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

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: $\rho ={\rho }_0\exp {(T}_0{/T)}^{1/2}.$ 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: $\rho \mathrm{}\left(E\right)=\rho {\mathrm{}\left(0\right)e}^{-x}$ and $\rho \mathrm{}\left(E\right)=\rho \mathrm{}\left(0\right)x/\sinh \mathrm{}\left(x\right),$ where $x\equiv eEl/kT,$ 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 $T.\mathrm{}$ 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.