Tunneling valley Hall effect induced by coherent geometric phase

The tunneling valley Hall effect allows the electrical generation of the transverse valley-dependent current in tunnel junctions, which is free from the Berry curvature but requires tilted Dirac cones. Here, we propose a tilt-free mechanism to realize the tunneling valley Hall effect, based on the geometric-phase coherent transmission through two combined electric barriers in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ lattices. It is shown that the back-reflected electrons at the barrier interface may acquire a valley-dependent geometric phase. The coherence of this geometric phase leads to the valley-dependent skew tunneling, which is responsible for the transverse valley current. We further demonstrate that this transverse valley Hall current can be electrically controlled by the gate voltages applied across the two combined barrier regions and is absent when the two barriers are of equal height. Our work opens an approach to generating the valley Hall effect, suggesting potential applications for geometric-phase devices and valleytronic devices.