Inelastic Coulomb scattering in a diffusive two-band system

A calculation of the inelastic scattering rate resulting from Coulomb scattering of conduction (s) electrons coupled to a narrow d band is given for a disordered system. The general analysis is carried out by extending the Keldysh formalism for nonequilibrium processes to the situation of two overlapping bands, including both intraband and interband elastic scattering, and interband Coulomb coupling. The general result is applied to the case of two overlapping bands, the s band by itself being in the ballistic regime. Then, when the interband elastic scattering vertex usd is negligible, the inelastic rate of the conduction electrons, at two dimensions, changes from the well-known T2lnT behavior at very low temperatures into a T3/2 law at higher ones. In three dimensions it behaves like T2, the ususal result for a ballistic band. When usd is sufficiently strong to render the motion of the s electrons diffusive, the inelastic rate is proportional to Td/2 at low temperatures, for d=2 and 3, as in the case of a single band. The implications of the results to the temperature-dependent resistivity of a two-band system and to experiments involving two quantum wells in heterostructures are discussed.