Effect of Fermi Surface Geometry on Electron-Electron Scattering

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 ${\epsilon }^2\left|\ln \epsilon \right|$ for small values of the parameter $\epsilon$ which is the electron energy relative to the Fermi energy $\mu$ measured in units of $\mu$. At finite temperatures the decay rate leads to an electrical resistivity proportional to $T^2\left|\frac{\ln \mathrm{kT}}{\mu }\right|$ 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 ${\epsilon }^2$ 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.