A computational Fourier series solution of the BenDaniel-Duke Hamiltonian for arbitrary shaped quantum wells

A new technique for solving the BenDaniel-Duke Hamiltonian using a Fourier series method is discussed. This method Fourier transforms the effective mass and potential profiles to calculate the eigenenergies and probability densities in transform space. Numerical solutions of the eigenenergies of a rectangular quantum well are compared to the finite difference, finite element, and transfer matrix methods. The eigenenergies of the envelope functions are computed and compared to the exact case made under a constant effective mass approximation for an asymmetric triangular and parabolic shaped quantum well. The necessity of using a variable effective mass in the BenDaniel-Duke Hamiltonian is shown by a comparison of the eigenenergies in the constant and variable effective mass cases. The Fourier series method is then used to analyze the effects of compositional gradients and electric fields on the eigenenergies and envelope functions for asymmetric coupled asymmetric triangular quantum wells