Materials with tilted Dirac cones, such as $8\text{−}Pmmn$ 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.

中文翻译：

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基于静电波导收缩的 8-Pmmn 硼烯谷过滤

具有倾斜狄拉克锥体的材料，例如$8\text{-}磷米米n$硼烯正在探索用于谷电子应用，因为非等效谷的倾斜方向不同。本文提出了一种基于静电波导的谷滤波装置。首先，从理论角度研究这些波导。内积是从沿波导轴的概率电流密度开始定义的。结果表明，具有实特征值的束缚模相互正交，并且相对于所有辐射模正交。下一步，通过利用这些正交性特性，引入了基于专门针对该问题的显式辛分区龙格-库塔时间步长方法的模拟过程。最后，将该方法应用于波导纳米收缩的情况，并证明该结构可以起到谷滤波器的作用。在能量域的某个窗口内，一个谷的透射率实际上为零，而另一个谷的透射率几乎为零。仔细研究了几个设计变量的影响，例如收缩的长度和宽度。此外，还评估了倾斜方向和波导轴之间未对准的影响，表明所提出的谷滤波设计对于高达几十度的偏差具有鲁棒性。此外，仿真结果表明，波导模式的色散关系不一定是单调的，这可能会由于干扰效应而引起传输函数的振荡。