Excitonic and shallow-donor states in semiconducting quantum wells: a fractional-dimensional space approach

Abstract

A systematic study of shallow-donor and excitonic states in semiconducting quantum wells within a fractional-dimensional space approach is presented. In this scheme, the Schrödinger equation is solved in a noninteger-dimensional space in which the interactions are assumed to occur in an isotropic effective environment, and the fundamental quantity is the parameter D, which defines the fractional dimension associated with the effective medium and the degree of anisotropy of the interactions. The fractional dimensionality of the isotropic effective space is derived via an unambiguous procedure in which one may obtain the exact solution for the energies of the actual physical system under consideration. Explicit calculations of the fractional-dimensional parameter are made in the case of shallow donors and excitons in finite-barrier GaAs - (Ga, Al)As quantum wells, with impurity and exciton binding energies found in good agreement with previous variational results and available experimental data.