Edge magnetization in Bernal-stacked trilayer zigzag graphene nanoribbons

We have used a tight-binding Hamiltonian of an ABA-stacked trilayer zigzag graphene nanoribbon with β-alignment edges to study the edge magnetizations. Our model includes the effect of the intralayer next-nearest-neighbor hopping, the interlayer hopping responsible for the trigonal warping and the interaction between electrons, which is considered by a single band Hubbard model in the mean field approximation. Firstly, in the neutral system we analyzed the two magnetic states in which both edge magnetizations reach their maximum value; the first one is characterized by an intralayer ferromagnetic coupling between the magnetizations at opposite edges, whereas in the second state that coupling is antiferromagnetic. The band structure, the location of the edge-state bands and the local density of states resolved in spin are calculated in order to understand the origins of the edge magnetizations. We have also introduced an electron doping so that the number of electrons in the ribbon unit cell is higher than in neutral case. As a consequence, we have obtained magnetization steps and charge accumulation at the edges of the sample, which are caused by the edge-state flat bands.