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Ferrovalley materials can achieve manipulation of the valley degree offreedom with intrinsic spontaneous valley polarization introduced by theirintrinsic ferromagnetism. A good ferrovalley material should possessperpendicular magnetic anisotropy (PMA), valence band maximum (VBM)/conductionband minimum (CBM) at valley points, strong ferromagnetic (FM) coupling andproper valley splitting. In this work, the monolayer $\mathrm{GdCl_2}$ isproposed as a potential candidate material for valleytronic applications by thefirst-principles calculations. It is proved that monolayer $\mathrm{GdCl_2}$ isa FM semiconductor with the easy axis along out of plane direction and strongFM coupling. A spontaneous valley polarization with a valley splitting of 42.3meV is produced due to its intrinsic ferromagnetism and spin orbital coupling(SOC). Although the VBM of unstrained monolayer $\mathrm{GdCl_2}$ is away fromvalley points, a very small compressive strain (about 1\%) can make VBM move tovalley points. We propose a possible way to realize anomalous valley Halleffect in monolayer $\mathrm{GdCl_2}$ by piezoelectric effect, not an externalelectric field, namely piezoelectric anomalous valley Hall effect (PAVHE). Thisphenomenon could be classified as piezo-valleytronics, being similar topiezotronics and piezophototronics. The only independent piezoelectric straincoefficient $d_{11}$ is -2.708 pm/V, which is comparable to one of classicalbulk piezoelectric material $\alpha$-quartz ($d_{11}$=2.3 pm/V). The biaxialin-plane strain and electronic correlation effects are considered to confirmthe reliability of our results. Finally, the monolayer $\mathrm{GdF_2}$ ispredicted to be a ferrovalley material with dynamic and mechanical stabilities,PMA, VBM at valley points, strong FM coupling, valley splitting of 47.6 meV,and $d_{11}$ of 0.584 pm/V.

A possible way to achieve anomalous valley Hall effect by piezoelectric effect in $\mathrm{GdCl_2}$ monolayer