  The realization of multifunctional two-dimensional (2D) materials is
fundamentally intriguing, such as combination of piezoelectricity with
topological insulating phase or ferromagnetism. In this work, a Janus monolayer
$\mathrm{SrAlGaSe_4}$ is built from 2D $\mathrm{MA_2Z_4}$ family with dynamic,
mechanical and thermal stabilities, which is piezoelectric due to lacking
inversion symmetry. The unstrained $\mathrm{SrAlGaSe_4}$ monolayer is a narrow
gap normal insulator (NI) with spin orbital coupling (SOC). However, the NI to
topological insulator (TI) phase transition can be induced by the biaxial
strain, and a piezoelectric quantum spin Hall insulator (PQSHI) can be
achieved. More excitingly, the phase transformation point is only about 1.01
tensile strain, and nontrivial band topology can hold until considered 1.16
tensile strain. Moreover, a Rashba spin splitting in the conduction bands can
exit in PQSHI due to the absence of a horizontal mirror symmetry and the
presence of SOC. For monolayer $\mathrm{SrAlGaSe_4}$, both in-plane and much
weak out-of-plane piezoelectric polarizations can be induced with a uniaxial
strain applied. The calculated piezoelectric strain coefficients $d_{11}$ and
$d_{31}$ of monolayer $\mathrm{SrAlGaSe_4}$ are -1.865 pm/V and -0.068 pm/V at
1.06 tensile strain as a representative TI. In fact, many PQSHIs can be
realized from 2D $\mathrm{MA_2Z_4}$ family. To confirm that, similar to
$\mathrm{SrAlGaSe_4}$, the coexistence of piezoelectricity and topological
orders can be realized by strain (about 1.04 tensile strain) in the
$\mathrm{CaAlGaSe_4}$ monolayer. Our works suggest that Janus monolayer
$\mathrm{SrAlGaSe_4}$ is a pure 2D system for PQSHI, enabling future studies
exploring the interplay between piezoelectricity and topological orders, which
can lead to novel applications in electronics and spintronics.
