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A derivation for the energy dependence of the density of band tail states in disordered materials
Solid State Communications, Vol. 41, No. 3, pp. 241-244. 1982. Printed in Great Britain.
0038-1098/82/030241-04502.00/0 Pergamon Press Ltd.
A DERIVATION FOR THE ENERGY DEPENDENCE OF THE DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS* J. Singh and A. Madhukar t Departments of Materials Science and Physics, University of Southern California, Los Angeles, CA 90007, U.S.A.
(Received 9 June 1981 by Z Tauc) Variational arguments are employed to present the first explicit derivation of the behavior of band tail states and localization length over most of the tail region of a random alloy..Exponential tails are shown to exist near the mobility edges, in conformity with the observations in disordered systems. The character of the states forming the band tails is discussed and, for random alloys, the localization length is shown to exhibit a minimum as a function of energy between band edge and mobility edge. IN SPITE of the central importance [ 1] of the origin and density of band tail states in random alloys and amorphous semiconductors, the subject has made little progress in theoretical understanding, even though considerable experimental information [1,2] has accumulated in recent years. For a random A - B alloy Lifshitz provided [3] intuitive and appealing arguments, valid for the behavior of the localized states near the edge of the spectrum (band edge). The region away from this edge towards the mobility edge has remained unexplained [4], even though experiments reveal this to be the major and observable region of band tails which show an essentially exponential behavior [2]. Attempts have been made to refine Lifshitz's argument, but are still limited to energies near the band tail edge [5, 6]. In this letter we present a derivation of the behavior of the density of band tail states covering a broader energy range. Near the edge of the spectrum it reduces to the LifshRz result and near the mobility edge it gives an essentially exponential behavior. The density of localized states is experimentally found to be several orders of magnitude smaller [2] than the density of extended states, implying that these states arise from special atomic configurations. Consequently one is forced to abandon mean field like theories and even numerical simulations to obtain the density of localized states. The arguments presented here are based on general variational principles and probability theory, and thus circumvent the limitations of the above noted approaches. The primary origin of localized electron states is the presence of potential fluctuations over the length scale of several atomic distances. These fluctuations are * Work supported by ONR (Contract No. N00014-77=C0397). t A.P. Sloan Foundation Fellow.
essentially due to variations in the physical parameters of the disordered system on the same scale. Depending upon the material these parameters could represent compositional (for alloys) or structural (bond-length, bond angle, etc.) disorder. These physical parameters and the nature of eigenfunctions near the band edges is important in determining the nature of potential fluctuations and their effects [7]. Here, we shall choose to stay within the framework of the random A - B alloy model and later compare this model to the elemental disorder case. To make explicit the contribution of the present work, we first briefly recall the original Lifshitz argument. Consider the spectra of the components A and B of a random A - B alloy shown in Fig. 1 (single band picture is assumed). The limits of the pure spectra are EmAin,E/tax and EBin and EBax. The region of interest is between Eamin and EBmi=(or Enmax and Eamax), when the mean concentration of A and B is ~a and Cg. We assume that no correlation between A and B atoms is present. To solve the problem near EAmin,Lifshitz asserted that energies near Eamin are due to states conf'med to regions (clusters) made up of purely A type atoms. The probability of a volume Vo having A type atoms with concentration CA when the mean concentration of A atoms is C~ is given by [3] P(Vo) = exp [-- Volvo(Ca In CA/C,~ + CB In CB/C°)], (1) where Vo is the atomic volume. Taking CA = 1 in the cluster of volume Vo and choosing the ground state [3] to maximize the probability, (since excited states would require a much larger volume of fluctuation) the total energy E of the ground state is given by,
E = Eami~+ B]R~, 241
(2)
242
DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS
nlE
, A
EMIN
B
A
E:MIN
EMAX
B
EMAX
E
Fig. 1. Individual electronic states spectra of the A and B components of the random A - B alloy. EAmin,EAmax and Eamin, EBm~ are the band limits for A and B. the second term being the kinetic energy of confining electron to Vo (~Roa). (B is a constant). Equation (2) provides Ro as a function of E. Since Roa = Vo and the density of states is proportional to P(Vo), equations (1) and (2) led Lifshitz to obtain, ~A
n(E) ~ e x p -
~
~oo
-3/2
-I
J
(3)
where C and Eo are constants determined by C~, EAmin and B. Equation (3) is valid only for E ~ Eamin and predicts the localization length, Ro(E), going to infinity at EAin. The density of states goes to zero at EAmin, as expected. A need for extending Lifshitz' arguments for energies away from EAin clearly exists. In this context, it is interesting to note that since the localization length must go to infinity at the mobility edge also (where the density of states in non-zero) any correct theory must, for the A - B alloy, show the localization length to exhibit a minimum as a function of energy between the band and mobility edges. Before we present our derivation of n(E) and the localization length over the whole tail region with the latter exhibiting the above noted behavior, two previous attempts to refme Lifshitz' argument are worthy of note. Friedberg and Luttinger [5] reformulated Lifshitz's problem in terms of a Brownian motion problem. Using a variational technique, they obtained corrections to the Lifshitz results, although still confining themselves to the regions of energy considered by Lifshitz. Halperin and Lax [6] used the variational principle to obtain the density of states for the problem of impurity bands. They confined themselves to states which arise in regions with primarily impurity atoms. In Lifshitz's treatment it was assumed that the fluctuations leading to localized states were made up of pure A type atoms. This condition is relaxed in the above two works. The exact make up of the fluctuation is obtained via a variational principle which maximizes the probability of fluctuations which give rise to a localized state at energies close to the Lifshitz limit. In the limit E -+ EAmin, the results reduce to Lifshitz's results, but away from EAmin,the results differ. In these considerations a central assumption is the
Vol. 41, No. 3
confinement of the electron wavefunction within the chosen cluster. This is clearly unphysical and too restrictive since the wavefunction, though centered on the potential fluctuation in the duster, will diffuse out into the surrounding medium. This essential physical effect becomes more and more significant as the energy moves away from the band edge towards the mobility edge since such states arise from smaller and shallower potential fluctuations. Our arguments, presented below, are based on this recognition which allows us to obtain n(E) over the whole tail region. We assume the density of localized states (and the localization length) are small enough so that there is no overlapping of localized states. This will be true except extremely close to the mobility edge. We also assume the average potential in the cluster to determine the localized state energy. This is true if the localized states extend over several atomic sites. The above assumption is thus valid towards the mobility edge and, in the Lifshitz problem close to the lowest energies. From the above discussion it follows that there are two length scales in the problem, viz. Ro, the core radius in which the potential fluctuation is confined and R, the extent of the wavefunction with R > Ro. Consequently the variational principle is applied in two steps, as discussed below. As seen from equation (1), the probability of finding a cluster of radius Ro for CA close to C° is just a Gaussian fluctuation. In our derivation we assume (as in [3-5]) that only the lowest energy state of the fluctuation need be considered. The reason for this is that excited states have a rapidly growing kinetic energy part and for any given energy they would require a larger volume of the same potential fluctuation. Since the probability (1) decreases exponentially the contribution of excited states will be small. The ground state is essentially determined by the lowest energy states (extrenum states) in the ordered A and B components. In general the wavefunction describing the localized state may be represented by ~0/~(3') ~ {exp (--~/[R)}f('y),
(4)
where exp (-- "t/R) represents the envelope of the wave and f(30 represents the structure in the envelope. For the ground state f('),) is determined by the extremum states. Given equation (4), the kinetic energy part of the total energy of a localized state can be written as 1]R 2. This, of course, simplifies the calculations since one can then fred the energy associated with a localized state by knowing the average potential in the well and the size of the well. However, to extend Lifshitz results one must develop a method by which the makeup of the potential well is determined using some criteria and not by assumption. This is straightforward if we allow the
Vol. 41, No. 3
243
DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS
concentration of A and B type atoms in the cluster to be determined by maximization of the probability in equation (1). This is achieved by the variational principle introduced by Lloyd and Best [8] and allows us to choose a special configuration from, in principle, an infinite set. Let x = CA -- C,~ be the concentration fluctuation in the volume Vo(~ R~), A = (E~Bin - - E ~ i n ) , : C°E~ in -4- C°E~Bin and E~ain = O. The total energy of the electron may then be written as,
10zl 10zc 10~s 1018 "7
1017
E(R,Ro, x) = (E--xA){1--(R°a]
'E
(5) The factor [1 -- (Rg[R3)I is the fraction of the wavefunction inside the potential well, and [1 -- (Ro[R)] 2 is / normalization factor. t The procedure to find n(E) is to minimize E in equation (5) so as to fred a relation between E, Ro and x. We now maximize P(Ro) in equation (1) which gives a relation between E and Ro or E and x which in turn gives P(E). Since the density of localized states is proportional to the probability of the potential fluctuations (only ground states of the wells are counted) an expression for n(E) readily follows. It is not possible to obtain an analytic solution for n(E) with all the terms of equation (5) included. However, if one retains terms to order R~/R 3 in the expansion, one may t'md n(E) analytically. We find that in this case, for E near E,
n(E) = no exp --
\---~o ]
(6)
and for E near Eamin we recover Lifshitz's result, equation (3). A numerical calculation of n(E) gives a behavior in which the power law dependence of the energy lies between 0.5 and 1.5 (see Fig. 2 solid line). It has been argued [1 ] that disorder in an elemental (or compound) semiconductor produces potential fluctuations which are essentially Gaussian. For Ca ~ C~, equation (1) gives a Gaussian probability distribution thus implying that the behavior of N(E) for E ""/~ obtained from equation (1) is valid for elemental amorphous semiconductors. However, one does not expect the Lifshitz result, equation (3) to appear since there is no lower energy cutoff as exists for the alloy problem. For E close to Eami,, N(E) derived here is expected to deviate from the behavior in such systems, however the method of derivation employed here is not dependent upon the specific form of equation (1). For Gaussian fluctuations for E far from Eamin, the exponent is found to be 0.5 for 3 dimension and 1.5 for 1 dimension, in agreement with previous work [5, 8, 9].
u
10Is
I 1015
LI_I
1014 1013
1012 -tOO 600 503 400 303 200 (meV)
100
Fig. 2. Exact density of states from the theory for CA = CB = 0.5;EBmin --gamin = 1.0 eV. The data points are results from [2]. The region below the dashed horizontal line is below the experimental observation limits. A comparison of the predicted behavior of N(E) with experimental results on a-Si can thus be meaningfully made, provided an estimate of Eo is available. Such an estimate is only available from experiments for a-Si [10], and we compare our calculations with experiments to check if the overall shape of the density of states is correct. We choose C° and C° = 0.5, EBmin -- EAmin = 1.0 or to get Eo equal to 60 meV (this is not a unique choice, of course). The density of states at E is chosen to be ~ 1021 cm-aeV -1, a value suggested by experiments. The solid in Fig. 2 shows the theoretically obtained density of states. The figure also shows the density of tail states inferred assuming that the absorption coefficient is proportional to the density of states over this narrow energy range (~ 0.2 eV). The agreement in the shape of the two results is rather good and suggests that the simple model calculation does reflect the correct physics of the band tail problem. The region of states below the horizontal line is too small to be observed experimentally so that a tapering off of the data points is expected. In the above calculations, only the ground state wavefunctions are used to determine the density of states. This allowed us to choose a simple I[R 2 dependence for kinetic energy and the relation N ( E ) = P ( V ) .
244
DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS
At energies close to if, the excited states arising from the potential fluctuations will also contribute. In this case, the above simplifications do not hold and a more complex method has to be used. However, it is simple to answer the question at which point the excited and ground states from different clusters (in space and composition) start contributing to the same energy with equal weight. By comparing variationally the relative contributions from the ground state and excited states, we fmd that up to ~ Eo/5 from E, the above treatment is valid. Beyond that the above simple procedure has to be substantially modified.
Vol. 41, No. 3
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9. I 0.
N.F. Mott & E.A. Davis. Electronic Processes in Non-Crystalline Materials. Oxford (1979). B. Abeles, C.R. Wronski, T. Tiedje & G.D. Cody, Solid State Commun. (in press). I.M. Lifshitz, Adv. Phys. 13,483 (1965). D. Thouless, Physics Reports 13C, 93 (1979) and references therein. R. Friedberg & J. Luttinger,Phys. Rev. B12, 4480 (1975). B.I. Halperin & M. Lax, Phys. Rev. 153,802 (1967). J. Singh, Phys. Rev. B23,4156 (1981). P. Lloyd & Best, J. Phys. C8, 3752 (1975). E.P. Gross, J. Stat. Phys. 17,265 (1977). T. Tiedje (to be published).
0038-1098/82/030241-04502.00/0 Pergamon Press Ltd.
A DERIVATION FOR THE ENERGY DEPENDENCE OF THE DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS* J. Singh and A. Madhukar t Departments of Materials Science and Physics, University of Southern California, Los Angeles, CA 90007, U.S.A.
(Received 9 June 1981 by Z Tauc) Variational arguments are employed to present the first explicit derivation of the behavior of band tail states and localization length over most of the tail region of a random alloy..Exponential tails are shown to exist near the mobility edges, in conformity with the observations in disordered systems. The character of the states forming the band tails is discussed and, for random alloys, the localization length is shown to exhibit a minimum as a function of energy between band edge and mobility edge. IN SPITE of the central importance [ 1] of the origin and density of band tail states in random alloys and amorphous semiconductors, the subject has made little progress in theoretical understanding, even though considerable experimental information [1,2] has accumulated in recent years. For a random A - B alloy Lifshitz provided [3] intuitive and appealing arguments, valid for the behavior of the localized states near the edge of the spectrum (band edge). The region away from this edge towards the mobility edge has remained unexplained [4], even though experiments reveal this to be the major and observable region of band tails which show an essentially exponential behavior [2]. Attempts have been made to refine Lifshitz's argument, but are still limited to energies near the band tail edge [5, 6]. In this letter we present a derivation of the behavior of the density of band tail states covering a broader energy range. Near the edge of the spectrum it reduces to the LifshRz result and near the mobility edge it gives an essentially exponential behavior. The density of localized states is experimentally found to be several orders of magnitude smaller [2] than the density of extended states, implying that these states arise from special atomic configurations. Consequently one is forced to abandon mean field like theories and even numerical simulations to obtain the density of localized states. The arguments presented here are based on general variational principles and probability theory, and thus circumvent the limitations of the above noted approaches. The primary origin of localized electron states is the presence of potential fluctuations over the length scale of several atomic distances. These fluctuations are * Work supported by ONR (Contract No. N00014-77=C0397). t A.P. Sloan Foundation Fellow.
essentially due to variations in the physical parameters of the disordered system on the same scale. Depending upon the material these parameters could represent compositional (for alloys) or structural (bond-length, bond angle, etc.) disorder. These physical parameters and the nature of eigenfunctions near the band edges is important in determining the nature of potential fluctuations and their effects [7]. Here, we shall choose to stay within the framework of the random A - B alloy model and later compare this model to the elemental disorder case. To make explicit the contribution of the present work, we first briefly recall the original Lifshitz argument. Consider the spectra of the components A and B of a random A - B alloy shown in Fig. 1 (single band picture is assumed). The limits of the pure spectra are EmAin,E/tax and EBin and EBax. The region of interest is between Eamin and EBmi=(or Enmax and Eamax), when the mean concentration of A and B is ~a and Cg. We assume that no correlation between A and B atoms is present. To solve the problem near EAmin,Lifshitz asserted that energies near Eamin are due to states conf'med to regions (clusters) made up of purely A type atoms. The probability of a volume Vo having A type atoms with concentration CA when the mean concentration of A atoms is C~ is given by [3] P(Vo) = exp [-- Volvo(Ca In CA/C,~ + CB In CB/C°)], (1) where Vo is the atomic volume. Taking CA = 1 in the cluster of volume Vo and choosing the ground state [3] to maximize the probability, (since excited states would require a much larger volume of fluctuation) the total energy E of the ground state is given by,
E = Eami~+ B]R~, 241
(2)
242
DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS
nlE
, A
EMIN
B
A
E:MIN
EMAX
B
EMAX
E
Fig. 1. Individual electronic states spectra of the A and B components of the random A - B alloy. EAmin,EAmax and Eamin, EBm~ are the band limits for A and B. the second term being the kinetic energy of confining electron to Vo (~Roa). (B is a constant). Equation (2) provides Ro as a function of E. Since Roa = Vo and the density of states is proportional to P(Vo), equations (1) and (2) led Lifshitz to obtain, ~A
n(E) ~ e x p -
~
~oo
-3/2
-I
J
(3)
where C and Eo are constants determined by C~, EAmin and B. Equation (3) is valid only for E ~ Eamin and predicts the localization length, Ro(E), going to infinity at EAin. The density of states goes to zero at EAmin, as expected. A need for extending Lifshitz' arguments for energies away from EAin clearly exists. In this context, it is interesting to note that since the localization length must go to infinity at the mobility edge also (where the density of states in non-zero) any correct theory must, for the A - B alloy, show the localization length to exhibit a minimum as a function of energy between the band and mobility edges. Before we present our derivation of n(E) and the localization length over the whole tail region with the latter exhibiting the above noted behavior, two previous attempts to refme Lifshitz' argument are worthy of note. Friedberg and Luttinger [5] reformulated Lifshitz's problem in terms of a Brownian motion problem. Using a variational technique, they obtained corrections to the Lifshitz results, although still confining themselves to the regions of energy considered by Lifshitz. Halperin and Lax [6] used the variational principle to obtain the density of states for the problem of impurity bands. They confined themselves to states which arise in regions with primarily impurity atoms. In Lifshitz's treatment it was assumed that the fluctuations leading to localized states were made up of pure A type atoms. This condition is relaxed in the above two works. The exact make up of the fluctuation is obtained via a variational principle which maximizes the probability of fluctuations which give rise to a localized state at energies close to the Lifshitz limit. In the limit E -+ EAmin, the results reduce to Lifshitz's results, but away from EAmin,the results differ. In these considerations a central assumption is the
Vol. 41, No. 3
confinement of the electron wavefunction within the chosen cluster. This is clearly unphysical and too restrictive since the wavefunction, though centered on the potential fluctuation in the duster, will diffuse out into the surrounding medium. This essential physical effect becomes more and more significant as the energy moves away from the band edge towards the mobility edge since such states arise from smaller and shallower potential fluctuations. Our arguments, presented below, are based on this recognition which allows us to obtain n(E) over the whole tail region. We assume the density of localized states (and the localization length) are small enough so that there is no overlapping of localized states. This will be true except extremely close to the mobility edge. We also assume the average potential in the cluster to determine the localized state energy. This is true if the localized states extend over several atomic sites. The above assumption is thus valid towards the mobility edge and, in the Lifshitz problem close to the lowest energies. From the above discussion it follows that there are two length scales in the problem, viz. Ro, the core radius in which the potential fluctuation is confined and R, the extent of the wavefunction with R > Ro. Consequently the variational principle is applied in two steps, as discussed below. As seen from equation (1), the probability of finding a cluster of radius Ro for CA close to C° is just a Gaussian fluctuation. In our derivation we assume (as in [3-5]) that only the lowest energy state of the fluctuation need be considered. The reason for this is that excited states have a rapidly growing kinetic energy part and for any given energy they would require a larger volume of the same potential fluctuation. Since the probability (1) decreases exponentially the contribution of excited states will be small. The ground state is essentially determined by the lowest energy states (extrenum states) in the ordered A and B components. In general the wavefunction describing the localized state may be represented by ~0/~(3') ~ {exp (--~/[R)}f('y),
(4)
where exp (-- "t/R) represents the envelope of the wave and f(30 represents the structure in the envelope. For the ground state f('),) is determined by the extremum states. Given equation (4), the kinetic energy part of the total energy of a localized state can be written as 1]R 2. This, of course, simplifies the calculations since one can then fred the energy associated with a localized state by knowing the average potential in the well and the size of the well. However, to extend Lifshitz results one must develop a method by which the makeup of the potential well is determined using some criteria and not by assumption. This is straightforward if we allow the
Vol. 41, No. 3
243
DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS
concentration of A and B type atoms in the cluster to be determined by maximization of the probability in equation (1). This is achieved by the variational principle introduced by Lloyd and Best [8] and allows us to choose a special configuration from, in principle, an infinite set. Let x = CA -- C,~ be the concentration fluctuation in the volume Vo(~ R~), A = (E~Bin - - E ~ i n ) , : C°E~ in -4- C°E~Bin and E~ain = O. The total energy of the electron may then be written as,
10zl 10zc 10~s 1018 "7
1017
E(R,Ro, x) = (E--xA){1--(R°a]
'E
(5) The factor [1 -- (Rg[R3)I is the fraction of the wavefunction inside the potential well, and [1 -- (Ro[R)] 2 is / normalization factor. t The procedure to find n(E) is to minimize E in equation (5) so as to fred a relation between E, Ro and x. We now maximize P(Ro) in equation (1) which gives a relation between E and Ro or E and x which in turn gives P(E). Since the density of localized states is proportional to the probability of the potential fluctuations (only ground states of the wells are counted) an expression for n(E) readily follows. It is not possible to obtain an analytic solution for n(E) with all the terms of equation (5) included. However, if one retains terms to order R~/R 3 in the expansion, one may t'md n(E) analytically. We find that in this case, for E near E,
n(E) = no exp --
\---~o ]
(6)
and for E near Eamin we recover Lifshitz's result, equation (3). A numerical calculation of n(E) gives a behavior in which the power law dependence of the energy lies between 0.5 and 1.5 (see Fig. 2 solid line). It has been argued [1 ] that disorder in an elemental (or compound) semiconductor produces potential fluctuations which are essentially Gaussian. For Ca ~ C~, equation (1) gives a Gaussian probability distribution thus implying that the behavior of N(E) for E ""/~ obtained from equation (1) is valid for elemental amorphous semiconductors. However, one does not expect the Lifshitz result, equation (3) to appear since there is no lower energy cutoff as exists for the alloy problem. For E close to Eami,, N(E) derived here is expected to deviate from the behavior in such systems, however the method of derivation employed here is not dependent upon the specific form of equation (1). For Gaussian fluctuations for E far from Eamin, the exponent is found to be 0.5 for 3 dimension and 1.5 for 1 dimension, in agreement with previous work [5, 8, 9].
u
10Is
I 1015
LI_I
1014 1013
1012 -tOO 600 503 400 303 200 (meV)
100
Fig. 2. Exact density of states from the theory for CA = CB = 0.5;EBmin --gamin = 1.0 eV. The data points are results from [2]. The region below the dashed horizontal line is below the experimental observation limits. A comparison of the predicted behavior of N(E) with experimental results on a-Si can thus be meaningfully made, provided an estimate of Eo is available. Such an estimate is only available from experiments for a-Si [10], and we compare our calculations with experiments to check if the overall shape of the density of states is correct. We choose C° and C° = 0.5, EBmin -- EAmin = 1.0 or to get Eo equal to 60 meV (this is not a unique choice, of course). The density of states at E is chosen to be ~ 1021 cm-aeV -1, a value suggested by experiments. The solid in Fig. 2 shows the theoretically obtained density of states. The figure also shows the density of tail states inferred assuming that the absorption coefficient is proportional to the density of states over this narrow energy range (~ 0.2 eV). The agreement in the shape of the two results is rather good and suggests that the simple model calculation does reflect the correct physics of the band tail problem. The region of states below the horizontal line is too small to be observed experimentally so that a tapering off of the data points is expected. In the above calculations, only the ground state wavefunctions are used to determine the density of states. This allowed us to choose a simple I[R 2 dependence for kinetic energy and the relation N ( E ) = P ( V ) .
244
DENSITY OF BAND TAIL STATES IN DISORDERED MATERIALS
At energies close to if, the excited states arising from the potential fluctuations will also contribute. In this case, the above simplifications do not hold and a more complex method has to be used. However, it is simple to answer the question at which point the excited and ground states from different clusters (in space and composition) start contributing to the same energy with equal weight. By comparing variationally the relative contributions from the ground state and excited states, we fmd that up to ~ Eo/5 from E, the above treatment is valid. Beyond that the above simple procedure has to be substantially modified.
Vol. 41, No. 3
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9. I 0.
N.F. Mott & E.A. Davis. Electronic Processes in Non-Crystalline Materials. Oxford (1979). B. Abeles, C.R. Wronski, T. Tiedje & G.D. Cody, Solid State Commun. (in press). I.M. Lifshitz, Adv. Phys. 13,483 (1965). D. Thouless, Physics Reports 13C, 93 (1979) and references therein. R. Friedberg & J. Luttinger,Phys. Rev. B12, 4480 (1975). B.I. Halperin & M. Lax, Phys. Rev. 153,802 (1967). J. Singh, Phys. Rev. B23,4156 (1981). P. Lloyd & Best, J. Phys. C8, 3752 (1975). E.P. Gross, J. Stat. Phys. 17,265 (1977). T. Tiedje (to be published).