Universal crossover in variable range hopping with Coulomb interactions

Using dimensional analysis, we show that the variable-range-hopping resistivity \ensuremath{\rho} of disordered systems with Coulomb interactions obeys the scaling form ln(\ensuremath{\rho}/${\mathrm{\ensuremath{\rho}}}_{0}$)=Af(T/${\mathit{T}}_{\mathit{x}}$), where f(x) is a universal function and A and ${\mathit{T}}_{\mathit{x}}$ are sample-dependent constants. A simple heuristic calculation in three dimensions yields an explicit form, for f(x), which exhibits a smooth-crossover from the Mott (f\ensuremath{\propto}${\mathit{x}}^{\mathrm{\ensuremath{-}}1/4}$) to the Efros-Shklovskii (f\ensuremath{\propto}${\mathit{x}}^{\mathrm{\ensuremath{-}}1/2}$) behaviors. Data on five different samples of compensated n-type CdSe are shown to collapse onto this single function.