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The realization of multifunctional two-dimensional (2D) materials isfundamentally intriguing, such as combination of piezoelectricity withtopological 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 lackinginversion symmetry. The unstrained $\mathrm{SrAlGaSe_4}$ monolayer is a narrowgap normal insulator (NI) with spin orbital coupling (SOC). However, the NI totopological insulator (TI) phase transition can be induced by the biaxialstrain, and a piezoelectric quantum spin Hall insulator (PQSHI) can beachieved. More excitingly, the phase transformation point is only about 1.01tensile strain, and nontrivial band topology can hold until considered 1.16tensile strain. Moreover, a Rashba spin splitting in the conduction bands canexit in PQSHI due to the absence of a horizontal mirror symmetry and thepresence of SOC. For monolayer $\mathrm{SrAlGaSe_4}$, both in-plane and muchweak out-of-plane piezoelectric polarizations can be induced with a uniaxialstrain 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 at1.06 tensile strain as a representative TI. In fact, many PQSHIs can berealized from 2D $\mathrm{MA_2Z_4}$ family. To confirm that, similar to$\mathrm{SrAlGaSe_4}$, the coexistence of piezoelectricity and topologicalorders 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 studiesexploring the interplay between piezoelectricity and topological orders, whichcan lead to novel applications in electronics and spintronics.

Piezoelectric quantum spin Hall insulator with Rashba spin splitting in Janus monolayer $\mathrm{SrAlGaSe_4}$