Carrier-induced antiferromagnet of graphene islands embedded in hexagonal boron nitride

Graphene islands with zigzag edges embedded in nitrogen-terminated vacancies in hexagonal boron nitride are shown to develop intrinsic magnetism and preferentially order antiferromagnetically. The magnetic moment of each graphene island is given by the numerical imbalance of carbon atoms on its two sublattices, which is in turn directly related to the size of the host defect. We propose a carrier-mediated model for antiferromagnetic coupling between islands and estimate Néel temperatures for these structures in excess of 100 K in some instances, with the possibility of attaining even higher temperatures at higher island densities. Our results suggest the possibility of designing molecular magnets via defect engineering of hexagonal boron nitride templates followed by trapping of carbon atoms in the defects.

Figure 1

Hexagonal array of graphene islands in an hBN lattice with $N$ primitive cells on edge. The primitive vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$ of hBN are of length $a_0=2.488$ Å. C atoms on the A and B sublattice are indicated by ${\mathrm{C}}_A$ and ${\mathrm{C}}_B$. The center-to-center distance between islands is $r_{nn}=N{a}_0/\sqrt{3}$. Dotted lines enclose the periodic simulation cell.

Figure 2

Energy difference between AFM and FM coupled C${}_4$ islands as a function of separation $N$ (solid squares) and Néel temperatures for AFM-coupled C${}_4$ islands (hollow squares). AFM coupling is distinctly favored at small island spacings with correspondingly higher Néel temperatures. The lines are a guide to the eye. Blue triangles are for a C${}_9$ island in an $N=12$ simulation cell, indicating that increasing the island size while keeping the island spacing constant can serve to tune the Néel temperature of the array.

Figure 3

Charge difference $\Delta {\rho }_{\mathrm{AFM}-\mathrm{FM}}$ between AFM and FM coupled C${}_4$ islands and relative spin density $\zeta =({\rho }_{\uparrow }-{\rho }_{\downarrow })/({\rho }_{\uparrow }+{\rho }_{\downarrow })$ for each case projected on the atomic plane for an $N=9$ cell (upper row) and $N=15$ cell (lower row). (Light) Yellow and (dark) blue regions correspond to values of $\Delta {\rho }_{\mathrm{AFM}-\mathrm{FM}}⩾{10}^{-4}e/{\AA }^2$ and $⩽-{10}^{-4}e/{\AA }^2$, respectively, as well as $\zeta ⩾{10}^{-4}$ and $⩽-{10}^{-4}$, respectively. As seen from the $\zeta$ plots, each C${}_4$ island spin-polarizes its BN neighbors due to its two unpaired electrons thereby inducing exchange interactions between islands. The $\Delta \rho$ plots show alternating belts of charge depletion and accumulation that are indicative of varying degrees of charge transfer between the islands and the hBN matrix depending upon the magnetic coupling between islands (FM or AFM). The charge transfer is mediated entirely via the $p_z$ orbitals of C, B, and N. The differences are more obvious for the $N=9$ case; for $N=15$ the differences are very small (near-degenerate AFM and FM configurations) and are localized to the immediate vicinity of the islands.

Figure 4

Decomposition of the total density of states of the (a) $N=6$ and (b) $N=15$ simulation cells into contributions from each graphene island and the B and N atoms. $\alpha$ and $\beta$ spin channels are indicated by solid and dotted lines, respectively. (For clarity, the $\beta$ channel is reflected in the $x$ axis; individual curves are rigidly shifted along the $y$ axis.) The shaded region indicates the LSDA band gap region ($\sim$4.66 eV) of a pure hBN layer.

Figure 5

Band structure of FM and AFM coupled islands for the (a) $N=6$ and (b) $N=15$ supercells. Solid red (dotted blue) lines denote $\alpha$ ($\beta$) spin channels. Only bands in the vicinity of the Fermi level are shown for clarity. For the $N=6$ supercell, the metastable FM case is metallic; the stable AFM case is semiconducting with some curvature to the defect levels. For the $N=15$ supercell, both FM and AFM cases are semiconducting. As shown in the insets in (b), the flat thick lines near the Fermi level are actually composed of four individual dispersionless defect levels which correspond to the peaks seen in Fig. 4 near the Fermi level; these defect levels are near-degenerate and separated by $\sim$1 meV. The shaded region indicates the LSDA band gap region ($\sim$4.66 eV) of a pure hBN layer.

Figure 6

Density of states for the C${}_4$ islands, and the B and N atoms in an $N=15$ supercell decomposed into contributions from $s$ and $p$ orbitals for each spin channel. As seen, only the $p_z$ orbitals contribute to states near the Fermi level. Therefore island-island coupling occurs primarily through electron transfer between the half-filled, empty, and filled $p_z$ orbitals of C, B, and N, respectively.