Electric-field-induced spin spiral state in bilayer zigzag graphene nanoribbons

We investigated the emergence of spin spiral ground state induced by the electric field in the bilayer zigzag graphene nanoribbons for the ferromagnetic edge states. To do that, we employed the generalized Bloch theorem to create flat spiral alignments for all the magnetic moments of carbon atoms at the edges within a constraint scheme approach. While the small ribbon width can preserve the ferromagnetic ground state, the large one shows the spiral ground state starting from a certain value of the electric field. We also pointed out that the spiral ground state is caused by the reduction of spin stiffness. In this case, the energy scale exhibits a subtle nature that can only be considered at the low temperature. For the last discussion, we also revealed that the spin spiral ground state appears more rapidly when the thickness increases. Therefore, we justify that the large ribbon width and large thickness can generate many spiral states induced by the electric field.