Valley filtering in 8-Pmmn borophene based on an electrostatic waveguide constriction

Materials with tilted Dirac cones, such as 8-Pmmn borophene, are being explored for valleytronic applications as the tilting direction is different for nonequivalent valleys. In this paper, a valley-filtering device based on electrostatic waveguides is proposed. First, these waveguides are examined from a theoretical point of view. An inner product is defined starting from the probability current density along the waveguide axis. It is shown that the bound modes with real eigenvalues are mutually orthogonal and orthogonal with respect to all radiating modes. In a next step, by exploiting these orthogonality properties, a simulation procedure is introduced based on an explicit, symplectic partitioned Runge-Kutta time-stepping method specifically adapted for this problem. Finally, this approach is applied to the situation of a waveguide nanoconstriction and it is demonstrated that this structure can function as a valley filter. Within a certain window in the energy domain, transmission is practically zero for one valley, while being almost perfect for the other one. The effect of several design variables, such as length and width of the constriction, is carefully investigated. Moreover, the effect of misalignment between the tilting direction and the waveguide axis is assessed, showing that the proposed valley-filtering design is robust against deviations up to several tens of degrees. In addition, the simulation results reveal that the dispersion relation of the waveguide modes is not necessarily monotonic, which can give rise to oscillations in the transmission function due to interference effects.