Graphene quantum dot with a Coulomb impurity: Subcritical and supercritical regime

We study the influence of confinement on the atomic collapse due to a Coulomb impurity placed at the center of a graphene quantum dot of radius $R$. We apply the zigzag or infinite-mass boundary condition and consider both a point-size and a finite-size impurity. As a function of the impurity strength $Z\ensuremath{\alpha}$, the energy spectra are discrete. In the case of the zigzag boundary condition, the degenerate (with respect to the angular momentum $m$) zero-energy levels are pulled down in energy as $Z\ensuremath{\alpha}$ increases, and they remain below $\ensuremath{\epsilon}=\ensuremath{-}Z\ensuremath{\alpha}$. Our results show that the energy levels exhibit a $1/R$ dependence in the subcritical regime $[Z\ensuremath{\alpha}<|km+1/2|$, $k=1\phantom{\rule{4pt}{0ex}}(\ensuremath{-}1)$ for the $K$ (${K}^{\ensuremath{'}}$) valley]. In the supercritical regime ($Z\ensuremath{\alpha}>|km+1/2|$) we find a qualitatively very different behavior where the levels decrease as a function of $R$ in a nonmonotonic manner. While the valley symmetry is preserved in the presence of the impurity, we find that the impurity breaks electron-hole symmetry. We further study the energy spectrum of zigzag quantum dots in gapped graphene. Our results show that as the gap increases, the lowest electron states are pushed into the gap by the impurity.

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