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For graphene nanoribbons with Rashba spin-orbit coupling, the peculiarmagnetic response due to the presence of a magnetization and geometricconfinement are analyzed within a tight-binding model. We observe a sizabletransverse susceptibility that can be considered as a gate voltage-inducedmagnetoelectric torque without the need of a bias voltage, with differentdirections for zigzag and armchair ribbons. The local torque generatesnon-collinear spin polarization between the two edges and/or along the ribbon,and the net torque averages to zero if the magnetization is homogeneous.Nevertheless, a nonzero net torque can appear in partially magnetizednanoribbons or in nanoflakes of irregular shapes. The equilibrium spin currentproduced by the spin-orbit coupling also appears in nanoribbons, but thecomponent flowing in the direction of confinement is strongly suppressed. Evenwithout the magnetization, an out-of-plane polarized chiral edge spin currentis produced, resembling that in the quantum spin Hall effect. Moreover, amagnetization pointing perpendicular to the edge produces a laminar flow ofedge charge currents, whose flow direction is symmetric (non chiral) orantisymmetric (chiral) between the two edges depends on whether themagnetization points in-plane or out-of-plane.

Magnetoelectric torque and edge currents in spin-orbit coupled graphene nanoribbons