Carrier-induced antisymmetric-symmetric tendencies of spin stiffness in zigzag graphene nanoribbons.

The generalized Bloch theorem was applied to calculate the spin stiffness and to consider its tendencies when introducing the doping in zigzag graphene nanoribbons. To reach the intentions, two different flat spin spiral formations were constructed by fixing the ferromagnetic and antiferromagnetic spin arrangements at the two different edges by applying a constraint scheme method. A spin stiffness was then calculated by means of a quadratic order function, which maps the total energy difference in the self-consistent calculations onto the Heisenberg Hamiltonian. We found a very high spin stiffness as predicted previously by the supercell calculation. We also showed that the antisymmetric-symmetric tendencies of spin stiffness are induced by the hole-electron doping. The dependence of ribbon widths of zigzag graphene nanoribbon on the spin stiffness was also provided with similar tendencies when the doping is taken into account.