Sub-electron charge relaxation via 2D hopping conductors

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We have extended Monte Carlo simulations of hopping transport incompletely disordered two-dimensional (2D) conductors to the processof external charge relaxation. In this situation, a conductor of areaL × W shunts an externalcapacitor C withinitial charge Qi. At low temperatures, the charge relaxation process stops at some ‘residual’charge value corresponding to the effective threshold of the Coulombblockade of hopping. We have calculated the root mean square (rms) valueQR of the residual charge for a statistical ensemble of capacitor-shunting conductorswith random distribution of localized sites in space and energy and randomQi, as a function of macroscopic parameters of the system. Rather unexpectedly,QR has turned out to depend only on some parameter combination: for negligible Coulomb interaction and for substantial interaction. (Hereν0 is the seed density oflocalized states, while κ is the dielectric constant.) For sufficiently large conductors, both functionsQR/e = F(X) follow the powerlaw F(X) = DX−β, but withdifferent exponents: β = 0.41 ± 0.01 for negligible and β = 0.28 ± 0.01 for significant Coulomb interaction. We have been able to derive this law analytically forthe former (most practical) case, and also explain the scaling (but not the exact value ofthe exponent) for the latter case. In conclusion, we discuss possible applications of thesub-electron charge transfer for ‘grounding’ random background charge in single-electrondevices.