Possible way to achieve anomalous valley Hall effect by piezoelectric effect in a GdCl2 monolayer

Ferrovalley materials can achieve manipulation of the valley degree of freedom with intrinsic spontaneous valley polarization introduced by their intrinsic ferromagnetism. A good ferrovalley material should possess perpendicular magnetic anisotropy (PMA), the valence band maximum (VBM) or conduction band minimum at valley points, strong ferromagnetic (FM) coupling, and proper valley splitting. In this work, the monolayer ${\mathrm{GdCl}}_{2}$ is proposed as a potential candidate material for valleytronic applications using the first-principles calculations. It is proved that monolayer ${\mathrm{GdCl}}_{2}$ is a FM semiconductor with the easy axis along the out-of-plane direction and strong FM coupling. A spontaneous valley polarization with a valley splitting of 42.3 meV is produced due to its intrinsic ferromagnetism and spin-orbit coupling. Such valley splitting is larger than the thermal energy at room temperature, suggesting the possibility of room-temperature device operations. Although the VBM of unstrained monolayer ${\mathrm{GdCl}}_{2}$ is away from the valley points, a very small compressive strain (about 1%) can make the VBM move to the valley points. We propose a possible way to realize the anomalous valley Hall effect in monolayer ${\mathrm{GdCl}}_{2}$ using the piezoelectric effect, not an external electric field, namely, the piezoelectric anomalous valley Hall effect. This phenomenon could be classified as piezovalleytronics, being similar to piezotronics and piezophototronics. The only independent piezoelectric strain coefficient ${d}_{11}$ is $\ensuremath{-}2.708$ pm/V, which is comparable to that of the classical bulk piezoelectric material $\ensuremath{\alpha}$-quartz (${d}_{11}=2.3$ pm/V). The biaxial in-plane strain and electronic correlation effects are considered to confirm the reliability of our results. Finally, monolayer ${\mathrm{GdF}}_{2}$ is predicted to be a ferrovalley material with dynamic and mechanical stabilities, PMA, the VBM at the valley points, strong FM coupling, valley splitting of 47.6 meV, and ${d}_{11}$ of 0.584 pm/V. Our work provides a possible way to achieve the anomalous valley Hall effect using the piezoelectric effect, which may stimulate further experimental works related to valleytronics.