Fractional-dimensional space and applications in quantum-confined semiconducting heterostructures

Abstract

We present a systematic study of excitonic and impurity states in semiconducting quantum wells within a fractional-dimensional space approach, in which the Schrödinger equation is solved in a noninteger-dimensional space where the interactions are assumed to occur in an isotropic effective environment. In this scheme, the fundamental quantity is the parameter D which defines the fractional dimension associated to the effective medium, and to the degree of anisotropy of the interactions. A direct procedure for determining the fractional dimensionality of the isotropic effective space is proposed in which one may obtain a reliable solution for the energies of the actual physical system under consideration. Explicit calculations of the fractional-dimensional D parameter are made in the case of excitons and impurities in infinite-barrier quantum wells, with exciton and impurity binding energies found in excellent agreement with previous variational results. Calculations are also performed for exciton binding energies in finite-barrier quantum wells with good agreement with recent experimental results.