Machine-learned model Hamiltonian and strength of spin–orbit interaction in strained Mg2X (X = Si, Ge, Sn, Pb)

Abstract

Machine-learned multi-orbital tight-binding (MMTB) Hamiltonian models have been developed to describe the electronic characteristics of intermetallic compounds Mg2Si, Mg2Ge, Mg2Sn, and Mg2Pb subject to strain. The MMTB models incorporate spin–orbital mediated interactions and they are calibrated to the electronic band structures calculated via density functional theory by a massively parallelized multi-dimensional Monte-Carlo search algorithm. The results show that a machine-learned five-band tight-binding (TB) model reproduces the key aspects of the valence band structures in the entire Brillouin zone. The five-band model reveals that compressive strain localizes the contribution of the 3s orbital of Mg to the conduction bands and the outer shell p orbitals of X (X = Si, Ge, Sn, Pb) to the valence bands. In contrast, tensile strain has a reversed effect as it weakens the contribution of the 3s orbital of Mg and the outer shell p orbitals of X to the conduction bands and valence bands, respectively. The π bonding in the Mg2X compounds is negligible compared to the σ bonding components, which follow the hierarchy $|\sigma_{sp}|\gt|\sigma_{pp}|\gt|\sigma_{ss}|$, and the largest variation against strain belongs to σpp. The five-band model allows for estimating the strength of spin–orbit coupling (SOC) in Mg2X and obtaining its dependence on the atomic number of X and strain. Further, the band structure calculations demonstrate a significant band gap tuning and band splitting due to strain. A compressive strain of $-10\%$ can open a band gap at the Γ point in metallic Mg2Pb, whereas a tensile strain of $+10\%$ closes the semiconducting band gap of Mg2Si. A tensile strain of $+5\%$ removes the three-fold degeneracy of valence bands at the Γ point in semiconducting Mg2Ge. The presented MMTB models can be extended for various materials and simulations (band structure, transport, classical molecular dynamics), and the obtained results can help in designing devices made of Mg2X.