  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.
