Finite difference method for Dirac electrons in circular quantum dots

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.