Crystal valley Hall effect

Time-reversal symmetry is thought to be a necessary condition for realizing the valley Hall effect. However, if time-reversal symmetry is broken, whether the valley Hall effect can be realized has not been explored. In this paper, based on symmetry analysis and first-principles electronic structure calculations, we demonstrate that the valley Hall effect without time-reversal symmetry can be realized in two-dimensional altermagnetic materials ${\mathrm{Fe}}_{2}{\mathrm{WSe}}_{4}$ and ${\mathrm{Fe}}_{2}{\mathrm{WS}}_{4}$. Due to crystal symmetry being required, the valley Hall effect without time-reversal symmetry is termed the crystal valley Hall effect. In addition, under uniaxial strain, both monolayer ${\mathrm{Fe}}_{2}{\mathrm{WSe}}_{4}$ and ${\mathrm{Fe}}_{2}{\mathrm{WS}}_{4}$ can realize the piezomagnetic effect. Under biaxial compressive stress, both monolayer ${\mathrm{Fe}}_{2}{\mathrm{WSe}}_{4}$ and ${\mathrm{Fe}}_{2}{\mathrm{WS}}_{4}$ will transform from the altermagnetic semiconductor phase to the bipolarized topological Weyl semimetal phase. Our paper not only provides another direction for exploring the valley Hall effect but also provides a good platform for exploring altermagnetic semiconductors and altermagnetic topological phase transitions.