Two-dimensional (2D) Janus materials due to their asymmetric structures show fascinating spintronic and piezoelectric properties making them a research hot spot in recent years. In this work, inspired by Janus group-III monochalcogenides, we propose $\mathrm{Zn}AXY$ ($A=\mathrm{Si}$, Ge, Sn, and $X/Y=\mathrm{S}$, Se, Te, $X<Y$) monolayers as a novel structure with broken inversion and mirror symmetry. By calculating cohesive energy and phonon dispersion, eight of nine possible $\mathrm{Zn}AXY$ monolayers are proved to be dynamically stable. In addition, thermal stability of these eight structures is confirmed by ab initio molecular dynamics simulations. The electronic band structures of $\mathrm{Zn}AXY$ monolayers indicate that all of them are indirect semiconductors with strong spin-orbit coupling effects. Lack of inversion symmetry gives rise to Zeeman-type spin splitting at the $K$ point of the conduction band with the highest value of 136 meV for ZnSiSeTe. Furthermore, out-of-plane asymmetry results in Rashba spin splitting (RSS) at the \ensuremath{\Gamma} point of the valence and conduction bands in the most compositions of Janus $\mathrm{Zn}AXY$ monolayers. Among them, ZnGeSTe with ${\ensuremath{\alpha}}_{R}^{{\mathrm{\ensuremath{\Gamma}}}_{V}}$ of $1.79\phantom{\rule{0.28em}{0ex}}\mathrm{eV}\phantom{\rule{0.28em}{0ex}}\AA{}$ and ${\ensuremath{\alpha}}_{R}^{{\mathrm{\ensuremath{\Gamma}}}_{c}}$ of $0.862\phantom{\rule{0.28em}{0ex}}\mathrm{eV}\phantom{\rule{0.28em}{0ex}}\AA{}$ and ZnSiSTe with ${\ensuremath{\alpha}}_{R}^{{\mathrm{\ensuremath{\Gamma}}}_{V}}$ of $1.537\phantom{\rule{0.28em}{0ex}}\mathrm{eV}\phantom{\rule{0.28em}{0ex}}\AA{}$ and ${\ensuremath{\alpha}}_{R}^{{\mathrm{\ensuremath{\Gamma}}}_{c}}$ of $0.756\phantom{\rule{0.28em}{0ex}}\mathrm{eV}\phantom{\rule{0.28em}{0ex}}\AA{}$ are found to be great materials for future spintronic applications. Interestingly, in addition to large RSS, Mexican hat dispersion is observed at the \ensuremath{\Gamma} point of the topmost valence band in these materials. Moreover, the calculated elastic coefficients for the hexagonal $\mathrm{Zn}AXY$ monolayers confirm the mechanical stability of the predicted structures. Finally, Janus $\mathrm{Zn}AXY$ monolayers possess high in-plane (up to 7.46 pm/V) and out-of-plane (up to 0.67 pm/V) piezoelectric coefficients making them appealing alternatives for prevalent piezoelectric materials.
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