Solutions to the Boltzmann equation for electrons in a metal: Energy dependence

Abstract

The solution to the Boltzmann equation for electrons in a metal is a distribution function which depends on energy and wave vector. This paper solves for the energy dependence by expanding the distribution function in sets of orthogonal functions, (a) energy polynomials, (b) Legendre polynomials in tanh ($\frac{{\epsilon }_k}{2k_{B}T}$), or (c) a combination of these two choices. To study only the effects of the energy dependence, the electrical and thermal conductivities were calculated for a class of isotropic models. For one of these models, the electrical resistivity is 37% lower than the Bloch-Grüneisen result at a temperature of 0.15 (in units of the Debye temperature, ${\Theta }_D$). For thermal resistivity, this method is consistent with the result of Klemens; i.e., at very low temperature the correction to the lowest-order result is 51%. Corrections are important at temperatures as high as 0.3 ${\Theta }_D$. These results show that the standard, i.e., simple variational, results for the temperature dependence of transport coefficients make significant errors. However, by the methods of this paper, accurate results can be obtained quite easily by computer, not only for simple isotropic models, but also for realistic metals. Results for transition metals are briefly mentioned; more complete calculations will be presented elsewhere.