Ferromagnetism in nitrogen-doped graphene

Inducing a robust long-range magnetic order in diamagnetic graphene remains a challenge. While nitrogen-doped graphene is reported to be a promising candidate, the corresponding exchange mechanism remains unclear and is essential to tune further and manipulate magnetism. Within the first-principles calculations, we systematically investigate the local moment formation and the concurrent interaction between various nitrogen-containing defect complexes. The importance of adatom diffusion on the differential defect abundance is discussed. The itinerant magnetism in the graphitic N defects is found to be fundamentally different from the more localized moments in the vacancy-containing N complexes. The magnetic interaction between the complexes is found to depend on the concentration, complex type, sublattice, separation, and orientation. We propose that the direct exchange mechanism between the delocalized magnetic moment originating from the itinerant $\pi$ electron at the prevalent graphitic complexes to be responsible for the observed ferromagnetism. We show that B codoping further improves ferromagnetism. Present results will assist in the microscopic understanding of the current experimental results and motivate new experiments to produce robust magnetism following the proposed synthesis strategy.

Figure 1

Schematic representation of defects in N-doped graphene. Apart from the lattice vacancy (a), chemisorbed (b), various graphitic [single graphitic N (c), meta-graphitic N pair ${\mathrm{N}}_{\mathrm{AA}}$ (d), paragraphitic N pair ${\mathrm{N}}_{{\mathrm{AB}}^{\prime }}$ (e), graphitic 3N (f), triazine (g)], and different vacancy-containing N complexes [pyridinic (h), pyrrolic (i)] are investigated. A and B represent the two graphene sublattices.

Figure 2

The dual origin of vacancy magnetism in graphene and its evolution with carrier doping. Positive (negative) concentration indicates electron (hole) doping. While the $\pi$ component of the magnetic moment is quenched under carrier doping, the $\sigma$ component remains protected. The inset represents a schematic energy diagram indicating defect-induced and spin-split localized $\pi$ and $\sigma$ states along with the bulk $\pi$ bands for the Jahn-Teller distorted $V_1\left(5\right|9)$ vacancy. This schematic picture explains the evolution of the magnetic moment of a lattice vacancy with varied carrier doping. While the magnetism is correctly described by the SCAN meta-GGA exchange-correlation functional, the PBE-GGA functional underestimates the moment.

Figure 3

Adatom diffusivity $D_{\mathrm{C}/\mathrm{N}}$ calculated within the harmonic approximation, $D_X=\frac{1}{4}{a}_{{}_X}^{\prime 2}{\mathrm{\Gamma }}_0{e}^{-E_{{}_X}^a/k_{B}T}$, where $a_{{}_X}^{\prime }$ is the jump distance for the $X=\mathrm{C}/\mathrm{N}$ diffusion, and $E_{{}_X}^a$ is the corresponding activation barrier. ${\mathrm{\Gamma }}_0(={\mathrm{\Pi }}_i{\nu }_i^{\mathrm{I}}/{\mathrm{\Pi }}_j{\nu }_j^{\mathrm{TS}})$ is the jump prefactor, where ${\nu }_i^{\mathrm{I}}$ and ${\nu }_j^{\mathrm{TS}}$ are the frequencies of harmonic vibrational modes corresponding to the structures at the initial and saddle points. Very high diffusivity of C and N adatoms on graphene, especially in the temperature range 800–1300 K relevant to growth and annealing conditions [24, 24, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], indicates self-healing of the vacancy defects and promotes the formation of graphitic N complexes. Both adatoms are absorbed at the bridge site, and the two different diffusion mechanisms are shown in the inset.

Figure 4

Spin-polarized projected density of states calculated using the $\left(10\times 5\right)\text{−}(\sqrt{3},3)a_0$ supercell for the graphitic and vacancy-containing N complexes. Substitutional N doping opens up a gap at the Dirac point $E_{\mathrm{D}}$. While the electronic structures are $n$ type for the graphitic defects, the vacancy-containing defects display $p$-type semiconducting behavior.

Figure 5

Magnetization density for graphitic and vacancy-containing N complexes: (a) triazine, (b) graphitic-3N, (c) trimerized pyridine, and (d) trimerized pyrrolic defect. Red and blue colors indicate the up and down spin density, respectively. While the magnetism in graphitic N complexes is mostly itinerant in nature, the magnetization density demonstrates that a very significant fraction of the moment is localized at the N sites for the vacancy-containing N complexes.

Figure 6

Interaction energy between the N complexes $E_{\mathrm{int}}^{{\mathcal{DD}}^{\prime }}$ in graphene is calculated in the low concentration limit (3 at. %) with varying distances and orientations along the zigzag and armchair directions. Independent of the interacting defect types, the magnetic coupling is observed to be sublattice dependent. The defects on the equivalent (opposite) sublattice are coupled ferromagnetically (antiferromagnetically). Such sublattice-dependent magnetic interaction will not lead to the observed ferromagnetism in nitrogen-doped graphene. In contrast, these complexes interact very differently in the high N-concentration limit, which is discussed in the text.

Figure 7

The calculated magnetization density reveals the corresponding magnetic interactions between the defects, which are highlighted with a triangle or circle. In the low N-concentration limit of 3 at. %, the graphitic defects follow the usual sublattice-dependent magnetic interaction. (a)–(d) The defects on the (opposite) same sublattice are (anti-) ferromagnetically coupled. However, the picture changes at the high N-concentration limit (12.5 at. %) and crucial exceptions to the usual sublattice-dependent magnetism occur. (e)–(g) In specific crystallographic directions of graphene lattice, the defects are coupled ferromagnetically, regardless of their sublattice. Such ferromagnetic interaction between the graphitic N defects is responsible for the observed ferromagnetism. (h),(i) The interaction between the T complex and the single graphitic B defect is always ferromagnetic, independent of the sublattice. Thus, B codoping promotes ferromagnetism in N-doped graphene and increases the saturation magnetization, which is consistent with the recent experimental observation [50]. Red and blue colors indicate the up and down spin density, respectively.