Parabolic-Cylinder Approach to Valley-Polarized Conductance in Tilted Anisotropic Dirac-Weyl Systems

Abstract

We develop a parabolic-cylinder approach to valley-polarized conductance in tilted anisotropic Dirac-Weyl systems, showing that the smooth-interface scattering problem can be reduced analytically to the Weber equation, which belongs to the same differential-equation class as the quantum harmonic oscillator. This reduction yields closed-form expressions for the angular transmission envelope and clarifies the distinct roles of the tilt components: the perpendicular tilt renormalizes the tunneling-envelope width, while the parallel tilt shifts the Fabry-Perot resonance structure differently in opposite valleys. Combined with the nonlinear mapping between the fixed device frame and the rotated barrier frame, this analytical structure provides a direct route from valley-dependent interface tunneling to net valley-polarized conductance. We apply the formalism to rotated electrostatic barriers and construct phase diagrams over barrier angle, tilt strength, width, height, and Fermi energy. The results reveal a robust optimum near t = 0.2 over the parameter range studied, identify the crossover from oscillatory to monotonic polarization regimes, and delineate practical operating windows for candidate materials including 8-Pmmn borophene and WTe2.

Figures (3)

• Figure 1: Harmonic oscillator mapping and valley-resolved transmission. (a) Nonlinear mapping from device-frame transverse momentum $k_y$ to barrier-frame conserved momentum $k_{\parallel}$ [Eq. \ref{['eq:nonlinear_mapping']}] for several barrier rotation angles $\phi$. At $\phi = 0$, the mapping is linear ($k_{\parallel} = k_y$); increasing $\phi$ introduces asymmetry that breaks time-reversal-protected valley degeneracy in the integrated conductance. (b) Parabolic cylinder function (PCY) transmission envelope [Eq. \ref{['eq:T_pcy']}] for a smooth $pn$ junction at different tilt parameters $t$. The tilt broadens the angular transmission window through the factor $(1 - t^2)^{-1}$ in the harmonic oscillator quantum number. (c) Device geometry: transport along $x$, electrostatic barrier along $y$ (green), tilt directed along $\pm y$ for valleys $K$/$K'$ (blue/red arrows), with a rotated barrier at angle $\phi$ (orange). (d--f) Valley-resolved transmission $T(\theta)$ in the device frame for tilt $t = 0.5$ at barrier rotation angles $\phi = 0^\circ$, $15^\circ$, and $25^\circ$. At $\phi = 0^\circ$, the valley profiles are mirror-symmetric and $\mathcal{P} = 0$. Increasing $\phi$ breaks this symmetry through the nonlinear coordinate mapping, yielding $\mathcal{P} = 0.58$ at $\phi = 15^\circ$ and $\mathcal{P} = 0.95$ at $\phi = 25^\circ$. Parameters: $E_{\mathrm{F}} = 100$ meV, $V_0 = 200$ meV, $d = 50$ nm.
• Figure 2: Valley polarization parameter dependences. (a) $\mathcal{P}$ versus barrier rotation angle $\phi$ for tilt values $t = 0.2$, $0.4$, $0.6$, and $0.8$. All curves start at $\mathcal{P} = 0$ for $\phi = 0$ (confirming the symmetry argument) and increase monotonically, with fine Fabry-Pérot oscillations visible at small $\phi$ for moderate tilts. (b) $\mathcal{P}$ versus tilt parameter $t$ for fixed barrier angles $\phi = 5^\circ$, $10^\circ$, $20^\circ$, and $30^\circ$. A common polarization peak appears near $t \approx 0.2$ for all angles, corresponding to the condition where the tilt-induced $k_{\parallel}$ shift matches the Fabry-Pérot fringe spacing $\sim\!\pi/d$. (c) Individual valley conductances $G_K$ and $G_{K'}$ versus $\phi$ for $t = 0.5$, showing the valley-selective filtering: $G_K$ increases while $G_{K'}$ is progressively suppressed with barrier rotation. (d) $\mathcal{P}$ versus barrier width $d$ for $t = 0.5$, $\phi = 20^\circ$. The oscillatory behavior reflects Fabry-Pérot resonances cycling through constructive/destructive conditions differently for each valley, with a general upward trend as wider barriers are more selective. (e) $\mathcal{P}$ versus barrier height $V_0$. Polarization vanishes for $V_0 < E_{\mathrm{F}}$ (no hole pocket inside the barrier) and saturates near unity for $V_0 \gtrsim 2E_{\mathrm{F}}$. (f) $\mathcal{P}$ versus Fermi energy $E_{\mathrm{F}}$ for $t = 0.5$, $\phi = 20^\circ$, $V_0 = 200$ meV. The peak near $E_{\mathrm{F}} \approx V_0/2$ coincides with low total conductance; practical operation requires $E_{\mathrm{F}}$ values offering both high $\mathcal{P}$ and appreciable $G_{\mathrm{tot}}$.
• Figure 3: Valley contrast map and phase diagram. (a) Device-frame transmission $T(\theta)$ for valleys $K$ (blue) and $K'$ (red) at $t = 0.5$, $\phi = 20^\circ$, yielding $\mathcal{P} = 0.78$. Valley $K$ dominates the negative-$\theta$ hemisphere with prominent Fabry-Pérot resonances, while $K'$ is confined to a narrow peak near $\theta \approx +60^\circ$. (b) Valley contrast $\Delta T = T_K - T_{K'}$ as a function of device-frame incidence angle $\theta$ and barrier rotation $\phi$ for $t = 0.5$. The diagonal structure reflects the systematic shift of valley transmission profiles with $\phi$. At $\phi = 0$, the pattern is antisymmetric about $\theta = 0$ (zero net polarization); increasing $\phi$ breaks this antisymmetry progressively. (c) Phase diagram of valley polarization $\mathcal{P}(t, \phi)$. Three regimes are evident: (i) negligible polarization at small $t$ and $\phi$; (ii) an oscillatory regime at low tilt ($t \lesssim 0.2$) where $\mathcal{P}$ alternates sign as individual Fabry-Pérot fringes shift between valleys; and (iii) a robust high-polarization regime ($\mathcal{P} > 0.8$) at moderate-to-large $t$ and $\phi$. The transition near $t \sim 0.2$ marks where the tilt-induced $k_{\parallel}$ shift exceeds the Fabry-Pérot fringe spacing. Parameters: $E_{\mathrm{F}} = 100$ meV, $V_0 = 200$ meV, $d = 50$ nm.