Biaxial strain modulated electronic structures of layered two-dimensional MoSiGeN₄ Rashba systems

The two-dimensional (2D) MA₂Z₄ family has received extensive attention in manipulating its electronic structure and achieving intriguing physical properties. However, engineering the electronic properties remains a challenge. Herein, based on first-principles calculations, we systematically investigate the effect of biaxial strains on the electronic structure of 2D Rashba MoSiGeN₄ (MSGN), and further explore how the interlayer interactions affect the Rashba spin splitting (RSS) in such strained layered MSGN systems. After applying biaxial strains, the band gap decreases monotonically with increasing tensile strains but increases when the compressive strains are applied. An indirect-direct-indirect band gap transition is induced by applying a moderate compressive strain (<5%) in the MSGN systems. Due to the symmetry breaking and moderate spin-orbit coupling (SOC), the monolayer MSGN possesses an isolated RSS near the Fermi level, which could be effectively regulated to the Lifshitz-type spin splitting (LSS) by biaxial strain. For instance, the LSS ← RSS → LSS transformation of the Fermi surface is presented in the monolayer and a more complex and changeable LSS ← RSS → LSS → RSS evolution is observed in bilayer and trilayer MSGN systems as the biaxial strain varies from -8% to 12%, which actually depends on the appearance, variation, and vanish of the Mexican hat band in the absence of SOC under different strains. The contribution of the Mo-d orbital hybridized with the N-p orbital in the highest valence band plays a dominant role in band evolution under biaxial strains, where the RSS → LSS evolution corresponds to the decreased Mo-d orbital contribution. Our study highlights the biaxial strain controllable RSS, in particular the introduction and even the evolution of LSS near the Fermi surface, which makes the strained MSGN systems promising candidates for future applications in spintronic devices.