Stability and accuracy control of $\mathbf{k \cdot p}$ parameters

The $\mathbf{k \cdot p}$ method is a successful approach to obtain band structure, optical and transport properties of semiconductors, and it depends on external parameters that are obtained either from experiments, tight binding or ab initio calculations. Despite the widespread use of the $\mathbf{k \cdot p}$ method, a systematic analysis of the stability and the accuracy of its parameters is not usual in the literature. In this work, we report a theoretical framework to determine the $\mathbf{k \cdot p}$ parameters from state-of-the-art hybrid density functional theory including spin-orbit coupling, providing a calculation where the gap and spin-orbit energy splitting are in agreement with the experimental values. The accuracy of the set of parameters is enhanced by fitting over several directions at once, minimizing the overall deviation from the original data. This strategy allows us to systematically evaluate the stability, preserving the accuracy of the parameters, providing a tool to determine optimal parameters for specific ranges around the ${\Gamma}$-point. To prove our concept, we investigate the zinc blende GaAs that shows results in excellent agreement with the most reliable data in the literature.