Finite difference method for solving the Schrödinger equation with band nonparabolicity in mid-infrared quantum cascade lasers

The nonparabolic Schrödinger equation for electrons in quantum cascade lasers (QCLs) is a cubic eigenvalue problem (EVP) which cannot be solved directly. While a method for linearizing this cubic EVP has been proposed in principle for quantum dots [ Hwang et al., Math. Comput. Modell., 40, 519 (2004) ] it was deemed too computationally expensive because of the three-dimensional geometry under consideration. We adapt this linearization approach to the one-dimensional geometry of QCLs, and arrive at a direct and exact solution to the cubic EVP. The method is then compared with the well established shooting method, and it is shown to be more accurate and reliable for calculating the bandstructure of mid-infrared QCLs.