Fig. 3. Strain induced valley polarization and piezomagnetism in hole doped monolayer V₂Se₂O.

a, Band structure evolution under different uniaxial strains along $\mathbf{a}$ direction (−5%, 0%, and −5%). Compared with the energy E (X) at X valley, the energy $E\left(\mathrm{Y}\right)$at Y valley monotanesouly shifts down. b Strain induced valley polarization, which defined as the energy difference between two valleys $P=E\left(\mathrm{X}\right)-E\left(\mathrm{Y}\right)$. c Diagram of strained induce net magnetization for different hole densities. d For a given hole density, the net magnetization increases linearly with strain but has an opposite direction for compressive and tensile strains in the region of small strains. When strain is large enough, all the carriers are polarized, and the net magnetization saturates. e For any given strain, the magnetization is always equal to the number of holes per unit cell when the doping is light. However, for heavy doping, the magnetization is almost a constant that depends on the strain.