Magnetic Fabry-Pérot interferometer for valley filtering in a honeycomb-dice model

We theoretically investigate the valley-dependent transmission of particles through a combined electric and magnetic barrier in the $\ensuremath{\alpha}\ensuremath{-}{T}₃$ model which interpolates between the honeycomb and the dice lattices. We show that the combination of the Fabry-P\'erot interferences and the magnetic field leads to a perfect transmission for one valley and a suppression of the transmission for the other valley. When only one of the barriers (magnetic or electric) is present, no valley-polarized current can be produced. By tuning the Fermi energy, this valley-dependent peculiar behavior can be used for valley filtering. Our results show that highly efficient valley filtering with maximum polarization can be achieved by controlling the value of the magnetic field and the electric barrier height and width.