Boundaries of Subcritical Coulomb Impurity Region in Gapped Graphene

The electronic energy spectrum of graphene electron subjected to a homogeneous magnetic field in the presence of a charged Coulomb impurity is studied analytically within two-dimensional Dirac-Weyl picture by using variational approach. The variational scheme we used is just based on utilizing the exact eigenstates of two-dimensional Dirac fermion in the presence of a uniform magnetic field as a basis for determining analytical energy eigenvalues in the presence of an attractive/repulsive charged Coulomb impurity. This approach allows us to determine under which conditions bound state solutions can or can not exist in gapped graphene in the presence of magnetic field. In addition, the effects of uniform magnetic field on the boundaries of subcritical Coulomb impurity region in the massless limit are also analyzed. Our analytical results show that the critical impurity strength decreases with increasing gap/mass parameter, and also that it increases with increasing magnetic field strength. In the massless limit, we investigate that the critical Coulomb coupling strength is independent of magnetic field, and its upper value for the ground-state energy is 0.752.