Quasi-exact solution to the Dirac equation for the hyperbolic secant potential

TL;DR

This paper investigates bound states of massless Dirac fermions in a hyperbolic secant potential, providing analytical solutions and identifying conditions for the emergence of bound modes relevant to graphene structures.

Contribution

It presents a quasi-exact analytical approach to solving the Dirac equation with a hyperbolic secant potential, including threshold conditions for bound state formation.

Findings

Bound states exist for both positive and negative energies.

A threshold potential strength is required for the first bound mode.

Analytical solutions are derived in specific cases.

Abstract

We analyze bound modes of two-dimensional massless Dirac fermions confined within a hyperbolic secant potential, which provides a good fit for potential profiles of existing top-gated graphene structures. We show that bound states of both positive and negative energies exist in the energy spectrum and that there is a threshold value of the characteristic potential strength for which the first mode appears. Analytical solutions are presented in several limited cases and supercriticality is discussed.