Hybrid superlattices of graphene and hexagonal boron nitride: A ferromagnetic semiconductor at room temperature

Carbon (C) -doped hexagonal boron nitride (hBN) has been experimentally reported to be ferromagnetic at room temperature. Substitution by C in hBN has also been reported to form islands of graphene. In this work, we derive a mechanistic understanding of ferromagnetism with graphene islands in hBN from first principles and the mean-field Hubbard model. We find a general property that in bipartite lattices where the sublattices differ in on-site energies, as in hBN, the ordering between local magnetic moments can be substantial and predominantly antiferromagnetic (AFM) if they are embedded in the same sublattice, unless dominated by Mott-like intersublattice spin separation due to strong localization. The dominant AFM order is rooted at spin-resolved spatial separation of lone pairs of nitrogen (N) and back-transferred electrons on boron (B) due to Coulomb repulsion, thus essentially implying a superexchange pathway. Subsequently, we propose a class of ferrimagnetically ordered interpenetrating superlattices of magnetic graphene islands in hBN, which can be chosen to be a ferromagnetic semiconductor or a half-metal, and we retain a net nonzero magnetic moment at room temperature.

Figure 1

Density of states (DOS) of pristine hBN from (a) DFT and (b) MFH models. Density of states projected on $2p_z$ orbital of C-atom substituted at (c,d) single N site, (e,f) single B site, (g,h) 3 N and 1 B sites, and (i,j) 3 B and 1 N sites, from spin-polarized DFT (c,e,g,i) and MFH models (d,f,h,j).

Figure 2

(a) Energy difference ($E_{\text{FM}}-E_{\text{AFM}}$) for two C1 sites (single substitution) and C4 islands for different island separation $d$ within a hydrogen-passivated hBN segment. Spin densities with separation (b,c) $d=0$, (d) $d=0.5$, (e) $d=1.5$, (f,g) $d=1$ between two C1 sites, and (h,i) for separation $d=1$ between two C4 islands. All contour plots except (d) are in the range $-0.001$ to +0.001. Part (d) is in the range $-0.0001$ to +0.0001.

Figure 3

Spin density of a honeycomb superlattice with (a) C4a (3B1N) and (d) C4b (3N1B), with $d=0$. Planar projection of Wannier functions representing $2p_z$ orbital of C along (b) C-N and (e) C-B bonds for honeycomb superlattices made of C4a and C4b islands, respectively. Spin-resolved Wannier function of $2p_z$ orbital of (c) the bridging N atoms between C4a (3B1N) islands and (f) the outer C atoms of a C4b (3N1B) island.

Figure 4

(a) Schematic representation of the spin-dependent hopping mechanism. Phase diagram in terms of total magnetization of the ground state calculated as a function of ($t-t^{\prime }$) and on-site energy $E_{\text{on-site}}$ for a pair of (b) C4a(3B1N) and (c) C4b(3N1B) islands with $d=0$ in a finite hBN segment at three different $U$ values (eV). For $U=4.0$ and 2.0 eV the FM-AFM crossover is shown by a black dotted line.

Figure 5

Schematic model of (a) AFM ordered C4a(3B1N) islands. (b) FM ordered C4a(3B1N) islands due to the extra local moment (C) at B site between the C4a islands. In (a,b) the dashed arrows represent back-transfer of electrons to neighboring sites. Spin density plot of honeycomb superlattice of C4a islands with $d=1$ in (c) the absence and (d) the presence of C in the intermediate B sites. Wannier function representing the $2p_z$ orbitals of (e) the two N atoms back-transferring to the C in the middle, and (f) the C atom itself, in the -N-C-N- zigzag pathway between C4a islands.

Figure 6

Occupancy of $2p_z$ orbitals of the two spins along zigzag path due to -N-C-N- connectivity between C4a(3B1N) islands calculated using (a) PBE and (c) HSE, as well as from MFH with (b) $U=3.0\mathrm{eV}$ and (d) $U=6.0\mathrm{eV}$. $\mathrm{\Delta }t$ set to 0.5 eV for (b) and (d). The occupation represented by the dotted lines in (b) corresponds to $\mathrm{\Delta }t=0.0\mathrm{eV}$.

Figure 7

(a) Representation of interpenetrating C4a-$X$ honeycomb-kagome superlattices for various intermediated substitution by C at $X$ (B or N) sites for $d=3$. Energy difference ($E_{\text{FM}}-E_{\text{AFM}}$) for different $X$ different separations of C4a islands (b) $d=1$, (c) $d=2$, (d) $d=3$, and (e) $d=4$. Red circles in (b,c) represent corresponding energy difference with an additional layer of hBN at $A-B$ stacking beneath the hybrid layer as reported for hBN bilayer. $E_{\text{FM}}-E_{\text{AFM}}<0$ implies FM order in a honeycomb superlattice.

Figure 8

Spin-polarized density of states (DOS) for (a) $X=3\text{N}$ and (b) $X=0\text{B}$ as described in Figs. 7 and 7 for $d=3$. (c)–(f) Schematic presentation of DOS to understand its evolution of half-metallic and FM-Sc phases with different choices of C4 islands (C4a or C4b) and $X$ (N or B).

Figure 9

Spin density and DOS for (a) C4b-C4a, (b) C9-C4a, (c) C9-C4b, and (d) C13-C4a honeycomb-kagome superlattices, where C9:6B3N, C4a:3B1N, C4b:3N1B, and C13:7B6N.