Extraction of eight-band k · p parameters from empirical pseudopotentials for GeSn

We extract the parameters for the eight-band k · p model for Ge 1 - x Sn x (x ≤ 0.15) from the calculation of the nonlocal empirical pseudopotential method with the modified virtual crystal approximation. The atomic pseudopotential form factors of Ge are improved such that the calculated Ge band structure has the commonly accepted bandgap and effective masses. The improved Ge parameters are used in proper interpolation to derive the parameters of GeSn for empirical pseudopotential calculation. The calculated band structures suggest that the Ge 1 - x Sn x alloys exhibit a transition between an indirect bandgap semiconductor and a direct one at Sn composition x c = 0.071. From the calculation, we extract the bandgap, the split-off energy, the interband momentum matrix element, and the effective masses of Ge 1 - x Sn x (x ≤ 0.15) as functions of x. From these results, we further derive the parameters used in the eight-band k · p model. These parameters are well expressed in quadratic form. The k · p model with the extracted parameters can give an interband tunneling current in a "pin" diode that is consistent with the current calculated by the empirical pseudopotential method.