The frequency dependent conductivity of the electron glass

Volume 156, number 9

PHYSICS LETTERS A

8 July 1991

The frequency dependent conductivity of the electron glass A. Hunt Earth Sciences Department, University ofCalifornia, Riverside, CA 92521, USA Received 25 October 1990; revised manuscript received 1 May 1991; accepted for publication 6 May 1991 Communicated by A.A. Maradudin

In a recent theory of the electron glass, the electrons near the Fermi level, EF, are perfectly ordered within domains of typical 2/ekT, with e the electronic charge, e the host dielectric constant and kTthe Boltzmann constant times the temperature. size r frequency is not too high, electrical conduction takes place along one-dimensional channels at the domain boundaries. A If the 0=e result for the ac conductivity in a pair approximation regime is derived and extended to the so-called multiple hopping regime. A critical frequency, w~,proportional to the dc conductivity, marks the onset ofsequentially correlated hopping. It is shown that the results for the low- and high-frequency conductivities join smoothly at w~Close correspondence to experiment is noted.

1. Model The system which has been called the “electron glass” is characterized by a competition between the effect of disorder, and those of Coulomb interactions. Electrons are localized on sites, which in solids are arranged randomly, but which may also be modelled as being distributed periodically [1]. The single electron, or “site” energies, E.=eØ,, are assumed uncorrelated, with a distribution which is independent of energy for ~W
2

competition between a defect entropy associated with a degeneracy of the domain corners, and a remainder term in the Madelung sum (which produces an increase in the free energy). The result obtained [21 for the domain size is r 2/~kT.The divergence at e T=O indicates that at0 =vanishing temperature the perfect ordering of electrons (within a small energy range, W’, near EF) extends to infinity. The justification for this ansatz is that the (negative) interaction energy of these electrons is sublinear in their concentration, while the additional disorder energy resulting from their periodic arrangement is linear in their concentration. If the energy range near EF is restricted sufficiently, the interactions must domi-

H= ~ n, eØ 1+

~

‘~

e

,

(1.1)

,~

with n, the occupation of site i, e a background dielectric constant, and 0, the random potential at site i.

The basis for the calculations [2] is the ansatz that the equilibrium thermodynamic configuration ofthe electrons within the Coulomb gap (near the Fermi energy) is described by temperature dependent domains of perfect ordering. The shapes of these domains are assumed square in 2-d and cubic in 3-d. The size of these domains is given by a parameter r0, which is then determined by a minimization of the free energy, Such an optimization is based on a ~.

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nate. An optimization of this competition yields the energy range, W’ and therefore the separation of those electrons arranged periodically. Although it has been known for some time that this model could gencrate the appropriate width, Egcx ~ of the Coulomb gap [3,4], it has only recently been recognized that it also generates the finite temperature density of states [2,5] and the dc conductivity [5]. We show here that this model also yields results for the ac conductivity which are in agreement with experiment. A further publication [6] will demonstrate that this model also reproduces the temperature and disorder energy, W, dependence of the modified Edwards— Anderson order parameter [7] as well as the specific

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—

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heat [8] and, qualitatively, the finite volume dependence [9] of the density of states at the Fermi level. A very important property of the domains is that electrons (or holes) in the corners where several domains join together have energies within kT of EF. (Although such a domain structure has not been directly seen in computer simulations, it has been noted that states with energies near EF are usually located in very close spatial proximity, in agreement with the theory given here, and in contradistinction to the prediction of Efros.) This is because the arrangement of the remaining charges about these positions is exactly antisymmetric producing a zero interaction energy [5]. Moreover, they can be translated [21 to the nearest site of opposite symmetry, which is occupied by a hole (electron) along (one-dimensional) domain boundaries without changing the energy of the system. This “defect” entropy leads to a contribution to S proportional to r0~’. (The implied proximity of states near EF = 0 has been confirmed in computer simulations.) On the other hand, the interaction energy per particle is given by [2] 1~] (1.2) U _Eg[l (rg/ro)~ 2/er with Eg~e 8. Putting these results together as U— TS, and optimizing .~Fwith respect to r0 yields2/ekT. (1.3) r0=e Since the states with zero interaction energy are lo,

—

.~=

cated in the corners of the domains, one gets [5] kTN(EF)=l/rg, or

2) d(kT)d_

(1.4) I

(1.5)

N(EF) = (e/e The energies of electronic states on domain boundaries are (in the ideal case of all equally sized domains) independent of the Coulomb interactions with the charges inside the domains. Thus, in dimension d, domain boundary surfaces form domains consistent with the structure of domains of dimensionality d— 1. The one dimensional domain boundaries along which transport occurs are indistinguishable from ideal one dimensional systems. Thus states within kT ofEF are distributed on the average at intervals r 0 on one dimensional “paths”, and the re-

8 July 1991

duction of N(EF) with stepped up dimensionality reflects the reduced density of these paths.

2. Calculation of a(w) Processes with relaxation times t= 1/w make the dominant contribution to the frequency dependent conductivity at frequency w. At high frequencies isolated pairs of states make the dominant contribution to a(w), while at low frequencies relaxation of very large portions of the network contributes significanfly to the conductivity. 2.1. Pair approximation At relatively high frequencies electrons jump back and forth between sites which form relatively compact pairs. On the time scale required for transitions on the domain boundaries, we can take the interiors of the domains to be largely stable so that the distributions, at least, of effective resistance and Capacitance values along the boundaries may be considered constant. The expressions for the resistance between sites i andj, R 0, and the capacitance C1joining site ito the external potential, are taken from the Miller—Abrahams—Pollak [10,11] impedance network, (e2/kT)vPhexp(—2r,J/a—E,,/kT) R~ —

Ii—

c

1

2/kT) exp(—E =

(e

1/kT)

,

(2.1)

where E0 is the largest of I I~IE, I, and I I, and where 4,~=E1—E~. The relaxation time for the pair i,j is then TiiRuCeffRuCic/(Ci+Cj). Pairs which values of Ceff as well astor,,~11w. contribute significantly the conductivity have large The previous treatment of the pair approximation for this system is due to Efros [12]. In this treatment, the density, ~pair~ ofsingly occupied pairs with relaxation times, = w = exp (2r,~/a), r~, = Ia ln ( vPh/w), (2.2) —

is flpair

[N(EF)kT] [N(EF)e2/arII]ar~.

(2.3)

The initial energy is restricted to be within kT of EF, and the final energy, to within kT of the initial en503

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ergy, or vice versa. The relaxation time does not involve the energies since they are so close to the Fermi energy. The typical capacitance is C_—ze2/kT. (2.4)

8 July 1991

tion) frequency proportional to the dc conductivity. On the average the capacitances of the initial sites are

The static polarizability P is r~,,so that the conductivity at frequency w is given by

(2.7) =e2//3kT. Since the hopping takes place in one dimension, we use the density ofstates, ale2, for one dimension. The density of pairs is then,

cr(w) = WCPflpair

flpair

=

(e4/e)co 1n3 ( vPh/W) [N2 (EF) ]a4.

(2.5)

This result is essentially the result for the Fermi glass with the substitution kT—+e2/er~.The density of pairs is independent of frequency, which renders it madequate here, except possibly at very high w, where the hopping length is much shorter than the separation of the electrons within the Coulomb gap. We consider hopping between sites on the one-dimensional chains, since all the sites within kT of EF are located on these chains. Transitions which bring an electron from one domain to the next are suppressed because of the increase in Coulomb interaction energy. Such transitions are important for the low frequency and dc conductivity and are sequentially correlated [13]. Transitions within 1 -d domain boundaries produce an increase in interaction energy which is much smaller than what would result if the electrons were allowed to jump to sites in the interiors of the 3-d domains. This allows an approximation that N(EF) be regarded as unchanged by the individual hopping processes. Thus we apply, with some reservations regarding their appropriateness (owing to the domain structure) the usual random pair statistics for variable range hopping systems, namely that the density ofpairs of sites within kT of the Fermi level, and with a given spatial separation is proportional to [N(EF )kT]2 It will be shown, however, that a formulation of the pair approximation which is consistent with percolation theory at low frequencies includes all sites within flkTof the Fermi level, i.e. Em
(2.6)

and pairs with energy differences flkT. The optimization of the final expression for the conductivity yields a value for fi which produces a smooth change in the distribution of contributing pairs to that appropriate for percolation at the critical (percola504

r~2( flakT/e2 ) ~a.

(2.8)

The factor r2a r2L~rin three dimensions is replaced by ~ a in one dimension. The factor r~2gives the density of one-dimensional paths on which pairs can be found. The static polarizability is r2, as usual. The relaxation time ofthe pair is increased by exp(fl), so that rWIa[ln(VPhIw)P] (2.9) Using eq. (2.9) in place of eq. (2.2), eq. (2.8) in place of eq. (2.3), and eq. (2.7) for the capacitance, gives for the ac conductivity, cr( w, fi), .

a(w,fl)=[a2(kT)3621e4]flw[ln(vPhlw)—fl]2. (2.10) Optimization with respect to fi yields a(w)=a2e2(kT/e2)3w1n3(vPh/w) (T/T 3wa ln3( Vph/W) (2.11) 0) We will consider this result again later, when we are in position to deal with the multiple hopping regime. =

.

2.2. Low-frequency conductivity The low-frequency regime has been discussed in considerable detail [13], so we are brief. First we demonstrate for w < w~that the pair approximation is no longer applicable. Then we outline the basic calculations. At w= vPhexp[— (T 2] =w~we find for the 0/T)U density of sites n~within flkT of the Fermi level n~=(e/e2)kTln(vPh/w~)=(ekT/e2)(T 1”2 0/T) =a~(T/T 2. (2.12) 0)” The average separation ofthese sites on 1-d paths is therefore =a(To/T)~’2.

(2.13)

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Only a fraction (TI T0)”2 of these sites are connected with sites for which the relaxation time is the inverse of the frequency; the remainder are connected to sites for which the relaxation times are shorter. The average hopping length is also

from variable range hopping [14]. All smaller resistom on the percolation path are shorted; their respective capacitors are merged into a single capacitance, C, for each maximally valued resistor. C is ~= ~ , (2.15)

r —a(T0 fT)”2

(2 14)

where is the average capacitance of a site in-

equivalent to percolation, so that for w < w~,a pair approximation is inappropriate. From percolation theory [5,14], the average separation, 1, on the percolation path, of processes with .r= vj,1 x x ‘T “T’’’2 is e ~ ~ 1=a(To/TY’ +oi’)/(d+ i =a(T 0/T)=ro,

eluded on the percolation path and n~,is the average number ofshorts per maximally valued resistor. is calculated [14] from the maximum energy allowed on the percolation path, Em = kT( T 12 0/ T)’ =(e2/kT)(T/To)U2. (2.16)

—

since a=v=l one dimension [15]. It has been in suggested by Summerfield [16] (on theoretical grounds) and by Almond and West [17] (on heuristic grounds) that the ubiquitous Ws behavior of a(w) in glassy systems should set on at a frequency w~cxadC. Here and elsewhere it has been demonstrated that the physical basis of the pair approximation automatically breaks down at w~cr.:adCkT/e in variable range hopping systems and in ionic conducting glasses, since the relevant pairs percolate. The result is that the loss peak (a-peak) in the dielectric constant can always be associated with the percolation of relevant pair processes (with ~ 11w). Due to the enhanced relaxation times of clusters of pair processes, however (as demonstrated below for the “electron glass”), the carriers do not “percolate”. The following discussion applies to the temporal evolution of cluster currents a time t after the application of a dc field, but the results can easily be transformed back into the frequency domain by letting t—ii/w. If an electnc field is applied at time t = 0, relaxation begins to occur in small regions. As t increases, the length scale of relaxation increases. For t> ‘r~ = wi’, individual relaxation processes can involve lengths of r0 or greater, as the slowest transitions are separated by this distance. The individual transitions with w0 w~must be sequentially correlated to minimize the free energy [13]. The charge passed by the rate limiting transitions is a product of the appropnate capacitance, C, associated with these transitions and the factor, F0, with F the external field. The calculation of C is familiar

(217) The quantity2n~,has been shown [14,18] to be nC ‘—‘(T0 iT)” so that C—e2 ‘kT (2 18 ‘~

/

—

The relaxation time of a cluster with N slowest individual rates .rc is .r N.r (2 19 —

N—

After a time t = .r~,equilibrium is established in clusters of extent N (2 20 r~ r 0. The typical growth ofthese clusters by one unit r0 on each end (in each time interval ;) assures a continuous source and sink of charge and dc current. The cluster growth generates a polarization current ~.

,

/ —‘ 2 ii —..~r01.r—~e ~

‘T~.(

/~%A

/ ,~r01r01.r~.

The number of clusters, path is proportional to

nN,

.

1

on a one-dimensional

(

nNccl/NrO.

.

2)

The density of one-dimensional chains contributing to the current is 1 /r~.Therefore the polarization current density at time t is 2

.j~(t) =

3

(e /kT)eFro(ro/r~) (1 /r0)

(.r~It) =F0e/t, ( ) .

so that (2.24) Eq. (2.24) for cr(w) is valid for frequencies on the order of the critical frequency, w~,or less. As this is 505

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a small frequency range and since the variation of a(w) is also small (a(wC)=ewC+adC=2ewC), we now consider the so-called multiple hopping regime, at frequencies just above w~.This regime is characterized by the formation of clusters of transitions, with just one “slowest” transition, but with other faster transitions, which may again be treated as shorts. 2.3. Multiple hopping regime It has been shown [19] that the structure for the result for the pair approximation holds as well in the multiple hopping regime, if the quantities C and P are modified to reflect the additional faster transitions (which are treated as shorts) in clusters of resistances. Below w~we have used the enhancement factor (T 2 to describe the scaling relationship be0/ T)” tween the number of shorts and the number of maximally valued resistors on typical randomly generated clusters of resistances. The relationship holds as well for a range of frequencies above w~.As the factorfl=ln(vPh/w) isa slowly varying function of w, (T 0/ T) 1/2 gives an approximate enhancement ofthe capacitance in a range of frequencies above w~,although the further w is removed from w~,the larger is the error that will result. The distance between maximally valued resistances on the percolation path has been shown to be 1= r0 here. As a consequence we use for the static polarizability, P, of a typical cluster in the multiple hopping regime, Pci. r~

(2.25) 2 ]2 (pair approxi-

enhancedbyover [a ( T0/ mation) two Fcc factors (T T)” 0/ T)’ 2/2 yields Multiplying eq. (2.11)= by three c(w) 0dc + (TI factors T 312ew(T0/T)” [ln (v ,Jw) ]3 (2.26) 0) p for the lower limit of applicability of the pair approximation (at the lower end of the multiple hopping regime). For w = w~,this expression agrees exactly with eq. (2.24) for the low-frequency conductivity. For w>w~ a(w)=adC+ew[l—(T/TO)

1/2

ln(w/w~)]3

.

(2 27) 506

8 July 1991

A plot of log cz( w) versus log w with a( w) given as in eq. (2.26) would yield a fairly straight line with slope 2 28 1 3 ( T1 T ‘/2 — —

I

k

—

0/

for frequencies larger than w~.Below w~such a log— log plot of a(w) versus w would obviously develop a curvature as a(w) ~ Since the low-frequency ac conductivity is temperature independent, we write the frequency dependence of a(w) in the multiple hopping regime as in eq. (2.28) emphasizing that an extrapolation to zero temperature yields a temperature independent contribution to the conductivity in the zero frequency limit as well. One must be careful because the cross-over (or percolation) frequency between the two regimes goes to zero as the temperature approaches zero, but so does the difference in the two results at frequencies slightly above the critical frequency. Application of the Kramers— Kronig relation e(w)=8

J

dw’ a(w’)/(w’2—w2)

(2.29)

to w)

~

=

ew

w< w ‘

C

=ew~(w/w~)s,w>w~, .

(2.30)

.

with s given in eq. (2.28) yields e(w)~8e/(l—s)+8eln(w~/w), ~const+8a(w/w~)~ , w>w~. (2.31) The exact representation of e(w) is quite complicated, but the transition of z( w) from a linear frequency dependence toproduces a slightlya smooth sublinear frequency dependence at w = w~ change from a logarithmic dependence to a slightly negative power in e(w). In neither e(w) nor a(w) should one expect to observe a striking transition at this frequency. These results are significant for the following reasons. First, an extrapolation to T= 0 gives a temperature-independent sample conductance and capacitance at any frequency, w. Second, both a(w) and e(w) are proportional to a, a background dielectric constant (which diverges as the conducting phase by an experiment is approached). performed Thesebyresults Paalanen are confirmed et al. [20]

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on Si: P just on the insulating side of the metal— insulator transition. An exact numerical comparison with experiment is not yet possible, primarily because the numerical

spondence with experiment. Moreover, no new problems arise with the comparison to experiment; the magnitude ew, derived here, is comparable to the magnitude, Cew/ln(oi0/w), as derived by Bhatt and

coefficient on the quantity t0 is not known with certainty. Nevertheless, it is clear that the agreement with experiment is superior to that of all existing theones. The closest result extant is due to Bhatt [21] and Bhatt and Ramakrishnan [22]. This treatment also yields the temperature independence and a proportionality to a background dielectric constant. However, these authors derive a slightly supralinear conductivity, w/ln(w0/w), and a double loganthmic divergence ofthe dielectric constant in the limit of zero frequency. In fact Paalanen et al. [20] find s= 0.9, i.e. a slightly sublinear conductivity. Moreover, careful examination of the experimental results for e(w) show that the divergence is stronger than double logarithmic, a fact, however, which is obvious, since the conductivity is sublinear. It is possible to derive a sublinear frequency dependence of the conductivity by a traditional pair approximation which does not invoke the opening of a Coulomb gap by Coulomb interactions. However, such treatments give necessarily a non-zero power on the temperature dependence, which must either diverge, or vanish in the limit T—~0.The calculation of Bhatt and Ramakrishnan [22] attempts to avoid this difficulty by using the Efros [1] density ofstates for long hops in the limit w—*0. However, this is not the most fundamental problem with extrapolation of a pair approximation to low frequencies, as it is believed here. As w—~0, the length of the hops of those electrons (in the pair approximation) which contribute most to a(w) becomes greater than the separation ofthe individual hops. This problem must be addressed in a theory of non-local relaxation. When Coulomb interactions are ignored, however, non-local relaxation involves a supralinear frequency dependence (in three dimensions) as well as a negative power on the temperature. Thus clear experimental and theoretical difficulties exist in the model of Bhatt and Ramakrishnan [22], while agreement with experiment is impossible if the effects of Coulomb interactionsare ignored. The most fundamental ofthese difficulties, namely the power s, is removed by the treatment of non-local relaxation here, bringsasitsa consistent value into theory much closer corre-

Ramakrishnan [22], with C a numerical constant somewhat larger than unity (derived from the Efros [11 density of states). One possible problem persists, however; in both the conductance and the capacitance of the sample a quadratic dependence on the temperature was noted in the finite temperature correction to the zero temperature results. The scale of this correction was not given by T0, as seems to be implied by any theory of the electron glass. Some possibilities for a finite temperature correction which might not involve merely the scale T0 are: (1) the effect of finite system size on hopping clusters; (2) the effect of finite system size on the divergence of a; (3) difficulty in extrapolation arising from the subtle transition from the multiple hopping regime to the critical (low frequency) regime; (4) a necessity to rescale the Coulomb gap width at finite temperature due to the “weakening” of the Coulomb interaction by thermalization and consequent increase in relative importance of disorder energy; (5) the rigid separation of electrons into the two groups, i.e. the neglect of electron states just outside the Coulomb gap. (3)— (5) are likely the most reasonable causes for this discrepancy, although others may also be possible. The fundamental agreement with the frequency dependence is more important than the as yet unresolved finite temperature correction; approaches which generate the latter behavior, but not the former, are inferior.

3. Conclusions A very simple model of the electron glass has been applied to explain the most striking experimental results for the frequency dependent transport properties in nearly delocalized Si: P. The role of sequential correlations in the electron glass is emphasized at low 2]). frequencies (below w~, = Vu,, exp [ (T0/ T)” The pair approximation for high-frequency ac —

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conduction is shown to join smoothly to the non-local low-frequency conduction process at the loss peak. In the lower limit of the applicability of the pair approximation (in the multiple hopping regime) a specific value for s is derived showing the tendency for s to approach one as the temperature is lowered.

References [l]A.L.Efros,J.Phys.C9(1976)2021. [2] A. Hunt, Philos. Msg. Lett. 62 ( 1990) 377. [3] G. Srinivasan, Phys. Rev. B 4 (1971) 2581. [4] A. Hunt and M. Pollak, J. Phys. C 18 (1986) 5325. [5]A.Hunt,Phys.Lett.A 151 (1990) 187. [6] A. Hunt, Adv. Phys., submitted. [ 71 J.H. Davies, P.A. Lee and T.M. Rice, Phys. Rev. Lett. 49 (1982) 758. [ 81 M. Mochena, dissertation, University of California, Riverside (1990). [9] S. Summerlield and P.N. Butcher, J. Phys. C 15 (1982) 7003.

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[ lo] A. Miller and E. Abrahams, Phys. Rev. 120 ( 1960) 945. in: Proc. on Amorphous and liquid semiconductors, eds. J. Stuke and W. Brenig (Halsted Press, New York, 1974) p. 127. [ 121 A.L. Efros, Philos. Mag. B 43 (1981) 829. [ 131 A. Hunt, Philos. Mag Lett. 62 ( 1990) 399. [ 141 A. Hunt, Philos. Mag. B, submitted. [ 15 ] D. Stauffer, Phys. Rep. 49 ( 1979) 1. [ 161 S. Summertield, Philos. Msg. B 52 ( 1985) 9. [ 171 D.P. Almond and A.R. West, Solid State Ion. 8 (1983) 159. [ 181 N. Apsley and H. Hughes, Philos. Mag. 31 (1975) 1325. [ 191 A. Hunt and M. Pollak, in: Hopping and related phenomena, eds. M. Pollak and H. Frietzsche (World Scientific, Singapore, 1990); M. Pollak and A. Hunt, in: Hopping transport in solids, eds. M. Polk& and B.I. Shklovskii (North-Holland, Amsterdam, 1990). [20] M. Paalanen, T.F. Rosenbaum, GA. Thomas and R.N. Bhatt, Phys. Rev. Lett. 51 (1982) 1896. [ 2 1] R.N. Bhatt, Philos. Mag. B 50 ( 1984) 189. [ 221 R.N. Bhatt and T.V. Ramakrishnan, J. Phys. C 17 ( 1984) L639.

[ 111 M. Pollak,