Engineering spin exchange in nonbipartite graphene zigzag edges

The rules that govern spin exchange interaction in pristine graphene nanostructures are constrained by the bipartite character of the lattice, so that the sign of the exchange is determined by whether magnetic moments are on the same sublattice or not. The synthesis of graphene ribbons with perfect zigzag edges and a fluoranthene group with a pentagon ring, a defect that breaks the bipartite nature of the honeycomb lattice, has been recently demonstrated. Here we address how the electronic and spin properties of these structures are modified by such defects, both for indirect exchange interactions as well as the emergent edge magnetism, studied both with density functional theory and mean-field Hubbard model calculations. In all instances we find that the local breakdown of the bipartite nature at the defect reverts the sign of the otherwise ferromagnetic correlations along the edge, introducing a locally antiferromagnetic intraedge coupling and, for narrow ribbons, also revert the antiferromagnetic interedge interactions that are normally found in pristine ribbons. Our findings show that these pentagon defects are a resource that permits us to engineer the spin exchange interactions in graphene-based nanostructures.

Figure 1

(a) Large graphene island with zigzag edges functionalized with a FG. (b) Energy spectrum, for eigenvalues close to the Fermi energy, showing four in-gap states, two per right/left edge. The twofold degeneracy in the smallest eigenvalues corresponds to the two lateral edges. (c), (d) are the wave functions labeled in (b), showing the edge nature of the in-gap states, in particular their bonding and antibonding character. Color in (c), (d) labels the sign, whereas the size indicates the magnitude of the component. The splitting between the two states in the edge allows us to map the system into a two-site tight-binding model, which is expected to develop antiferromagnetic correlations. The width of the island was chosen so that the bulk is gapped. The dimensions of the island are shown in (a), width of 60 unit cells and four carbon atoms per semi-edge.

Figure 2

(a) Map of the nonlocal spin susceptibility ${\chi }_{i{j}_0}$, when $j_0$ is fixed at the green atom at the top-left edge, and $i$ runs over the entire island. The color reflects the sign of ${\chi }_{i{j}_0}$, and the size the magnitude. (b), (c), (d) show the spatial profile of the eigenstate $v$ of ${\chi }_{ij}$, so that ${\chi }_{ij}{v}_i=\lambda v_j$, with $\lambda$ the largest eigenvalue, that characterizes the magnetic density right below the critical temperature. The color represents the sign of the component $v_i$, and the size its magnitude. (c) corresponds to the ribbon without FG, showing conventional intraedge ferromagnetic and interedge antiferromagnetic correlations. (b), (d) show the antiferromagnetic correlations in the same edge due to the FG group, while the interedge correlations depend on the size of the island (b), (d).

Figure 3

Isocontours of the spin density obtained with DFT calculations, for two different systems (a) and (b), showing in both cases intraedge antiferromagnetic correlations. The color stands for the sign of the magnetic moment. Calculations were performed with $\mathrm{CAM}-\mathrm{B}3\mathrm{LYP}/6-31\mathrm{g}(\mathrm{d},\mathrm{p})$, and the isosurfaces plotted correspond to isovalue 0.005.

Figure 4

(a) Magnetization profile of various self-consistent configurations obtained within the mean-field Hubbard model. Red and blue stand for the sign of the magnetic moment. The size of the symbols stands for the magnitude of the magnetic moment. (b) Evolution of the energy difference between the four configurations shown in panel (a) as a function of the size of the ribbon $L$. The lowest energy configurations are always the 1010 or the 1001, the ones showing intraedge antiferromagnetic correlations. (c) shows the energy difference between 1001 and 1100 as the island becomes thicker for $L=5$, whereas (d) shows the energy difference as a function of U $(W=L=5)$. Exponential extrapolation of (c) to $W=\infty$ gives a total energy difference of 0.007 [t], 19 meV for $t=2.7$ eV

Figure 5

(a) Magnetization profile for structure with several fluoranthene groups obtained within the mean-field Hubbard model $U=\left|t\right|$. Red and blue stand for the sign of the magnetic moment. The size of the symbols stands for the magnitude of $m_i$. (a) shows the ground-state configuration, whereas (b) an excited state.