Ferromagnetism in nitrogen doped graphene
Rohit Babar Mukul Kabir





Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India	

mukul.kabir@iiserpune.ac.in 
Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India
Centre for Energy Science, Indian Institute of Science Education and Research, Pune 411008, India


 
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 endures 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 defect complexes. The importance of adatom diffusion on the differential defect abundance is discussed. The individual nitrogen complexes that contribute toward itinerant and a local magnetic moment are identified. The magnetic interaction between the complexes is found to depend on the concentration, complex type, sublattice, distance, and orientation. We propose that the direct exchange mechanism between the delocalized magnetic moment originating from the itinerant π-electron at the prevalent graphitic complexes to be responsible for the observed ferromagnetism. We show that B co-doping further improves ferromagnetism. Present results will assist in the microscopic understanding of the current experimental results and motivate experiments to produce robust magnetism following the proposed synthesis strategy.

Ferromagnetism in nitrogen doped graphene
    Mukul Kabir
    December 30, 2023
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§ INTRODUCTION


The absence of intrinsic magnetism in graphene limits its application in spintronics and is being actively persuaded since it was first isolated. <cit.> However, the magnetism can be induced in graphene due to the p_z electron imbalance in bipartite hexagonal sublattice, <cit.> electron localization at the zigzag edge, <cit.> interaction with a spin-orbit coupled ferromagnetic substrate, <cit.> structural defect creation, <cit.> and heteroatom doping. <cit.> However, a long-range magnetic order in graphene can only emerge through the exchange coupling between these induced local moments. 

Hydrogen absorption on graphene creates an imbalance in p_z electrons between the two sublattices, which in turn generates a net magnetic moment. <cit.> Further, the spin-polarized state is observed to extend over several nanometers through the scanning tunneling microscopy (STM) experiment. This spatially extended magnetic state provides a channel for direct coupling between the induced magnetic moments at large distances. The half-filled flat band at the Fermi level produces localized magnetic moments at the zigzag edge of a graphene flake, <cit.> and a long-range order has been predicted. <cit.>  However,  the edge magnetism is predicted to be very intricate as it is strongly affected by the shape and size of the flakes, charge doping, chemical functionalization at the edge, and on-site Coulomb interaction. <cit.> Proximity-induced exchange-coupled ferromagnetic order sets in graphene sheet at room temperature while places on an atomically flat ferromagnetic thin film insulator. <cit.> The smallest structural defect, a lattice vacancy in graphene generates a semi-localized magnetic moment due to the π electron imbalance in addition to a σ contribution from the dangling bond. <cit.>  Thus, proton irradiated graphene and graphene with a network of point defects show magnetic ordering. <cit.> Similarly, the larger lattice defects such as grain boundaries with dangling bonds are theoretically predicted to have magnetic moment. <cit.> However, the experimental evidence of a magnetic ordering remains scarce and controversial. <cit.> While a room temperature ferromagnetism is predicted, <cit.> defected graphene with vacancies and other sp^3 defects lead to paramagnetism and devoid of any magnetic ordering down to low temperature. <cit.> Furthermore, in the cases where ferromagnetic signals have been detected, the origin of such long-range ordering remained unexplained. The presence of d-electrons in transition-metal impurities induces a local moment in pristine and defected graphene. <cit.> However, only a paramagnetic ordering has been observed in this transition-metal doped graphene samples that were investigated through the X-ray magnetic circular dichroism measurement. <cit.> 

In contrast, the substitutional and sp^3 impurities have attracted much attention with intriguing but elusive magnetic properties. 
The itinerant sp-electron Stoner ferromagnetism in a narrow impurity band is fundamentally different from the magnetism induced by the d-band of transition metals. <cit.> Such sp-electron ferromagnetism may lead to a higher Curie temperature than the usual dilute magnetic semiconductors. The adatom impurities like hydrogen and fluorine exhibit strong absorption and induce magnetic moments <cit.> due to sublattice imbalance in π-electrons following Lieb's theorem. <cit.> While the hydrogenated graphene show magnetic order at 5 K, <cit.> the fluorine doped graphene reveals paramagnetic behaviour. <cit.> In this context, the substitutional N-doped graphene attracted much attention, which exhibits robust ferromagnetism. <cit.> However, the corresponding Curie temperature and saturation magnetization are reported to show a strong dependence on the growth, annealing strategies, and nitrogen concentration, which hinder a careful microscopic analysis. A variety of nitrogen complexes are generated during the non-equilibrium synthesis, while N-concentration and the type of defect complex determines the electronic and transport properties. <cit.> Recently, the room-temperature ferromagnetism was reported in boron and nitrogen-doped turbostratic carbon films grown via pulsed laser deposition technique, while the saturation magnetization is observed to increase with increasing N-doping. <cit.> However, even with such efforts, the nature and microscopic mechanism of magnetic interaction are not well understood in N-doped graphene, which is necessary to tune and manipulate magnetism toward its successful applications. 



Within the first-principles calculations, here we systematically investigate the electronic and magnetic properties of various N-defect complexes in graphene. Further, how these defects interact with each other and lattice vacancies are studied to understand the mechanism responsible for the observed ferromagnetism. We report that the long-range magnetic interactions show a strong dependence on the N-concentration, complex type and on which sublattice they reside, distance, and orientation. We observe that the magnetism in the graphitic and vacancy containing N-complexes are fundamentally very different, and propose that the observed ferromagnetic order originates from the direct exchange between the extended magnetic states of the graphitic defects. Further, we investigate the crucial role of boron co-doping in the ferromagnetism. 





§ COMPUTATIONAL DETAILS

The spin-polarized density functional theory based calculations <cit.> were performed within the projector augmented wave formalism, <cit.>, and the wave functions were expanded in a plane-wave basis with 600 eV kinetic energy cut-off. The exchange-correlation energy was described using the conventional Perdew-Burke-Ernzerhof (PBE) form for the generalized gradient approximation (GGA). <cit.> The structures were fully relaxed until all the forces became smaller than 0.015 eV/Å  threshold using the PBE functional. The Brillouin zone was sampled using a Γ-centered 8×6×1 k-grid for the structural optimization, and a much dense Γ-centered 24×18×1 k-grid was used for the evaluation of density of states (DOS) and magnetization density. The subsequent N-doping was carried with the 4×3(√(3),3)a_0 and 10×5(√(3),3)a_0 rectangular supercells, where a_0 is the C–C bond length in graphene.  The periodic images of the supercell along the direction perpendicular to the graphene sheet are separated by at least 12 Å distance. It is known that the self-interaction error inherent to the conventional GGA functional results in spurious electron delocalization and incorrect magnetic solutions for adatom doping and vacancy defects in graphene. <cit.>  In this regard, we recalculated the electronic and magnetic properties using the Heyd-Scuseria-Ernzerhof (HSE06) <cit.> hybrid functional as well as using the strongly constrained and appropriately normed (SCAN) meta-GGA functional. <cit.> While the inclusion of fractional exact exchange for HSE06 hybrid functional improves the description of the electronic structure and the magnetization density, it is computationally expensive. In contrast, since the semilocal SCAN meta-GGA density functional satisfies all the known exact constraints for the exchange-correlation energy and the self-interaction error is reduced, it should better describe the thermodynamic, electronic and magnetic properties of materials.  Indeed the SCAN functional predicts these properties correctly for diverse materials with a similar computational cost to the conventional GGA. <cit.> In the present context of defect magnetism and the concurrent interaction between them, we further validate the applicability of the SCAN functional. The results are obtained within the SCAN functional if it is not otherwise stated. 






§ RESULTS AND DISCUSSION

Before we investigate the magnetic interactions between the moment generating defect complexes in N-doped graphene, we first address the local magnetism produced by these individual defects. In addition to the lattice vacancies, the common defects in N-doped graphene are chemisorbed, graphitic and vacancy containing nitrogen complexes  (Figure <ref>). <cit.> As the treatment of exchange-correlation may affect the electronic and magnetic description of these defects, here we compare the PBE, hybrid HSE06, and SCAN meta-GGA functionals. 



 §.§ Magnetism of individual defects

Single vacancy. Magnetic properties of single vacancy defect in graphene are still studied and debated. <cit.> The Jahn-Teller distorted V_1(5|9) vacancy [Figure <ref>(a)] generates defect induced localized σ and π states, V_σ and V_π (Figure <ref>). Thus, the magnetic moment of V_1(5|9) vacancy has two components. <cit.> The π-electron imbalance in the bipartite graphene lattice should generate 1 μ_B moment according to the Lieb's theorem. Additionally, the singly occupied dangling σ-bond should provide with another 1 μ_B moment, leaving the total vacancy moment to 2 μ_B. However, the π-magnetism is suppressed within the conventional GGA description of exchange-correlation due to improper delocalization of electrons in the V_π orbital. Thus, the magnetic moment of V_1(5|9) is underestimated to 1.5 μ_B within the PBE exchange-correlation functional (Figure <ref>, Table <ref>), similar to that predicted earlier. <cit.> Further, the moment depends on the supercell size, and thus on the vacancy concentration. <cit.> The picture is slightly improved while a fraction a__  HF of Hartree-Fock exact exchange is considered in the HSE06 functional and 1.6 μ_B moment is predicted at the vacancy for a__ HF=0.25. A further increase in a__  HF to 0.4 ensures a 2 μ_B moment at the vacancy. In contrast, treating the exchange-correlation with SCAN meta-GGA functional describe the magnetism correctly with 2 μ_B moment for the vacancy (Figure <ref>, Table <ref>). 




Similarly, the corresponding electronic structure depends on the exchange-correlation functional. While a metallic solution emerges within the PBE and HSE06 functionals, the SCAN functional correctly predicts a gapped band structure.  This result is consistent with the diverging resistivity observed in irradiation induced defected graphene with about 10^12 cm^-2 vacancy concentration, which indicate insulating behaviour. <cit.> This was latter theoretically corroborated in graphene with high vacancy concentrations and an insulating solution with a large bandgap was proposed. <cit.>  Further, while the defect induced V_σ state shows a large spin-splitting above 2 eV originating from the JT reconstruction, the splitting for the V_π is small, about 250 meV (Figure <ref>). This prediction is consistent with the recent STM experiment, where the spin-polarized π density of states (DOS) peaks at the defect site are observed to be separated by 20-60 meV. <cit.> It should be noted here that in such experiments the V_π spin-splitting may be affected by a metastable vacancy geometry or by its interaction with the substrate. 








We find the vacancy magnetism to be robust against carrier doping with a realistic concentration, <cit.> as the position of the Fermi level is tuned (Figure <ref>). The graphene with a neutral V_1(5|9) defect is intrinsically hole doped, and one extra electron on a large 10×5  (√(3), 3)a_0 supercell corresponds to 1.89×10^13 cm^-2 carrier density. The extra electron occupies the unfilled V_π^↓ state destroying the π-electron imbalance in the two bipartite sublattices. Thus,  the π-component of the vacancy moment is completely quenched (Figure <ref>). In contrast, the σ-component of the magnetic moment is protected against carrier doping. Increasing the electron doping beyond 1.89×10^13 cm^-2 does not further affect the magnetic moment (Figure <ref>) At higher electron density the unoccupied bulk π-states get filled, while the unoccupied V_σ^↓ state lies much higher in energy and is impossible to populate with any realistic carrier density. The carrier concentration dependent differential charge density corroborates this picture. <cit.> The scenario with hole doping is different only qualitatively. The σ-component is unaffected, and hole doping does not quench the π-moment completely indicating the particle-hole asymmetry in defected graphene  (Figure <ref>). The semi-localized V_π^↑ state is only partially depopulated and a fractional π-moment survives. <cit.> Therefore, while the π magnetism in graphene vacancy could be completely quenched with sufficient carrier doping, the spin-1/2 σ-magnetism is robust. 


In the context of differential defect abundance in N-doped graphene, it is relevant to investigate the C and N adatom diffusion on graphene lattice. The C and N adatoms are absorbed at the bridge site with 1.51 and 0.9 eV binding energy. We calculate the activation barrier for adatom diffusion within climbing image nudged elastic band method. <cit.> In agreement with the previous calculations, <cit.> the calculated activation barriers are found to be 0.43 and 0.86 eV, respectively, for C and N diffusion (Supplemental Material). <cit.> Such low activation barrier results into a very high diffusivity D_ C/N at the experimental growth and annealing temperatures (Figure <ref>), and the adatoms will quickly recombine with the graphene vacancies. Therefore, due to high adatom mobility, it is expected that the graphene vacancies will be annihilated to decrease in concentration and from graphitic N-complexes, while annealed at high-temperature or non-equilibrium growth conditions. <cit.> This prediction is consistent with the experimental observation of pyridinic to graphitic defect conversion during postannealing at high-temperature. <cit.>  Further, the diffusion mediated self-healing has been observed in scanning transmission electron microscopy. <cit.>

N-adatom. The chemisorbed N-adatom [Figure <ref>(b)] is a p-type defect with the impurity state above the Fermi level. A localized magnetic moment of 1 μ_B is generated from the in-plane N-orbitals (Table <ref>). However, due to the high N diffusion on graphene (Figure <ref>), such adatom defect may not be present and effectively contribute to magnetism in graphene samples that are prepared and annealed at high-temperature. In contrast, high N diffusion will further promote the formation of graphitic N-complexes. 

Graphitic nitrogen defects. The appearance of Dirac point (DP, E_ D) in graphene band structure is very sensitive to external perturbations, and the breaking of sublattice symmetry induces a substantial gap at the DP. <cit.> Thus, all the graphitic N-defects are expected to generate a gap at E_ D. Single graphitic nitrogen [Figure <ref>(c)] introduces an impurity state near the Fermi level and induces a 0.60 eV gap at E_ D. An overall semiconducting electronic structure emerges and the  results are consistent with the STM measurement. <cit.> Further, an exchange splitting of 18 meV is observed between the majority and minority spin channels, which results in itinerant π-magnetism along with a small 0.06 μ_B moment localized at the N-site. However, the total moment generated by this defect strongly depends on the treatment of exchange-correlation energy (Table <ref>) and defect concentration shown in Supplemental Material. <cit.>  While the SCAN functional correctly generates 1 μ_B moment, the moment strongly depends on the fractional exact exchange a__ HF in HSE06 functional. While the conventional HSE06 functional (a__ HF=0.25) underestimates the moment (Table <ref>), it gradually increases with increasing a__ HF and generates 1 μ_B moment for a__ HF=1.  In contrast, in agreement with an earlier report, <cit.>  the conventional PBE functional fails to produce any spin-splitting and a non-magnetic solution emerges.  Therefore, the SCAN meta-GGA functional provides the correct magnetic picture over other exchange-correlational functionals.








The graphitic complexes with two nitrogens N_ AA and N_ AB' have been experimentally identified [Figure <ref>(d) and (e)]. <cit.> Due to the sublattice asymmetry in these n-type complexes, the DP is destroyed, and within the SCAN functional a 0.31 and 0.50 eV gap emerges at the E_ D for N_ AA and N_ AB' complexes, respectively.  The overall electronic structure is found to be defect specific as a half-metallic (semiconducting) solution is observed for the N_ AA  (N_ AB') defect in the large 10×5  (√(3), 3)a_0 supercell. The overall results here are in agreement with the experimental results as the character of electronic DOS near the Fermi level is consistent with the STS measurements. <cit.> 

The exchange-split bands generate itinerant magnetism and a small moment of 0.06–0.08 μ_B is generated at the N-sites, which are coupled ferromagnetically (antiferromagnetically) while the nitrogen atoms occupy the same (different) sublattice. This fact results into 2 and 0 μ_B moments for the N_ AA and N_ AB' pairs, respectively (Table <ref>). While the results from the different exchange-correlation functionals are compared, a similar picture emerges as discussed for the single graphitic N-defect. 








While the triazine (T) complex [Figure <ref>(g)] is thermodynamically slightly more favorable over the graphitic-3N (G3N) complex [Figure <ref>(g)], both of these defects are formed at the non-equilibrium growth conditions.<cit.> The graphitic-3N complex could also be produced when a highly mobile carbon adatom diffuses into the lattice vacancy in the trimerized pyridinic and pyrrolic  complexes in the bulk regions of graphene. In both the defects [Figure <ref>(f) and Figure <ref>(g)], all three nitrogen atoms occupy the same sublattice leading to a sublattice asymmetry, which in turn destroys the DP and consequently open up a gap at the E_ D (Figure <ref>).  The overall electronic structure is half-metallic for the T-complex, and the same is found to be semiconducting for the G3N-complex  (Figure <ref>). It is important to note that such half-metallicity induced asymmetric electron transport for different spin components could have significant implication in spin-based devices. To gain further insight into the magnetism, we investigate the differential charge density and the corresponding magnetization density (Figure <ref>). The C→N π-electron transfer creates Lieb's imbalance and an exchange split π-band emerges, which is responsible for the itinerant magnetism that is confirmed form the magnetization density [Figure <ref>(a) and (b)]. In addition, a hybridized nitrogen impurity state is observed near the Fermi level, which is responsible for a tiny N-moment of ∼ 0.05 μ_B. Note the C-atom at the centre of G3N-defect acquire a comparatively larger localized moment of 0.25 μ_B [Figure <ref>(b)]. These together result into a total moment of 1 μ_B  for these graphitic defect complexes (Table <ref>). 


 


Thus, in general, the electronic and magnetic structure of the different graphitic defect complexes are remarkably similar. First, all of them open up a gap at the E_ D due to sublattice asymmetry, which is consistent with the the angle-resolved photoemission spectroscopy results.  <cit.> Second, the exchange-split π-band generates itinerant magnetism, while the localized moment at the N-site is small about 0.03–0.09 μ_B.  While the overall electronic structure of the graphitic complexes are defect specific, all of them are found to be n-type, which are in overall agreement with the STS results. <cit.>
 

Vacancy containing nitrogen defects. Now we explore the defect complexes wherein the N atoms are incorporated in a vacancy. The trimerized pyridine (TPY), where three N atoms occupy the same graphene sublattice surrounding a vacancy [Figure <ref>(h)] is energetically favoured over the mono and dimerized pyridines. <cit.> The 1.33 Å C-N bonds are consistent with experimental values for pyridinic complexes. The N atoms form an equilateral triangle with 2.61 Å sides, which is larger than the triangular unrelaxed vacancy with 2.46 Å. Further, this complex is found to be p-type and the Bader analysis indicates a 0.9-1.12 e/N charge transfer to nitrogen from carbon. 


The magnetism in such vacancy containing nitrogen defects is rather intricate, and a detailed investigation into the electronic DOS (Figure <ref>) and  the magnetization density [Figure <ref>(c)] unravel the origin of 1 μ_B moment for this complex. Unlike the bare vacancy, the σ component to the moment is completely ruled out due to the presence of occupied σ-state that originates from the undercoordinated N atoms (Figure <ref>). In contrast, the moment is of entirely π character and originates from the localized N-p_z and itinerant C-p_z states with an exchange-splitting of 11 meV.  As a result, a sizeable localized moment of 0.15 μ_B is observed at the N-sites for this complex and a fraction of the total moment originates form the itinerant magnetism [Figure <ref>(c)]. In contrast to the SCAN functional, the PBE results provide a very contrasting picture. First, the moment is substantially underestimated (Table <ref>), and further, the moment originates from the localized σ-state at the N-sites without any π contributions.  However, the STM and STS investigations of the monomerized and trimerized pyridine reveal an electron density distribution that is consistent with the p_z states of the vacancy defect and thus corroborate the results obtained with the SCAN functional. <cit.> 

The trimerized pyrrolic (TPI) complex is formed when a single N atom is absorbed in a divacancy to form a pentagon and rest of the undercoordinated C atoms are also substituted by N in Figure <ref>(i). The pentagonal C-N bonds (1.41 Å) are larger in comparison with the other C-N bonds (1.33 Å). Similar to the TPY-complex and bare vacancy, this complex has p-type character. The electronic DOS calculated within the PBE and SCAN functionals show qualitatively similar features, and the unoccupied σ-states originating from the N-impurities appear just above the Fermi level (Figure <ref>). The calculated magnetization density indicates that a large fraction ∼ 80% of the total 1 μ_B moment is localized on the nitrogen atoms and only a small fraction originates from the delocalized π-band [Figure <ref>(d)]. Further, due to the proximity of the unoccupied σ-states to the Fermi level, a slight perturbation through the charge doping will adversely affect this localized σ component of the moment. 

Therefore, the magnetism in the vacancy containing N-complexes is fundamentally different from the graphitic defects. While the magnetism in graphitic defects is entirely itinerant, a very significant fraction of the moment is localized at the N-sites in vacancy containing N-complexes.  Thus, in the absence of a strong exchange path, one would not expect that such localized magnetic moments in these vacancy containing defect complexes to contribute significantly to the high-temperature ferromagnetism. Indeed, the role of this particular defect on the observed ferromagnetism has remained controversial.  <cit.> 




 §.§ Magnetic interactions

Although nitrogen-doped graphene is reported to show ferromagnetism, the microscopic mechanism is not yet understood and resulted in conflicting arguments. <cit.>  The appearance of a diverse Curie temperature with varied growth and annealing conditions makes the microscopic analysis very complicated as the differential defect abundance may fluctuate drastically in the samples. A long-range magnetic order in nitrogen doped graphene can only emerge when the moment generating defects are exchange coupled. Thus, we investigate the magnetic interactions between the abundant defect complexes.


The single vacancy in graphene lattice generates a robust semi-localized moment.  However, the interaction between the vacancy moments is sublattice dependent. Vacancies in the same sublattice are coupled ferromagnetically, whereas they prefer antiferromagnetic alignment when placed on different sublattices. Further, the magnetic interaction sharply falls to zero below 1 nm distance due to semi-localized nature of the moment. <cit.> Thus, a randomly distributed vacancy in graphene will not lead to any magnetic ordering, which is consistent with the results predicting paramagnetism. <cit.> Moreover, it should be noticed in this context that, the existence of a vacancy in the N-doped graphene is rare due to the very high C and N adatom mobility at the high-temperature growth conditions (Figure <ref>), which leads to self-healing of vacancy or the formation of graphitic N-defect.      

Single graphitic nitrogen defect is similar to the pristine vacancy. For two substitutional atoms on the (different) same sublattice, the spin-configuration is (anti-) ferromagnetic. Moreover, the interaction strength decays fast with increasing separation and follows an inverse power law, and the long-range interaction may develop through Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism. <cit.> Thus, a statistically random distribution of single graphitic nitrogen will not result in high-temperature ferromagnetism that has been reported so far. <cit.> 






A very complex picture emerges when we explore the interactions between the larger defects. Here, we address two distinct regimes of N-doping that is motivated by several experimental observations. <cit.> It has been unanimously pointed out in the literature that ferromagnetism emerges at a critical N-concentration around 5 at. % and above which the saturation magnetization and coercive field increase further with increasing the concentration. Thus, we consider N-concentration below 5 at. % and in the 8-13 at. % range, and investigate how the concentration influences the magnetic interaction. In this regard, we calculate the interaction energy E^𝒟𝒟'_ int between two magnetic defects 𝒟 and 𝒟'   as, E^𝒟𝒟'_ int = (E^𝒟𝒟'_ m -  E^𝒟𝒟'_ nm) - (E^𝒟_ m -  E^𝒟_ nm) - (E^𝒟'_ m -  E^𝒟'_ nm) and shown in the Figure <ref>. The energies corresponding to the magnetic and non-magnetic solutions for the interacting defects are E^𝒟𝒟'_ m and E^𝒟𝒟'_ nm, while E^𝒟/𝒟'_ m and E^𝒟/𝒟'_ nm represent magnetic and non-magnetic energies of the isolated defects.    

First, we discuss the long-range magnetic interaction between the graphitic defects in the low N-concentration limit and investigate how it is affected by the spatial separation and direction between the defects (Figure <ref>). Two interacting T-T, G3N-G3N, and mixed T-G3N defects in a (10×5)-(√(3),3)a_0 supercell represents 3 at. % N-concentration. At such a low concentration and for all these types of defect interactions, we notice some general trends [Figure <ref>(a) and <ref>(b)].  (i) These defects mostly prefer to occupy the same graphene sublattice, (ii) owing to the itinerant nature of the individual moments in these graphitic defects, the interaction between them is long-range, and (iii) the nature of magnetic interaction between the defects is entirely dictated by their respective sublattice [Figure <ref>(a)–(d)]. For the defects on the equivalent (opposite) graphene sublattice, the magnetic interaction is found to be ferromagnetic FM (antiferromagnetic AFM). The picture remains unaltered regardless of the spatial separation and relative orientation between the defects along the armchair and zigzag directions (Figure <ref>). Although the nature of the magnetic moment in these defects is completely different, the sublattice dependent magnetic interaction is in complete agreement with results predicted by the RKKY interaction between the localized moments on the bipartite graphene lattice. <cit.> While these defects mostly prefer to occupy the same sublattice 𝒟_ A𝒟'_ A over the 𝒟_ A𝒟'_ B configuration, during the non-equilibrium growth process the defects are equivalently distributed over the two graphene sublattice. Therefore, the sublattice dependent FM and AFM interaction will restrict the observation of FM order at such a low concentration of nitrogen. This picture is consistent with the experimental observation that FM ordering is not found below 5 at. % N-concentration. <cit.> 






Now we turn our attention onto high N-concentration, and two interacting T-T, G3N-G3N, and mixed T-G3N defects in a (4×3)-(√(3),3)a_0 supercell correspond to 12.5 at. % N-doping. The sublattice dependent magnetism for such high N-concentration changes dramatically. In contrast to the AFM interaction, a FM interaction with spontaneous magnetization emerges while the defects that are placed on the different sublattice [Figure <ref>(e)–(g)].  However, such FM structure is found when the defects are interacting along a particular direction, which is along the zigzag direction for  T-T interaction [Figure <ref>(e)] and along the armchair direction for the G3N-G3N [Figure <ref>(f)] , and mixed T-G3N [Figure <ref>(g)] defect interactions. Thus, at high N-concentration, the graphitic T and G3N defects are responsible for the observed ferromagnetism. Further, the concentration of the graphitic T-complex may be increased by using the triazine-based precursor during the growth using the chemical vapour deposition technique. <cit.> Moreover, a suitable postannealing strategy may convert the vacancy containing pyridinic and pyrrolic defects to graphitic G3N-complex. <cit.> Therefore, we conclude that such sample preparation strategies may give rise to robust ferromagnetism with improved Curie temperature.  This discussion is consistent with the recent observation of room-temperature ferromagnetism in turbostratic N-doped graphene films, where the T-defect is found to be the most abundant complex. <cit.> 
  

Further, in the high concentration limit of 8.3 at. % N-doping, we investigate the interactions between the two N_ XX defects, and between the T-complex and a single graphitic N-defect. Both these interactions follow the usual sublattice dependent magnetic couplings. While the N_ AA- N_ AA and T_ AAA-N_ A interactions are found to be ferromagnetic, the N_ AA- N_ BB and T_ AAA-N_ B are observed to be antiferromagnetic. Therefore, such defect-defect interactions are not responsible for the observed ferromagnetism in the N-doped graphene.
   
In contrast to the graphitic N-defects, the vacancy containing complexes do not contribute to ferromagnetism. Moreover, in accordance with the experimental observation, <cit.> we have discussed earlier that the C-adatom is highly mobile on the graphene lattice (Figure <ref>), and thus the vacancy containing TPY-defect complex will be converted into the graphitic G3N-complex during high-temperature growth and annealing. Therefore, the concentration of this defect will be negligibly small. However, for completeness, we have investigated the magnetic interaction between two TPY-complexes in both the low and high concentration limits. Unlike the graphitic T- and G3N-complexes and independent of N-concentration, the TPY-TPY magnetic coupling is AFM (FM) for defects on the opposite (equivalent) sublattice. Further, the picture is independent of defect orientation and separation [Figure <ref>(c)]. Thus, this vacancy containing defect will not play any role in the ferromagnetism in N-doped graphene.       



 §.§ Effects of boron co-doping

Motivated by the recent observation of room-temperature ferromagnetism in boron and nitrogen doped turbostratic carbon films, <cit.> we further investigate the effect of B co-doping on the ferromagnetism. Notably, we consider a small B co-doping in the limit of high N-concentration that is responsible for the observed ferromagnetism. Due to very-low B-concentration, we have only considered graphitic single B-defect and ignored the possibility of the larger B-complexes. We also ignored the chemisorbed B-defect, as in this configuration, the B-adatom is expected to be very mobile similar to the C and N adatoms (Figure <ref>), and consequently form single graphitic B-defect. 


In contrast to the single N-defect, the graphitic B-defect leads to p-type behaviour due to electron deficiency. The C–B bonds are found to elongated (1.49 Å) compared to the C–N bonds (1.41 Å) in the graphitic N-defect and a B → C charge transfer takes place. However, similar to the single graphitic N-defect, the B-defect also results in an itinerant magnetism with 1 μ_B moment, where a very small 0.03 μ_B moment is localized at the B-site. In contrast to the repulsive and sublattice dependent N-N magnetic interactions, the N-B interaction is found to be attractive, and a completely compensated AFM structure emerges, which is independent of the sublattice.  Further, we investigate the interaction between this graphitic B-defect and the prevalent graphitic N-defects, T and G3N complexes with 1 μ_B moment each. Surprisingly, the B_ A/B affects the overall picture significantly. The interaction between these graphitic defects and the B_ A/B is found to be attractive, which is opposite to their interaction with single graphitic N-defect. Independent of their respective sublattice, the interacting T_ AAA-B_ A/B [Figure <ref>(h)–(i)] and G3N_ AAA-B_ A/B configurations generate a total moment of 2 μ_B, while for all cases, the compensated AFM magnetic structure is found to be  energetically much unfavourable. A closer look at the magnetization density reveals that the resultant 2 μ_B moment arises due to the sublattice imbalance of the itinerant C-p_z electrons. This finding is consistent with the recent experimental observation of increased saturation magnetization in N-doped graphene due to B co-doping. <cit.> Therefore, we argue that small B co-doping will enhance ferromagnetism in nitrogen-doped graphene.   



§ SUMMARY

The microscopic origin of ferromagnetism that is observed in N-doped graphene has remained elusive. <cit.> The difficulty to comprehend arises due to the complex dependence of magnetism on N-content and on the corresponding differential defect distribution originating from the varied growth and annealing details. In this context, we systematically investigate the moment formation and the intricate magnetic interaction between various defect complexes within the first-principles calculations. The C and N adatom diffusion on graphene plays a very important role in healing vacancy containing defects and generating graphitic N-complexes. The magnetism in the graphitic N-complexes are found to be fundamentally different from the vacancy containing complexes. While an itinerant magnetism emerges in the graphitic N-complexes, a very significant fraction of the moment is localized at the N-sites for the later. By comparing the results with PBE and HSE06 exchange-correlation functional and available experimental data, we conclude that the SCAN semilocal meta-GGA density functional provides a better description of the electronic structure and magnetism in defected graphene. We further demonstrate that the vacancy magnetism is robust against carrier doping. 

The magnetic interaction between the defects are found to be intricate and complex, which depends on the defect type, N-concentration, defect orientation and the corresponding distance between them. Regardless of the defect type and at low N-concentration, the magnetic interaction shows a generic sublattice dependence independent of defect orientation and distance. In agreement with the RKKY interaction leading to sublattice dependent magnetic interaction in bipartite graphene lattice, <cit.> the long-range coupling is found to be FM (AFM) for the defects on the equivalent (opposite) sublattice. Thus, at low N-content, the statistical distribution of defects on both graphene sublattice will not produce FM ordering. This result is consistent with experimental observation. <cit.> Interestingly, the picture drastically changes at the high N-concentration owing to a direction dependent and sublattice independent FM interaction with spontaneous magnetization for the interaction between the graphitic T and G3N defects. While we reject the other type of defects through a comprehensive investigation, we conclude that graphitic T and G3N defects are predominantly responsible for the observed ferromagnetism. Further, we predict that an experimental strategy to increase the concentration of these graphitic defects will produce robust ferromagnetism with improved Curie temperature. Finally, we demonstrate that a  small amount of boron co-doping will further enhance the ferromagnetism in N-doped graphene. Thus, while the present results help us to understand the scattered and often controversial experimental results, an overall microscopic understanding and a concurrent experimental synthesis strategy provided here will motivate new experiments to control magnetism in nitrogen and boron doped graphene.


We acknowledge stimulating discussions with S. B. Ogale and R. Mandal.  M.K. acknowledges funding from the Science and Engineering Research Board through Ramanujan Fellowship and the Department of Science and Technology through Nano Mission project SR/NM/TP-13/2016. The supercomputing facilities at the Centre for Development of Advanced Computing, Pune; Inter University Accelerator Centre, Delhi; and at the Center for Computational Materials Science, Institute of Materials Research, Tohoku University are greatly acknowledged.













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[Novoselov et al.(2004)Novoselov, Geim, Morozov, Jiang, Zhang, Dubonos, Grigorieva, and Firsov]Novoselov666
  author author K. S. Novoselov, author A. K. Geim,
  author S. V. Morozov, author D. Jiang, author
  Y. Zhang, author S. V. Dubonos, author I. V.  Grigorieva,  and author A. A.  Firsov, http://science.sciencemag.org/content/306/5696/666 journal
  journal Science volume 306, pages 666 (year 2004)NoStop
[Tucek et al.(2018)Tucek,
  Blonski, Ugolotti, Swain,
  Enoki, and Zboril]C7CS00288B
  author author J. Tucek, author P. Blonski,
  author J. Ugolotti, author A. K. Swain, author
  T. Enoki,  and author
  R. Zboril, http://dx.doi.org/10.1039/C7CS00288B journal journal Chem. Soc. Rev. volume 47, pages 3899 (year 2018)NoStop
[Boukhvalov et al.(2008)Boukhvalov, Katsnelson, and Lichtenstein]PhysRevB.77.035427
  author author D. W. Boukhvalov, author M. I. Katsnelson,  and author A. I. Lichtenstein, https://link.aps.org/doi/10.1103/PhysRevB.77.035427 journal
  journal Phys. Rev. B volume 77,  pages 035427 (year 2008)NoStop
[McCreary et al.(2012)McCreary, Swartz, Han, Fabian, and Kawakami]PhysRevLett.109.186604
  author author K. M. McCreary, author A. G. Swartz, author W. Han,
  author J. Fabian,  and author R. K. Kawakami, https://link.aps.org/doi/10.1103/PhysRevLett.109.186604 journal journal Phys. Rev. Lett. volume 109, pages 186604 (year
  2012)NoStop
[González-Herrero et al.(2016)González-Herrero, Gómez-Rodríguez, Mallet, Moaied,
  Palacios, Salgado, Ugeda,
  Veuillen, Yndurain, and Brihuega]Herrero437
  author author H. González-Herrero, author J. M.  Gómez-Rodríguez, author P. Mallet, author M. Moaied, author J. J.  Palacios, author C. Salgado, author M. M. Ugeda,
  author J.-Y. Veuillen, author F. Yndurain,  and author I. Brihuega, http://science.sciencemag.org/content/352/6284/437 journal
  journal Science volume 352, pages 437 (year 2016)NoStop
[Nakada et al.(1996)Nakada,
  Fujita, Dresselhaus, and Dresselhaus]PhysRevB.54.17954
  author author K. Nakada, author M. Fujita,
  author G. Dresselhaus,  and  author M. S. Dresselhaus,  https://link.aps.org/doi/10.1103/PhysRevB.54.17954 journal journal Phys. Rev. B volume
  54, pages 17954 (year 1996)NoStop
[Fernández-Rossier and Palacios(2007)]PhysRevLett.99.177204
  author author J. Fernández-Rossier and author J. J.  Palacios, https://link.aps.org/doi/10.1103/PhysRevLett.99.177204 journal journal Phys. Rev. Lett. volume 99, pages 177204 (year
  2007)NoStop
[Yazyev and Katsnelson(2008)]PhysRevLett.100.047209
  author author O. V. Yazyev and author M. I. Katsnelson, https://link.aps.org/doi/10.1103/PhysRevLett.100.047209 journal journal Phys. Rev. Lett. volume 100, pages 047209 (year
  2008)NoStop
[Kabir and Saha-Dasgupta(2014)]PhysRevB.90.035403
  author author M. Kabir and author T. Saha-Dasgupta, https://link.aps.org/doi/10.1103/PhysRevB.90.035403 journal
  journal Phys. Rev. B volume 90,  pages 035403 (year 2014)NoStop
[Magda et al.(2014)Magda,
  Jin, Hagymási, Vancsó, Osváth, Nemes-Incze,
  Hwang, Biró, and Tapasztó]nature13831
  author author G. Z. Magda, author X. Jin, author I. Hagymási, author
  P. Vancsó, author
  Z. Osváth, author
  P. Nemes-Incze, author
  C. Hwang, author L. Biró,  and author L. Tapasztó, http://dx.doi.org/10.1038/nature13831 journal journal Nature volume 514, pages
  608 (year 2014)NoStop
[Ganguly et al.(2017)Ganguly, Kabir, and Saha-Dasgupta]PhysRevB.95.174419
  author author S. Ganguly, author M. Kabir,   and author T. Saha-Dasgupta,  https://link.aps.org/doi/10.1103/PhysRevB.95.174419 journal journal Phys. Rev. B volume
  95, pages 174419 (year 2017)NoStop
[Slota et al.(2018)Slota,
  Keerthi, Myers, Tretyakov,
  Baumgarten, Ardavan, Sadeghi,
  Lambert, Narita, Müllen,  and Bogani]s41586-018-0154-7
  author author M. Slota, author A. Keerthi,
  author W. K. Myers, author E. Tretyakov, author
  M. Baumgarten, author
  A. Ardavan, author H. Sadeghi, author C. J.  Lambert, author A. Narita, author K. Müllen,
   and author L. Bogani, https://doi.org/10.1038/s41586-018-0154-7 journal journal Nature volume 557, pages
  691 (year 2018)NoStop
[Swartz et al.(2012)Swartz,
  Odenthal, Hao, Ruoff, and  Kawakami]nn303771f
  author author A. G. Swartz, author P. M. Odenthal, author Y. Hao,
  author R. S. Ruoff,  and  author R. K. Kawakami, https://doi.org/10.1021/nn303771f journal journal ACS Nano volume 6, pages
  10063 (year 2012)NoStop
[Wang et al.(2015)Wang,
  Tang, Sachs, Barlas, and  Shi]PhysRevLett.114.016603
  author author Z. Wang, author C. Tang, author R. Sachs, author
  Y. Barlas,  and author
  J. Shi, https://link.aps.org/doi/10.1103/PhysRevLett.114.016603 journal journal Phys. Rev. Lett. volume 114, pages 016603 (year
  2015)NoStop
[Leutenantsmeyer et al.(2017)Leutenantsmeyer, Kaverzin, Wojtaszek, and  van Wees]2053-1583-4-1-014001
  author author J. C. Leutenantsmeyer, author A. A. Kaverzin, author M. Wojtaszek,
   and author B. J. van Wees,  http://stacks.iop.org/2053-1583/4/i=1/a=014001 journal
  journal 2D Mater. volume 4, pages 014001 (year 2017)NoStop
[Esquinazi et al.(2003)Esquinazi, Spemann, Höhne, Setzer, Han, and Butz]PhysRevLett.91.227201
  author author P. Esquinazi, author D. Spemann,
  author R. Höhne, author A. Setzer, author
  K.-H. Han,  and author
  T. Butz, https://link.aps.org/doi/10.1103/PhysRevLett.91.227201 journal journal Phys. Rev. Lett. volume 91, pages 227201 (year
  2003)NoStop
[Yazyev and Helm(2007)]PhysRevB.75.125408
  author author O. V. Yazyev and author L. Helm,  10.1103/PhysRevB.75.125408 journal journal Phys. Rev. B volume 75, pages 125408 (year 2007)NoStop
[Červenka et al.(2009)Červenka, Katsnelson, and Flipse]nphys1399
  author author J. Červenka, author M. I. Katsnelson,  and author C. F. J.  Flipse, http://dx.doi.org/10.1038/nphys1399
  journal journal Nat. Phys. volume 5, pages 840 (year 2009)NoStop
[Ugeda et al.(2010)Ugeda,
  Brihuega, Guinea, and Gómez-Rodríguez]PhysRevLett.104.096804
  author author M. M. Ugeda, author I. Brihuega,
  author F. Guinea,  and author J. M. Gómez-Rodríguez,  10.1103/PhysRevLett.104.096804 journal
  journal Phys. Rev. Lett. volume 104,  pages 096804 (year 2010)NoStop
[Nair et al.(2012)Nair,
  Sepioni, Tsai, Lehtinen,
  Keinonen, Krasheninnikov, Thomson, Geim, and Grigorieva]10.1038/nphys2183
  author author R. R. Nair, author M. Sepioni,
  author I.-L. Tsai, author O. Lehtinen, author
  J. Keinonen, author
  A. V. Krasheninnikov, author
  T. Thomson, author A. K. Geim,  and author I. V. Grigorieva, http://dx.doi.org/10.1038/nphys2183 journal journal Nat. Phys. volume 8, pages
  199 (year 2012)NoStop
[Nair et al.(2013)Nair,
  Tsai, Sepioni, Lehtinen,
  Keinonen, Krasheninnikov, Castro Neto, Katsnelson, Geim, and  Grigorieva]ncomms3010
  author author R. R. Nair, author I. L. Tsai,
  author M. Sepioni, author O. Lehtinen, author
  J. Keinonen, author
  A. V. Krasheninnikov, author
  A. H. Castro Neto, author
  M. I. Katsnelson, author
  A. K. Geim,  and author
  I. V. Grigorieva, http://dx.doi.org/10.1038/ncomms3010 journal journal Nat. Commun. volume 4, pages
  2010 (year 2013)NoStop
[Zhang et al.(2016)Zhang,
  Li, Huang, Li, Qiao, Wang, Yin, Bai,
  Duan, and He]PhysRevLett.117.166801
  author author Y. Zhang, author S.-Y. Li,
  author H. Huang, author W.-T. Li, author
  J.-B. Qiao, author
  W.-X. Wang, author
  L.-J. Yin, author K.-K. Bai, author W. Duan,  and author L. He, https://link.aps.org/doi/10.1103/PhysRevLett.117.166801 journal journal Phys. Rev. Lett. volume 117, pages 166801 (year
  2016)NoStop
[Krasheninnikov et al.(2009)Krasheninnikov, Lehtinen, Foster,
  Pyykkö, and Nieminen]PhysRevLett.102.126807
  author author A. V. Krasheninnikov, author P. O. Lehtinen, author A. S. Foster, author P. Pyykkö,   and author R. M. Nieminen,  https://link.aps.org/doi/10.1103/PhysRevLett.102.126807 journal journal Phys. Rev. Lett. volume 102, pages 126807 (year
  2009)NoStop
[Blonski et al.(2017)Blonski, Tucek, Sofer, Mazanek, Petr, Pumera, Otyepka, and Zbořil]jacs.6b12934
  author author P. Blonski, author J. Tucek,
  author Z. Sofer, author V. Mazanek, author
  M. Petr, author M. Pumera, author M. Otyepka,  and author R. Zbořil, https://doi.org/10.1021/jacs.6b12934
  journal journal J. Am. Chem. Soc. volume 139, pages 3171 (year
  2017)NoStop
[Akhukov et al.(2012)Akhukov, Fasolino, Gornostyrev, and  Katsnelson]PhysRevB.85.115407
  author author M. A. Akhukov, author A. Fasolino,
  author Y. N. Gornostyrev,  and  author M. I. Katsnelson, https://link.aps.org/doi/10.1103/PhysRevB.85.115407 journal
  journal Phys. Rev. B volume 85,  pages 115407 (year 2012)NoStop
[Wang et al.(2009)Wang,
  Huang, Song, Zhang,
  Ma, Liang, and Chen]nl802810g
  author author Y. Wang, author Y. Huang,
  author Y. Song, author
  X. Zhang, author Y. Ma, author J. Liang,  and author Y. Chen,  https://doi.org/10.1021/nl802810g journal journal Nano Lett. volume 9, pages
  220 (year 2009)NoStop
[Matte et al.(2009)Matte,
  Subrahmanyam, and Rao]jp903397u
  author author H. S. S. R.  Matte, author K. S. Subrahmanyam,  and author C. N. R.  Rao, https://doi.org/10.1021/jp903397u journal journal J. Phys. Chem. C volume 113, pages 9982 (year
  2009)NoStop
[Sepioni et al.(2010)Sepioni, Nair, Rablen, Narayanan, Tuna, Winpenny, Geim, and Grigorieva]PhysRevLett.105.207205
  author author M. Sepioni, author R. R. Nair,
  author S. Rablen, author J. Narayanan, author
  F. Tuna, author R. Winpenny, author A. K.  Geim,  and author I. V.  Grigorieva, https://link.aps.org/doi/10.1103/PhysRevLett.105.207205 journal journal Phys. Rev. Lett. volume 105, pages 207205 (year
  2010)NoStop
[Ney et al.(2011)Ney,
  Papakonstantinou, Kumar, Shang, and Peng]10.1063/1.3628245
  author author A. Ney, author P. Papakonstantinou, author A. Kumar, author N.-G. Shang,   and author N. Peng, https://doi.org/10.1063/1.3628245 journal journal Appl. Phys. Lett. volume 99, pages 102504 (year 2011)NoStop
[Nafday et al.(2016)Nafday,
  Kabir, and Saha-Dasgupta]PhysRevB.93.045433
  author author D. Nafday, author M. Kabir,   and author T. Saha-Dasgupta,  https://link.aps.org/doi/10.1103/PhysRevB.93.045433 journal journal Phys. Rev. B volume
  93, pages 045433 (year 2016)NoStop
[Eelbo et al.(2013)Eelbo,
  Wa śśniowska, Thakur, Gyamfi, Sachs, Wehling, Forti, Starke, Tieg, Lichtenstein, and Wiesendanger]PhysRevLett.110.136804
  author author T. Eelbo, author M. Wa śśniowska, author
  P. Thakur, author M. Gyamfi, author B. Sachs, author T. O.  Wehling, author S. Forti, author U. Starke,
  author C. Tieg, author
  A. I. Lichtenstein,  and author R. Wiesendanger, https://link.aps.org/doi/10.1103/PhysRevLett.110.136804 journal journal Phys. Rev. Lett. volume 110, pages 136804 (year
  2013)NoStop
[Edwards and Katsnelson(2006)]0953-8984-18-31-016
  author author D. M. Edwards and author M. I. Katsnelson, http://stacks.iop.org/0953-8984/18/i=31/a=016
  journal journal J. Phys.: Condens. Matt.  volume 18, pages 7209 (year
  2006)NoStop
[Hong et al.(2011)Hong,
  Cheng, Herding, and Zhu]PhysRevB.83.085410
  author author X. Hong, author S.-H. Cheng,
  author C. Herding,  and  author J. Zhu, https://link.aps.org/doi/10.1103/PhysRevB.83.085410 journal
  journal Phys. Rev. B volume 83,  pages 085410 (year 2011)NoStop
[Kim and Cho(2013)]PhysRevB.87.174435
  author author H.-J. Kim and author J.-H. Cho,  https://link.aps.org/doi/10.1103/PhysRevB.87.174435 journal journal Phys. Rev. B volume
  87, pages 174435 (year 2013)NoStop
[Lieb(1989)]PhysRevLett.62.1201
  author author E. H. Lieb, https://link.aps.org/doi/10.1103/PhysRevLett.62.1201
  journal journal Phys. Rev. Lett. volume 62, pages 1201 (year
  1989)NoStop
[Ito et al.(2015)Ito,
  Christodoulou, Nardi, Koch,
  Kläui, Sachdev, and Müllen]ja512897m
  author author Y. Ito, author C. Christodoulou,
  author M. V. Nardi, author N. Koch, author
  M. Kläui, author
  H. Sachdev,  and author
  K. Müllen, https://doi.org/10.1021/ja512897m journal journal J. Am. Chem. Soc. volume 137, pages 7678 (year 2015)NoStop
[Liu et al.(2016)Liu,
  Shen, Sun, Li, Liu, Ren, Li, Gao,
  Chen, Liu, Sun, Tang, Cheng, and Du]Liu2016
  author author Y. Liu, author Y. Shen, author L. Sun, author
  J. Li, author C. Liu, author W. Ren, author F. Li, author L. Gao, author
  J. Chen, author F. Liu, author Y. Sun, author N. Tang, author H.-M. Cheng,  and author Y. Du, http://dx.doi.org/10.1038/ncomms10921 journal journal Nat. Commun. volume 7, pages
  10921 (year 2016)NoStop
[Li et al.(2015)Li,
  Li, Zhao, Lei, Li, Bai, Ren, and Xu]LI2015460
  author author J. Li, author X. Li, author P. Zhao, author
  D. Y. Lei, author W. Li, author J. Bai, author Z. Ren,  and  author X. Xu, http://www.sciencedirect.com/science/article/pii/S0008622314011749
  journal journal Carbon volume 84, pages 460  (year
  2015)NoStop
[Miao et al.(2016)Miao,
  Wang, Liu, Wei, Xu, and Fei]Miao2016
  author author Q. Miao, author L. Wang, author Z. Liu, author
  B. Wei, author F. Xu,  and author W. Fei, http://dx.doi.org/10.1038/srep21832
  journal journal Sci. Rep. volume 6, pages 21832 (year
  2016)NoStop
[Khurana et al.(2013)Khurana, Kumar, Kotnala, Nautiyal, and Katiyar]C3NR34291C
  author author G. Khurana, author N. Kumar,
  author R. K. Kotnala, author T. Nautiyal,  and author R. S. Katiyar, http://dx.doi.org/10.1039/C3NR34291C journal journal Nanoscale volume 5, pages
  3346 (year 2013)NoStop
[Zhao et al.(2011)Zhao,
  He, Rim, Schiros,
  Kim, Zhou, Gutiérrez,
  Chockalingam, Arguello, Pálová, Nordlund, Hybertsen,
  Reichman, Heinz, Kim,
  Pinczuk, Flynn, and Pasupathy]Zhao999
  author author L. Zhao, author R. He, author K. T. Rim, author
  T. Schiros, author K. S. Kim, author H. Zhou, author C. Gutiérrez, author S. P. Chockalingam, author C. J. Arguello, author L. Pálová, author D. Nordlund, author M. S. Hybertsen, author D. R. Reichman, author T. F. Heinz,
  author P. Kim, author
  A. Pinczuk, author G. W. Flynn,  and author A. N. Pasupathy, http://science.sciencemag.org/content/333/6045/999 journal
  journal Science volume 333, pages 999 (year 2011)NoStop
[Usachov et al.(2011)Usachov, Vilkov, Grüneis, Haberer, Fedorov, Adamchuk, Preobrajenski, Dudin, Barinov,
  Oehzelt, Laubschat, and Vyalikh]nl2031037
  author author D. Usachov, author O. Vilkov,
  author A. Grüneis, author D. Haberer, author
  A. Fedorov, author V. K. Adamchuk, author A. B.  Preobrajenski, author P. Dudin, author A. Barinov, author M. Oehzelt, author C. Laubschat,  and author D. V. Vyalikh, https://doi.org/10.1021/nl2031037
  journal journal Nano Lett. volume 11, pages 5401 (year
  2011)NoStop
[Wang et al.(2012)Wang,
  Maiyalagan, and Wang]cs200652y
  author author H. Wang, author T. Maiyalagan,   and author X. Wang, https://doi.org/10.1021/cs200652y journal journal ACS Catal. volume 2, pages
  781 (year 2012)NoStop
[Lv et al.(2012)Lv,
  Li, Botello-Méndez, Hayashi, Wang, Berkdemir, Hao, Elías, Cruz-Silva, Gutiérrez, Kim, Muramatsu,
  Zhu, Endo, Terrones,
  Charlier, Pan, and Terrones]srep00586
  author author R. Lv, author Q. Li, author A. R. Botello-Méndez, author T. Hayashi, author
  B. Wang, author A. Berkdemir, author Q. Hao, author A. L.  Elías, author R. Cruz-Silva, author H. R.  Gutiérrez, author Y. A.  Kim, author H. Muramatsu, author J. Zhu,
  author M. Endo, author
  H. Terrones, author
  J.-C. Charlier, author
  M. Pan,  and author
  M. Terrones, http://dx.doi.org/10.1038/srep00586 journal journal Sci. Rep. volume 2, pages
  586 (year 2012)NoStop
[Zabet-Khosousi et al.(2014)Zabet-Khosousi, Zhao, Pálová,
  Hybertsen, Reichman, Pasupathy, and Flynn]ja408463g
  author author A. Zabet-Khosousi, author L. Zhao, author L. Pálová, author M. S. Hybertsen, author D. R. Reichman, author A. N. Pasupathy,  and author G. W. Flynn, https://doi.org/10.1021/ja408463g journal
  journal J. Am. Chem. Soc. volume
  136, pages 1391 (year 2014)NoStop
[Tison et al.(2015a)Tison, Lagoute,
  Repain, Chacon, Girard,
  Rousset, Joucken, Sharma,
  Henrard, Amara, Ghedjatti,  and Ducastelle]10.1021/nn506074u
  author author Y. Tison, author J. Lagoute,
  author V. Repain, author C. Chacon, author
  Y. Girard, author S. Rousset, author F. Joucken, author D. Sharma, author L. Henrard, author H. Amara, author A. Ghedjatti,  and author F. Ducastelle, https://doi.org/10.1021/nn506074u
  journal journal ACS Nano volume 9, pages 670 (year
  2015a)NoStop
[Fujimoto and Saito(2011)]PhysRevB.84.245446
  author author Y. Fujimoto and author S. Saito, https://link.aps.org/doi/10.1103/PhysRevB.84.245446
  journal journal Phys. Rev. B volume 84, pages 245446 (year
  2011)NoStop
[Lherbier et al.(2013)Lherbier, Botello-Méndez, and Charlier]nl304351z
  author author A. Lherbier, author A. R. Botello-Méndez,  and author J.-C.  Charlier, https://doi.org/10.1021/nl304351z
  journal journal Nano Lett. volume 13, pages 1446 (year
  2013)NoStop
[Deng et al.(2011)Deng,
  Pan, Yu, Cui, Jiang, Qi, Li, Fu,
  Ma, Xue, Sun, and  Bao]cm102666r
  author author D. Deng, author X. Pan, author L. Yu, author
  Y. Cui, author Y. Jiang, author J. Qi, author W.-X.  Li, author Q. Fu,
  author X. Ma, author
  Q. Xue, author G. Sun,  and author X. Bao, https://doi.org/10.1021/cm102666r
  journal journal Chem. Mater. volume 23, pages 1188 (year
  2011)NoStop
[Mondal et al.(2018)Mondal,
  Babar, Kabir, and Ogale]unpublished
  author author R. Mondal, author R. Babar,
  author M. Kabir,  and author S. Ogale, @noop 
  journal journal Unpublished  (year
  2018)NoStop
[Kresse and Hafner(1993)]PhysRevB.47.558
  author author G. Kresse and author J. Hafner, https://link.aps.org/doi/10.1103/PhysRevB.47.558
  journal journal Phys. Rev. B volume 47, pages 558 (year 1993)NoStop
[Kresse and Furthmüller(1996)]PhysRevB.54.11169
  author author G. Kresse and author J. Furthmüller, https://link.aps.org/doi/10.1103/PhysRevB.54.11169 journal
  journal Phys. Rev. B volume 54,  pages 11169 (year 1996)NoStop
[Blöchl(1994)]PhysRevB.50.17953
  author author P. E. Blöchl, https://link.aps.org/doi/10.1103/PhysRevB.50.17953
  journal journal Phys. Rev. B volume 50, pages 17953 (year
  1994)NoStop
[Perdew et al.(1996)Perdew,
  Burke, and Ernzerhof]PhysRevLett.77.3865
  author author J. P. Perdew, author K. Burke,   and author M. Ernzerhof, https://link.aps.org/doi/10.1103/PhysRevLett.77.3865 journal
  journal Phys. Rev. Lett. volume 77,  pages 3865 (year 1996)NoStop
[Ronchi et al.(2017)Ronchi,
  Datteo, Perilli, Ferrighi,
  Fazio, Selli, and Di Valentin]10.1021/acs.jpcc.7b02306
  author author C. Ronchi, author M. Datteo,
  author D. Perilli, author L. Ferrighi, author
  G. Fazio, author D. Selli,  and author C. Di Valentin, https://doi.org/10.1021/acs.jpcc.7b02306 journal journal J. Phys. Chem. C volume 121, pages 8653 (year 2017)NoStop
[Valencia and Caldas(2017)]PhysRevB.96.125431
  author author A. M. Valencia and author M. J. Caldas, https://link.aps.org/doi/10.1103/PhysRevB.96.125431
  journal journal Phys. Rev. B volume 96, pages 125431 (year
  2017)NoStop
[Heyd et al.(2003)Heyd,
  Scuseria, and Ernzerhof]10.1063/1.1564060
  author author J. Heyd, author G. E. Scuseria,
   and author M. Ernzerhof,  http://dx.doi.org/10.1063/1.1564060 journal journal J. Chem. Phys. volume 118, pages 8207 (year 2003)NoStop
[Sun et al.(2015)Sun,
  Ruzsinszky, and Perdew]PhysRevLett.115.036402
  author author J. Sun, author A. Ruzsinszky,   and author J. P. Perdew,  https://link.aps.org/doi/10.1103/PhysRevLett.115.036402 journal journal Phys. Rev. Lett. volume 115, pages 036402 (year
  2015)NoStop
[Sun et al.(2016)Sun,
  Remsing, Zhang, Sun,
  Ruzsinszky, Peng, Yang,
  Paul, Waghmare, Wu,
  Klein, and Perdew]Sun2016
  author author J. Sun, author R. C. Remsing,
  author Y. Zhang, author Z. Sun, author
  A. Ruzsinszky, author
  H. Peng, author Z. Yang, author A. Paul, author U. Waghmare,
  author X. Wu, author
  M. L. Klein,  and author
  J. P. Perdew, http://dx.doi.org/10.1038/nchem.2535 journal journal Nat. Chem. volume 8, pages
  831 (year 2016)NoStop
[Isaacs and Wolverton(2018)]PhysRevMaterials.2.063801
  author author E. B. Isaacs and author C. Wolverton, https://link.aps.org/doi/10.1103/PhysRevMaterials.2.063801 journal journal Phys. Rev. Materials volume 2, pages 063801 (year
  2018)NoStop
[sup()]supple
  @noop  note Supplemental MaterialNoStop
[Chen et al.(2009)Chen,
  Cullen, Jang, Fuhrer, and  Williams]PhysRevLett.102.236805
  author author J.-H. Chen, author W. G. Cullen,
  author C. Jang, author
  M. S. Fuhrer,  and author
  E. D. Williams, https://link.aps.org/doi/10.1103/PhysRevLett.102.236805 journal journal Phys. Rev. Lett. volume 102, pages 236805 (year
  2009)NoStop
[Lherbier et al.(2012)Lherbier, Dubois, Declerck, Niquet, Roche, and Charlier]PhysRevB.86.075402
  author author A. Lherbier, author S. M.-M. Dubois, author X. Declerck,
  author Y.-M. Niquet, author S. Roche,  and author
  J.-C. Charlier, https://link.aps.org/doi/10.1103/PhysRevB.86.075402 journal
  journal Phys. Rev. B volume 86,  pages 075402 (year 2012)NoStop
[Ulybyshev and Katsnelson(2015)]PhysRevLett.114.246801
  author author M. V. Ulybyshev and author M. I. Katsnelson, https://link.aps.org/doi/10.1103/PhysRevLett.114.246801 journal journal Phys. Rev. Lett. volume 114, pages 246801 (year
  2015)NoStop
[Das et al.(2008)Das,
  Pisana, Chakraborty, Piscanec, Saha, Waghmare, Novoselov, Krishnamurthy, Geim,
  Ferrari, and Sood]nnano.2008.67
  author author A. Das, author S. Pisana,
  author B. Chakraborty, author S. Piscanec, author
  S. K. Saha, author
  U. V. Waghmare, author
  K. S. Novoselov, author
  H. R. Krishnamurthy, author
  A. K. Geim, author
  A. C. Ferrari,  and author
  A. K. Sood, http://dx.doi.org/10.1038/nnano.2008.67 journal journal Nat. Nano. volume 3, pages
  210 (year 2008)NoStop
[Babar and Kabir(2018)]PhysRevB.98.075439
  author author R. Babar and author M. Kabir,  https://link.aps.org/doi/10.1103/PhysRevB.98.075439 journal journal Phys. Rev. B volume
  98, pages 075439 (year 2018)NoStop
[Yndurain(2014)]PhysRevB.90.245420
  author author F. Yndurain, https://link.aps.org/doi/10.1103/PhysRevB.90.245420
  journal journal Phys. Rev. B volume 90, pages 245420 (year
  2014)NoStop
[Henkelman et al.(2000)Henkelman, Uberuaga, and Jónsson]1.1329672
  author author G. Henkelman, author B. P. Uberuaga,  and author H. Jónsson, https://doi.org/10.1063/1.1329672 journal journal J. Chem. Phys. volume
  113, pages 9901 (year 2000)NoStop
[Lehtinen et al.(2003)Lehtinen, Foster, Ayuela, Krasheninnikov, Nordlund, and Nieminen]PhysRevLett.91.017202
  author author P. O. Lehtinen, author A. S. Foster, author A. Ayuela,
  author A. Krasheninnikov,
  author K. Nordlund,  and  author R. M. Nieminen, https://link.aps.org/doi/10.1103/PhysRevLett.91.017202 journal journal Phys. Rev. Lett. volume 91, pages 017202 (year
  2003)NoStop
[Hou and Terakura(2015)]jp512886t
  author author Z. Hou and author K. Terakura,  https://doi.org/10.1021/jp512886t journal journal J. Phys. Chem. C volume 119, pages 4922 (year 2015)NoStop
[Zan et al.(2012)Zan,
  Ramasse, Bangert, and Novoselov]nl300985q
  author author R. Zan, author Q. M. Ramasse,
  author U. Bangert,  and  author K. S. Novoselov, https://doi.org/10.1021/nl300985q journal journal Nano Lett. volume 12, pages
  3936 (year 2012)NoStop
[Robertson et al.(2013)Robertson, Montanari, He, Allen, Wu, Harrison, Kirkland, and Warner]nn401113r
  author author A. W. Robertson, author B. Montanari, author K. He,
  author C. S. Allen, author Y. A. Wu, author
  N. M. Harrison, author
  A. I. Kirkland,  and author
  J. H. Warner, https://doi.org/10.1021/nn401113r journal journal ACS Nano volume 7, pages
  4495 (year 2013)NoStop
[Ohta et al.(2006)Ohta,
  Bostwick, Seyller, Horn,  and Rotenberg]Ohta951
  author author T. Ohta, author A. Bostwick,
  author T. Seyller, author K. Horn,  and author
  E. Rotenberg, http://science.sciencemag.org/content/313/5789/951 journal
  journal Science volume 313, pages 951 (year 2006)NoStop
[Giovannetti et al.(2007)Giovannetti, Khomyakov, Brocks,
  Kelly, and van den
  Brink]PhysRevB.76.073103
  author author G. Giovannetti, author P. A. Khomyakov, author G. Brocks,
  author P. J. Kelly,  and  author J. van den Brink, https://link.aps.org/doi/10.1103/PhysRevB.76.073103 journal
  journal Phys. Rev. B volume 76,  pages 073103 (year 2007)NoStop
[Zhou et al.(2008)Zhou,
  Siegel, Fedorov, Gabaly,
  Schmid, Neto, Lee, and  Lanzara]nmat2154b
  author author S. Y. Zhou, author D. A. Siegel,
  author A. V. Fedorov, author F. E. Gabaly, author
  A. K. Schmid, author
  A. H. C. Neto, author
  D. H. Lee,  and author
  A. Lanzara, https://doi.org/10.1038/nmat2154b journal journal Nat. Mater. volume 7, pages
  259 (year 2008)NoStop
[Tison et al.(2015b)Tison, Lagoute,
  Repain, Chacon, Girard,
  Rousset, Joucken, Sharma,
  Henrard, Amara, Ghedjatti,  and Ducastelle]nn506074u
  author author Y. Tison, author J. Lagoute,
  author V. Repain, author C. Chacon, author
  Y. Girard, author S. Rousset, author F. Joucken, author D. Sharma, author L. Henrard, author H. Amara, author A. Ghedjatti,  and author F. Ducastelle, https://doi.org/10.1021/nn506074u
  journal journal ACS Nano volume 9, pages 670 (year
  2015b)NoStop
[Park et al.(2013)Park,
  Wadehra, Wilkins, and Castro Neto]PhysRevB.87.085441
  author author H. Park, author A. Wadehra,
  author J. W. Wilkins,  and  author A. H. Castro Neto,  https://link.aps.org/doi/10.1103/PhysRevB.87.085441 journal journal Phys. Rev. B volume
  87, pages 085441 (year 2013)NoStop
[Kondo et al.(2012)Kondo,
  Casolo, Suzuki, Shikano,
  Sakurai, Harada, Saito,
  Oshima, Trioni, Tantardini,  and Nakamura]PhysRevB.86.035436
  author author T. Kondo, author S. Casolo,
  author T. Suzuki, author T. Shikano, author
  M. Sakurai, author Y. Harada, author M. Saito, author M. Oshima, author M. I.  Trioni, author G. F.  Tantardini,  and author J. Nakamura, https://link.aps.org/doi/10.1103/PhysRevB.86.035436 journal
  journal Phys. Rev. B volume 86,  pages 035436 (year 2012)NoStop
[Scopel et al.(2016)Scopel,
  Paz, and Freitas]SCOPEL20165
  author author W. Scopel, author W. S. Paz,   and author J. C. Freitas,  http://www.sciencedirect.com/science/article/pii/S0038109816300278
  journal journal Solid State Commun. volume 240, pages 5  (year
  2016)NoStop
[Saremi(2007)]PhysRevB.76.184430
  author author S. Saremi, https://link.aps.org/doi/10.1103/PhysRevB.76.184430
  journal journal Phys. Rev. B volume 76, pages 184430 (year
  2007)NoStop
[Brey et al.(2007)Brey,
  Fertig, and Das Sarma]PhysRevLett.99.116802
  author author L. Brey, author H. A. Fertig,   and author S. Das Sarma, https://link.aps.org/doi/10.1103/PhysRevLett.99.116802 journal journal Phys. Rev. Lett. volume 99, pages 116802 (year
  2007)NoStop
[Lu et al.(2013)Lu,
  Lo, Lin, Zhang, Lu, Liu, Tseng, Lee,
  Liang, and Li]10.1021/nn402102y
  author author Y.-F. Lu, author S.-T. Lo, author J.-C. Lin, author
  W. Zhang, author J.-Y. Lu, author F.-H.  Liu, author C.-M. Tseng, author Y.-H. Lee,
  author C.-T. Liang,  and  author L.-J. Li, https://doi.org/10.1021/nn402102y journal journal ACS Nano volume 7, pages
  6522 (year 2013)NoStop





