  Quantum Spin Hall (QSH) insulators represent a quintessential example of a
topological phase of matter, characterized by a conducting edge mode within a
bulk energy gap. The pursuit of a tunable QSH state stands as a pivotal
objective in the development of QSH-based topological devices. In this study,
we employ first-principles calculations to identify three strain-tunable QSH
insulators based on monolayer MAlGaTe4 (where M represents Mg, Ca, or Sr).
These monolayers exhibit dynamic stability, with no imaginary modes detected in
their phonon dispersion. Additionally, they possess piezoelectric properties,
rendering them amenable to strain-induced tuning. While MgAlGaTe4 is a normal
insulator under zero strain, it transitions into the QSH phase when subjected
to external strain. Conversely, CaAlGaTe4 and SrAlGaTe4 already exhibit the QSH
phase at zero strain. Intriguingly, upon the application of biaxial strain,
these two compounds undergo phase transitions, encompassing metallic (M),
normal/trivial insulator (NI), and topological insulator (TI) phases, thereby
illustrating their strain-tunable electronic and topological properties. (Ca,
Sr)AlGaTe4, in particular, undergo M-TI/TI-M transitions under applied strain,
while MgAlGaTe4 additionally experiences an M-NI/NI-M transition, signifying it
as a material featuring a metal-insulator transition (MIT). Remarkably, the
observation of metal-trivial insulator-topological insulator transitions in
MgAlGaTe4 introduces it as a unique material platform in which both MIT and
topological phase transitions can be controlled through the same physical
parameter. Our study thus introduces a novel material platform distinguished by
highly strain-tunable electronic and topological properties, offering promising
prospects for the development of next-generation, low-power topological
devices.
