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: ρ=ρ₀exp(T₀/T)1/2. The samples included in this study have characteristic temperatures T₀ 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: ρ(E)=ρ(0)e^-x and ρ(E)=ρ(0)x/sinh(x), where x≡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 T₀ 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-T₀--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.