Reliable k⋅p band structure calculation for nanostructures using finite elements

The k⋅p envelope function method is a popular tool for the study of electronic properties of III–V nanostructures. The equations are usually transferred to real-space and solved using standard numerical techniques. The powerful and flexible finite element method was seldom employed due to problems with spurious solutions. The method would be favorable for the calculation of electronic properties of large strained nanostructures as it allows a flexible representation of complex geometries. In this paper, we show our consistent implementation of the k⋅p envelope equations for nanostructures of any dimensionality. By including Burt-Foreman operator ordering and ensuring the ellipticity of the equations, we are able to calculate reliable and spurious solution free subband structures for the standard k⋅p 4×4, 6×6 and 8×8 models for zinc-blende and wurtzite crystals. We further show how to consistently include strain effects up to second order by means of the Pikus-Bir transformation. Finally, we analyze the performance of our implementation using benchmark examples.