Abstract
The origin of the ferromagnetism in metalfree graphitic materials has been a decadeold puzzle. The possibility of longrange magnetic order in graphene has been recently questioned by the experimental findings that point defects in graphene, such as fluorine adatoms and vacancies, lead to defectinduced paramagnetism but no magnetic ordering down to 2 K. It remains controversial whether collective magnetic order in graphene can emerge from point defects at finite temperatures. This work provides a new framework for understanding the ferromagnetism in hydrogenated graphene, highlighting the key contribution of the spinpolarized pseudospin as a “mediator” of longrange magnetic interactions in graphene. Using firstprinciples calculations of hydrogenated graphene, we found that the unique ‘zeroenergy’ position of Hinduced quasilocalized states enables notable spin polarization of the graphene’s sublattice pseudospin. The pseudospinmediated magnetic interactions between the Hinduced magnetic moments stabilize the twodimensional ferromagnetic ordering with Curie temperatures of T_{c} = n_{H} × 34,000 K for the atom percentage n_{H} of H adatoms. These findings show that atomicscale control of hydrogen adsorption on graphene can give rise to a robust magnetic order.
Introduction
The twodimensional (2D) magnetism in graphene has attracted considerable attention because of its exceptional promise in graphenebased spintronics^{1,2,3,4,5,6,7,8,9,10,11,12}. Several experiments showed that the ferromagnetic order in graphitic materials originates from the carbon πelectron systems rather than from magnetic impurities^{13,14,15}. Further studies have shown that point defects in graphitic materials contribute to carbonbased ferromagnetism^{16,17,18,19,20,21}. Recently, proximityinduced ferromagnetism^{22,23,24,25} was also demonstrated for a singlelayer graphene placed on an insulating magnetic substrate. Despite decades of research on carbonbased magnetism, it remains unclear under what conditions longrange magnetic order can emerge from point defects in graphitic materials. Contrary to the previous notion that graphene ferromagnetism arises from defectinduced magnetic moments^{16,17,18,19,20,21}, Nair et al.^{5} recently demonstrated that point defects in graphene, such as fluorine adatoms and vacancies, lead to notable paramagnetism but no magnetic ordering down to liquid helium temperatures. The maximum response of the induced paramagnetism was limited to one moment per approximately 1,000 carbon atoms. The lack of collective magnetic order in graphene was attributed to the absence of longrange magnetic interactions between the wellseparated magnetic moments^{5}, with the implication that previously reported roomtemperature ferromagnetism in graphitic materials^{3,13,14,15,26,27} might originate from undetected magnetic impurities or particles.
Recent scanning tunneling microscope experiments^{11} have provided direct evidence that individual hydrogen atoms adsorbed on graphene induce magnetic moments, creating opportunities for atomicscale control of graphene ferromagnetism^{11,12}. As in the graphene systems with fluorine adatoms or carbon vacancies^{5}, however, the magnetic response in hydrogenated graphene is limited because the phase separation into pure graphene and fully hydrogenated graphene (called graphane) parts is thermodynamically more stable^{28}. The ferromagnetism at reasonably high temperatures in such a magnetically dilute system thus requires longrange magnetic interactions between the Hinduced magnetic moments, as well as controlled hydrogenation of graphene under nonequilibrium conditions.
Using spinpolarized density functional theory (DFT) calculations, we show that hydrogenated graphene not only hosts Hinduced localized spins but also responds to them by forming spinpolarized pseudospin as a “mediator” of longrange magnetic interactions in graphene. In hydrogenated graphene, it is wellknown that the C–H σbond formation effectively induces a “vacancy” in the πelectron system^{2,29,30,31}, creating a quasilocalized state occupied by an electron^{11,32,33} and an associated magnetic moment (Fig. 1a). The Hinduced ‘vacancy’ state lies almost at the Diracpoint energy^{11,30,31}, unlike the cases of fluorine adatoms and carbon vacancies with the corresponding defect state at lower energy^{2,30}. The unique energy position of the Hinduced ‘vacancy’ state in the halffilled πelectron system enables strong pseudospinmediated magnetic interactions. Hereafter, we refer to the Hinduced “vacancy” state as an “Avacancy” or “Bvacancy”, depending on which sublattice of the graphene contains the adsorption site.
Results and Discussion
Direct versus “mediated” magnetic interactions in hydrogenated graphene
Figure 1b depicts two different types of magnetic interaction in hydrogenated graphene: (i) The direct (DR) exchange interaction J_{DR}, which arises from the overlap between the wavefunctions of “vacancies”, leads to ferromagnetic (FM) interaction between “vacancies” in the same sublattices^{11,18}. Because of the slow decay of the wavefunctions, the interaction range is relatively large (Fig. 2a). For two nearby “vacancies” in opposing sublattices (not shown in Fig. 1b), the electronic coupling between the “vacancy” states leads to shortrange antiferromagnetic (AFM) interaction or even quenching of the defectinduced magnetic moments if the distance is too small^{11,18,34}. (ii) In addition to the “conventional” magnetic interactions based on the wavefunction overlap, we show that “unconventional” longrange interactions exist that involve a delocalized “mediator” between the localized spins (zoomedin view in Fig. 1b). The “mediator” (its physical nature will be discussed later) interacts with individual localized spins in a meanfield sense. The “mediated” magnetic interaction can be interpreted as a longrange pairwise interaction, whose strength is inversely proportional to the number of sites Λ in the graphene. The longrange (LR) interaction is FM for a pair of “vacancies” in the same sublattice \((\frac{{J}_{LR}^{FM}}{{\rm{\Lambda }}})\), while it is AFM for a pair in opposing sublattices \((\frac{{J}_{LR}^{AFM}}{{\rm{\Lambda }}})\).
To distinguish the longrange “mediated” magnetic interaction from the direct exchange interaction, we performed DFT calculations of hydrogenated graphene using judiciously chosen Hadatom positions (see Methods and Supplementary Table S1). The H adatoms in a graphene supercell exist either as an isolated H or as an H pair of a single type, which was chosen among the three types of pairs denoted by pair 1, 2, and 3 in the inset of Fig. 2a. The direct exchange interaction beyond 20 Å was neglected. First, we consider only the “Avacancies”. The magnetic interaction energy is then given by
where the symbol (i, j) in the first term denotes the H pairs of a given type, and \({J}_{DR}^{{\rm{pair}}}\) is the corresponding direct exchange interaction. A local magnetic moment \({m}_{i}^{A}\) at site i on the A sublattice can take either 1 (spin up) or –1 (down). The spinpolarized DFT calculations were performed in the S_{z} = S subspace; thus, \(S=\frac{1}{2} \sum _{i}{m}_{i}^{A} \). We considered two different spin configurations, {m^{(1)}} and {m^{(2)}}, for which some of the FMcoupled (i, j) pairs in {m^{(1)}} were changed to the AFMcoupled (i, j) pairs in {m^{(2)}} (Supplementary Table S1). From the number of FMtoAFM spin flips (N_{flip}), the energy difference is given by \({E}^{(1)}{E}^{(2)}=\,2{N}_{{\rm{f}}{\rm{l}}{\rm{i}}{\rm{p}}}\,\,{J}_{DR}^{{\rm{p}}{\rm{a}}{\rm{i}}{\rm{r}}}{J}_{LR}^{FM}\frac{{({\sum }_{i}{m}_{i}^{A(1)})}^{2}{({\sum }_{i}{m}_{i}^{A(2)})}^{2}}{2{\rm{\Lambda }}}\). By defining \({\rm{\Delta }}{E}^{\ast }=\frac{{E}^{(1)}{E}^{(2)}}{2{N}_{{\rm{flip}}}}\) and \({\rm{\Delta }}{M}^{\ast }=\frac{{({\sum }_{i}{m}_{i}^{A(1)})}^{2}{({\sum }_{i}{m}_{i}^{A(2)})}^{2}}{4{N}_{{\rm{flip}}}{\rm{\Lambda }}}\), we obtain a simple linear relation between them:
Indeed, the DFT data in Fig. 2b lie almost on straight lines with \({J}_{LR}^{FM}\) = 2.2 eV, which indicates that in addition to the direct exchange interaction (first term in Eq. 1), a longrange “mediated” interaction exists that is scaled with \(\frac{1}{{\rm{\Lambda }}}\) (second term in Eq. 1).
Longrange magnetic interactions mediated by spinpolarized pseudospin
We next turn to the question of what mediates the longrange magnetic interaction in graphene. We temporarily ignore the spin degree of freedom and focus on the lowenergy graphene states in the presence of a finite density of “Avacancies”. Each “Avacancy” induces a ‘zeroenergy’ quasilocalized state, whose charge density is distributed only on the B sublattice sites^{32}. The electron hopping between sites on the opposing sublattices of graphene then makes the “Avacancy” state selectively hybridized with the \( {\rm{A}}\rangle \) sublattice state (Supplementary Fig. S1). As a result, the lowenergy graphene state near the Fermi energy becomes “polarized” to the remaining \( {\rm{B}}\rangle \) pseudospin. We now consider the real spin polarization. Because their constituent orbitals are on the same sublattice, the direct exchange interaction between the spinup “Avacancy” state and the \( {\rm{B}}\rangle \) pseudospin induces parallel magnetization on the \( {\rm{B}}\rangle \) pseudospin. To demonstrate this effect, we considered two H_{A} impurities in a graphene supercell (Fig. 3a–c). For the FM state in Fig. 3a, the spinup polarized \( {\rm{B}}\rangle \) (yellow) is induced as the spinpolarized pseudospin (SPPS), which fills the graphene (see Fig. 3c and legend for more details). The spin polarization of \( {\rm{B}}\rangle \) does not violate Lieb’s theorem^{35} because the local magnetic moments of the “Avacancies” are reduced by the hybridization with \( {\rm{A}}\rangle \). We note that as a secondary effect, the spinup \( {\rm{B}}\rangle \) induces the antiparallel magnetization (blue) on the opposite sublattice in Fig. 3a due to exchange polarization^{18}. For the AFM state in Fig. 3b, however, the \( {\rm{B}}\rangle \) pseudospin is nonspinpolarized due to the spin updown symmetry of the two “Avacancies”, which results in spin polarization only around the impurities.
Figure 3d illustrates the mechanism of the SPPSmediated FM interaction between the “vacancies” in the same sublattices. Here, we assume many “Avacancies”, although only two of them are shown in the schematic. For the local magnetic moments \(\{{m}_{i}^{A}\}\), \(\sum _{i}{m}_{i}^{A} > 0\) was assumed; thus, the \( {\rm{B}}\rangle \) pseudospin is spinup polarized. Regarding individual localized spins, each spin m^{A} interacts with the spinup \( {\rm{B}}\rangle \) through the direct exchange interaction, which lowers the system’s energy when m^{A} takes the same spin direction as that of the SPPS (i.e., m^{A} = 1), with an energy gain proportional to \({J}_{LR}^{FM}\frac{{\sum }_{i}{m}_{i}^{A}}{{\rm{\Lambda }}}\). Therefore, the SPPS effectively mediates the pairwise FM interaction with the coupling strength of \(\frac{{J}_{LR}^{FM}}{{\rm{\Lambda }}}\).
We now consider the mixture of H_{A} and H_{B} in graphene. The spin ground state has \(\sum _{i}{m}_{i}^{A}= {H}_{A} \) and \(\sum _{i}{m}_{i}^{B}=\, {H}_{B} \) at zero temperature^{18}, where H_{A} and H_{B} are the number of H adatoms at the graphene’s A and B sublattices. Assuming an antiparallel net magnetization on opposite sublattices at finite temperatures, the lowenergy graphene state for the occupied electrons is characterized by the spinup \( {\rm{B}}\rangle \) and spindown \( {\rm{A}}\rangle \) (Supplementary Fig. S2). The spinflipped counterparts constitute the unoccupied pseudospin state above the Fermi energy (Fig. 3e). In addition to the FM interaction between a pair of “vacancies” in the same sublattices, a superexchange interaction exists between the occupied (or unoccupied) “vacancy” state and the unoccupied (or occupied) SPPS of the same spin. The electron hopping between opposite sublattices results in the sublatticedependent hybridization between the localized spins and the SPPS with the effective coupling \(\alpha =\frac{{t}^{\ast }}{\sqrt{{\rm{\Lambda }}}}\), where t^{*} = 3.3 eV in the DFT calculations (Supplementary Fig. S1). The superexchange interaction is a secondorder interaction proportional to 𝛼^{2} and thus scales with \(\frac{1}{{\rm{\Lambda }}}\). For \(\sum _{i}{m}_{i}^{A} > 0\) and \(\sum _{i}{m}_{i}^{B} < 0\), the SPPSmediated interaction energetically favors m^{B} = −1 and m^{A} = 1, hence effectively leading to the AFM interaction of \(\frac{{J}_{LR}^{AFM}}{{\rm{\Lambda }}}\) for a pair of “vacancies” in opposing sublattices.
Meanfield ferromagnetism in graphene
By combining the two types of SPPSmediated interactions, we obtain the interaction energy for a given spin state \(\{{m}_{i}^{A},\,{m}_{j}^{B}\}\),
Our DFT calculations show that the AFM interaction is stronger than the FM interaction, with a ratio of \(\frac{{J}_{LR}^{AFM}}{{J}_{LR}^{FM}}\) = 4.3 (see Methods for details). The longrange nature of the SPPSmediated interactions allows us to use a meanfield approximation with \({E}_{LR}^{MFA}\) = \((\tfrac{{J}_{LR}^{FM}{H}_{A}}{2{\rm{\Lambda }}}\langle {m}^{A}\rangle \tfrac{{J}_{LR}^{AFM}{H}_{B}}{2{\rm{\Lambda }}}\langle {m}^{B}\rangle )\sum _{i}{m}_{i}^{A}\) − \((\tfrac{{J}_{LR}^{FM}{H}_{B}}{2{\rm{\Lambda }}}\langle {m}^{B}\rangle \tfrac{{J}_{LR}^{AFM}{H}_{A}}{2{\rm{\Lambda }}}\langle {m}^{A}\rangle )\sum _{i}{m}_{i}^{B}\), in which the ensembleaveraged spins, 〈m^{A}(T)〉 and 〈m^{B}(T)〉, at temperature T are calculated selfconsistently (Methods). We define two characteristic temperatures, \({T}_{F}=\frac{{n}_{H}{J}_{LR}^{FM}}{4{k}_{B}}\) and \({T}_{AF}=\frac{{n}_{H}{J}_{LR}^{AFM}}{4{k}_{B}}\), where k_{B} is the Boltzmann constant, and n_{H} is the atom percentage of H adatoms. Figure 4a shows the magnetization as a function of the reduced temperature T/T_{F} for the different probabilities, P_{A} and P_{B}, of having H_{A} and H_{B} on the graphene layer. The magnetization per H adatom is \(m=\frac{{H}_{A}{H}_{B}}{{H}_{A}+{H}_{B}}={P}_{A}{P}_{B}\) at T = 0, which is consistent with Lieb’s theorem^{35}. A slight imbalance with P_{A} = 0.51 and P_{B} = 0.49, for example, induces m = 0.02 μ_{B}/atom, which corresponds to the magnetization per weight of 0.1 Am^{2}/kg at n_{H} = 1 at. % (Supplementary Fig. S3). The preferential H adsorption for one sublattice might be possible using a suitable substrate for graphene such as hexagonal boron nitride^{36} or exploiting the effect of the stacking order of multilayer graphene^{18}. Near the Curie temperature T_{c}, the magnetization M ~ (T_{c} – T)^{β} has the critical exponent β = 0.5, regardless of P_{A} and P_{B} (Fig. 4b), as expected from meanfield theory^{37}.
Unlike the magnetization in Fig. 4a, the Curie temperature weakly depends on P_{A} and P_{B}, which enables the enhanced magnetization while maintaining the high T_{c}. The T_{c} is \({T}_{c}={T}_{F}+\sqrt{{({P}_{A}{P}_{B})}^{2}{T}_{F}^{2}+4{P}_{A}{P}_{B}{T}_{AF}^{2}}\) and has a maximum at P_{A} = P_{B} with \({{T}_{c}}^{max}={T}_{F}+{T}_{AF}\) = n_{H} × 34,000 K (Fig. 4c). To achieve roomtemperature ferromagnetism (Supplementary Fig. S3), it is necessary to introduce a relatively high concentration (~1 at. %) of H adatoms, which are magnetically active and not quenched by forming Hadatom “dimers”^{11,18,34}. Partial hydrogenation of graphene with high n_{H} is challenging because of the phase separation of hydrogenated graphene^{28}. Therefore, controlled hydrogenation under nonequilibrium conditions is required to realize roomtemperature FM graphene.
We note that the AFM interaction contributes to \({{T}_{c}}^{max}={T}_{F}+{T}_{AF}\) as much as the FM interaction. This may appear to contradict the notion that the AFM interaction typically reduces the T_{c}; for carriermediated FM semiconductors (e.g., Mndoped GaAs), the T_{c} is determined by competition between the FM and AFM interactions^{38}, i.e., \({T}_{c}={T}_{F}{T}_{AF}\). For hydrogenated graphene, however, the SPPSmediated FM and AFM interactions do not compete (Eq. 3); the spinup \( {\rm{B}}\rangle \) pseudospin in Fig. 3e, for example, stabilizes the spinup polarized H_{A} through the direct exchange interaction, and the same SPPS simultaneously stabilizes the spindown H_{B} through the superexchange interaction. The FM and AFM interactions thus “cooperate”, rather than compete, in hydrogenated graphene.
Summary and Conclusions
In summary, we have shown that the SPPSmediated, longrange magnetic interactions give rise to a robust magnetic order in hydrogenated graphene, which can be stable even at room temperature for the H concentration of n_{H} ~ 1 at. %. The “cooperativeness” of the FM and AFM interactions contributes to the high T_{c}. The realization of the robust 2D ferromagnetism in such a magnetically and electrically dilute system is unusual; the 2D magnetism in hydrogenated graphene is described by meanfield theory owing to the intrinsic longrange nature of the SPPSmediated interactions. Our new finding of the mechanism underlying graphene ferromagnetism should have enormous implications for understanding and atomicscale control of graphenebased magnetism, which is an important step toward bringing the vision described in “Painting magnetism on a canvas of graphene (ref.^{12})” closer to reality.
Methods
Spinpolarized densityfunctional theory (DFT) calculations
We calculated total energies and spin densities of partially hydrogenated graphene using the generalized gradient approximation (GGAPBE^{39}) to DFT, as implemented in the Vienna abinitio Simulation Package^{40}. The DFT calculations employed the projector augmented wave method^{41,42} with an energy cutoff of 500 eV for the planewave part of the wave function. For the Brillouin zone integration, we used a Γcentred grid containing enough k points, as dense as the 60 × 60 grid for the twoatom unitcell of graphene.
Calculations of the SPPSmediated magnetic interactions
We first performed spinpolarized DFT calculations for H adatoms on the same sublattice of graphene to distinguish the longrange “mediated” FM interactions from the direct exchange interactions. To this end, the H positions in a graphene supercell were chosen so that only the direct pair interactions for the H pairs of a single type are involved in Eq. 1. The H pairs were selected among the three types of pairs denoted by pair 1, 2, and 3 in Fig. 2a. Different H concentrations were considered by changing either the number of H adatoms on a graphene supercell (N_{H} = 2, 3 or 4) or the number of the C atoms in a supercell with Λ = 96, 150, 216, 300, 384, 432, and 864 (see Supplementary Table S1 and legend for more details). Each data point in Fig. 2b was obtained from the energy difference for two different spin configurations that are listed in Supplementary Table S1. For the cases of N_{flip} = 0, we arbitrarily selected the “N_{flip} values” in \({\rm{\Delta }}{E}^{\ast }=\frac{{E}^{(1)}{E}^{(2)}}{2{N}_{{\rm{flip}}}}\) and \({\rm{\Delta }}{M}^{\ast }=\frac{{({\sum }_{i}{m}_{i}^{A(1)})}^{2}{({\sum }_{i}{m}_{i}^{A(2)})}^{2}}{4{N}_{{\rm{flip}}}{\rm{\Lambda }}}\) to plot the data in the same figure; note that these data also lie on a line with the yintercept of zero as expected.
To extract the interaction strength of the longrange AFM interaction \({J}_{LR}^{{\rm{AFM}}}\), we considered two H adatoms on the opposing sublattices of the 864atom graphene supercell. The two H adatoms in the supercell are well separated to ensure that the direct exchange interaction is negligible. Two spin configurations, {m^{(1)}} = (up, down) and {m^{(2)}} = (up, up), were considered. Then, from Eq. 3, the energy difference is \({E}_{LR}^{(1)}{E}_{LR}^{(2)}=\,\frac{2{J}_{LR}^{AFM}}{{\rm{\Lambda }}}\), which was calculated from the spinpolarized DFT calculations to determine \({J}_{LR}^{{\rm{AFM}}}\).
Finitetemperature magnetism of hydrogenated grapheme
We used a meanfield approximation with \({E}_{LR}^{MFA}\) = \((\tfrac{{J}_{LR}^{FM}{H}_{A}}{2{\rm{\Lambda }}}\langle {m}^{A}\rangle \tfrac{{J}_{LR}^{AFM}{H}_{B}}{2{\rm{\Lambda }}}\langle {m}^{B}\rangle )\sum _{i}{m}_{i}^{A}\) − \((\tfrac{{J}_{LR}^{FM}{H}_{B}}{2{\rm{\Lambda }}}\langle {m}^{B}\rangle \tfrac{{J}_{LR}^{AFM}{H}_{A}}{2{\rm{\Lambda }}}\langle {m}^{A}\rangle )\sum _{i}{m}_{i}^{B}\) to obtain the ensemble averaged spins at temperature T. We introduced two characteristic temperatures, \({T}_{F}=\frac{{n}_{H}{J}_{LR}^{FM}}{4{k}_{B}}\) and \({T}_{AF}=\frac{{n}_{H}{J}_{LR}^{AFM}}{4{k}_{B}}\), where k_{B} is the Boltzmann constant, and n_{H} is the H concentration, \({n}_{H}=\frac{{H}_{A}+{H}_{B}}{\Lambda }\). For \({P}_{A}=\frac{{H}_{A}}{{H}_{A}+{H}_{B}}\) and \({P}_{B}=\frac{{H}_{B}}{{H}_{A}+{H}_{B}}\), the coupled equations for \(\langle {m}^{{\rm{A}}}\rangle \) and \(\langle {m}^{{\rm{B}}}\rangle \) are given by \(\langle {m}^{A}\rangle =\,\tanh [\frac{2}{T}({T}_{F}{P}_{A}{T}_{AF}{P}_{B}\eta (T)){\langle m\rangle }^{A}]\) and \(\langle {m}^{B}\rangle =\,\tanh [\frac{2}{T}({T}_{F}{P}_{B}{T}_{AF}{P}_{A}\frac{1}{\eta (T)})\langle {m}^{B}\rangle ]\), where \(\eta (T)\) is the ratio of the magnetization on the opposing sublattices. \(\eta (T)\) was determined numerically as a function of temperatures. We found that \(\eta \) is always negative; for P_{A} = P_{B}, \(\eta \) is constant with \(\eta =\,1\), while for the case of P_{A} ≠ P_{B}, it gradually increases in magnitude with increasing T. From the condition of no net magnetization on each sublattice at T_{c}, we obtained \(\eta ({T}_{c})=\frac{1}{2}\frac{{T}_{F}}{{T}_{AF}}(1\frac{{P}_{A}}{{P}_{B}})\) \(\sqrt{\frac{{P}_{A}}{{P}_{B}}+\frac{1}{4}{(\frac{{T}_{F}}{{T}_{AF}})}^{2}{(1\frac{{P}_{A}}{{P}_{B}})}^{2}}\) and \({T}_{c}={T}_{F}+\sqrt{{({P}_{A}{P}_{B})}^{2}{T}_{F}^{2}+4{P}_{A}{P}_{B}{T}_{AF}^{2}}\).
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Acknowledgements
This work was supported by the DGIST R&D Program of the Ministry of Science and ICT of Korea (Grant No. 18BT02).
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J.K. conceived the project, H.K. did the calculations, all authors analyzed the data and wrote the manuscript.
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Correspondence to Joongoo Kang.
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Kim, H., Bang, J. & Kang, J. Robust ferromagnetism in hydrogenated graphene mediated by spinpolarized pseudospin. Sci Rep 8, 13940 (2018). https://doi.org/10.1038/s41598018319340
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