths of one step contribute to J(w) a quantity of order Mc(w), it is reasonable to ignore all paths the sum of whose contribution is less than aMc(6o), where a is of order 10-3; this leads to the equation for N(6o) c(6o)N(¢°) < a(1 - c(6o)).
(15)
Two extreme cases are worth considering separately. At very high frequencies, c(6o) will be small and eq. (15) will be satisfied for N(6o) = 1, i.e. only paths involving a single step need to be considered. This is the case if c(6o) is less than a, and so N(6o) = 1
if 6o > max (up/a).
(16)
At very low frequencies, on the other hand, c(6o) is so close to unity that 1 c(6o) < a . In such a case, eq. (15) becomes
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F. Halpern/Electric conductzvity o f dzsordered dielectrics. I
N(~) ~-,max ((2@/w2 ) In (2@/w 2 ))
if w "~ (2a) ½ max
vp.
(17)
P This latter condition allows paths containing many more steps than the condition one might have expected, namely that if the mean time taken for a path of N(w) steps is much longer than the period of the field, and so ifN(6o)/Uq >2>2~r/6o, the contribution from the path is unimportant. Physically, the reason why this latter condition is not sufficient is that, while the contribution from each individual path of this length may be small, the total number of possible paths increases rapidly with the number of steps.' The result contained in eq. (15) could, in principle, be used as the basis for a cluster approximation for o(~o). For charge carriers starting at state p, one would only need to consider the cluster of sites accessible within N(~o) transitions from p. This is, in fact, the only form of cluster approximation valid at all non-zero frequencies and for all possible physical systems. However, it is not a very useful form, since it usually involves many more states than are in fact required, while consideration of different paths is, in general, a very complicated method for calculating o(w). Fortunately, in most systems there is a distribution of transition rates Upq, and in such cases a much more useful form of cluster approximation can be defined.
3. Cluster approximations The results derived in section 2 show that in order to calculate a(w) it is unnecessary to consider paths containing more than N ( ~ ) steps for any given carder. However, this does not imply that all possible paths containing N(w) steps are important. It can happen, for instance, that the only important paths are those involving only transitions between a pair of states; this is the case when the frequently-used pair approximation is valid2). More generally, it often happens that all the important paths of N(~o) steps are confined to transitions between clusters of states that contain much less than the total number of states accessible within N(w) steps from a given state, and are independent of the state within the cluster from which the particle starts. In such a case, one can calculate a(co) by considering only transitions within such small clusters, a problem which is conceptually simpler for a random system and which is mathematically much easier to solve. In this section, we examine the conditions for the validity of such cluster approximations, and how the relevant clusters can be chosen.
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As a preliminary point, we wish to clarify the relationship between the path probability approach used in section 2 and the calculation of a(60). Our approach calculates the current due to all transitions, and not just the extra current produced by the applied field. Thus, we take into account a lot of terms that are of zero order in the field, which must cancel out, and it is not obvious that those paths contributing large terms to J(60) are the most important in determining o(60). The reason that they usually are can be found by considering the difference between the currents j+-(t) produced by applying fields + E 0 at time t=0. The difference between the transition rates for any given step are of first order in the field, and so for a path containing many steps only the differences for one step at a time need be taken into account. (An exact mathematical formulation of this point is simple but tedious, and so will not be presented here.) Hence, the paths important in determining J+(60) - J-(60), and hence a(60), will be the same as those that make the major contribution to J(w) if the difference between the transition rates from p to q, for each pair of states, in the presence of fields + Eo is more or less proportional to Vpq. Since a field Eo changes the difference in energy of the states by Eolapq, the difference in transition rates will (by the principle of detailed balance) usually be proportional to Eo lapq Vpq. Now, as remarked earlier, lapq varies much less rapidly than Vpq with the distance between states p and q. Hence, lapq will be of the same order of magnitude for all transitions with appreciable values of Upq, and it is the values of the Vpq that determine the importance of a given path. Thus, the paths that are important for determining o(60) are, in general, the ones that contribute the largest terms to ./(co). The contribution to J(60) from a particle following a given path p q r . . . uv is just Iduv~lpq ~ q r • • • l~uv ( W ) . Even if this path contains less than N(60) steps its contribution can have a small modulus for one of two reasons~ Firstly, one of the transitions in the path, say that from r to s, may have a much smaller probability than other transitions from r, so that vrs is much less than Vr; we call such transitions minor ones, while those with Vrs of comparable magnitude to vr will be called major transitions. Alternatively, even for a major transition, the transition rate Vrs may be much less than the frequency 6o. Any transition, major or minor, for which Vrs ~ 60 will be called a slow transition, while all other transitions will be termed fast ones; in addition, a state such that all transitions from it are slow ones will be called a slow state. We note that the classification of a transition as fast or slow depends on the frequency, but its classification as major or minor does not. In general, we might expect that all paths containing several minor or slow transitions will make negligible contributions to J(co) and o(60).
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However, J(6o) involves not only the moduli of the ~rs(6o) but also their phases, and also the changes in dipole moment grs which may be positive or negative. As a result, the contributions to J(6o) from successive steps (i. e. from successive sub-paths) along a given path may cancel. For instance, a particle following a closed path makes a zero contribution to fo/(r)dt, and so contributes nothing to J(6o) for 6o = 0. Thus, if all paths with an unlimited number of steps that contain only major transitions return to their starting point, these paths will not contribute to J(0). In such a case, the major contribution to the dc conductivity o(0) will come from paths containing minor transitions. This problem is a well-known one in the calculation of the dc conductivity of random systems, and has been discussed by various authorsT'9'l°). At finite frequencies, the factors ~brs(6o) are complex, so that there will not normally be an exact cancellation of the contribuffons from different steps or sub-paths in a closed path, but if 6o is small the imaginary points of the ffrs(6o) are also small, and the cancellation will be nearly complete. Physically, this means that a closed path that on average is completed within a small fraction of a cycle makes a negligible contribution to f~j(r) exp ( - i6ot)dt, i.e., to J(w). For these reasons, it is difficult to decide which paths are the most important purely on the basis of major and minor transitions. Fortunately, however, the situation in connection with fast and slow transitions is very different. Physically, the meaning of a slow state is that a particle is unlikely to leave it, along any path, during a single cycle of the field. Thus, whatever the signs of the #rs and phases of the complex ~krs, a path reaching a slow state r and stopping there will in general make a much larger contribution to J(6o) than any of the paths continuing from state r. Mathematically, if r is a slow state, Cr(6o) is so small that all the paths involving transitions from r make contributions of negligible modulus, regardless of the signs of the tars or phases of the ~krs. For similar reasons, paths involving slow transitions from a fast state can also be neglected. There are only two exceptions to this situation. Firstly, if all states are slow, as will be the case at sufficiently high frequencies since the vr have a finite maximum, one cannot ignore the contribution from slow steps. However, only paths containing one such step will be important, as those involving two steps will make contributions of modulus proportional to c(6o) 2 , which is much less than c(6o). This is, of course, just the situation in which N(w) = I, as eq. (16) is satisfied. The other exception is that in which a slow state has a very high occupation probability: this will happen, for instance, at low temperatures if one state has an energy appreciably lower than that of its neighbours. In this case, the contribution to o(6o) from paths starting at such a slow state is not negligible (as the number of particles making a slow transition from
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this state in one period may.exceed that of particles making a fast transition in one period from a state with a low probability o f occupation): however, the contribution from paths containing transitions from this state at any other step will be negligible. In order to define a useful cluster approximation, we consider a system in which there is a distribution o f transition rates Vrs and v r. Let us choose a frequency v 0, and examine whether it is possible to construct a continuous path o f transitions from a state at one end o f the system to a state at the other end, such that at each step in it the transition rate Vrs is greater than v 0. If v o is sufficiently large, e.g. if v0 > m a x ISr, this will n o t be possible, while if Vo is small enough, e.g. if Vo < min Vrs, such a path can be constructed. Hence a maximum value of v 0 exists for which such a construction is possible: we denote this by vc, and call it the critical percolation frequency. While the value of v c can be determined b y percolation theory, we note that ours is not the classical bond percolation problem~°), since in general Vrs ~ Vsr, b u t is one of directed bond percolation. F o r values o f v o greater than v c, all the paths consisting exclusively o f transitions with Vrs > v o will consist o f transitions between states within discrete, finite clusters. Hence, at frequencies co appreciably greater than v c, the paths consisting only o f fast transitions will contain only transitions within small clusters o f states, which we call the relevant clusters at this frequency. The states on the edge o f such a cluster will either be slow ones or will have one or more major transitions leading back into the cluster and only slow transitions to states outside it. Our analysis then shows that in general a(co) can be calculated b y considering only the transitions between states within the relevant clusters. F o r the exceptional case in which all transitions are slow, so that only single-step paths need to ',be taken into account, the calculation o f a(co) is trivial n, ~4 ). F o r the other exceptional case, in which some slow states have a very high occupation probability, the sitution is as follows. If some transition from such a slow state r leads to a fast state, state r will be on the boundary o f a relevant cluster, while if more than one transition from r leads to a fast state it may be on the boundary o f several clusters. Since paths that pass through this slow-state without either starting or finishing at it are o f no importance, the contributions from this state to the conductivity o f each relevant cluster can be considered separately, provided that the correct v r is used in evaluating these contributions. Physically, the effect of this v r is to ensure the correct probability of a particle from state r entering a given cluster. Similarly, the contributions to a(co) from transitions from state r to other slow rates are additive, provided that the correct v r is used: for these, o f course, only single step paths are important.
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V. Halpern/Electric conductivity of disordered dielectrics. I
4. Discussion The cluster approximation that we have derived in the previous section is much simpler and more general than those proposed by Pollak 2' 16). However, in order to see the practical implications of our results, it is of interest to apply them to the model systems that he considers. The essential feature o f these systems is that the transition rates between any two states p and q decrease exponentially with the d i s t a n c e Rpq between them, a feature that is typical of many real systems 6). Thus the condition that l)pq ~ PO can only be satisfied if Rpq <~ro, where the value o f ro depends on Vo. For frequency co Pollak and Geballe 2 ) define a distance rco such that w h e n Rpq = r6o, ppq q- Vqp = W. Hence, if Rpq ~ rco at least one o f the transitions between p and q is fast, while ifRpq is appreciably greater than rco both transitions are slow. In this case, our condition for the validity o f the pair approximation is that for every pair o f states with Rpq <~rco (so that at least one of them is a fast state), there are no other fast transitions from either p or q. This will certainly be satisfied if there are no other states within distance rco of eitherp or q, which is just Pollak's condition2'~6), but can also be satisfied under other conditions if the transition rates depend also on energy differences, for instance. Similarly, when this condition is not fulfilled, a sufficient condition for defining a relevant cluster at frequency w is to include in it all states within distance r e o f one of the states in it, which is equivalent to Pollak's criterion for this case 16). Here too, if Vpq and l~qp are not necessarily equal, the relevant clusters may be smaller than this. Incidentally, for any system with transition rates o f this form, as one reduces the density and so increases Rpq one will eventually reach a stage at which Rpq > r~o for each pair o f states, for arbitrary non-zero frequency w. Hence at this stage (where all states are slow and only single-step paths need be considered), if not earlier, the pair approximation is valid. This is the reason why, as shown by Butcher and Morys ~2) from their detailed analysis, the pair approximation is always valid in the low-density limit. With regard to the temperature dependence o f the size of the relevant clusters, one expects the transition rates vpq to increase with increasing temperature. Consequently, the size of the relevant clusters at a given frequency will tend to increase as the temperature increases. A specific example o f this feature has been proposed by Pollak ~6) as a possible explanation of the temperature dependence of o(w). However, it is not clear how valid is his use o f a form o f critical path analysis under these conditionsaT), and detailed numerical calculations on typical random systems are needed. A simple m e t h o d for performing such calculations is described in the accompanying paper.
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For inhomogeneous systems, the cluster approximations have some very interesting implications. One can readily imagine that in some materials there will be clusters of sites comparatively close to each other, with states of comparable energies, which are somewhat separated spatially, and perhaps in energy, from other localized states in their neighbourhood. Such clusters of states could be associated, for instance, with partial phase separation, or with a surface or interface. If this does happen, we expect steps between sites within such clusters to be much more probable than steps to sites outside them. If the maximum transition probability for a step from a site within such a cluster to one outside it is Vrn, then for frequencies much greater than vm the system will behave like a dielectric medium with isolated inclusions having a finite conductivity, the sort of system which can be analyzed in terms of the Maxwell-Wagner analysisla). At frequencies below Vm, however, charge will be able to hop from these clusters to the rest of the material, so that the Maxwell-Wagner analysis will not apply (unless account is taken of the hopping of charge carriers in the dielectric medium). A similar situation may arise with regard to sites surrounding a defect, but with clusters too small for the Maxwell-Wagner analysis to be meaningfully applied. One example of such a situation is a cation vacancy near a doubly-charged impurity cation in an alkali-halide crysta119). Transitions of the vacancy between the equivalent cation sites nearest to the impurity may then have a transition rate v~ much higher than the rate v2 typical of transitions from the first to the second shell of cation sites surrounding the impurity. In such a case, which is most probable at low temperatures, our analysis shows that only transitions within the first shell of cation sites make a significant contribution to o(60) at frequencies 60 appreciably larger than v 2. Similarly, at lower frequencies it may be necessary to consider transitions involving the first two shells of sites, but not more distant shellslg). Our analysis enables us to define explicitly, a priori and not just after analyzing the experimental results, the frequencies at which different shells of sites need to be taken into account at a given temperature.
5. Conclusions For any system containing charge carriers in localized states, the only significant contributions to the ac conductivity o(60) at frequeiacy 60 from carriers making transitions between such states arise from paths containing not more than N(60) steps. At sufficiently high frequencies, N(60) = 1, while N(60) increases with decreasing 60 and tends to infinity as 60 tends
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to zero. The different transitions can conveniently be classified as fast or slow at frequency ¢o according to whether their transition rates are at least comparable with ¢o or are much less than ¢o, respectively. The case N(¢o) -- 1 corresponds to frequencies for which all transitions are slow. At lower frequencies, not all paths of up to N(¢o) steps are important, but only those containing not more than one slow step. For many systems, including amorphous semiconductors and most dielectric media, it is possible to define a critical percolation frequency vc such that it is possible to construct a continuous path of transitions across the system with transition rates greater than Vo only if Vo is less than vc. In this case, at frequencies ¢o much greater than vc all the important paths consist of transitions between states within isolated clusters which contain all the states linked by paths consisting only of fast steps (plus an initial slow step in certain cases, as explained at the end of section 3). Hence, at such frequencies o(co) can be calculated by considering separately each of the relevant clusters of states. Such cluster approximations, of which the well-known pair approximation is a special case, simplify very considerably the practical calculation of o(¢o) for any given system. Moreover, as shown at the end of section 4, a number of general features of the temperature and frequency dependence of o(¢o) can be derived from the corresponding dependence of the size of tl~e relevant clusters.
Acknowledgements The author thanks the Israel Academy of Sciences and Humanities and the Royal Society for a travel grant to enable him to work at Chelsea College, where much of this research was done, and Professor A. K. Jonscher for his hospitality there. He also thanks Dr. R. M. Hill for many useful discussions.
References 1) 2) 3) 4) 5) 6)
D. D. Thornburg, J. Electronic Mater. 2 (1973) 495. M. Pollak and T. H. Geballe, Phys. Rev. 122 (1961) 1742. M. Pollak, Phil. Mag. 23 (1971) 519. G. E. Pike, Phys. Rev. B6 (1972) 1572. P. Emin, Phys. Rev. Letters 32 (1974) 303. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Oxford University Press, London, 1971 ). 7) V. Ambeguokar, B. I. Halperin and J. S. Langer, Phys. Rev. B4 (1971) 2612. 8) N. F. Mott, Phil. Mag. 19 (1969) 835. 9) A. Miller and E. Abrahams, Phys. Rev. 120 (1960) 745.
E Halpern/Electric conductivity, of disordered dielectrics. I 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
S. Kirkpatriek, Rev. Mod. Phys. 45 (1973) 574. P. N. Butcher, J. Phys. C 5 (1972) 1817. P. N. Butcher and P. L. Morys, J. Phys. C 6 (1973) 2147. H. Scher and M. Lax, Phys. Rev. B7 (1973) 4502. E. J. Moore, J. Phys. C 7 (1974) 339. N. G. McCrum, B. E. Read and G. Williams, Anelastic and Dielectric Effects in Polymeric Solids (John Wiley, London, 1967). Chap. 5. M. PoUak, Phys. Rev. 138A (1965) 1822. C. H. Seager and G. E. Pike, Phys. Rev. B10 (1974) 1435. L. K. H. van Beck, Progr. Semicond. 7 (1967) 69. G. D. Watkins, Phys. Rev. 113 (1959) 91.
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T H E AC C O N D U C T I V I T Y O F D I S O R D E R E D A N D D I E L E C T R I C SOLIDS I. CLUSTER APPROXIMATIONS
V. HALPERN* Department of Physics, Chelsea College, Pulton Place, London SW6 5PR, England and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel Received 7 February 1975
The ac conductivity is considered of systems in which the current is due to the motion of charge carriers between localized states. It is shown first that for any given frequency co, a number N(w) can be defined such that carrier paths containing more than N(co) steps make a negligible contribution to the conductivity a(w) at that frequency. For sufficiently high frequencies, N(w) equals umty, while as 09 decreases N(co) increases. An analysis of the possible paths in terms of transitions that are slow or fast relative to the frequency 60 leads, in many cases, to a cluster approximation that is valid for frequencies appreciably greater than a critical percolation frequency. The number of states in each of the dusters that are relevant to the calculation of 0(09) decreases as co increases.
1. I n t r o d u c t i o n In m a n y solids, such as a m o r p h o u s s e m i c o n d u c t o r s 1) and suitably d o p e d s e m i c o n d u c t o r s at low temperatures2), the dc electrical c o n d u c t i v i t y is low, and the ac c o n d u c t i v i t y o(co) increases rapidly with increasing f r e q u e n c y co. It is generally believed t h a t the ac c o n d u c t i v i t y in m a n y o f these s y s t e m s is d u e t o t h e h o p p i n g o f charge carders, usually e l e c t r o n s o r holes, b e t w e e n localized states s i t u a t e d at r a n d o m p o s i t i o n s and having r a n d o m energies. This h o p p i n g is a c c o m p a n i e d b y t h e a b s o r p t i o n o r emission o f e i t h e r a single p h o n o n 2, 3) or o f several p h o n o n s 4' 5). S u c h h o p p i n g m a y also be responsible f o r the dc c o n d u c t i v i t y at finite t e m p e r atures 6, 7 ), b u t a larger c o n t r i b u t i o n t o this dc c o n d u c t i v i t y can s o m e t i m e s c o m e f r o m carriers e x c i t e d i n t o band states, as in the e x p e r i m e n t s o f Pollak a n d Geballe 2 ). T h e dc c o n d u c t i v i t y has b e e n studied b o t h in *Present address: Department of Physics, Bar-Ilan University, Ramat-Gan, Israel. 323
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terms of the most probable hop between pairs of states 8) and in terms of an infinite network of resistancesT'9'l°), but most studies of the ac conductivity have concentrated on the hopping between pairs of states a'~x), which is known as the pair approximation. This approximation has the advantages of being exact in the limits of high frequencies and of low densities at any given frequency12), and of being readily visualised physically. Approaches to the ac conductivity which do not make use of this approximation, such as those of Scher and Lax 13) and Moore14), generally involve decoupling procedures, or the partial summation of infinite sets of diagrams, the physical significance and validity of which is not so easy to understand. In this paper, we show that the pair approximation is only a special case of a more general cluster approximation that can be defined for frequencies greater than a critical percolation frequency Pc. A related problem is the ac dielectric loss e" (w), in solid dielectric media such as alkali halide crystals and polymers. Here, too, the loss is usually attributed to the motion of charge carriers, such as vacancies or dipolar groups of ions, between discrete localized states. However, such transitions may involve a consequent rearrangement of the system, and may sometimes be better described as changes in the state of the system rather than as the motion of individual charge carrierslS). The dielectric loss in these systems can be treated by the same formal analysis as that used for the ac conductivity of amorphous semiconductors, with a suitable interpretation of terms such as charge carrier, localized state, etc. It is only for the sake of simplicity that we use in this paper the model of single charge carriers moving between discrete localized states. In order to derive our cluster approximations, we find it convenient to consider the possible paths of a charge carrier between different states. While this approach is not as convenient for purposes of calculation as the rate equation approach used by Pollak and Geballe 2) and by Butcher11), inter alia, it provides a much simpler and more easily understood method of deriving formal results about cluster approximations. In an accompanying paper, we describe and present examples of a convenient general technique for calculating a(60) when the cluster approximation is valid. Specifically, in this paper we consider the contribution of different possible hopping paths to the current produced in the system at time t in response to a step voltage applied at time t = 0, and hence their contribution to the ac conductivity a(60). In section 2, we show that, for any given infinite frequency 6o, it is possible to define a number N(60) such that the contribution to o(60) from all paths of more than N(60) steps is negligible. This number N(60) increases as 60 decreases, and becomes infinite as 60 tends to zero. Such a result can be used as the basis for a cluster approximation in the analysis of the conductivity of general systems.
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However, the size of cluster indicated by this analysis is often much greater than necessary, and in section 3 we show how to define a more useful cluster approximation for frequencies greater than a critical percolation frequency vc. The size of the cluster relevant to the calculation of o(co) depends on w, and in section 4 we show that our approach leads to a much more general form of Pollak's conditions for cluster approximations16). In the latter part of this section we consider some general results that follow from the dependence on frequency and temperature of the size of the relevant clusters in various systems, while a summary of our main conclusions is presented in section 5.
2. General theory It is convenient to consider a system which is initially in equilibrium with no net current flowing in it, in the absence of an external electric field. At time t = 0, a step voltage is applied to the system, so that at all subsequent times there is a small constant applied field Eo. Let j (t) be the current density produced in response to the application of this field. Then the conductivity a(¢o) is related to the Fourier transforms, J(co) and E(¢o) respectively, of the current density and field by the equation J(~) = a(~)E(o~) = o(~)g0/i~.
(1)
We now define a number of properties of the states of the system. Let the contribution to the dipole moment of the system in the direction of the applied field from a particle in state p be/ap, and let ~pa = ~a - #p
(2)
be the change in the dipole moment of the system in this direction when a particle moves from state p to state q. We define the transition rate for such a move, Vpq, by requiring that the probability that a particle in state p at time t moves to state q by time t + 8t be VpqSt; such a definition avoids the need for different treatments for particles obeying different statistics, a problem that is considered in the accompanying paper. We denote by vp the total transition rate out of state p,
Vp = ~qVpq and define a dipole m o m e n t M such that, for all statesp in the system,
(3)
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V.
q
Halpern/Electric conductivity of disordered dielectrics. I
Vpq tlapq [ ~< Mvp.
(4)
Such an upper b o u n d M exists since for large values of [tZpqI the transition probability ~pq decreases very rapidly with I/apql. For tunnelling, for instance, where I#pql is proportional to the distance tunnelled, vpq decreases exponentially with increasing IlapqI. Finally, let qJpq(s)ds be the probability that a particle in state p at time t = 0 remains there until time t = s and then makes a transition to state q within the time interval (s, s + ds): thus
qJpq(t) = Vpq exp ( - u p t), and its Fourier transform,
(5)
~pq (w) is given by
qJpa(~o) = Vpq/(Vp + ico).
(6)
We now consider the contribution,/p (t), to the current density at time t in the direction of the applied field from a particle that was initially in state p at time t = 0. The contribution t o / p (t) from all paths involving only a single transition or step is just
jO)(t) = ~ Cpq(t)lapq,
(7)
q
and these contribute to ,/(co) a term J(~) (¢o), the modulus of which satisfies the inequality IJ~l)(¢o)l < ~ q
II~pqIvpq/Ivp 4- i6ol ~Mvp/lup 4- i6ol.
(8)
It is convenient to define
cp(¢o) = up/Ivp + i¢ol.
(9a)
and c ( ~ ) = max P
Cp(~).
(9b)
Thus, for all states p,
Ij~l)(co)l~
(I0)
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The contribution t o / ( t ) from all paths of t w o steps starting from p is t
/(2)(t) = ~ ~ f ~pq(S)~qr(t-s)dslzqr , q
(11)
r o
and its Fourier transform is j(2)(6o)
= ~ ~ q r
idqrVqrPpq (Vp q- i¢o)-1 (Vq + i6o)-1.
(12)
Hence IJ[2)(6o)1 < M Y
q
<~M Y
vpqvq
Ivp ÷ i6ol-~ Ivq + i6ol -!
c(6o)Vpq/lup+ icbl
(13)
q
Similarly, the contribution o f all paths of exactly n steps to J(6o) satisfies
IJptn)(6o)l <
Mc(6O) n .
(14)
If 6o is non-zero, c(6o) < 1 since up is always finite, and so the contribution from all paths o f more than N steps to J(6o) tends to zero as N tends to infinity. Thus, in calculation of J(6o) and v(6o) it is only necessary to consider paths involving no more than N(6o) steps. Since paths of one step contribute to J(w) a quantity of order Mc(w), it is reasonable to ignore all paths the sum of whose contribution is less than aMc(6o), where a is of order 10-3; this leads to the equation for N(6o) c(6o)N(¢°) < a(1 - c(6o)).
(15)
Two extreme cases are worth considering separately. At very high frequencies, c(6o) will be small and eq. (15) will be satisfied for N(6o) = 1, i.e. only paths involving a single step need to be considered. This is the case if c(6o) is less than a, and so N(6o) = 1
if 6o > max (up/a).
(16)
At very low frequencies, on the other hand, c(6o) is so close to unity that 1 c(6o) < a . In such a case, eq. (15) becomes
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F. Halpern/Electric conductzvity o f dzsordered dielectrics. I
N(~) ~-,max ((2@/w2 ) In (2@/w 2 ))
if w "~ (2a) ½ max
vp.
(17)
P This latter condition allows paths containing many more steps than the condition one might have expected, namely that if the mean time taken for a path of N(w) steps is much longer than the period of the field, and so ifN(6o)/Uq >2>2~r/6o, the contribution from the path is unimportant. Physically, the reason why this latter condition is not sufficient is that, while the contribution from each individual path of this length may be small, the total number of possible paths increases rapidly with the number of steps.' The result contained in eq. (15) could, in principle, be used as the basis for a cluster approximation for o(~o). For charge carriers starting at state p, one would only need to consider the cluster of sites accessible within N(~o) transitions from p. This is, in fact, the only form of cluster approximation valid at all non-zero frequencies and for all possible physical systems. However, it is not a very useful form, since it usually involves many more states than are in fact required, while consideration of different paths is, in general, a very complicated method for calculating o(w). Fortunately, in most systems there is a distribution of transition rates Upq, and in such cases a much more useful form of cluster approximation can be defined.
3. Cluster approximations The results derived in section 2 show that in order to calculate a(w) it is unnecessary to consider paths containing more than N ( ~ ) steps for any given carder. However, this does not imply that all possible paths containing N(w) steps are important. It can happen, for instance, that the only important paths are those involving only transitions between a pair of states; this is the case when the frequently-used pair approximation is valid2). More generally, it often happens that all the important paths of N(~o) steps are confined to transitions between clusters of states that contain much less than the total number of states accessible within N(w) steps from a given state, and are independent of the state within the cluster from which the particle starts. In such a case, one can calculate a(co) by considering only transitions within such small clusters, a problem which is conceptually simpler for a random system and which is mathematically much easier to solve. In this section, we examine the conditions for the validity of such cluster approximations, and how the relevant clusters can be chosen.
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As a preliminary point, we wish to clarify the relationship between the path probability approach used in section 2 and the calculation of a(60). Our approach calculates the current due to all transitions, and not just the extra current produced by the applied field. Thus, we take into account a lot of terms that are of zero order in the field, which must cancel out, and it is not obvious that those paths contributing large terms to J(60) are the most important in determining o(60). The reason that they usually are can be found by considering the difference between the currents j+-(t) produced by applying fields + E 0 at time t=0. The difference between the transition rates for any given step are of first order in the field, and so for a path containing many steps only the differences for one step at a time need be taken into account. (An exact mathematical formulation of this point is simple but tedious, and so will not be presented here.) Hence, the paths important in determining J+(60) - J-(60), and hence a(60), will be the same as those that make the major contribution to J(w) if the difference between the transition rates from p to q, for each pair of states, in the presence of fields + Eo is more or less proportional to Vpq. Since a field Eo changes the difference in energy of the states by Eolapq, the difference in transition rates will (by the principle of detailed balance) usually be proportional to Eo lapq Vpq. Now, as remarked earlier, lapq varies much less rapidly than Vpq with the distance between states p and q. Hence, lapq will be of the same order of magnitude for all transitions with appreciable values of Upq, and it is the values of the Vpq that determine the importance of a given path. Thus, the paths that are important for determining o(60) are, in general, the ones that contribute the largest terms to ./(co). The contribution to J(60) from a particle following a given path p q r . . . uv is just Iduv~lpq ~ q r • • • l~uv ( W ) . Even if this path contains less than N(60) steps its contribution can have a small modulus for one of two reasons~ Firstly, one of the transitions in the path, say that from r to s, may have a much smaller probability than other transitions from r, so that vrs is much less than Vr; we call such transitions minor ones, while those with Vrs of comparable magnitude to vr will be called major transitions. Alternatively, even for a major transition, the transition rate Vrs may be much less than the frequency 6o. Any transition, major or minor, for which Vrs ~ 60 will be called a slow transition, while all other transitions will be termed fast ones; in addition, a state such that all transitions from it are slow ones will be called a slow state. We note that the classification of a transition as fast or slow depends on the frequency, but its classification as major or minor does not. In general, we might expect that all paths containing several minor or slow transitions will make negligible contributions to J(co) and o(60).
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V. Halpern/Electric conductivity of disordered dtelectrics. I
However, J(6o) involves not only the moduli of the ~rs(6o) but also their phases, and also the changes in dipole moment grs which may be positive or negative. As a result, the contributions to J(6o) from successive steps (i. e. from successive sub-paths) along a given path may cancel. For instance, a particle following a closed path makes a zero contribution to fo/(r)dt, and so contributes nothing to J(6o) for 6o = 0. Thus, if all paths with an unlimited number of steps that contain only major transitions return to their starting point, these paths will not contribute to J(0). In such a case, the major contribution to the dc conductivity o(0) will come from paths containing minor transitions. This problem is a well-known one in the calculation of the dc conductivity of random systems, and has been discussed by various authorsT'9'l°). At finite frequencies, the factors ~brs(6o) are complex, so that there will not normally be an exact cancellation of the contribuffons from different steps or sub-paths in a closed path, but if 6o is small the imaginary points of the ffrs(6o) are also small, and the cancellation will be nearly complete. Physically, this means that a closed path that on average is completed within a small fraction of a cycle makes a negligible contribution to f~j(r) exp ( - i6ot)dt, i.e., to J(w). For these reasons, it is difficult to decide which paths are the most important purely on the basis of major and minor transitions. Fortunately, however, the situation in connection with fast and slow transitions is very different. Physically, the meaning of a slow state is that a particle is unlikely to leave it, along any path, during a single cycle of the field. Thus, whatever the signs of the #rs and phases of the complex ~krs, a path reaching a slow state r and stopping there will in general make a much larger contribution to J(6o) than any of the paths continuing from state r. Mathematically, if r is a slow state, Cr(6o) is so small that all the paths involving transitions from r make contributions of negligible modulus, regardless of the signs of the tars or phases of the ~krs. For similar reasons, paths involving slow transitions from a fast state can also be neglected. There are only two exceptions to this situation. Firstly, if all states are slow, as will be the case at sufficiently high frequencies since the vr have a finite maximum, one cannot ignore the contribution from slow steps. However, only paths containing one such step will be important, as those involving two steps will make contributions of modulus proportional to c(6o) 2 , which is much less than c(6o). This is, of course, just the situation in which N(w) = I, as eq. (16) is satisfied. The other exception is that in which a slow state has a very high occupation probability: this will happen, for instance, at low temperatures if one state has an energy appreciably lower than that of its neighbours. In this case, the contribution to o(6o) from paths starting at such a slow state is not negligible (as the number of particles making a slow transition from
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this state in one period may.exceed that of particles making a fast transition in one period from a state with a low probability o f occupation): however, the contribution from paths containing transitions from this state at any other step will be negligible. In order to define a useful cluster approximation, we consider a system in which there is a distribution o f transition rates Vrs and v r. Let us choose a frequency v 0, and examine whether it is possible to construct a continuous path o f transitions from a state at one end o f the system to a state at the other end, such that at each step in it the transition rate Vrs is greater than v 0. If v o is sufficiently large, e.g. if v0 > m a x ISr, this will n o t be possible, while if Vo is small enough, e.g. if Vo < min Vrs, such a path can be constructed. Hence a maximum value of v 0 exists for which such a construction is possible: we denote this by vc, and call it the critical percolation frequency. While the value of v c can be determined b y percolation theory, we note that ours is not the classical bond percolation problem~°), since in general Vrs ~ Vsr, b u t is one of directed bond percolation. F o r values o f v o greater than v c, all the paths consisting exclusively o f transitions with Vrs > v o will consist o f transitions between states within discrete, finite clusters. Hence, at frequencies co appreciably greater than v c, the paths consisting only o f fast transitions will contain only transitions within small clusters o f states, which we call the relevant clusters at this frequency. The states on the edge o f such a cluster will either be slow ones or will have one or more major transitions leading back into the cluster and only slow transitions to states outside it. Our analysis then shows that in general a(co) can be calculated b y considering only the transitions between states within the relevant clusters. F o r the exceptional case in which all transitions are slow, so that only single-step paths need to ',be taken into account, the calculation o f a(co) is trivial n, ~4 ). F o r the other exceptional case, in which some slow states have a very high occupation probability, the sitution is as follows. If some transition from such a slow state r leads to a fast state, state r will be on the boundary o f a relevant cluster, while if more than one transition from r leads to a fast state it may be on the boundary o f several clusters. Since paths that pass through this slow-state without either starting or finishing at it are o f no importance, the contributions from this state to the conductivity o f each relevant cluster can be considered separately, provided that the correct v r is used in evaluating these contributions. Physically, the effect of this v r is to ensure the correct probability of a particle from state r entering a given cluster. Similarly, the contributions to a(co) from transitions from state r to other slow rates are additive, provided that the correct v r is used: for these, o f course, only single step paths are important.
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V. Halpern/Electric conductivity of disordered dielectrics. I
4. Discussion The cluster approximation that we have derived in the previous section is much simpler and more general than those proposed by Pollak 2' 16). However, in order to see the practical implications of our results, it is of interest to apply them to the model systems that he considers. The essential feature o f these systems is that the transition rates between any two states p and q decrease exponentially with the d i s t a n c e Rpq between them, a feature that is typical of many real systems 6). Thus the condition that l)pq ~ PO can only be satisfied if Rpq <~ro, where the value o f ro depends on Vo. For frequency co Pollak and Geballe 2 ) define a distance rco such that w h e n Rpq = r6o, ppq q- Vqp = W. Hence, if Rpq ~ rco at least one o f the transitions between p and q is fast, while ifRpq is appreciably greater than rco both transitions are slow. In this case, our condition for the validity o f the pair approximation is that for every pair o f states with Rpq <~rco (so that at least one of them is a fast state), there are no other fast transitions from either p or q. This will certainly be satisfied if there are no other states within distance rco of eitherp or q, which is just Pollak's condition2'~6), but can also be satisfied under other conditions if the transition rates depend also on energy differences, for instance. Similarly, when this condition is not fulfilled, a sufficient condition for defining a relevant cluster at frequency w is to include in it all states within distance r e o f one of the states in it, which is equivalent to Pollak's criterion for this case 16). Here too, if Vpq and l~qp are not necessarily equal, the relevant clusters may be smaller than this. Incidentally, for any system with transition rates o f this form, as one reduces the density and so increases Rpq one will eventually reach a stage at which Rpq > r~o for each pair o f states, for arbitrary non-zero frequency w. Hence at this stage (where all states are slow and only single-step paths need be considered), if not earlier, the pair approximation is valid. This is the reason why, as shown by Butcher and Morys ~2) from their detailed analysis, the pair approximation is always valid in the low-density limit. With regard to the temperature dependence o f the size of the relevant clusters, one expects the transition rates vpq to increase with increasing temperature. Consequently, the size of the relevant clusters at a given frequency will tend to increase as the temperature increases. A specific example o f this feature has been proposed by Pollak ~6) as a possible explanation of the temperature dependence of o(w). However, it is not clear how valid is his use o f a form o f critical path analysis under these conditionsaT), and detailed numerical calculations on typical random systems are needed. A simple m e t h o d for performing such calculations is described in the accompanying paper.
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For inhomogeneous systems, the cluster approximations have some very interesting implications. One can readily imagine that in some materials there will be clusters of sites comparatively close to each other, with states of comparable energies, which are somewhat separated spatially, and perhaps in energy, from other localized states in their neighbourhood. Such clusters of states could be associated, for instance, with partial phase separation, or with a surface or interface. If this does happen, we expect steps between sites within such clusters to be much more probable than steps to sites outside them. If the maximum transition probability for a step from a site within such a cluster to one outside it is Vrn, then for frequencies much greater than vm the system will behave like a dielectric medium with isolated inclusions having a finite conductivity, the sort of system which can be analyzed in terms of the Maxwell-Wagner analysisla). At frequencies below Vm, however, charge will be able to hop from these clusters to the rest of the material, so that the Maxwell-Wagner analysis will not apply (unless account is taken of the hopping of charge carriers in the dielectric medium). A similar situation may arise with regard to sites surrounding a defect, but with clusters too small for the Maxwell-Wagner analysis to be meaningfully applied. One example of such a situation is a cation vacancy near a doubly-charged impurity cation in an alkali-halide crysta119). Transitions of the vacancy between the equivalent cation sites nearest to the impurity may then have a transition rate v~ much higher than the rate v2 typical of transitions from the first to the second shell of cation sites surrounding the impurity. In such a case, which is most probable at low temperatures, our analysis shows that only transitions within the first shell of cation sites make a significant contribution to o(60) at frequencies 60 appreciably larger than v 2. Similarly, at lower frequencies it may be necessary to consider transitions involving the first two shells of sites, but not more distant shellslg). Our analysis enables us to define explicitly, a priori and not just after analyzing the experimental results, the frequencies at which different shells of sites need to be taken into account at a given temperature.
5. Conclusions For any system containing charge carriers in localized states, the only significant contributions to the ac conductivity o(60) at frequeiacy 60 from carriers making transitions between such states arise from paths containing not more than N(60) steps. At sufficiently high frequencies, N(60) = 1, while N(60) increases with decreasing 60 and tends to infinity as 60 tends
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to zero. The different transitions can conveniently be classified as fast or slow at frequency ¢o according to whether their transition rates are at least comparable with ¢o or are much less than ¢o, respectively. The case N(¢o) -- 1 corresponds to frequencies for which all transitions are slow. At lower frequencies, not all paths of up to N(¢o) steps are important, but only those containing not more than one slow step. For many systems, including amorphous semiconductors and most dielectric media, it is possible to define a critical percolation frequency vc such that it is possible to construct a continuous path of transitions across the system with transition rates greater than Vo only if Vo is less than vc. In this case, at frequencies ¢o much greater than vc all the important paths consist of transitions between states within isolated clusters which contain all the states linked by paths consisting only of fast steps (plus an initial slow step in certain cases, as explained at the end of section 3). Hence, at such frequencies o(co) can be calculated by considering separately each of the relevant clusters of states. Such cluster approximations, of which the well-known pair approximation is a special case, simplify very considerably the practical calculation of o(¢o) for any given system. Moreover, as shown at the end of section 4, a number of general features of the temperature and frequency dependence of o(¢o) can be derived from the corresponding dependence of the size of tl~e relevant clusters.
Acknowledgements The author thanks the Israel Academy of Sciences and Humanities and the Royal Society for a travel grant to enable him to work at Chelsea College, where much of this research was done, and Professor A. K. Jonscher for his hospitality there. He also thanks Dr. R. M. Hill for many useful discussions.
References 1) 2) 3) 4) 5) 6)
D. D. Thornburg, J. Electronic Mater. 2 (1973) 495. M. Pollak and T. H. Geballe, Phys. Rev. 122 (1961) 1742. M. Pollak, Phil. Mag. 23 (1971) 519. G. E. Pike, Phys. Rev. B6 (1972) 1572. P. Emin, Phys. Rev. Letters 32 (1974) 303. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Oxford University Press, London, 1971 ). 7) V. Ambeguokar, B. I. Halperin and J. S. Langer, Phys. Rev. B4 (1971) 2612. 8) N. F. Mott, Phil. Mag. 19 (1969) 835. 9) A. Miller and E. Abrahams, Phys. Rev. 120 (1960) 745.
E Halpern/Electric conductivity, of disordered dielectrics. I 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
S. Kirkpatriek, Rev. Mod. Phys. 45 (1973) 574. P. N. Butcher, J. Phys. C 5 (1972) 1817. P. N. Butcher and P. L. Morys, J. Phys. C 6 (1973) 2147. H. Scher and M. Lax, Phys. Rev. B7 (1973) 4502. E. J. Moore, J. Phys. C 7 (1974) 339. N. G. McCrum, B. E. Read and G. Williams, Anelastic and Dielectric Effects in Polymeric Solids (John Wiley, London, 1967). Chap. 5. M. PoUak, Phys. Rev. 138A (1965) 1822. C. H. Seager and G. E. Pike, Phys. Rev. B10 (1974) 1435. L. K. H. van Beck, Progr. Semicond. 7 (1967) 69. G. D. Watkins, Phys. Rev. 113 (1959) 91.
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