Energy spectrum of electrons in a parabolic quantum well in a strong magnetic field

We study, in the ν=1 quantum limit, the spin- and the charge-density excitation spectra of electrons confined in a parabolic quantum well in a strong magnetic field applied perpendicular to the electron gas. The electron-electron interaction is treated in the Hartree-Fock approximation. The inter-Landau-level dispersion relations show similar features with those calculated for a two-dimensional electron gas. On the contrary, for the intra-Landau-level charge-density excitations we found that, as the thickness of the electron layer inside the well increases, the dispersion presents a soft mode. From this we conclude that, as the two-dimensional density of electrons in the parabolic well increases, the system undergoes a phase transition, probably to a charge-density-wave state.