To stably employ multiband k·p model for analyzing the band structure in semiconductor heterostructures without spurious solutions (SSs), the Hermitian forward and backward difference (HFBD) scheme for finite difference method (FDM) is presented. The HFBD is the discretization scheme that eliminates the difference instability and employs the Burt-Foreman Hermitian operator ordering without geometric asymmetry. The difference instability arises from employing Foreman's strategy (FS). FS removes SSs caused by unphysical bowing in bulk dispersion curve meanwhile the HFBD is the only difference scheme that can accurately adapt for it. In comparison with other recent strategies, the proposed method in this paper is as accurate and reliable as FS, along with preserving the rapidness and simplicity of FDM. This difference scheme shows stable convergence without any SSs under variable grid size. Therefore, a wide range of experiment-determined band parameters can be applied to large-scale stable simulation with this method regardless of the SSs they originally generate.

We present a two-band finite difference method for the bandstructure calculation of quantum cascade lasers (QCLs) based on the equivalent two-band model of the nonparabolic Schrödinger equation. Particular backward and forward difference forms are employed in the discretization procedure instead of the common central difference form. In comparison with the linearization approach of the nonparabolic Schrödinger equation, the method is as accurate and reliable as the linearization approach, while the velocity of the method is faster and the matrix elements are more concise, therefore making the method more practical for QCLs simulations.

A Lagrange-Hermite finite element method for the eight-band k·p model is developed. We demonstrate that besides the incompletion of k·p basis functions, the ill representation of first-order derivatives can also bend the conduction band structure down and lead to the highly oscillatory solutions. Our method simultaneously solves these two problems and achieves robust stability and high accuracy in real-space numerical calculation. The more physical asymmetric operator ordering is employed and the connection problem in abrupt interface is resolved by using an approximately abrupt interface. The situation of smooth interface used to explain the discrepancies between experiment and simulation of abrupt interface is also calculated by our method, and the result suggests that the influence of the interface smoothing should be considered in the short period superlattices or quantum structures of the narrow well.