Electron localization in graphene quantum dots

We study theoretically a localized state of an electron in a graphene quantum dot with a sharp boundary. Due to Klein's tunneling, the relativistic electron in graphene cannot be localized by any confinement potential. In this case the electronic states in a graphene quantum dot become resonances with finite trapping time. We consider these resonances as the states with complex energy. To find the energy of these states we solve the time-independent Schrodinger equation with outgoing boundary conditions at infinity. The imaginary part of the energy determines the width of the resonances and the trapping time of an electron within quantum dot. We show that if the parameters of the confinement potential satisfy a special condition, then the electron can be strongly localized in such quantum dot, i.e., the trapping time is infinitely large. In this case the electron localization is due to interference effects. We show how the deviation from this condition affects the trapping time of an electron. We also analyze the energy spectra of an electron in a graphene quantum ring with a sharp boundary. We show that in this case the condition of constructive interference can be tuned by varying internal radius of the ring, i.e., parameters of confinement potential.