Resonant Coupling of Landau Levels via LO Phonons in Polar Semiconductors and its Effects on the Landau-Level Raman Scattering from Semiconductor Plasmas

The Green's-function method for studying the resonant coupling of Landau levels via LO phonons is reexamined, and the approximations made therein are justified. The coupling between the $n=2\mathrm{}$ Landau level and the $n=1\mathrm{}$ Landau level plus one-LO-phonon continuum is investigated under the condition that the cyclotron frequency ${\omega }_c$ is nearly equal to the LO-phonon frequency ${\omega }_0$. Since the $n=1\mathrm{}$ Landau level has been resonantly coupled to the $n=0\mathrm{}$ Landau level plus one-LO-phonon continuum under the same condition, novel features of the electron energy spectrum appear. It is shown that the pinning effect that occurs for the $n=1\mathrm{}$ Landau level exists also for the $n=2\mathrm{}$ level. The cross section for the inelastic light scattering from electron plasmas in polar semiconductors under the influence of a dc magnetic field is calculated approximately with the inclusion of electron-phonon polar interaction. The effect of both Coulomb interactions and electron-phonon interactions on the momentum matrix elements occurring in the calculation is neglected, but is kept in the evaluation of the correlation function of a generalized pair operator. The results of scattering from single-particle excitations between the $n=0\mathrm{}$ and $n=2\mathrm{}$ Landau levels exhibits the effect of the resonant coupling between the $n=2\mathrm{}$ and $n=1\mathrm{}$ levels via LO phonons. However, for scattering from collective modes such as the coupled LO-phonon-magnetoplasma mode and the Bernstein modes, the possibility of observing similar resonant coupling effects is ruled out.