Symmetry-assisted anomalous Hall conductivity in a ${\mathrm{CrS}}_2\text{−}{\mathrm{CrBr}}_3$ heterostructure

The symmetry-assisted anomalous valley Hall effect offers exciting prospects in valleytronics and spintronics. Here, using density functional theory and symmetry analysis, we propose an underlying mechanism for the fundamental origin of enhanced anomalous Hall conductivity ${\sigma }_{xy}$ in the ${\mathrm{CrS}}_2$ layer in the ${\mathrm{CrS}}_2/{\mathrm{CrBr}}_3$ heterostructure. Notably, we find that the magnetic proximity effect (MPE) induces an identical nature of spin magnetic moments from the magnetic ${\mathrm{CrBr}}_3$ to nonmagnetic ${\mathrm{CrS}}_2$ layer. For out-of-plane spin components $\left(0^{\circ }\le \theta <{90}^{\circ }\right)$, in addition to orbital magnetic moments, induced magnetization and correction from Berry curvatures $\left[{\Omega }_z\left(k\right)\right]$ leads to an additional contribution to ${\sigma }_{xy}$ in the ${\mathrm{CrS}}_2$ layer. Under in-plane magnetization directions, both time-reversal symmetry $\left(T\right)$ and out-of-plane mirror symmetry $\left(M_Z\right)$ are broken; however, their joint operation is preserved. This joint symmetry operation guarantees Kramers spin degeneracy at the K and $K^{\prime }$ valleys, resulting in vanishing ${\sigma }_{xy}$. The exchange coupling between spins at valleys and magnetization directions induces an angular profile of ${\sigma }_{xy}$ identical to the magnetic moment nature in the ${\mathrm{CrBr}}_3$ layer. Our study reveals a potential approach to predict the variation of ${\sigma }_{xy}$ in valley-polarized heterostructures through symmetry-assisted MPE and sheds light on understanding recent experimental findings in valleytronics.