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 $F_J\mathrm{}\left(k\right)\mathrm{}$ and polynomials ${\sigma }_n\left(\epsilon \right)$ in the energy $\frac{(\epsilon -\mu )}{k_{B}T}$. The former are orthonormal on the Fermi surface, and the latter are orthonormal with weight function $-\frac{\partial f}{\partial \epsilon }$. 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 × 1 matrix gives the usual variational solution where ${\phi }_k$ is assumed proportional to ${\nu }_{\mathrm{kx}}$ for electrical conductivity and $(\epsilon -\mu ){\nu }_{\mathrm{kx}}$ for thermal conductivity. Explicit equations are given for the matrix elements $Q_{Jn,J^{\prime }{n}^{\prime }}$ of the scattering operator for the case of phonon scattering, and a perturbation formula for $\rho$ is given which is accurate for weak anisotropy. The matrix elements are simple integrals over spectral functions ${\alpha }^2(\pm ,J,J^{\prime })F\left(\Omega \right)$ which generalize the electron-phonon spectral function ${\alpha }^{2}F\left(\Omega \right)$ 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.