Spin-polarized superconducting phase in semiconducting system with next-nearest-neighbor hopping on the honeycomb lattice

The particles in the honeycomb lattice with on-site $s$-wave pairing exhibit many interesting behaviors, which can be described in the framework of the Hubbard model. Among others, at the half-filling, some critical value $\mathrm{\mid }{U}_c\mathrm{\mid }$ of pairing interaction $U<0$ exists that, for $U<U_c$, the superconducting phase becomes unstable. Introduction of the nonzero hopping $t^{\mathrm{\prime }}$ between next-nearest-neighbor sites strongly modifies the physical properties of the system. Here, we discuss the behavior of the system for $t^{\mathrm{\prime }}\mathrm{\ne }0$ (at the ground state), where the hopping between next-nearest neighbors leads to change of the order of phase transition between superconducting and normal phases from discontinuous to continuous one in the external magnetic field $h$. We show that this behavior is strongly dependent on $t^{\mathrm{\prime }}$ and associated with the Dirac cones in the non-interacting band structure of the system. For non-zero magnetic field and for some range of $t^{\mathrm{\prime }}$, a spin-polarized superconducting phase occurs in the ground state phase diagram (only at the half-filling and for $h\mathrm{\ne }0$).