Landau levels above the optical-phonon continuum in two and three dimensions

The position of the Landau levels perturbed by the electron-phonon interaction is calculated for energies above the LO-phonon continuum in two and three dimensions. In addition to the Landau levels at nħ${\omega }_c$ (${\omega }_c$ is the cyclotron frequency) a series of levels are found at ħ${\omega }_0$+lħ${\omega }_c$ (${\omega }_0$ is the LO-phonon frequency). The degeneracy between the levels nħ${\omega }_c$ and ħ${\omega }_0$+lħ${\omega }_c$ is lifted by the LO-phonon interaction. Our results are valid for excitation energies smaller than 2ħ${\omega }_0$, for zero temperature, and for weak polar materials.