Alternating spin splitting of electronic and magnon bands in two-dimensional altermagnetic materials

Unconventional antiferromagnetism dubbed as altermagnetism was first discovered in rutile structured magnets, which is featured by spin splitting even without the spin-orbital coupling effect. This interesting phenomenon has been discovered in more altermagnetic materials. In this work, we explore two-dimensional altermagnetic materials by studying two series of two-dimensional magnets, including MF4 with M covering all 3d and 4d transition metal elements, as well as TS2 with T = V, Cr, Mn, Fe. Through the magnetic symmetry operation of RuF4 and MnS2, it is verified that breaking the time inversion is a necessary condition for spin splitting. Based on symmetry analysis and first-principles calculations, we find that the electronic bands and magnon dispersion experience alternating spin splitting along the same path. This work paves the way for exploring altermagnetism in two-dimensional materials.