Transition from one- to two-dimensional fluctuating variable-range-hopping conduction in microstructures

We study numerically variable-range-hopping conduction in long narrow channels of finite widths as a function of temperature, chemical potential, and the width of the channel. We find clear evidence of the transition from a one-dimensional Mott variable-range-hopping law of $\ln R\sim T^{-\frac{1}{2}}$ to a two-dimensional behavior of $\ln R\sim T^{-\frac{1}{3}}$ in the average resistance $R$ as the width of the channel increases. Consistent with experimental data, calculated absolute fluctuations in the channel resistance are substantially suppressed with the increase in width. Relative resistance fluctuations seem to be much more insensitive to the dimensionality of the system. Our results are in excellent agreement with the available experimental data and show conclusively that variable range hopping is the dominant low-temperature transport mechanism in these narrow channel systems.