  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 structures of 2D Rashba
MoSiGeN4 (MSGN), and further explore how the interlayer interactions affect the
Rashba spin splitting in such strained layered MSGNs. 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 MSGNs. Due to the symmetry breaking and
moderate spin-orbit coupling (SOC), the monolayer MSGN possess an isolated
Rashba spin splitting (R) near the Fermi level, which could be effectively
regulated to the Lifshitz transition (L) by biaxial strain. For instance, a
L-R-L transformation of Fermi surface is presented in monolayer and a more
complex and changeable L-R-L-R evolution is observed in bilayer and trilayer
MSGNs as the biaxial strain vary from -8% to 12%, which actually depend on the
appearance, variation, and vanish of the Mexican hat band in the absence of SOC
under different strains. The contribution of Mo-dz2 orbital hybridized with
N-pz orbital in the highest valence band plays a dominant role on the band
evolution under biaxial strains, where the R-L evolution corresponds to the
decreased Mo-dz2 orbital contribution. Our study highlights the biaxial strain
controllable Rashba spin splitting, in particular the introduction and even the
evolution of Lifshitz transition near Fermi surface, which makes the strained
MSGNs as promising candidates for future applications in spintronic devices.
