Ï Magnetism of Carbon Monovacancy in Graphene by Hybrid Density

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# Magnetism of Carbon Monovacancy in Graphene by Hybrid Density Functional Calculations Costanza Ronchi, Martina Datteo, Daniele Perilli, Lara Ferrighi, Gianluca Fazio, Daniele Selli, and Cristiana Di Valentin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b02306 • Publication Date (Web): 31 Mar 2017 Downloaded from http://pubs.acs.org on April 4, 2017

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The Journal of Physical Chemistry

π Magnetism of Carbon Monovacancy in Graphene by Hybrid Density Functional Calculations

Costanza Ronchi, Martina Datteo, Daniele Perilli, Lara Ferrighi, Gianluca Fazio, Daniele Selli, Cristiana Di Valentin*

Dipartimento di Scienza dei Materiali, Università di Milano Bicocca, via R. Cozzi 55 20125 Milano Italy

Abstract Understanding magnetism in defective graphene is paramount to improve and broaden its technological applications. A single vacancy in graphene is expected to lead to a magnetic moment with both a σ (1 µB) and a π (1 µB) component. Theoretical calculations based on standard LDA or GGA functional on periodic systems report a partial quenching of the π magnetization (0.5 µB) due to the crossing of two spin split bands at the Fermi level. In contrast, STS experiments (Zhang, Y; Li, S.Y.; Huang, H.; Li, W. T.; Qiao, J. B.; Wang, W. X.; Yin, L. J.; Bai, K. K.; Duan, W.; He, L. Phys. Rev. Lett. 2016, 117, 166801) have recently proved the existence of two defect spin states that are separated in energy by 20-60 meV. In this work we show that self-interaction corrected hybrid functional methods (B3LYP-D*) are capable of correctly reproducing this finite energy gap and, consequently, provide a π magnetization of 1 µB. The crucial role played by the exact exchange is highlighted by comparison with PBE-D2 results and by the magnetic moment dependence with the exact exchange portion in the functional used. The ground state ferromagnetic planar solution is compared to the antiferromagnetic and to the diamagnetic ones, which present an out-of-plane distortion. Periodic models are then compared to graphene nanoflakes of increasing size (up to C383H48). For large models, the triplet spin configuration (total magnetization 2µB) is the most stable, independently of the functional used, which further corroborates the conclusions of this work and puts an end to the long-debated issue of the magnetic properties of an isolated C monovacancy in graphene.

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Corresponding author: [email protected] ACS Paragon Plus Environment

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Introduction Researchers are currently very keen in trying to add magnetism to the several extraordinary capabilities of graphene using various strategies.1,2,3,4,5 The origin of the magnetism induced by the presence of lattice defects has been largely debated by both the experimental6,7,8,9,10,11,12,13,14 and computational15,16,17,18,19,20,21,22,23 materials science communities during the last few years. The carbon monovacancy24 has been proposed as the candidate defect carrying a magnetic moment. Very recently, an accurate scanning tunneling spectroscopy (STS) experiment25 provided clear evidence of π magnetism in the vicinity of the vacancy site in graphene, as suggested before.13 However, the theoretical description of what could appear as a simple defect is far from being univocally determined.15-23 Several theoretical studies exist in the literature,15-23 which present different views and, sometimes, opposite conclusions. It has been recognized that the evaluation of the magnetic moment of a graphene model containing a monovacancy is very delicate.19,23 The convergence with the supercell size, the smearing width and the k-point sampling density, just to mention some of the factors, was found to be hard to achieve.19,23 The topic of structural defects in graphene has been revised in a number of review articles,24,26,27 where, however, most of the attention is focused on the ability of this fabulous material to reconstruct around the defect leading to interesting mechanical properties. From the electronic structure and spin properties point of view, the theoretical investigations of the C monovacancy followed one another for the last decade,15-23 first indicating a non magnetic ground state,15 and then pointing to a magnetic one with computed magnetic moments ranging from 1 to 2 µB, depending on the computational method and setup of calculations. As Wang et al.19 point out before, this widely spread range of values is confusing and does not favour the rationalization of experimental findings that are not fully coherent, either. The magnetic moment arises as a consequence of the presence of two unpaired electrons per vacancy in the form of a localized dangling bond in the σ network and of some delocalized density in the π states. This is because one of the undercoordinated C atoms (c in Figure 1) remains with a dangling bond (sp2 hybridization with an unpaired electron), whereas the other two undercoordinated C atoms (a and b in Figure 1) get closer to form an elongated C-C bond (with length between 1.9-2.1 Å, depending on the model and method, as reported in Table S1). The spin density in the π delocalized band is caused by the removal of one π electron when the C atom is missing. Two opposite views are proposed in the theoretical literature according to which, in the limit of the isolated vacancy, the magnetization should tend to i) 1 µB or to ii) 2 µB. The first view18 considers that, with increasing graphene model size and lowering of the vacancy concentration, the ACS Paragon Plus Environment

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magnetization deriving from the dangling bond in the σ network is maintained as a very local atomic state, while the π magnetization is quenched as a consequence of a half electron transfer from the vπα band to the vπβ band, which become perfectly degenerate at the infinite limit. Oppositely, the second view19,21,23 proposes an increasing occupation of the vπα band at the expenses of the vπβ with increasing graphene size, leading to a complete filling at the limit of an isolated C monovacancy. The considerations above were basically derived from periodic calculations, which are found to be very sensitive to the repetition unit, to the smearing width and to the k-point sampling, leading to such contrasting conclusions. Nanoflakes have also been considered19,20 as models for an isolated vacancy. Again, the size is a crucial factor. Different studies seem to agree towards a 2 µB ground state, at least at the infinite limit.19,20 Note, however, that up to a nanoflake diameter size of 4 nm, a non magnetic 0 µB solution was found to be more stable than the 2 µB one.19 We wish to stress that none of the works mentioned above paid attention to the type of density functional employed. In these studies, either the local density approximation (LDA) or the generalized gradient approximation (GGA) functionals were applied. However, it is well known that such functionals suffer for the self-interaction problem inherent in density functional theory (DFT), which can seriously affect the degree of electron localization/delocalization and consequently the magnetization, too. In the present work, we undoubtedly and decisively show that the use of hybrid functionals, presenting a portion of exact exchange that reduces the self-interaction error, leads always to a total magnetization of 2 µB, independently of the model (i.e. periodic or finite) used to simulate the C monovacancy in graphene, independently of the model size (i.e. vacancy concentration, distance from edges), and even of the k-point mesh density. Thus, the current hybrid functional study overcomes all the previous works and puts a definitive end to the long-debated issue on the correct magnetic ground state and on its associated magnetic moment, for a single isolated C vacancy defect in graphene. Computational details The simulation package CRYSTAL14 (CRY14)28 was employed both for periodic and molecular calculations. We exploited the hybrid B3LYP29,30 functional including long-range van der Waals interactions with the Grimme’s correction in its D*31,32 formulation and the standard GGA/PBE33 ACS Paragon Plus Environment

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with Grimme’s (D2) correction.31 The Kohn−Sham orbitals were described with localized atomic Gaussian basis sets: C 6-31(d1) and also H 5-11(p1) in the case of molecular models. For periodic CRY14 calculations, three supercell models were used: 6×6 (71 C atoms), 8×8 (127 C) and 16×16 (511 C), with Monkhorst-Pack k-point grids34 of 12×12×1, 10×10×1, or 4×4×1 with B3LYP-D*, and of 12×12×1, 10×10×1 or 8×8×1 with PBE-D2, respectively, except for calculations in Figure 2. A denser 30×30×1 k-point grid was used to compute the total (DOS) and projected (PDOS) densities of states with the 6×6 supercell. The band structures were obtained for 4×4, 8×8 and 16×16 supercells with 30×30×1, 15×15×1 and 10×10×1 k-point grids. No smearing width is used in the CRY14 calculations, except for some calculations in the Supporting Information where the smearing was varied from 0.001 to 0.410 eV.The hexagonal lattice parameter of pure graphene is computed to be 2.463 Å for B3LYP-D* and 2.472 Å for PBE-D2 and was kept fixed in all the calculations. For the molecular calculations with CRY14, three nanoflake models were used: C95H24, C215H36 and C383H48, with diameter size of 1.9, 2.9 and 3.9 nm, respectively. The QuantumESPRESSO (QE)35 code was employed to perform periodic calculations. The PBED2 functional and Vanderbilt ultrasoft pseudopotentials36 were used with energy cutoffs of 30 and 240 Ry (for plane wave basis set and charge density grids). We used 6×6, 8×8, 12×12 (287 C) and 16×16 supercells with Monkhorst-Pack k-point grids of 14×14×1, 14×14×1, 12×12×1 and 6×6×1 respectively, except for calculations in Figure 2. A cold smearing37 width of 0.007 eV was used for all the supercells considered, except for some calculations in the Supporting Information where the smearing was varied from 0.001 to 0.680 eV. For the 6×6, 8×8 and 12×12 supercells, denser k-point meshes of 24×24×1, 24×24×1 and 14×14×1, respectively, were used to calculate DOS and PDOS. For the 16×16 supercell the k-point mesh has not been increased. Periodic replica were separated by 20 Å of vacuum in the direction perpendicular to the surface to avoid interaction between images. The pure graphene lattice parameter used is 2.467 Å. The Gaussian09 (G09)38 code was employed to perform molecular calculations. The B3LYP functional was used with the inclusion of the Grimme’s D339 correction to dispersion forces. The basis 6-31(d1) was used for all atoms. For the molecular calculations with G09, two nanoflake models were used: C95H24 and C215H36. All the calculations reported in this work were performed including spin polarization when it was necessary. Results and Discussion

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The carbon monovacancy is created by removing one C atom in the graphene lattice, which costs about 8 eV, as detailed in Table S2 for different supercells and functionals considered in this work. Depending on the local point group symmetry imposed to the defective lattice, different electronic solutions can be obtained, as described in Figure 1: for D3h symmetry a ferromagnetic quintet (Q, the highest possible spin configuration with four unpaired electrons); for the C2v symmetry a ferromagnetic triplet (T, with two unpaired electrons); and for the Cs symmetry an antiferromagnetic open shell singlet (Sos, with two unpaired electron with opposite spin) or a diamagnetic closed shell singlet (Scs, with no unpaired electrons) solution. The Q solutions are always much higher in energy (> 1.5 eV) and therefore they have not been considered further in the analysis. Results for all the other three electronic configurations in the case of supercell models of increasing size are reported in Table 1, using both the dispersion-corrected B3LYP29,30 functional (B3LYP-D*)31,32 and the dispersion corrected PBE33 functional (PBE-D2).31 The T solution is always the most stable, although the Sos is only slightly above in energy, by less than 0.08 eV. We may note that only the T solutions are characterized by a planar geometry, since the dz parameter (reported and defined in Table 1) is an indicator of the out-of-plane distortion caused by a lift of one undercoordinated C atoms (c in Figure 1 and in Figure S1 in the Supporting Information). T and Sos are very close in energy because going from T to Sos, and keeping the geometry fixed to that of T, there is only an energy cost to flip the spin of one of the two unpaired electrons. This cost corresponds the loss of the exchange energy (estimated to be +0.10 eV for the 6×6 model). Note, however, that Sos has been fully optimized and, thus, the final energy difference reported in Table 1 is slightly smaller (+0.08 eV for the same 6×6 model). The relaxed Sos structures are not planar (dz = 0.18-0.46 Å, depending on the supercell with B3LYP-D* in Table 1; values for PBE-D2 are also reported). Even less planar are the Scs optimized structures (dz = 0.55-0.70 Å, depending on the supercell with B3LYP-D* in Table 1; values for PBE-D2 are also reported). In this last case (Scs) a robust lift of the undercoordinated atom is necessary in order to obtain a good overlap between the sp2 orbital and the vπ state, resulting in the pairing up of the two electrons and, thus, in the complete quenching of the magnetization. Further structural details for the three optimized spin configurations are presented in Tables S1, S3 and S4. In the case of the B3LYP-D* functional, the computed magnetic moment for the electronic ground state is fully coherent with a T spin configuration, since it corresponds to 2 µB after spin relaxation. This does not hold for the PBE-D2 calculations, where the total relaxed magnetic moments span the range from 1.59 to 1.49 µB, going from the smallest to the largest supercell model considered. Analogously, for the Sos spin configuration, with B3LYP-D* the magnetic moments is 0 µB, as expected for an antiferromagnetic solution, whereas with PBE-D2 it is 0.32 µB, which is a sort of ACS Paragon Plus Environment

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uncompensated antiferromagnetism (value obtained with 6×6 cell model since with the 8×8 and 16×16 cells the solution collapsed on the T spin configurations during optimization). The PBE-D2 magnetic moment value is found to depend on the k-point sampling of the Brillouin zone, although it is at convergence already with a 8×8 k point mesh (see upper panel of Figure 2) with all cell sizes, and on the smearing, although it is at convergence already below ∼0.003 eV (see Figure S2 in the Supporting Information), in the case of the CRYSTAL1428 (CRY14) calculations, where a localized atomic basis set is used. Since previous works in the literature reported much more delicate dependency in the case of plane-wave calculations, we have performed PBE-D2 calculations with the QuantumESPRESSO (QE) package,35 too. We confirm that (i) low smearing values (see Figure S2) and (ii) large k-point grids (see lower panel of Figure 2) must be used to compute stable and converged magnetic moments. Using larger supercells magnetic moments converge to slightly lower values with CRY14 code and, after some larger oscillations, also with QE code. The partial quenching of the magnetic moment of defective graphene with DFT/GGA calculations has been often previously22,23 reported and rationalized in terms of a partial charge transfer from the vπα to the vπβ band, since both bands cross the Fermi level, as it is clearly observable, for the present study, in the band structures in the lower panels of Figure 3 and in the density of states (DOS) in the lower left panel of Figure 4 (for PBE-D2). The vπα and vπβ bands become closer for increasing cell size, but for all the three sizes considered, they both cross the Fermi level, leading to a partial magnetization passivation. The DOS for the ferromagnetic or T solution with the other cell models considered in this study are reported, together with the corresponding spin plots, in Figure S3 in the Supporting Information. What is striking is that a very different situation emerges from the B3LYP-D* calculations, shown in the upper panels of Figure 3. Here, the vπα band is always below the Fermi level, whereas the vπβ band is always above the Fermi level, getting closer in energy for increasing supercell size, but still clearly separated in energy (30 meV for the 16×16 cell). This means that in the case of B3LYPD* calculations, no charge transfer takes place, which is the reason for no partial π magnetic moment quenching. This result is in excellent agreement with the recent STS experiment proving that the π magnetization is caused by a splitting of about 20-60 meV between two spin-polarized vacancy π states (two distinct peaks in the spectra).25 Thus, B3LYP-D*, at odds with PBE-D2, is capable of reproducing this very delicate but extremely relevant experimental quantity. This is because of a more accurate treatment of the exchange energy and of the partial correction for the self-interaction error for hybrid functional methods.40,41 The unpaired electron in the vπα benefits ACS Paragon Plus Environment

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from the exchange with the other unpaired electron in the sp2 orbital in the σ plane. No π passivation, as observed in B3LYP-D* calculations, results in more exchange energy, which is the reason why a total 2 µB magnetic moment is always obtained with hybrid functional calculations. A higher degree of magnetization in the B3LYP-D* calculations with respect to PBE-D2 counterpart is also evident when comparing spin density plots in Figure S4 in the Supporting Information. Here, we can note that the magnetization involves more atoms in the B3LYP-D* calculations than in the PBE-D2. Additionally we observe that the degree of spin delocalization is not dependent on the model size but it involves, in the case of B3LYP-D*, up to the third next neighbor to the C vacancy. We investigated further whether there is a trend in the degree of magnetization with the increasing portion of exact exchange (EXX) in the hybrid functional. Indeed, this trend exists and it is evident when observing the computed magnetic moments in Figure 5. We considered five values of % of exact exchange, going from 0 to 20% (B3LYP-D*). The magnetic moment for 0% of EXX is similar to that computed with PBE-D2 (1.47 vs 1.59 µB), but increases rapidly and converges to 2 µB already for 15% of exact exchange contribution to the exchange functional. These results prove the crucial role played by the exact exchange in determining the correct magnetic moment of the defective graphene model. Even a small contribution of exact exchange is capable of largely increasing the resulting free magnetization. We tested other hybrid functionals to prove that this is not a peculiar behavior of B3LYP. We considered both HSE06-D242, B3PW9129,43 and PBE0.44 B3PW91 was chosen because it was found to be particularly suitable for metal and magnetic system.45 All these hybrid functional methods provide a magnetization of 2 µB, in agreement with B3LYP-D* calculations. The band structures for the 8×8 cell model are reported in Figure 6 and can be compared to the corresponding B3LYPD* one in Figure 3. Although we have considered supercell models of rather large size (up to 16×16), we are still far from the isolated vacancy, as it is clear from the degree of dispersion of the vπα band in Figure 3. For this reason, we decided to investigate nanoflakes models of increasing size (C95H24, C215H36, C383H48), as shown in Figure 7. The nanoflakes are saturated by H atoms and respect the hexagonal symmetry of graphene. One C atom is removed from the center of the nanoflake and the model is then fully relaxed in the three spin configurations (T, Sos and Scs). Note that, for finite models, in contrast with what observed for periodic systems, there is always full correspondence between the spin multiplicity and the resulting total magnetic moment: T 2 µB; Sos and Scs 0 µB. This is


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because finite systems are molecular models and must have a fixed magnetization. For these calculations, both the CRY14 and Gaussian09 (G09)38 codes were used. The T spin configuration leads to fully planar structures with a similar spin density distribution, as observed above for the periodic models: one unpaired electron is hosted in the sp2 orbital of one of the three undercoordinated C atoms (c in Figure 1), whereas the other unpaired electron is in a π state (vπ) in the vicinity of the defect, although some tails of opposite spin densities are registered at the edges of the nanoflake (see Figure 7). The latter is a well-known effect at the edges of nanoribbons46 and is due to the finite model size. Although it is very interesting, it is beyond the scope of the present work and it would not have affect the electronic states of a truly isolated C monovacancy. We note that the degree of spin localization around the defect does not depend much on the size of the nanoflake, involving essentially up to the third next neighbor to the vacancy site. Structural details, such as bond distances, are reported in Table S1. The T solution, presenting a C2v symmetry, is characterized by an enhanced shortening of the bond between the two undercoordinated C atom (a and b in Figure 1), with respect to periodic calculations. HOMOαLUMOβ gaps for the T solution of three nanoflakes are reported in Table S5: the larger the flake, the smaller the gap. The Sos and Scs configurations are not planar, in line with what previously observed in periodic calculations. However, for nanoflakes, the distortion is much more pronounced and does not only involve the out-of-plane shift of the undercoordinated C atom c (see Figure 1 and the right side of Figure S1 in the Supporting Information) but it also involves a tilting of the flake. We have determined a structural parameter to measure the degree of tilting, which is the dihedral angle τ, as depicted on the right side of Figure S1). The values of τ are reported in Table 2, whereas other structural details can be found in Tables S3 and S4. The flake tilting is progressively less pronounced going from the smallest to the largest flake models, as one can visually catch observing the side views in Figure 7. The tilting is larger for Scs than for Sos because of the requirement, in the former case, of an overlap between the sp2 and the vπ states, allowing the pairing up of the two electrons in a bonding orbital, analogously to what discussed above for periodic systems. In the following, we analyze the relative stability of the three different spin configurations of defective nanoflakes. We start from the B3LYP-D* results in Table 2. For the small C95H24 model, the B3LYP-D* ground state is the Sos, by a tiny amount of energy, -0.03 eV (-0.02 eV with G09). This is because, although the Sos is higher in energy when computed on the T optimized structure (the electron flip costs +0.07 eV), after geometry relaxation, it gains a stabilization of -0.10 eV (ACS Paragon Plus Environment

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0.09 eV with G09). The Scs is only +0.07 eV above the T (+0.10 eV with G09). The cost to pair up the two electron in a bonding state seem to be much smaller in the case of the nanoflake than for any of the supercell models discussed above (> 0.3 eV in Table 1). However, for larger flake sizes, the situation changes. The smallest nanoflake presents a peculiarly high distortion. Already for the medium size flake, the ground state is a T spin configuration, with the Sos only +0.05 eV and the Scs +0.26 eV above the T. This latter energy difference, ∆E(Scs-T), is more consistent with what computed for periodic models. The trend is maintained (see Figure S5 in the Supporting Information) and the value of ∆E(Scs-T) even further increases for the largest flake model, up to 0.99 eV (see Table 2), which is rather high, and much higher than what observed for periodic systems (about 0.3-0.35 eV in Table 1). Thus, the larger the nanoflake, the larger the relative stability of the T solution with a full 2 µB magnetization with respect to the other solutions. Therefore, we can extrapolate that the isolated C monovacancy defect would be characterized by a 2 µB magnetic moment. We now analyze the results obtained with the PBE-D2 method (see Table 2). Here we may notice a peculiar stability of the Scs solution, which makes it the ground state for the small C95H24 model by 0.19 eV with respect to the T (-0.17 eV with G09). However, increasing the nanoflake size, the ∆E(Scs-T) becomes first less negative (-0.09 eV) and then even positive (+0.03 eV). The trend is shown in Figure S5 and is in agreement with what previously reported by Wang et al.,19 although in their work Scs is still the ground state even for a flake with a 4 nm diameter. Therefore, although the result starts to converge only at a large nanoflake size, we may safely conclude that even with PBED2 functional the isolated C monovacancy is expected to present full 2 µB magnetic moment, in line with B3LYP-D* results. Conclusions To summarize and conclude this study, we first want to stress that a comparative investigation of the C monovacancy in graphene, confronting a GGA functional, i.e. PBE-D2, and a hybrid functional, i.e. B3LYP-D*, has clearly evidenced that the former is not suited or accurate enough to study these magnetic systems. Although PBE-D2 is capable of reproducing a magnetic ground state, the magnetic moment is found to be partially quenched (from 2 to about 1.5 µB) because of a spurious charge transfer from the vπα band to the vπβ one, which can be evidenced by both the band structure or the density of states. The partial passivation induced by the charge transfer is not observed when a hybrid functional is applied (this is not true only for B3LYP-D*, but we have also tested HSE06-D2, B3PW91 and PBE0, as discussed above) because the two bands are found not to


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cross the Fermi level, resulting in a fully occupied vπα band and a fully unoccupied vπβ one (total magnetization 2 µB). Notably, such splitting was recently experimentally determined to be in the range of 20-60 meV,25 in excellent agreement with the value computed by B3LYP-D* of 30 meV (for the 16×16 cell). The B3LYP-D* magnetization is not sensible to the supercell size and the k-point mesh density, as it is observed, both in the present and in previous works, for PBE-D2. The hybrid functional results are thus very robust and were also successfully tested against antiferromagnetic (Sos) and diamagnetic (Scs) solutions, whose structures present a peculiar out-of-plane distortion of one of the undercoordinated C atom. This is not only a comparative study of different types of functionals, but also of different models. We investigated results obtained with supercells of increasing size with those obtained with nanoflakes of increasing size. The goal is to determine trends that allow extrapolating the magnetic properties of an ideally isolated C monovacancy. Again, B3LYP-D* data are very consistent and all models provide a T ground state with a net 2 µB magnetization. The partial passivation observed when PBE-D2 is applied to periodic systems cannot occur when applied to finite ones, because here we have orbitals and not bands. Nevertheless, for small nanoflakes PBE-D2 still clashes with B3LYP-D*, favoring the closed-shell solution. This crucial controversy is solved by increasing the model size: the open shell triplet configuration becomes the most stable, as for B3LYP-D*, with a net magnetization of 2 µB. This last result confirms once more the conclusions of this work and puts an end to the long-debated issue of the magnetic properties of an isolated C monovacancy in graphene.

Authors information C. R., M. D. and D. P. have equally contributed to this work. Associated content Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: Tables reporting structural details, Table reporting formation energies, Table reporting HOMOαLUMOβ energy differences for nanoflakes, schematic representation of dz and τ, graph of the magnetic moment dependence with smearing, density of states for T with different cell sizes, spin plots and graph of the ∆E(Scs-T) dependence with the flake size.


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Acknowledgments The authors wish to thank Lorenzo Ferraro for his constant technical help. This research activity is supported by the Italian MIUR through the national grant Futuro in ricerca 2012 RBFR128BEC "Beyond graphene: tailored C-layers for novel catalytic materials and green chemistry" and by the CINECA

supercomputing

center

through

the

computing

LI05p_GRV4CUPT

and

IscrB_CMGEC4FC grants.


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Tables Table 1 – Magnetization (µ), out-of-plane distortion (dz) and relative energy (ΔE) referred to the T, for T, Sos and Scs spin configurations of VG for increasing size cells (6×6, 8×8 and 16×16), as obtained with B3LYPD*/CRY14, PBE-D2/CRY14, and PBE-D2/QE (values in parenthesis). dz is defined as the difference between the z cartesian coordinate of the undercoordinated C atom c and the average value for all the other carbon atoms in the cell, as schematically shown on the left side of Figure S1 in the Supporting Information. Spin Multiplicity

Cell

6x6 (71 C)

8x8 (127 C)

16x16 (511 C)

a

B3LYP-D*

PBE-D2

µ (µB)

dz (Å)

∆E (eV)

µ (µB)

dz (Å)

∆E (eV)

T

2.00

0.00

0.00

1.59 (1.55)

0.00 (0.00)

0.00 (0.00)

Sos

0.00

0.18

0.08

0.32 (0.64)

0.38 (0.38)

0.02 (0.03)

Scs

0.00

0.55

0.33

0.00 (0.00)

0.48 (0.52)

0.07 (0.15)

T

2.00

0.00

0.00

1.54 (1.49)

0.00 (0.00)

0.00 (0.00)

Sos

0.00

0.38

0.07

- (-)

Scs

0.00

0.62

0.34

0.00 (0.00)

0.55 (0.55)

0.11 (0.19)

T

2.00

0.00

0.00

1.49 (1.25)

0.00 (0.00)

0.00 (0.00)

Sos

0.00

0.46

0.05

- (-)

Scs

0.00

0.70

0.31

0.00 (0.00)

a

a

- (-)

- (-)

a

a

0.67 (0.55)

- (-)

- (-)

a

a

0.11 (0.23)

During relaxation SOS collapses in T.

Table 2 – Magnetization (µ), out-of-plane distortion (τ) and relative energy (ΔE) referred to the T state, for T, Sos and Scs configurations of VG modeled by increasing size flakes (C95H24, C215H36, C383H48), as obtained with B3LYP-D*/CRY14 and B3LYP-D3/G09 (values in parenthesis). τ is defined as the dihedral angle involving the undercoordinated C atom c and other three C atoms in the flake, as shown in color on the right side of Figure S1 in the Supporting Information. B3LYP-D* Flake

C95H24

C215H36

C383H48

a

PBE-D2

Spin Multiplicity

µ (µB)

τ (°)

∆E (eV)

µ (µB)

T

2.00 (2.00)

0.00 (0.00)

0.00

2.00 (2.00)

∆E (eV)

τ (°)

a

0.00 (0.00) a

0.00 (0.00) a

Sos

0.00

8.23 (8.55)

-0.03(-0.02)

Scs

0.00

12.48(12.39)

0.07 (0.10)

0.00 (0.00)

11.26(11.49)

-0.19 (-0.17)

T

2.00 (2.00)

0.00 (0.00)

0.00 (0.00)

2.00 (2.00)

0.00 (0.00)

0.00 (0.00)

a

Sos

0.00 (-)

Scs

0.00 (0.00)

T

6.42 (-)

a

a

- (0.00)

a

- (0.00)

a

- (0.06)

-0.10 (-)

a

0.05 (-)

0.00 (-)

9.30 (-)

12.08(11.86)

0.26 (0.25)

0.00 (0.00)

10.63(10.66)

-0.09 (0.01)

2.00

0.00

0.00

2.00

0.00

0.00

Sos

0.00

5.22

0.05

0.00

7.61

-0.02

Scs

0.00

11.55

0.99

0.00 (0.00)

10.23

0.03

During relaxation SOS collapses in Scs.


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Figures

Figure 1 – Schematic top and side view representations of the C monovacancy defect in graphene with spin multiplicity of quintet (Q), triplet (T), closed-shell (Scs) and open-shell (Sos) singlets. The local point group symmetry is shown together with the type of distortion involved: JT (Jahn-Teller) and non-planar. Undercoordinated carbon atoms (a, b and c) are in red. Out-of-plane C atom (c) is in blue. Distances and angles in the optimized periodic and finite models are detailed in Tables S1, S3 and S4.


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Figure 2 – Magnetic moment values for the C monovacancy (for increasing cell size) as a function of the kpoints grid density, as obtained with PBE-D2/CRY14 and B3LYP-D*/CRY14 (top panel) or with PBED2/QE (bottom panel).


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Figure 3 – Band structure of the ground T state for the C monovacancy in 4×4, 8×8 and 16×16 cell models as obtained with B3LYP-D*/CRY14 (top panel) and PBE-D2/CRY14 (bottom panel). α bands are in red and β bands are in blue. The zero energy is set at the Fermi level (dashed line).


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Figure 4 – Total (DOS) and Projected (PDOS) Density of States for the C monovacancy in the 6×6 cell, as obtained with B3LYP-D*/CRY14 (top panels) and PBE-D2/QE (bottom panels): ferromagnetic (T) spin configuration on the left and antiferromagnetic (Sos) configuration on the right. The zero energy is set at the Fermi level (dashed line). Corresponding spin density plots (isovalue=0.005) are shown in the insets (+ density in yellow and – density in blue). The atomic spin density value in a.u. is shown for the undercoordinated atom c.


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Figure 5 – Magnetic moment values for the C monovacancy (in 6×6 or 8×8 cells) as a function of the exact exchange (EXX) percentage (CRY14 calculations). The 20% exact exchange functional corresponds to the B3LYP-D* used throughout this work.


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Figure 6 – Band structure of the ground T state for the C monovacancy in 8×8 cell models as obtained with HSE06-D2/CRY14 (left), B3PW91/CRY14 (center) and PBE0/CRY14 (right). α bands are in red and β bands are in blue. The zero energy is set at the Fermi level (dashed line).


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Figure 7 – Top and side views of the ball-and-stick representations for the nanoflake models of different size with diameter of 1.9, 2.9 and 3.9 nm, respectively. For the open shell configurations the B3LYP-D*/CRY14 spin plots (isovalue=0.005) are also shown in the top view (+ density in yellow and – density in blue)


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TOC GRAPHIC


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Ï Magnetism of Carbon Monovacancy in Graphene by Hybrid Density

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Ï Magnetism of Carbon Monovacancy in Graphene by Hybrid Density