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

Machine-learned multi-orbital tight-binding (MMTB) Hamiltonian models have been developed to describe the electronic characteristics of intermetallic compounds $\rm Mg_2Si, Mg_2Ge, Mg_2Sn$, and $\rm Mg_2Pb$ 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 (DFT) by a massively parallelized multi-dimensional Monte-Carlo search algorithm. The results show that a machine-learned five-band tight-binding 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 $\rm Mg$ to the conduction bands and the outer shell $p$ orbitals of $\rm X~(X=Si,Ge,Sn,Pb)$ to the valence bands. In contrast, tensile strain has a reversed effect as it weakens the contribution of the $3s$ orbital of $\rm Mg$ and the outer shell $p$ orbitals of $\rm X$ to the conduction bands and valence bands, respectively. The $\pi$ bonding in the $\rm Mg_2X$ compounds is negligible compared to the $\sigma$ bonding components, which follow the hierarchy $|\sigma_{sp}|>|\sigma_{pp}|>|\sigma_{ss}|$, and the largest variation against strain belongs to $\sigma_{pp}$. The five-band model allows for estimating the strength of spin-orbit coupling (SOC) in $\rm Mg_2X$ and obtaining its dependence on the atomic number of $\rm X$ 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 $\Gamma$ point in metallic $\rm Mg_2Pb$, whereas a tensile strain of $+10\%$ closes the semiconducting band gap of $\rm Mg_2Si$. A tensile strain of $+5\%$ removes the three-fold degeneracy of valence bands at the $\Gamma$ point in semiconducting $\rm Mg_2Ge$.