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

The temperature dependence of the resistivity $\ensuremath{\rho}$ has been calculated, assuming Brooks-Herring scattering, for a degenerately doped semiconductor having $\ensuremath{\nu}$ 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 $\frac{\ensuremath{\Delta}\ensuremath{\rho}}{{\ensuremath{\rho}}_{0}}=\ensuremath{\gamma}{(\frac{T}{{T}_{D}})}^{2}$, where ${\ensuremath{\rho}}_{0}$ is the resistivity at $T=0$, and $\ensuremath{\gamma}$ is a function only of the dimensionless parameter $\frac{{a}_{0}{k}_{0}}{{\ensuremath{\nu}}^{\frac{4}{3}}}$. 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 $\ensuremath{\nu}=1$, $\ensuremath{\gamma}$ is always negative for degenerate doping and the values obtained are in quantitative agreement with available experimental data. For $\ensuremath{\nu}>1$, $\ensuremath{\gamma}$ 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.