Spectra of electronic excitations in graphene near Coulomb impurities

TL;DR

This paper analyzes the electronic excitation spectra in gapped graphene with Coulomb impurities, constructing self-adjoint Hamiltonians and identifying supercritical charges that induce bound states at the spectrum's edge.

Contribution

It introduces a family of self-adjoint Hamiltonians for Coulomb impurities in graphene and derives conditions for supercritical charges causing bound states.

Findings

Constructed self-adjoint Hamiltonians for all impurity charges.

Derived equations for supercritical impurity charges.

Identified conditions for bound states at the spectrum boundary.

Abstract

We study the problem of the electron excitation spectrum in the presence of point-like and regularized Coulomb impurities in gapped graphene. To this end, we use the Dirac model and in the point-like case theory of self-adjoint extensions of symmetric operators. In the point-like case, we construct a family of self-adjoint Hamiltonians describing the excitations for any charge of an impurity. Spectra and (generalized) eigenfunctions for all such Hamiltonians are found. Then, we consider the spectral problem in the case of a regularized Coulomb potential of impurities for a special regularization. We study exact equations for charges of impurities that may generate bound states with energy that coincides with the upper boundary of the negative branch of the continuous spectrum (supercritical charges) and calculate these charges.