VASP2KP: k⋅p Models and Landé g-Factors from ab initio Calculations
Sheng 盛 Zhang 章, Haohao Sheng, Zhi-Da 志达 Song 宋, Chenhao Liang, Yi 毅 Jiang 蒋, Song 松 Sun 孙, Quansheng 泉生 Wu 吴, Hongming 红明 Weng 翁, Zhong 忠 Fang 方, Xi 希 Dai 戴, Zhijun Wang
Abstract
The k ⋅ p method is significant in condensed matter physics for the compact and analytical Hamiltonian. In the presence of magnetic field, it is described by the effective Zeeman’s coupling Hamiltonian with Landé g -factors. Here, we develop an open-source package VASP2KP (including two parts: vasp2mat and mat2kp ) to compute k ⋅ 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mi mathvariant="bold-italic">π</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mover accent="true"> <mml:mi mathvariant="bold-italic">p</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> <mml:mo>+</mml:mo> <mml:mstyle displaystyle="true"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mo stretchy="false">[</mml:mo> <mml:mover accent="true"> <mml:mi mathvariant="bold-italic">s</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> <mml:mo>×</mml:mo> <mml:mo>∇</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">r</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> , spin operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mi mathvariant="bold-italic">s</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:math> , time reversal operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>T</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:math> , and crystalline symmetry operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>R</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:math> 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 ⋅ 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 .