Non-Ohmic effects in hopping conduction in doped silicon and germanium between 0.05 and 1 K

Abstract

We have studied non-Ohmic effects in hopping conduction in moderately compensated ion-implanted Si:P, B (both $n$- and $p$-type) and neutron-transmutation-doped Ge:Ga,As over the temperature range 0.05--0.8 K and up to moderately strong electric fields. In the limit of small fields, where the current is proportional to applied voltage, the resistivities of these materials are approximated over a wide temperature range by the model of variable range hopping with a Coulomb gap: $\ensuremath{\rho}={\ensuremath{\rho}}_{0}\mathrm{exp}{(T}_{0}{/T)}^{1/2}.$ The samples included in this study have characteristic temperatures ${T}_{0}$ in the range 1.4--60 K for silicon, and 22--60 K for germanium. We have compared our data to exponential and ``hyperbolic-sine'' field-effect models of the electrical nonlinearity: $\ensuremath{\rho}(E)=\ensuremath{\rho}{(0)e}^{\ensuremath{-}x}$ and $\ensuremath{\rho}(E)=\ensuremath{\rho}(0)x/\mathrm{sinh}(x),$ where $x\ensuremath{\equiv}eEl/kT,$ and to an empirical hot-electron model. The exponential field-effect model tends to be a good representation for the samples with high ${T}_{0}$ at low $T.$ The sinh model can match the data only at low fields. The hot-electron model fits our data well over a wide range of power in the low-${T}_{0}$--high-$T$ regime. We discuss the quantitative implications of these results for the application of these materials as thermometers for microcalorimeters optimized for high-resolution spectroscopy.