Inducing and optimizing magnetism in graphene nanomeshes

Using first-principles calculations, we explore the electronic and magnetic properties of graphene nanomesh (GNM), a regular network of large vacancies, produced either by lithography or nanoimprint. When removing an equal number of $A$ and $B$ sites of the graphene bipartite lattice, the nanomesh made mostly of zigzag (armchair) -type edges exhibit antiferromagnetic (spin unpolarized) states. In contrast, in situations of sublattice symmetry breaking, stable ferri(o)magnetic states are obtained. For hydrogen-passivated nanomesh, the formation energy is dramatically decreased, and ground state is found to strongly depend on the vacancies shape and size. For triangular-shaped holes, the obtained net magnetic moments increase with the number difference of removed $A$ and $B$ sites in agreement with Lieb’s theorem for even $A+B$. For odd $A+B$ triangular meshes and all cases of nontriangular nanomeshes, including the one with even $A+B$, Lieb’s theorem does not hold anymore, which can be partially attributed to the introduction of armchair edges. In addition, large triangular-shaped GNMs could be as robust as nontriangular GNMs, providing a possible solution to overcome one of the crucial challenges for the $sp$ magnetism. Finally, significant exchange-splitting values as large as $\sim 0.5$ eV can be obtained for highly asymmetric structures evidencing the potential of GNM for room-temperature carbon-based spintronics. These results demonstrate that a turn from zero-dimensional graphene nanoflakes throughout one-dimensional graphene nanoribbons with zigzag edges to GNM breaks localization of unpaired electrons and provides deviation from the rules based on Lieb’s theorem. Such delocalization of the electrons leads the switch of the ground state of a system from an antiferromagnetic narrow gap insulator discussed for graphene nanoribons to a ferromagnetic or nonmagnetic metal.

Figure 1

[(a)–(c)] Schematics of the calculated crystalline structures for balanced nonpassivated circular-shaped C${}_{3:3}$, C${}_{6:6}$, and C${}_{12:12}$ GNM structures, respectively; (d) the same for unbalanced nonpassivated triangular-shaped T${}_{6:7}$ GNM structure. Edge carbon atoms are in blue and red to represent $A$ and $B$ sites, respectively. For convenience, positions of removed atoms are indicated in orange.

Figure 2

H-passivated GNMs with triangular shapes: (a) T${}_{6:7}^{\mathrm{H}}$, (b)T${}_{10:12}^{\mathrm{H}}$, (c)T${}_{15:18}^{\mathrm{H}}$, (d) T${}_{21:25}^{\mathrm{H}}$; with a circular shape: (e) C${}_{12:12}^{\mathrm{H}}$; with a rhombic shape: (f) R${}_{24:24}^{\mathrm{H}}$; with a sector shape: (g) S${}_{19:21}^{\mathrm{H}}$; and with a pentagon shape: (h) P${}_{18:21}^{\mathrm{H}}$. The corresponding net magnetic moments for each structure are also indicated.

Figure 3

(a) Total magnetic moment (${\mu }_B$/cell) (left) and spin-splitting (right) as a function of ${\Delta }_{AB}$ for various GNM geometries, where P${}_{18:19}^{\prime \mathrm{H}}$ is transformed from a pentagon structure (P${}_{18:21}^{\mathrm{H}}$) by adding two $A$ atoms to the two opened hexagons. The result of the Lieb’s theorem prediction is also given for comparison. The even number of $A+B$ structures T${}_{10:12}^{\mathrm{H}}$ and T${}_{21:25}^{\mathrm{H}}$ are shown in red to indicate the good agreement with Lieb’s theorem prediction. (b) Energy difference between ferrimagnetic and paramagnetic states.

Figure 4

Density of states for triangular GNMs of (a) T${}_{6:7}^{\mathrm{H}}$, (b) T${}_{10:12}^{\mathrm{H}}$, (c) T${}_{15:18}^{\mathrm{H}}$, (d) T${}_{21:25}^{\mathrm{H}}$, (e) C${}_{3:3}^{\mathrm{H}}$, (f) R${}_{24:24}^{\mathrm{H}}$, (g)P${}^{{\prime }_{18:19}^{\mathrm{H}}}$, (h) S${}_{19:21}^{\mathrm{H}}$, and (i) P${}_{18:21}^{\mathrm{H}}$; the peaks around Fermi level are marked with arrows. It can be seen that only the $p$${}_z$ state contributes to the moment from (c${}^{\prime }$) projected density of states of one edge atom in the T${}_{15:18}^{\mathrm{H}}$ GNM.

Figure 5

Spin density (${\mu }_B$/Å${}^2$) distribution for the two types of graphene nanomesh with localized (a) and delocalized (b) unpaired electrons. Localization and delocalization for these structures can be clearly seen in Fig. 4b and (4c), respectively.