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Electric field control of the indirect magnetic coupling through a short graphene nanoribbon

In the paper we consider the system composed of two magnetic planes attached to zigzag terminations of the graphene nanostructure being an ultrashort fragment of the armchair nanoribbon. We investigate theoretically an indirect coupling between these magnetic planes mediated by charge carriers as a function of external in-plane electric field. The calculations are based on a tight-binding model supplemented with Hubbard term to account for coulombic interactions. For selected sizes of the graphene nanostructure, particularly high sensitivity of the coupling to the electric field is found. This leads to the possibility of control over coupling magnitude and continuous switching of its sign between antiferromagnetic and ferromagnetic one. Such a phenomenon is demonstrated in the numerical results and its origin is analysed. The robustness of this effect against armchair edge deformation and variation of exchange energy between magnetic planes and spins of charge carriers is discussed in detailed way.