Confinement of an electron with effective mass depending on its position inside a semiconductor Al$_{\mathrm{x}}$Ga$_{\mathrm{1-x}}$As medium

The confinement of an electron inside a two-dimensional semiconductor structure of Al$_{\mathrm{x}}$Ga$_{\mathrm{1-x}}$As which has been deposited on a substrate is studied. In the structure, that has a circular form with radii $\rho_{\mathrm{0}}$, the concentration x in a point depends of the distance from this point to the center of the circumference, and this dependence appears in the potential energy of the electron and in its effective mass too. Considering different forms in which concentration x varies respect $\rho $, we solve the Schr\"{o}dinger equation in polar coordinates ($\rho $,$\theta )$ using a basis formed with the first zeros of Bessel functions of different order m. Taking into account in the Schr\"{o}dinger equation the dependence of the effective mass of the electron on its radial coordinate $\rho $, we work a numerical solution and we obtain the ground state energy, the energy for some excited states, and the probability density of those states.