Transport in Nearly-Free-Electron-Model Metals. I. Point-Defect Scattering

We have constructed a general method for calculating the electron mean free path of small-band-gap multivalent metals via a nearly-free-electron model (NFEM) in which only one pair of Brillouin-zone planes is considered at a time. The expansion coefficients of the appropriate Bloch wave functions are concentrated about the zone planes with discontinuities in value or slope at the planes themselves. Our procedure is to postulate trial solutions for the Boltzmann equation explicitly containing these concentrated functions as well as functions smoothly distributed over the Fermi surface. The ratio of band-gap energy to Fermi energy forms a natural expansion parameter. Form factors for Fermi-surface scattering are expressed as power series in the square of the scattering vector; only a few terms suffice to give quite satisfactory representations for most form factors. Analytic solutions are developed for a two-term expansion. As pointed out by Ziman, the s-type and p-type states on opposite sides of the zone gap have markedly different mean free paths. The mean free path is not only dependent on →k, but also direction dependent; the component in the zone plane is less affected by the discontinuities than the component perpendicular to the zone plane. It follows that the relaxation time is both →k dependent and a tensor.