VASP2KP: $ k\cdot p $ models and Landé $g$-factors from $ab-initio$ calculations

The $k\cdot p$ method is significant in condensed matter physics for the compact and analytical Hamiltonian.In the presence of magnetic field, it is described by effective Zeeman's coupling Hamiltonian with Landé $g$-factors. Here, we develop an $pen-source$ package VASP2KP (including two parts: vasp2mat and mat2kp) to compute $k\cdot p$ parameters and Landé $g$-factors directly from the wavefunctions provided by the density functional theory (DFT) as implemented in Vienna $ab initio$ Simulation Package (VASP). First, we develop a VASP patch vasp2mat to compute matrix representations of the generalized momentum operator $\boldsymbol{\hat{\pi}}=\boldsymbol{\hat{p}}+\frac{1}{2mc^2}\left(\boldsymbol{\hat{s}}\times\nabla V(\boldsymbol{r})\right) $, spin operator $\boldsymbol{\hat{s}}$, time reversal operator $\hat{T}$ and crystalline symmetry operators $\hat{R}$ on the DFT wavefunctions. Second, we develop a python code mat2kp to obtain the unitary transformation $U$ that rotates the degenerate DFT basis towards the standard basis, and then automatically compute the $k\cdot p$ parameters and $g$-factors. The theory and the methodology behind VASP2KP are described in detail. The matrix elements of the operators are derived comprehensively and computed correctly within the Projector Augmented Wave method. We apply this package to some materials, $e.g$.Bi$_2$Se$_3$, Na$_3$Bi, Te, InAs and 1H-TMD monolayers. The obtained effective model's dispersions are in good agreement with the DFT data around the specific wave vector, and the $g$-factors are consistent with experimental data. The VASP2KP package is available at https://github.com/zjwang11/VASP2KP.