Unified treatment of temperature, concentration, and electric-field dependences of variable-range-hopping conductivity

The conductivity of ${\mathrm{Ge}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Au}}_{\mathit{x}}$ thin films was studied at temperatures in the range 0.019Tx${\mathit{T}}^{\mathrm{*}}$/T behavior. The x dependence of ${\mathit{T}}^{\mathrm{*}}$ accurately followed the theory of Shklovskii and Efros, which includes a Coulomb gap in the electronic density of states. At bias fields exceeding ∼${10}^4$ V/m, nonlinearity was observed that can be accounted for by introducing the effect of the bias field into a calculation of the optimized hopping exponent.