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We have extended Monte Carlo simulations of hopping transport in completelydisordered 2D conductors to the process of external charge relaxation. In thissituation, a conductor of area $L \times W$ shunts an external capacitor $C$with initial charge $Q_i$. At low temperatures, the charge relaxation processstops at some "residual" charge value corresponding to the effective thresholdof the Coulomb blockade of hopping. We have calculated the r.m.s$.$ value $Q_R$of the residual charge for a statistical ensemble of capacitor-shuntingconductors with random distribution of localized sites in space and energy andrandom $Q_i$, as a function of macroscopic parameters of the system. Ratherunexpectedly, $Q_{R}$ has turned out to depend only on some parametercombination: $X_0 \equiv L W \nu_0 e^2/C$ for negligible Coulomb interactionand $X_{\chi} \equiv LW \kappa^2/C^{2}$ for substantial interaction. (Here$\nu_0$ is the seed density of localized states, while $\kappa$ is thedielectric constant.) For sufficiently large conductors, both functions$Q_{R}/e =F(X)$ follow the power law $F(X)=DX^{-\beta}$, but with differentexponents: $\beta = 0.41 \pm 0.01$ for negligible and $\beta = 0.28 \pm 0.01$for significant Coulomb interaction. We have been able to derive this lawanalytically for the former (most practical) case, and also explain the scaling(but not the exact value of the exponent) for the latter case. In conclusion,we discuss possible applications of the sub-electron charge transfer for"grounding" random background charge in single-electron devices.Comment: 12 pages, 5 figures. In addition to fixing minor typos and updating  references, the discussion has been changed and expande

Sub-electron charge relaxation via 2D hopping conductors