4 AC–conductivity in ion conducting glasses
4.2 Extension of the EMA to dilute hard–core systems
is the difference of the real part of the dielectric permittivity at low frequencies (which is governed by the ionic contributions) and the electronic (or more generally: non–
ionic) contributions at high frequencies"(el.)1 0. The proportionality factor is of the order of one. The relation can be made plausible when regarding it as an electrodynamic analog of the MAXWELLrelation-1'-1=G-11, where!xis identified with-1,-1 withdcandG-11 with. If"(T;c)follows a Curie type law, that is"(T;c)/c=T, eq. (4.3) reduces to eq. (4.2). In general, however, it is difficult to measure "(T;c). In most cases electrode polarization effects make it impossible to find a saturation of
"
0
(!;T;c)in the limit!!0, and determining"(el.)1
0
requires some extrapolation of the data. For this reason SIDEBOTTOM [Sid00] instead used the condition
D 0
(!x)
!
= 2D
1 (4.5)
to determine!xand to perform the scaling of thex–axis. New results put emphasize on nonuniversal features of the conductivity spectra dependent on the glass composi-tion [Gos00, Rol00]. This suggests the scaling of the conductivity data to be an effective one that can be applied only in a limited range of concentrations, where the range of validity depends on the class of materials considered. For example the effective scal-ing seems to work much better for germanate and thioborate glasses than for borate or tellurite glasses, cf. [Sid99, Gos00, Sid01].
We conclude that the form of the conductivity spectra has both generic contri-butions and contricontri-butions dependent on the chemical details of the materials under consideration. In this work we are not interested in the influence of chemical details of specific materials and the subtle differences in the conductivity spectra they may lead to, but in generic mechanisms and procedures to describe them. Thus we will focus on the universal features of the conductivity spectra that can be seen in scaling behavior.
4.2 Extension of the EMA to dilute hard–core systems
The explanation of the temperature–concentration effective scaling for the ac–conduc-tivity is a challenging task. To our knowledge only one theoretical ansatz for it has been proposed so far [Sch00c]. Since in the model proposed there the dc–conductivity is an input parameter and this input parameter is the only entry point of the concen-tration into the model, it is hardly suited to describe concenconcen-tration effects.
The EMA as developed in chapter 3 is a one–particle theory and thus not capable of describing concentration effects. However, we will show in section 4.2.1 that the mas-ter equation in the case of a dilute hard–core system can be reduced approximately to an effective one–particle equation which can be treated by the Effective Medium Approximation described in chapter 3. This approach has been developed in collab-oration with diploma student F. Scheffler and has been described also by F. Scheffler in [Sch00a]. An analog procedure for a different class of systems has been proposed earlier by Bleibaumet al.[Ble97].
4.2.1 Derivation of an effective one–particle master equation
We consider a hard–core lattice gas withnPclassical particles with site energy disorder obeying a given density of states. Since we want to derive an effective one–particle approximation, we mark one particle as tracer. The master equation describes the time development of the probabilityPm(t;n )to find the tracer at timetat lattice site
mwhile all other particles are distributed according to the occupation numbersn =
(n
1
;:::;n
N ).
We denote by l;l+Æ the hopping rate of a single particle from lattice site l to a nearest–neighbor sitel+Æand bywl;l+Æ(n)the transition probability of the occupation number vectornton(l;Æ)which is obtained when the occupation numbernlandnl+Æ are exchanged. The master equation reads
@P
(n)is symmetric even if the one–particle hopping ratel;l+Æ is not symmetric.
The equation ensures that the following condition holds true:
8t: P
m
(t;n) = 0; if nm = 1: (4.7)
Since we are only interested in the tracer–diffusion, but not in the movement of the other particles, we sum up the master equation over allnto obtain the one–particle probability
Due to the summation overfngwe may in the second sum of eq. (4.6) substituten(l;Æ) bynand thus the sum vanishes. In the first sum we can use eq. (4.7) and so we get the
4.2 Extension of the EMA to dilute hard–core systems describes the compound probability for finding the tracer at timetat lattice sitem+Æ and simultaneously another particle at lattice sitem. For a dilute system this correlator may be factorized,
h n
Since the particles2;:::;nP are in equilibrium (when one neglects the disturbance by the tracer particle itself), we can substitute the averaged occupation number by its equilibrium value the FERMIdistribution, and finally arrive at the effective one–particle master equation
dp This equation has a simple interpretation: It is the one–particle master equation with a hopping rate that has been multiplied by(1-f(Em+Æ)). This factor is the probability that on the average a sitem+Æis not yet occupied by another particle, since only in this case the tracer may jump frommtom+Æ.
In the many–particle master equation the detailed balance condition reads
Thus the one–particle master equation with the effective ratesi;j(1-fj)has the cor-rect equilibrium distributionp(ieq)
=f
i
=Nproportional to the FERMIdistribution.
4.2.2 Density of States
It is experimentally well known that the activation energy strongly depends on the concentrationc[Ing87]. For a wide range of materials a logarithmic dependence
Eact / -lnc (4.16)
has been found. From a critical path analysis it can be shown that an exponentially decreasing Density of States (DOS) leads to a dependence of the form of eq. (4.16) (for a systematic analysis see [Maa92]). Thus we will use an exponentially decreasing DOS
(E) =
4.3 The self–consistency equation
With this preparatory work done we can now treat the self–consistency equation (3.20) for the dilute hard–core system. To this end we note the symmetrized form of the effective one–particle rate from eq. (4.13) according to eq. (2.7a)
(E;E (which is determined by f(E) = c, for details see appendix E.4.1).
Since eq. (4.18) has the factorization property (3.35a), the frequency–energy scaling eq. (3.38) holds and it is sufficient to consider the EMA self–consistency equation (3.20) forE=0, The fugacity is very small for low temperatures. In order to make a numerical scheme that solves eq. (4.19) for(z;0)robust, it is advantageous to define
a
and to substitute x = exp0E0 for E0. After this substitutions the self–consistency equation reads
For details on the limits of this equation and how to compute the frequency–dependent diffusion coefficient, see appendix E.4.