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

The temperature dependence of the resistivity ρ has been calculated, assuming Brooks-Herring scattering, for a degenerately doped semiconductor having ν valleys in the conduction band, all with the same isotropic effective mass m*. For temperatures well below the degeneracy temperature TD, we derive the expression Δρρ₀=γ(TTD)², where ρ₀ is the resistivity at $T=0$, and γ is a function only of the dimensionless parameter a₀k₀ν4/3. Here k₀ is the Fermi wave number if all the electrons were in a single valley, and a₀ is the first Bohr radius in the material. For ν=1, γ is always negative for degenerate doping and the values obtained are in quantitative agreement with available experimental data. For ν>1, γ 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.