Time dependent conductivity in disordered systems

Abstract

The time dependent current in disordered systems under a step applied voltage for a planar symmetry is deduced according to the continuous time random walk approximation. Known dielectric response functions like Cole-Cole, Davidson-Cole, Havriliak-Negami and a few others are used as hopping time distribution functions in order to generate conductive responses. A theoretical relation exists between the dielectric and the conductive response which is the same one prevailing between the time derivative of the creep and the relaxation function, as found long ago by Gross J.Appl. Phys.,18, 212 1947 . A truncated version of the Widder method, in connection with MapleTM software facilities, was employed to obtain graphical primitives of Laplace transforms. I IntroductionIn the opening talk Electret Research -Stages in its Development" 1 of the 5th International Symposium on Electret, ISE 5, held at Heidelberg, honoring his 80th birthday, Prof Gross reviewed the electret research at its very early stages. This testimony is particularly relevant on account of the role Prof Gross played in the development of the eld. He succeeded experimentally in distinguishing between hetero and homocharges appearing in poled dielectrics 2 . Heterocharge is the charge with opposite polarity to that residing on the nearby electrode, while the homocharge has the same sign. The former is attributed to polarization and the latter to space charge injection. Polarization in most cases is a linear e ect while space charge is not. See also 3 , and for the latest developments in this topic, see 4 .In dielectrics, polarization obeys a superposition principle 1,5 , by which the material presents a delayed response to changes of the poling electric eld. The simplest response of the polarization to a step variation in the electric eld is characterized by an exponential increase the so called Debye response , but this is rarely observed experimentally. A n yway the delayed response of the polarization to an alternating applied voltage gives rise to dielectric losses. In this kind of measurement the real and the imaginary components are related through the Kramers-Kronig 3 relations. To these relations Prof. Gross arrived working independently 6 .Here we will be concerned with a new type of delayed response: that a orded by current o wing in disordered materials. That is, absorption currents appear not only due to dipoles during the localized orientation process but also to the unbound motion of a drifting carrier in a disordered medium.The theoretical study of transport in a disordered medium was considerably advanced in a series of works which culminated with the famous article by S c her and Montroll SM 7-10 , using the Continuous Time Random Walk approach CTRW . In this treatment, the carrier performs a random walk in a regular lattice, biased by an electric eld, and the disordered character of the medium re ects itself in the hopping time distribution function HTDF , which n o w deviates from a simple exponential. For time dependent electric eld...

Limit theorems for randomly coarse grained continuous-time random walks

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