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Valley polarization transition driven by biaxial strain in Janus $\mathrm{GdClF}$ monolayer

The valley degrees of freedom of carriers in crystals is useful to process information and perform logic operations, and it is a key factor for valley application to realize the valley polarization. Here, we propose a model that the valley polarization transition at different valley points (-K and K points) is produced by biaxial strain. By the first-principle calculations, we illustrate our idea with a concrete example of Janus $\mathrm{GdClF}$ monolayer. The predicted $\mathrm{GdClF}$ monolayer is dynamically, mechanically and thermally stable, and is a ferromagnetic (FM) semiconductor with perpendicular magnetic anisotropy (PMA), valence band maximum (VBM) at valley points and high Curie temperature ($T_C$). Due to its intrinsic ferromagnetism and spin orbital coupling (SOC), a spontaneous valley polarization will be induced, but the valley splitting is only -3.1 meV, which provides an opportunity to achieve valley polarization transition at different valley points by strain. In considered strain range ($a/a_0$: 0.94$\sim$1.06), the strained GdClF monolayer has always energy bandgap, strong FM coupling and PMA. The compressive strain is in favour of -K valley polarization, while the tensile strain makes for K valley polarization. The corresponding valley splitting at 0.96 and 1.04 strain are -44.5 meV and 29.4 meV, which are higher than the thermal energy of room temperature (25 meV). Due to special Janus structure, both in-plane and out-of-plane piezoelectric polarizations can be observed. It is found that the direction of in-plane piezoelectric polarizations can be overturned by strain, and the $d_{11}$ at 0.96 and 1.04 strain are -1.37 pm/V and 2.05 pm/V. Our works pave the way to design the ferrovalley material as multifunctional valleytronics and piezoelectric devices by strain.