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 hybridize with the Landau levels. The charge at which the lowest Landau level $m = -1, N = 1$ crosses increases $E = 0$ with enhancing magnetic field. Landau level scaling anomaly occurs when the charge is larger than the critical charge $\beta\approx0.6$ and this critical charge is independent of the magnetic field.