# Finite difference method for Dirac electrons in circular quantum dots

## TL;DR

This paper introduces a finite difference method for solving the Dirac equation in circular quantum dots, effectively handling boundary conditions and avoiding fermion doubling issues.

## Contribution

It presents a simple, reliable finite difference approach that sets boundary conditions at the system edge and sweeps inward, preventing spurious solutions.

## Key findings

- Method effectively solves Dirac eigenproblems in circular geometries.
- Prevents fermion doubling and spurious oscillations in solutions.
- Applicable to rotationally symmetric quantum systems.

## Abstract

A simple and reliable finite difference approach is presented for solution of the Dirac equation eigenproblem for states confined in rotationally symmetric systems. The method sets the boundary condition for the spinor wave function components at the external edge of the system and then sweeps the radial mesh in search for the energies for which the boundary conditions are met inside the flake. The sweep that is performed from the edge of the system towards the origin allows for application of a two-point finite difference quotient of the first derivative, which prevents the fermion doubling problem and the appearance of the spurious solutions with rapid oscillations of the wave functions in space.