New method for solving Boltzmann's equation for electrons in metals

Abstract

A new set of basis functions is introduced, consisting of products of Fermi-surface harmonics ${F}_{J}(k)$ and polynomials ${\ensuremath{\sigma}}_{n}(\ensuremath{\epsilon})$ in the energy $\frac{(\ensuremath{\epsilon}\ensuremath{-}\ensuremath{\mu})}{{k}_{B}T}$. The former are orthonormal on the Fermi surface, and the latter are orthonormal with weight function $\ensuremath{-}\frac{\ensuremath{\partial}f}{\ensuremath{\partial}\ensuremath{\epsilon}}$. In terms of this set the exact semiclassical Boltzmann equation takes a particularly simple form, giving a matrix equation which can probably be truncated at low order to high accuracy. The connection with variational methods is simple. Truncating at a 1 \ifmmode\times\else\texttimes\fi{} 1 matrix gives the usual variational solution where ${\ensuremath{\varphi}}_{k}$ is assumed proportional to ${\ensuremath{\nu}}_{\mathrm{kx}}$ for electrical conductivity and $(\ensuremath{\epsilon}\ensuremath{-}\ensuremath{\mu}){\ensuremath{\nu}}_{\mathrm{kx}}$ for thermal conductivity. Explicit equations are given for the matrix elements ${Q}_{Jn,{J}^{\ensuremath{'}}{n}^{\ensuremath{'}}}$ of the scattering operator for the case of phonon scattering, and a perturbation formula for $\ensuremath{\rho}$ is given which is accurate for weak anisotropy. The matrix elements are simple integrals over spectral functions ${\ensuremath{\alpha}}^{2}(\ifmmode\pm\else\textpm\fi{},J,{J}^{\ensuremath{'}})F(\ensuremath{\Omega})$ which generalize the electron-phonon spectral function ${\ensuremath{\alpha}}^{2}F(\ensuremath{\Omega})$ used in superconductivity theory. Analogies are described between Boltzmann theory and Eliashberg theory for ${T}_{c}$ of superconductors. The intimate relations between high-temperature resistance and the $s$- or $p$-wave transition temperature are made explicit.