Quasi-continuous Charge Transfer via 2D Hopping

We have extended our Monte Carlo simulations of hopping transport in completely disordered 2D conductors to the process of external charge relaxation. In this situation, the conductor shunts an external capacitor $C$ with initial charge $Q_i \sim e$. As the charge relaxes due to random hops of electrons through the conductor, so does the electric field $E =Q_R \left( t \right) / C L$ applied to it. At $T \rightarrow 0$, the charge relaxation process stops at some ``residual" charge value $Q_R < e$ corresponding to the effective Coulomb blockade of hopping. We have calculated the r.m.s. value of $Q_R$ (for the statistical ensemble of conductors with random distribution of localized sites) as a function of parameters of the system, and have found that for conductors with sufficiently large area $L \times W \gg a^2$ (where $a$ is the localization radius) it is a universal function of the ratio $(LW/a^2)/C$ for negligible electron- electron interaction and of the ratio $(LW/a^2)/(\chi C)^{2}$ for substantial interaction. (Here $\chi = e^{2} \nu_0 a/ \kappa$ is the dimensionless strength of the Coulomb interaction with $\nu_0$ the density of states and $\kappa$ the dielectric constant.)