Electron with a position-dependent effective mass inside a semiconductor medium of variable concentration

In this paper we manage the concept of position-dependent effective mass, for an electron confined in a semiconductor, which was deposited two-dimensionally with a variable concentration on a substrate. We assume that the semiconductor structure has a circular form with ρ0 radius and that the concentration varies linearly, quadratically, and harmonically with the radial coordinate ρ. The electron has potential energy due to the semiconductor. In our work, we introduced a relation for the concentration, the potential energy of the electron, and the position-dependent effective mass, which are related to the geometry followed to deposit the semiconductor. In order to illustrate the management of the position-dependent effective mass, we chose as semiconductor Al_xGa_1-x As and numerically solved Schrödinger equation independently of the time in polar coordinates (ρ, θ, where we considered the dependency of the mass with respect to position to get the ground-state energy and the energy of some excited energy states for the three forms in which concentration varies. Estimations reveal the differences when we consider an average mass (a constant mass) or when we take into account the position-dependent mass, observing that from certain values of the ρ0 radius of the structure, the energies that are found when considering the dependent mass are higher than the ones that are obtained with an average mass. Based on results from our study, we can conclude that we can model any type of potential, working with a specific geometry when depositing the semiconductor.