Electron states in the field of charged impurities in two-dimensional Dirac systems

We review the theoretical and experimental results connected with the electron states in two-dimensional Dirac systems paying a special attention to the atomic collapse in graphene. Two-electron bound states of a Coulomb impurity are considered too. A rather subtle role of a magnetic field in the supercritical charge problem in graphene is discussed. The electron states in the field of two equally charged impurities are studied and the conditions for supercritical instability to occur are determined. It is shown that the supercriticality of novel type is realized in gapped graphene with two unlikely charged impurities. For sufficiently large charges of impurities, it is found that the wave function of the occupied electron bound state of the highest energy changes its localization from the negatively charged impurity to the positively charged one as the distance between the impurities increases. The specifics of the atomic collapse in bilayer graphene is considered and it is shown that the atomic collapse in this material is not related to the phenomenon of the fall-to-center.