Tight-binding model study of substrate induced pseudo-spin polarization and magnetism in mono-layer graphene

We present here a tight-binding model study of generation of magnetism and pseudo-spin polarization in monolayer graphene arising due to substrate, impurity and Coulomb correlation effects. The model Hamiltonian contains the first-, second- and third-nearest-neighbor hopping integrals for π electrons of graphene besides substrate induced gap, impurity interactions and Coulomb correlation of electrons. The Hubbard type Coulomb interactions present in both the sub-lattices A and B are treated within the mean-field approximation. The electronic Green's functions are calculated by using Zubarev's technique and hence the electron occupancies of both sub-lattices are calculated for up and down spins separately. These four temperature dependent occupancies are calculated numerically and self-consistently. Then we have calculated the temperature dependent pseudo-spin polarization, ferromagnetic and anti-ferromagnetic magnetizations. We observe that there exists pseudo-spin polarization for lower Coulomb energy, u<2.2t1 and pseudo-spin polarization is enhanced with substrate induced gap and impurity effect. For larger Coulomb energy u>2.5t1, there exists pseudo-spin polarization (p); while ferromagnetic (m) and antiferromagnetic (pm) magnetizations exhibit oscillatory behavior. With increase of the substrate induced gap, the ferromagnetic and antiferromagnetic transition temperatures are enhanced with increase of the substrate induced gap; while polarization (p) is enhanced in magnitude only.