Charged vacancy in graphene: Interplay between Landau levels and atomic collapse resonances

The interplay between a magnetic field and the Coulomb potential from a charged vacancy on the electron states in graphene is investigated within the tight-binding model. The Coulomb potential removes locally Landau level degeneracy, while the vacancy introduces a satellite level next to the normal Landau level. These satellite levels are found throughout the positive-energy region, but in the negative-energy region, they turn into atomic collapse resonances. Crossings between Landau levels with different angular quantum number $m$ are found. Unlike the point impurity system in which an anticrossing occurs between Landau levels of the same $m$, in this work anticrossing is found between the normal Landau level and the vacancy-induced level. The atomic collapse resonance hybridizes with the Landau levels. The charge at which the lowest Landau level $m=-1,N=1$ crosses $E=0$ increases with enhancing magnetic field. A Landau level scaling anomaly occurs when the charge is larger than the critical charge $\beta \approx 0.6$ and this critical charge is independent of the magnetic field.