Designing van der Waals layers for ferromagnetic quantum spin Hall phase

Most theoretically predicted and experimentally confirmed quantum spin Hall effects (QSHEs) are limited to zero-moment magnets. In this study, we theoretically anticipate the QSHE in ferromagnets and the quantum anomalous Hall effect (QAHE) in antiferromagnets through van der Waals stacked layers, where the magnetization direction is confined to the in-plane. We introduce a search rule for constructing ferromagnetic (FM) quantum spin Hall insulator (QSHI) and antiferromagnetic (AFM) quantum anomalous Hall insulator (QAHI) and further predict a real material of the LuN2/GaS/LuN2 van der Waals multilayer by first-principles calculations. Our results demonstrate the superposition and elimination of Chern numbers under mirror symmetry and time-reversal symmetry, and they further indicate that FM QSHI and AFM QAHI can be converted through deflection magnetization. These findings broaden the material class for QSHE and QAHE and propose a scheme for constructing FM QSHI and AFM QAHI.