A New Class of Solvable Two-dimensional Scalar Potentials for Graphene

TL;DR

This paper introduces a systematic method using asymmetric SUSY-like relations to find new exactly solvable two-dimensional scalar potentials relevant for graphene, expanding the class of solvable models.

Contribution

It presents a novel approach based on asymmetric SUSY intertwining relations to construct solvable two-dimensional scalar potentials for the Dirac equation.

Findings

Extended the class of analytically solvable two-dimensional models.

Established a method to generate partner potentials with non-trivial coordinate dependence.

Applicable to materials like graphene with external electrostatic potentials.

Abstract

In the present paper, a systematic approach is presented for solution of two-dimensional massless Dirac equation with external electrostatic potential applied. This approach is based on the new - asymmetric - form of SUSY-like intertwining relations. It allows to build a wide variety of pairs of SUSY-partner external scalar potentials. If one of them is simple enough to be solvable, its partner is also solvable although it may have a non-trivial dependency on both coordinates. Physically, this kind of problems is related to the description of graphene and some other materials with external potential. Solvability obtained by means of asymmetric form of SUSY intertwining relations allows to extend the class of analytically solvable two-dimensional models.