Two-dimensional Dirac particles in a P\"oschl-Teller waveguide

TL;DR

This paper provides exact solutions for the 2D Dirac equation with P"oschl-Teller potential, revealing eigenfunctions, eigenvalues, and conditions for bound states, with applications to 2D Dirac-Weyl electron waveguides.

Contribution

It introduces exact analytical solutions for the 2D Dirac equation with P"oschl-Teller potential, including asymmetric cases, and derives a universal bound state condition.

Findings

Eigenfunctions expressed via Heun confluent functions

Eigenvalues determined by transcendental equations

Universal bound state condition for symmetric potentials

Abstract

We obtain exact solutions to the two-dimensional (2D) Dirac equation for the one-dimensional P\"oschl-Teller potential which contains an asymmetry term. The eigenfunctions are expressed in terms of Heun confluent functions, while the eigenvalues are determined via the solutions of a simple transcendental equation. For the symmetric case, the eigenfunctions of the supercritical states are expressed as spheroidal wave functions, and approximate analytical expressions are obtained for the corresponding eigenvalues. A universal condition for any square integrable symmetric potential is obtained for the minimum strength of the potential required to hold a bound state of zero energy. Applications for smooth electron waveguides in 2D Dirac-Weyl systems are discussed.