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 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 3sorbital of Mg to the conduction bands and the outer shellporbitals 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 3sorbital of Mg and the outer shellporbitals of X to the conduction bands and valence bands, respectively. Theπbonding in the Mg₂X compounds is negligible compared to theσbonding components, which follow the hierarchy|σsp|>|σpp|>|σss|, and the largest variation against strain belongs toσ. 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.