Broken inversion symmetry in atomic structure can lead to the emergence of specific functionalities at the nanoscale. Therefore, realizing 2D materials in Janus form is a growing field, which offers unique features and opportunities. In this paper, we investigate the structural, vibrational, elastic, piezoelectric, and electronic properties of Janus $\mathrm{Bi}XY$ ($X=\mathrm{S}$, Se, Te and $Y=\mathrm{F}$, Cl, Br, I) monolayers based on first-principle methods. The structural optimization and vibrational frequency analysis reveal that all of the proposed structures are dynamically stable. Additionally, ab initio molecular dynamics simulations verify the thermal stability of these structures even at elevated temperatures. The mechanical response of the Janus $\mathrm{Bi}XY$ crystals in the elastic regime is investigated in terms of in-plane stiffness and the Poisson ratio, and the obtained results ascertain their mechanical flexibility. The piezoelectric stress and strain coefficient analysis demonstrates the appearance of strong out-of-plane piezoelectricity, which is comparable with the Janus transition metal dichalcogenide monolayers. The calculated electronic band structures reveal that except for BiTeF, all Janus $\mathrm{Bi}XY$ monolayers are indirect band gap semiconductors, and their energy band gaps span from the infrared to the visible part of the optical spectrum. Subsequently, large Rashba spin splitting is observed in electronic band structures when the spin-orbit coupling is included. The obtained results point out Janus 2D $\mathrm{Bi}XY$ structures as promising materials for a wide range of applications in nanoscale piezoelectric and spintronics fields.
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