Effect of Fermi Surface Geometry on Electron-Electron Scattering

Abstract

In order to investigate the influence of Fermi-surface geometry on the lifetime of an electron due to interactions with other electrons, we have performed a calculation (using Fermi's "Golden Rule") of the energy- and temperature-dependent lifetime of an electron on a cylindrical Fermi surface. At zero temperature, the dominant energy dependence of the inverse lifetime or the decay rate is ${\ensuremath{\epsilon}}^{2}|\mathrm{ln}\ensuremath{\epsilon}|$ for small values of the parameter $\ensuremath{\epsilon}$ which is the electron energy relative to the Fermi energy $\ensuremath{\mu}$ measured in units of $\ensuremath{\mu}$. At finite temperatures the decay rate leads to an electrical resistivity proportional to ${T}^{2}|\frac{\mathrm{ln}\mathrm{kT}}{\ensuremath{\mu}}|$ instead of the ${T}^{2}$ dependence characteristic of a spherical Fermi surface. In addition, the similar calculation (using Fermi's "Golden Rule") for a spherical Fermi surface has been done exactly at zero temperature. The magnitude of the correction to the well-known ${\ensuremath{\epsilon}}^{2}$ term has been obtained. Furthermore, in an appendix, written with N. D. Mermin, the dominating influence of the density of states on the wave-vector dependence of the susceptibility is demonstrated.