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

TL;DR

This paper introduces an analytical approach using parabolic-cylinder functions to study valley-polarized conductance in tilted anisotropic Dirac-Weyl systems, providing insights into how tilt components influence tunneling and resonance effects.

Contribution

It develops a novel analytical method reducing the scattering problem to the Weber equation, enabling explicit expressions for valley-dependent conductance in tilted Dirac materials.

Findings

Identifies optimal tilt parameter near t=0.2 for maximum polarization.

Maps the transition from oscillatory to monotonic polarization regimes.

Provides phase diagrams for valley polarization across various device parameters.

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.