Tunneling conduction in disordered dissipative systems

We consider in this paper the incoherent transport via tunneling of an electron among randomly distributed impurity centers in a heat bath. The heat bath is modeled as a collection of harmonic oscillators. A general expression is first derived using standard instanton methods for the distance dependence of the rate of tunneling between two spatially separated impurity centers coupled to a heat bath. We find that there are two leading terms in the tunneling rate: (1) the standard e^-r/r0 from the wave function overlap and (2) the dissipation correction e-^ηr2, η the friction of the medium. Using this rate, we solve the pair approximation to the master equation for incoherent tunneling transport among N randomly distributed impurity sites and obtain the time dependent diffusion coefficient and the site return Green function. From the long-time limit of the return Green function, we show that as a result of dissipation, there is at long times strictly no diffusion when d=1,2 at low impurity concentrations. A crossover region from nondiffusive to diffusive transport is shown to exist when d=3 that is determined by the magnitude of the friction, η. We discuss the relationship between these results and classical percolation.