Coexistence of a two- and three-dimensional Landau states in a wide parabolic quantum well

Abstract

Shubnikov--de Haas oscillations are measured in wide parabolic quantum wells with five to eight subbands in a tilted magnetic field. We find two types of oscillations. The oscillations at low magnetic fields are shifted toward higher field with the tilt angle increasing and can be attributed to two-dimensional Landau states. The position of the oscillations of the second type does not change with increasing the tilt angle which points to a three-dimensional character of these Landau states. We calculate the level broadening due to the elastic scattering rate $\ensuremath{\Gamma}=\ensuremath{\Elzxh}/2\ensuremath{\tau},$ where $\ensuremath{\tau}$ is the quantum time, and the energy separation between two-dimensional subbands, ${\ensuremath{\Delta}}_{\mathrm{ij}}{=E}_{j}\ensuremath{-}{E}_{i},$ in a parabolic well. For all levels we obtain ${\ensuremath{\Gamma}}_{j}\ensuremath{\sim}{\ensuremath{\Delta}}_{\mathrm{ij}},$ which means that the levels overlap, supporting the observation of three-dimensional Landau states. Surprisingly, we find that the lowest subband, which has a smaller energy separation from the higher level, does not overlap with these subbands and forms a two-dimensional state.