Polarizability of a crystal with impurities

The expression for the complex frequency- and wavevector-dependent longitudinal electronic polarizability due to the presence of a weak static disorder (e.g., impurities) in a crystal with an arbitrary band structure is derived. The quantum kinetic equation in the self-consistent-field approximation is solved, expanding the one-particle density operator in powers of the screened static imperfection field and a weak perturbing electric field. The polarizability is determined by the induced electronic charge density quadratic in the imperfection field and linear in the perturbing field, averaged over the statistical distribution of imperfections. The obtained expression, which accounts properly for the collective effects in the electronic plasma, takes also into account the polar coupling of the plasma with longitudinal optical phonons. The conductivity in the optical limit (vector q yields 0) is calculated, and the correspondence with one-band effective-mass approximation is established.