Coulomb gap, Coulomb blockade, and dynamic activation energy in frustrated single-electron arrays

We have used modern supercomputer facilities to carry out extensive numerical simulations of statistical properties of 1D and 2D arrays of single-electron islands with random background charges, in the limit of small island self-capacitance. In particular, the spectrum of single-electron addition energies shows a clear Coulomb gap that, in 2D arrays, obeys the Efros-Shklovskii theory modified for the specific electron-electron interaction law. The Coulomb blockade threshold voltage statistics for 1D arrays is very broad, with r.m.s. width $\delta V_t$ growing as $<V_t>\propto N^{1\mathrm{/}2}$ with the array size $N$. On the contrary, in square 2D arrays of large size the distribution around $<V_t>\propto N$ becomes relatively narrow $(\delta V_t/< V_t> \propto 1/N)$,andthedc$I$-$V$ curves are virtually universal. At low voltages, the slope $G_0(T)$ of $I$-$V$ curves obeys the Arrhenius law. The corresponding activation energy $U_0$ grows only slowly with $N$ and is considerably lower than the formally calculated “lowest pass” energy $E_{max}$ of the potential profile, thus indicating the profile “softness”.