Bilayer stacking𝐴-type altermagnet: A general approach to generating two-dimensional altermagnetism

In this paper, we propose a concept of bilayer stacking 𝐴-type altermagnet (BSAA), in which two identical ferromagnetic monolayers are stacked with antiferromagnetic coupling to form a two-dimensional (2D) 𝐴-type altermagnet. By solving the stacking model, we derive all BSAAs for all layer groups and draw three key conclusions: (i) Only 17 layer groups can realize intrinsic 𝐴-type altermagnetism. All 2D 𝐴-type altermagnets must belong to these 17 layer groups, which will be helpful to search for 2D 𝐴-type altermagnet. (ii) It is impossible to connect the two sublattices of BSAA using 𝑆3⁢𝑧 or 𝑆6⁢𝑧, a constraint that is also applicable to all 2D altermagnets. (iii) 𝐶2⁢𝛼 is a general stacking operation to generate BSAA for an arbitrary monolayer. Our theory not only can explain the previously reported twisted-bilayer altermagnets, but also can provide more possibilities to generate 𝐴-type altermagnets. Our research has significantly broadened the range of candidate materials for 2D altermagnets. Based on conclusion (i), the bilayer NiZrCl6 is predicted to exhibit intrinsic 𝐴-type altermagnetism. Additionally, we use twisted-bilayer NiCl2 and CrOCl as supplementary examples of BSAA. Furthermore, utilizing symmetry analysis and first-principles calculation, we scrutinize their spin-momentum locking characteristic to substantiate their altermagnetic properties.