A solution of the effective-mass Schrödinger equation in general isotropic and nonparabolic bands for the study of two-dimensional carrier gases

In this paper we develop a suitable method for solving the effective-mass Schrödinger equation for two-dimensional electron and hole gases in semiconductor structures such as quantum wells using a general nonparabolic band structure. We present two different ways to treat barriers, the first being the exact solution and the second a suitable option when the band structure is not determined inside the gap. As a first application, this procedure was implemented to solve the effective-mass Schrödinger equation for holes in Si and Ge using an analytical valence-band model. Analyzing the results obtained enabled us to demonstrate the importance of nonparabolicity in energy quantization in these systems and to discuss the suitability of each of these two procedures for dealing with barriers.