A SIMPLE MODEL OF ac HOPPING CONDUCTIVITY

The physics of ac hopping conduc2ivity in disordered solids is discussed in the light of a recently proposed model. Several new comments are made. One of the most interesting facts about conductivity in disordered solids is the surprising similarity between ionic and electronic conductivity. The same kinds of non-linearities in strong electric fields are observed, and ionic and electronic conductivity have similar temperature and frequency dependence /1,2/. In a recent paper by the author it was suggested that the latter similarities are observed because ionic and electronic hopping conductivity have the same jump frequency distribution p(y) . In the ionic case, the jump frequency Y is given by yaexp(-BAE) where 6 is the inverse temperature and AE is the energy barrier. In the case of tunneling electrons, Y is essentially given by y aexp(-ar) where a is the decay parameter for the wavefunctions and r is the jump distance. If AE resp. r are randomly varying, one finds in both cases that p(y) varies as l/u . From a phenomenological point of view, ionic and electronic conductivity in disordered solids are thus claimed both to be characterized by randomly varying free energy barriers. In order to check this hypothesis, the frequency-dependent conductivity o ( w ) must be calculated. In the CTRW approximation one finds /3-6/ Here q and n are charge resp. density of the charge carriers, a2 is the mean square jump distance in the x-direction, d is the dimensionality, k is the Boltzmann constant, T is the temperature, y , and Y, are the smallest and largest jump frequencies, and = Y,/Y, By letting y o go to infinity the following expression is obtained where T = l/y, /3/ Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985851 C8-344 JOURNAL DE PHYSIQUE This formula reproduces the qualitative features of ac hopping conductivity. In particular one finds an approximate power law of the conductivity at high frequencies, Re u(w) = wS , where s e 1 2/ln(wr) /3/ In the research field of ionic conductivity in glasses, the idea of randomly varying ion jump activation energies has been around since the fifties. At that time Stevels and Taylor suggested this as a reasonable hypothesis /7,8/ but they never actually calculated a(*). The idea was not generally accepted however, because it was thought to be inconsistent with the frequent observation of temperature-independent dielectric loss peaks /9,10/. But this argument is not correct. Equation (2) does indeed imply temperature-independent dielectric loss peaks. At the same time the activation energy for ac conductivity is smaller than that for dc conductivity (compare the discussion of the temperature-dependence of o(w) in ref. /3/). Thus there is no paradox, the point is that the relevant variable to consider is the conductivity and not the dielectric loss, at least when activation energies are discussed. It has been known experimentally for several years that there is a connection between the dc conductivity and the dielectric loss peak frequency W O /11,12/. Namikawa finds that most amorphous solids satisfy the following relation /12/ where p is a numerical constant of order one, E~ is the vacuum permittivity, and A E is the dielectric loss. The above model fits nicely into eq. (3). From eq. (2) one finds that is given by When W O is determined numerically, eq. (3) is found to be satisfied with p = 0,42.