Application of coordinate transformation and finite differences method in numerical modeling of quantum dash band structure

In this paper we propose an efficient and simple method for the band structure calculation of semiconductor quantum dashes. The method combines a coordinate transformation (mapping) based on an analytical function and the finite differences method (FDM) for solving the single-band Schrödinger equation. We explore suitable coordinate transformations and propose those, which might simultaneously provide a satisfactory fit of the quantum dash heterointerface and creation of an appropriate computational domain which encloses the quantum dash structure. After mapping of the quantum dash and the rest of computational domain, the Schrödinger equation is solved by the FDM in the mapped space. For the proposed coordinate transformations, we investigate and analyze applicability, robustness and convergence of the method by varying the FDM grid density and size of the computational domain. We find that the method provides sufficient accuracy, stability and flexibility with respect to the size and shape of the quantum dash and above all, extreme simplicity, which is promising and essential for an extension of the method to the multiband Schrödinger equation case.