Valley polarization in two-dimensional tetragonal altermagnetism

The altermagnetism caused by alternating crystal environment provides a unique opportunity for designing new type of valley polarization. Here, we propose a way to realize valley polarization in two-dimensional (2D) tetragonal altermagnetism by regulating the direction of magnetization. The valley polarization along with spin polarization will arise when the orientation of magnetization breaks the ${C}_{4z}$ lattice rotational symmetry, particularly in the conventional in-plane $x$ or $y$ directions. When the direction of magnetization switches between the $x$ and $y$ direction, the valley and spin polarizations will be reversed. This is different from the widely studied valley polarization, which occurs in out-of-plane hexagonal magnetic materials with valley physics at the -K/K point. Followed by first-principles calculations, our proposal is demonstrated in a 2D Janus tetragonal altermagnetic ${\mathrm{Fe}}_{2}{\mathrm{MoS}}_{2}{\mathrm{Se}}_{2}$ monolayer with good stability but very small valley splitting of 1.6 meV. To clearly see the feasibility of our proposal, an unrealistic material ${\mathrm{Ru}}_{2}{\mathrm{MoS}}_{2}{\mathrm{Se}}_{2}$ is used to show large valley splitting of 90 meV. In fact, our proposal can be readily extended to 2D tetragonal ferromagnetic (FM) materials, for example FM ${\mathrm{Fe}}_{2}{\mathrm{I}}_{2}$ monolayer. Our findings can enrich the valley physics, and provide new type of valley materials.