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 C4z 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 Fe2MoS2Se2 monolayer with good stability but very small valley splitting of 1.6 meV. To clearly see the feasibility of our proposal, an unrealistic material Ru2MoS2Se2 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 Fe2I2 monolayer. Our findings can enrich the valley physics, and provide new type of valley materials.