Quantum dots and cuboid quantum wells in fractal dimensions with position-dependent masses

Abstract

In this study, we have constructed a generalized Schrödinger equation in fractal dimensions characterized by an effective potential which is generated by a position-dependent mass. Our analysis is based on the fractal anisotropy and product-like fractal measure approach introduced by Li and Ostoja-Starzewski in their formulation of continuum media. Our analysis is based on the von Roos phenomenological approach and is characterized by a parametric ordering ambiguity α, β, γ which arises in the Schrödinger Hamiltonian as a result of the non-unique representation of the kinetic energy term. We have selected numerical values of von Roos parameters different from the trivial case, and we have solved the corresponding generalized Schrödinger equation for different forms of the position-dependent mass. This study was proved to be practical to analyze several effective physical properties of nanostructures including quantum dots and quantum box that are relevant to overall photonics and exciton spectroscopy in quantum-well structures.