van der Waals interaction in magnetic bilayer graphene nanoribbons

We study the interaction energy between two graphene nanoribbons by first-principles calculations, including van der Waals interactions and spin polarization. For ultranarrow zigzag nanoribbons, the direct stacking is even more stable than the Bernal stacking, competing in energy for wider ribbons. This behavior is due to the magnetic interaction between edge states. We relate the reduction of the magnetization in zigzag nanoribbons with increasing ribbon width to the structural changes produced by the magnetic interaction, and we show that when deposited on a substrate, zigzag bilayer ribbons remain magnetic for larger widths.

Figure 1

(a) Schematic packing of zigzag bilayer graphene nanoribbons: AA (above), AB${}_{\beta }$ (middle), and AB${}_{\alpha }$(below). The dark gray layer corresponds to the bottom layer and the light gray (blue) layer correspond to the upper layer. Hydrogen atoms are denoted by white balls. The ribbon width $N$ is given by the number of dimers (zigzag chains) from edge to edge. (b) Magnetic configurations of the edge states in $b$-ZGNRs. Interlayer (intralayer) coupling can be either ferromagnetic [FM($\mathit{fm}$)] or antiferromagnetic [AFM(afm)]. Notice that one single edge is always ferromagnetic, i.e., all the spins along the same edge are parallel.

Figure 2

Binding energy (BE) of $b$-ZGNR as a function of the ribbon width $N$ after full relaxation. It is noteworthy that the AB${}_{\alpha }$ and AA stackings have the largest absolute values of the binding energies.

Figure 3

(a) Structural distortions of the zigzag ribbon with $N=10$ for the most stable stackings AA and AB${}_{\alpha }$ in the AFM-afm configurations after full relaxation. The systems with higher binding energy (in absolute value) consist of nonplanar, distorted, graphene ribbons. (b) Interlayer distances at the center $h_c$ (empty diamonds) and edges $h_e$ (full diamonds) for $b$-ZGNRs with AA (black) and AB${}_{\alpha }$ [blue (gray)] stackings in the AFM-afm configuration as a function of the ribbon width.

Figure 4

Magnetization of the $b$-ZGNR as a function of the ribbon width $N$ after full relaxation.

Figure 5

(a) Band structure of a $b$-ZGNR with $N=8$ in the most stable stackings, AA and AB${}_{\alpha }$, and AFM-afm magnetic configuration. The Fermi energy is set to zero. Note the narrowing of the gap. (b) Gaps vs ribbon widths for the two most stable stackings and magnetic configurations, namely AA and AB${}_{\alpha }$, all with AFM-afm.

Figure 6

Geometry of the $b$-ZGNR with $N=10$ deposited on graphene. We have chosen a cut perpendicular to the graphene plane in order to highlight the edge deformations. The local magnetic moments on the edge atoms are also depicted. Note that the magnetic moments are almost as large as in a single ribbon.