The Layer Hall Effect without External Electric Field

The layer Hall effect is an intriguing phenomenon observed in magnetic topological layered materials, where the Hall response arises from the opposite deflection of electrons on top and bottom layers. To realize layer Hall effect, space-time $\mathcal{PT}$ symmetry is typically broken by applying an external electric field. In this work, we propose a new mechanism to realize the layer Hall effect by introducing inequivalent exchange fields on both surfaces of a topological insulator thin film, in the absence of an electric field. This approach yields a distinct Hall response compared to the conventional electric-field-induced layer Hall effect, particularly with respect to the Fermi level. Taking the topological insulator Sb$_2$Te$_3$ as a concrete example, we demonstrate the feasibility of inducing the layer Hall effect only by coupling the top and bottom surfaces of Sb$_2$Te$_3$ with different magnetic insulators. Notably, we show that both built-in electric-field-induced and inequivalent exchange-fields-induced layer Hall effects can be achieved by tuning the stacking order between Sb$_2$Te$_3$ and the magnetic layers. Given the well-established experimental techniques for fabricating topological insulator thin films, our work offers a viable pathway for realizing layer Hall effect without external electric field.