Temperature Dependence of the Resistivity of Degenerately Doped Semiconductors at Low Temperatures

The temperature dependence of the resistivity $\rho$ has been calculated, assuming Brooks-Herring scattering, for a degenerately doped semiconductor having $\nu$ valleys in the conduction band, all with the same isotropic effective mass $m^{*}$. For temperatures well below the degeneracy temperature $T_D$, we derive the expression $\frac{\Delta \rho }{{\rho }_0}=\gamma {\left(\frac{T}{T_D}\right)}^2$, where ${\rho }_0$ is the resistivity at $T=0\mathrm{}$, and $\gamma$ is a function only of the dimensionless parameter $\frac{a_0{k}_0}{{\nu }^{\frac{4}{3}}}$. Here $k_0$ is the Fermi wave number if all the electrons were in a single valley, and $a_0$ is the first Bohr radius in the material. For $\nu =1\mathrm{}$, $\gamma$ is always negative for degenerate doping and the values obtained are in quantitative agreement with available experimental data. For $\nu >1\mathrm{}$, $\gamma$ may be either positive or negative, but for all Si and Ge data available, its magnitude is smaller than that obtained from experimental results, suggesting the existence of a contribution to the resistivity due to intervalley electron-electron scattering.