Landau-level population inversion in crossed electric and quantizing magnetic fields

The problem of the electron distribution in crossed electric and quantizing magnetic fields is treated for free electrons in a parabolic band. The master equation for the diagonal elements of the density matrix is solved, neglecting the $k_z$ dependence of the electron distribution. It is shown that population inversion between Landau levels can be achieved if scattering by optical phonons and a quasielastic scattering process are taken into account.