Inequality between optical and DC conductivities in simple metals

Abstract On the basis of the generalized Drude formulae derived from Wigner operator equations for quantum electron–ion plasmas, it is proven analytically that the low-frequency limit of the optical conductivity is lower than the DC conductivity for simple metals under the condition of weak electron–ion scattering. These findings show a contrast with the traditional belief that the two conductivities are equal. The inequality originates from the difference in the inelastic electron–ion scattering rate driven by an inhomogeneous optical electric field (with nonzero wavenumber k ) and that by a homogeneous DC field ( k = 0). These two conductivities become equal and reduce to the Ziman formula in the classical ion regime at temperatures sufficiently higher than the Debye temperature. We then apply the theory to the solid phase by taking into account one- and multi-phonon scattering in the ionic structure factors. At cryogenic temperatures such that thermal phonon is negligible, the DC conductivity becomes infinitely large, whereas the optical conductivity remains finite due to the relaxation of photo-excited electrons coupled with zero-point lattice vibrations. Numerical results for solid Pb are presented and compared with experimental data.