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.