Generalized eigenproblem method for surface and interface states: The complex bands of GaAs and AlAs

A proper calculation of the complex band structure is essential for accurately obtaining the energy levels of quantum wells, or the resonances of resonant tunneling diodes. Most present empirical tight-binding calculations are based upon the nearest-neighbor ${\mathit{sp}}^3$s* model, and determine the complex band structure via a transfer-matrix-type equation. This procedure will fail at certain values of the in-plane wave vector ${\mathbf{k}}_{\mathrm{\parallel }}$ or for certain parameter sets; other methods are unsuitable since they do not fully address this problem. Additionally, the nearest-neighbor ${\mathit{sp}}^3$s* model typically does a rather poor job reproducing the X-valley transverse effective mass. More complete calculations thus require an improved method for finding the complex bands and a more complete underlying tight-binding model. Here we develop a method which easily handles those ${\mathbf{k}}_{\mathrm{\parallel }}$ at, or parameter sets for, which other approaches fail and implement it in the second-near neighbor ${\mathit{sp}}^3$s* model to find the complex bands of GaAs and AlAs. We also give the change of basis necessary to transform the equations into a real system, thus allowing for a more efficient calculation. © 1996 The American Physical Society.