Infrared and dc conductivity in metals with strong scattering: Nonclassical behavior from a generalized Boltzmann equation containing band-mixing effects

Abstract

Metals with high resistivity (∼ 100 μΩ cm) seem to show weaker variation of resistivity (as a function of temperature and perhaps also static disorder) than predicted by semiclassical (Bloch-Boltzmann) theory (SBT). We argue that the effect is not closely related to Anderson localization, and therefore does not necessarily signify a failure of the independent collision approximation. Instead we propose a failure of the semiclassical acceleration and conduction approximations. A generalization of Boltzmann theory is made which includes quantum (interband) acceleration and conduction, as well as a complete treatment of interband-collision effects (within the independent-collision approximation). The interband terms enhance short-time response to $\overrightarrow{\mathrm{E}}$ fields (because the theory satisfies the exact $f$-sum rule instead of the semiclassical approximation to it). This suggests that the additional conductivity, as expressed phenomenologically by the shunt resistor model, is explained by interband effects. The scattering operator is complex, its imaginary parts being related to energy-band renormalization caused by the disorder. Charge conservation is respected and thermal equilibrium is restored by the collision operator. The theory is formally solved for the leading corrections to SBT, which have the form of a shunt resistor model. At infrared frequencies, the conductivity mostly obeys the Drude law $\sigma \left(\omega \right)\sim \sigma \mathrm{}\left(0\right)\mathrm{}{(1-i\omega \tau )}^{-1\mathrm{}}$, except for one term which goes as ${(1-i\omega \tau )}^{-2\mathrm{}}$.