Electron-electron interaction in graphene at finite Fermi energy

TL;DR

This paper derives the wave equation for two-electron interactions in graphene at various Fermi energies, revealing localized states at negative Fermi energies and analyzing wave packet evolution during scattering.

Contribution

It provides explicit solutions for electron interactions in graphene at arbitrary Fermi energy, including localized states and wave packet dynamics, which were not previously detailed.

Findings

Localized quasi-stationary peaks appear at $E_F<0$

Wave packets decay into successive packets over time

Total outgoing wave packet norm equals incoming norm

Abstract

The wave equation describing the interaction of two electrons in graphene at arbitrary value of the Fermi energy $E_F$ is derived. For the solutions of this equation, we have found the explicit forms of the density and the current which obey the continuity equation. We have traced the evolution of the wave packet during a scattering process. It is shown that the long-leaving localized quasi-stationary peak may appear at $E_F<0$ . Then this peak decays into a set of wave packets following each other. At $t\rightarrow\infty$ a total norm of all outgoing wave packets equals to that of incoming wave packet. At $E_F=0$ the localized state does not appear. For $E_F<0$ there is an infinite set of the localized solutions with the finite norms.