All-Electrostatic Valley Filtering by Barrier Rotation in Tilted Dirac/Weyl Semimetals

Abstract

Charge carriers in Dirac/Weyl semimetals with tilted anisotropic energy dispersion exhibit valley-dependent refraction and reflection at electrostatic barrier interfaces. Here, we show that an angled barrier interface provides a purely electrostatic route to valley filtering, producing finite valley-polarized conductance. We develop a generalized transfer-matrix formalism for the tilted, anisotropic Dirac Hamiltonian, extended to treat electrostatic barriers at arbitrary angles, and calculate the transmission in the rotated-barrier frame. We also present simulated valley-resolved trajectories in a finite device geometry, which clearly show that one valley is selectively transmitted, whereas the other is predominantly reflected by the angled barrier, without secondary effects such as real or pseudo-magnetic fields.

Figures (3)

• Figure 1: Proposed valley filter device and operating principle. (a) Three-terminal device schematic with source and drain contacts, a global back gate, and a tilted top gate defining the angled barrier of height $V_0$. (b) Energy-band diagram along the transport direction showing tilted Dirac cones in each region; the barrier region cones are shifted by $V_0$ and the Fermi energy $E_{\mathrm{F}}$ is indicated. (c) Top-view illustration of the device with a straight barrier ($\alpha = 0$): $K$ (blue, dotted) and $K'$ (red, dotted) carriers refract symmetrically at the oblique Klein tunneling angle $\theta$, producing no net valley polarization in the integrated conductance. (d) Tilted barrier ($\alpha = 20^\circ$): the angled barrier selectively transmits one valley while reflecting the other, giving rise to valley-polarized conductance.
• Figure 2: Valley-resolved conductance as a function of barrier height $V_0$. (a) Straight barrier ($\alpha = 0^\circ$), with the device geometry illustrated in (e): $G_K$ and $G_{K'}$ calculated by the transfer matrix method (TMM; $K$: blue solid, $K'$: orange dashed) and the semiclassical approach ($K$: green solid, $K'$: red dashed), confirming zero valley polarization ($P_{K,K'} = 0$). (b) Tilted barrier ($\alpha = 20^\circ$), with the device geometry illustrated in (d): above $V_0 \approx E_{\mathrm{F}} = 80$ meV, $K$ and $K'$ conductances split clearly, demonstrating the valley filtering effect. (c) Valley-dependent conductance difference $\Delta G/G_0$ for $\alpha = 20^\circ$, showing the onset of valley polarization in the $n$-$p$-$n$ regime. (d), (e) Device schematics for the tilted and straight barrier configurations, respectively. Parameters: $v_x/v_{\mathrm{F}} = 0.86$, $v_y/v_{\mathrm{F}} = 0.69$, $|w|/v_{\mathrm{F}} = 0.32$, $\phi_{\mathrm{t}} = 90^\circ$ (8-Pmmn borophene), $W = 300$ nm, $L = 300$ nm, $d = 100$ nm, $E_{\mathrm{F}} = 80$ meV.
• Figure 3: Semiclassical carrier trajectories from valley degeneracy to valley filtering. Each trajectory opacity is proportional to its transmission probability; blue: $K$ ($s = +1$), red: $K'$ ($s = -1$). Black lines indicate barrier boundaries. (a) Graphene, $\alpha = 0^\circ$: degenerate $K$ and $K'$ trajectories with symmetric Veselago focusing. (b) Borophene, $\alpha = 0^\circ$: the tilt induces valley-dependent refraction with asymmetric caustic patterns, but no net valley filtering. (c) Borophene, $\alpha = 20^\circ$: the tilted barrier selectively transmits $K$ (blue) while reflecting $K'$ (red), clearly demonstrating the valley filtering effect. Parameters: $v_x/v_{\mathrm{F}} = 0.86$, $v_y/v_{\mathrm{F}} = 0.69$, $|w|/v_{\mathrm{F}} = 0.32$, $\phi_{\mathrm{t}} = 90^\circ$ (8-Pmmn borophene), $W = 300$ nm, $L = 300$ nm, $d = 100$ nm, $E_{\mathrm{F}} = 80$ meV, $V_0 = 160$ meV.