Magnetism of Single Vacancies in Rippled Graphene - The Journal of

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Magnetism of Single Vacancies in Rippled Graphene E. J. G. Santos,†,‡,⊥ S. Riikonen,§ D. Sánchez-Portal,†,‡ and A. Ayuela*,†,‡ †

Centro de Física de Materiales CFM-MPC, Centro Mixto CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, 20080 San Sebastián, Spain ‡ Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain § Laboratory of Physical Chemistry, Department of Chemistry, University of Helsinki, P.O. Box 55, FI-00014, Finland ⊥ School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, 02138, Cambridge, Massachusetts, USA ABSTRACT: Using first-principles calculations, the dependence in the properties of the monovacancy of graphene under rippling controlled by an isotropic strain was determined, with a particular focus on spin moments. At zero strain, the vacancy shows a spin moment of 1.5 μB that increases to ∼2 μB when the graphene is in tension. The changes are more dramatic under compression in that the vacancy becomes nonmagnetic when graphene is compressed more than 2%. This transition is linked to the structural changes that occur around vacancies and is associated with formation of ripples. For compressions slightly greater than 3%, this rippling leads to formation of a heavily reconstructed vacancy structure consisting of two deformed hexagons and pentagons. Our results suggest that any magnetism induced by vacancies that occurs in graphene can be controlled by applying strain.

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ince its discovery,1 graphene has been shown to possess a number of remarkable electronic and elastic properties.2 Its electronic properties are mainly associated with the pz orbitals of carbon, while its elastic properties are associated with the sp2 orbitals. Graphene can sustain elastic deformations as large as 20%, and it is typically under a strain of several percent when deposited on surfaces.3 This strain could be used to engineer the electronic properties of graphene. Unfortunately, a very high tensile force must be applied to defect-free graphene to modify the electronic structure because the spectrum of planar graphene remains metallic up to 10% stretching.4−6 However, there are not many experimental conditions under which layers of graphene remain strictly planar. Due to the soft transversal phonons of 2D graphene, at certain temperatures the layer shrinks and ripples are formed. These ripples are present in free-standing layers of pristine graphene7,8 and for layers deposited on substrates.9,10 The rippling of graphene induces midgap states11 in the electronic structure that disappear when relaxations in the layers are carefully taken into account.12 The relaxations that accompany the rippling of the layer and their influence on the electronic structure are particularly important in layers that are under a high degree of compression.12 Strong rippling of this kind can also be induced by adsorbates13 and defects14 such as that recently discovered for OH impurities.8,15,16 Understanding the interaction between defectinduced deformations and ripples is one of the main challenges now faced in the study of the electronic structure of graphene. Theoretical calculations have shown that substitutional doping17−20 and the presence of defects21,22 in graphene and in graphitic materials in general produce magnetism that is of interest in the potential use of these materials in spintronics. In © 2012 American Chemical Society

experiments, irradiation23−26 and ion bombardment27 of carbon-based materials create vacancies which indeed are linked with magnetic signals. The vacancies21,28,29 and edges30,31 present in graphene layers have been the focus of detailed theoretical studies. Although there have been studies of both the changes in the electronic structure induced by rippling and defects in graphene, to our knowledge there have been no studies of the magnetism of rippled graphene. Such a study is of particular interest when considered alongside the effect of strain on the structural and electronic properties of graphene. Here, we combine the two perspectives and present a study of defects in graphene under rippling. Existing in graphene either suspended or placed on a substrate such as SiO2 the intrisic ripples are mimicked in calculations by a regular pattern under isotropic strain. We focus on a particular type of defect, namely, related to carbon monovacancies, and find that these show a rich diversity of structural and spin solutions as a function of strain. Note that we fully relaxed both the defects and the height and phase of ripples. For zero strain, each vacancy has a solution with spin moment of 1.5 μB. However, nonmagnetic solutions become stable under a moderate compression of less than 2% strain. This transition in magnetic moment may be traced to the modification in the local atomic structure of the defect as ripples develop in the layer. In calculations for compressions greater than 3%, the vacancy structure departs from its usual geometry under rippling, and Received: January 26, 2012 Revised: February 24, 2012 Published: March 2, 2012 7602

dx.doi.org/10.1021/jp300861m | J. Phys. Chem. C 2012, 116, 7602−7606

The Journal of Physical Chemistry C

Article

Figure 1. (A) Graphene with vacancies under an isotropic compression of 1.2%. As an example we use the 10 × 10 unit cell (highlighted). (Inset) Geometry of the vacancies and atomic labels. Local bending at the vacancies is described by the angle θ12 between the pentagon and the plane defined by three C atoms around the vacancy, i.e., two equivalent atoms labeled 1 and a third atom labeled 2. Note the rippling of graphene sheet with vacancies at the saddle points. (B) Different vacancy structure for a compression slightly under 3%. It has two distorted hexagons and pentagons, while the central C atom shows strong sp3 hybridization.

Figure 2. (A) Maximum amplitude h (Å) of rippling for 10 × 10 graphene supercells versus strain in units of the equilibrium lattice constant ao. Filled circles denote defective graphene, while empty circles denote the pristine layer that has been recovered by adding a carbon atom to the vacancy and relaxing. (Insets) Corrugation patterns: bright/yellow (dark/green) areas indicate the higher (lower) regions. (B) Bending angle θ12 (see definition in Figure 1) as a function of strain. (Inset) Behavior of the bending angles with size of supercell for different values of strain. Local induced deformation clearly depends on the presence of vacancies and on the size of the supercell.

lowers its energy by ∼200 meV: atoms of type 1 (see the inset of Figure 1) reconstruct to form a pentagon with the neighboring atoms, while atom 2 is left with the dangling bond responsible for spin polarization. Although here we consider a nonplanar structure under strain, our findings at low compressions are similar to results previously obtained for flat graphene.37,38 However, atom 2 in our rippled structure is progressively lifted from the graphene surface by as much as ∼1.0 Å for strain slightly below 3%, just before the occurrence of a strong vacancy reconstruction. The data of Figure 1B reveal a particularly striking result. In contrast to the 2D vacancy, for which previous authors invariably described a pentagon-goggles structure, we find that the vacancy is strongly reconstructed under rippling. For compressive strain greater than 3%, a different defect structure is produced, consisting of two pairs of heavily distorted hexagons and pentagons and resembling the transition state of a planar vacancy movement. We shall now focus on the description of this region of transition for compression up to 2.8%. Figure 2A shows the maximum amplitude of the rippling h as a function of strain for defective graphene (shown by the solid symbols). The defects from each rippled structure were removed by adding carbon atoms to the vacancy sites and relaxing the structure. We obtained rippled pristine graphene with maximum amplitudes as shown in Figure 2 (open symbols). The amplitudes and topographic patterns (insets in Figure 2A) show little difference between pristine or defective graphene, particularly for compressive strains greater than ∼1%.

this reconstruction results in formation of a different defect structure that has a central atom with strong sp3 hybridization. All these structural changes are linked with the appearance of ripples as the layer is compressed, similar to those observed in recent experiments.32−34 We herein explore the relationship between the appearance of rippling, the presence of vacancies, and changes in the electronic and magnetic properties of graphene. We perform density functional calculations from first principles using the SIESTA code.35 We use the generalized gradient approximation,36 and integration over the Brillouin zone uses a well-converged Monkhorst-Pack k sampling that is equivalent to 70 × 70 points for the graphene unit cell, without imposing any symmetry. We use a C basis set of DZP numerical atomic orbitals obtained from solution of the atomic pseudopotential at slightly excited energies.35 Conjugate gradient structural optimizations leads to forces converged to better than 0.05 eV/Å. The cell vectors are expanded or compressed, but all atomic positions are fully relaxed so the layer ripples and distances between carbon atoms are inhomogeneous. Structure versus Strain. While under tension the layer remains perfectly flat; small compressions produce spontaneous rippling of graphene with a characteristic deformation around the vacancy. Figure 1 illustrates the rippled geometry after relaxing one of our models for a free-standing defective graphene layer under an isotropic compression of 1.2%. At zero strain, the monovacancy tends to undergo a distortion that 7603



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Figure 3. Variation in total energy per atom (A), spin moment (B), and bond length (C) versus strain for three different configurations of vacancy in the 10 × 10 supercell: high-spin flat (triangles), low-spin flat (squares), and low-spin rippled (circles). (Inset in A) Change in total energy with respect to the low-spin solutions at about zero strain. Filled and empty symbols in C represent the 1−2 and 1−1 distances, respectively (see the inset of Figure 1 for atomic labels). Marks given by stars refer to the average distances as the geometry departs from the usual vacancy to the structure in Figure 1B. Note that the magnetism disappears at a strain of ∼2% when allowing for out-of-plane deformations.

magnetic and electronic properties of the vacancies. We first consider the case of zero strain, i.e., at the equilibrium lattice constant. The usual reconstruction37,38 is accompanied by a decrease in the 1−1 distance in Figure 1 and an increase in the 1−2 distances. The two dangling bonds in the type 1 atoms are thus saturated, while atom 2 remains uncoordinated. It is the polarization of the corresponding dangling bond that is the main reason for the appearance of a spin moment of ∼1.5 μB associated with the carbon monovacancy. However, we were able to stabilize another reconstruction characterized by a different structural distortion, in which the 1−2 distance decreases, while the 1−1 distance increases [see the local structures and distances in Figure 3B and 3C]. This structure has a larger spin moment of ∼1.82 μB. We refer to this latter structure as the high-spin (HS) configuration and to the former as the low-spin (LS) configuration. At zero strain the LS structure is more stable than the HS by around 250 meV. The behavior of these two structures as a function of strain is summarized in Figure 3. When applying tension to the layer both the HS and the LS structures remain flat and are almost degenerate. In both cases the spin moments show a slight increase. Conversely, when the layer is compressed, the HS structure becomes unstable. Indeed, it is only possible to stabilize the HS configuration under compression provided that the layer is constrained to remain flat. Even for flat graphene, the energy difference between the HS and the LS states increases significantly with compression, and for strains greater than 1.5% the HS configuration spontaneously transforms into LS-flat (i.e., low-spin constrained to be flat). In both structures, the spin moments decrease as the layer is compressed. The change is greater for the LS-flat configuration, which varies from 1.98 μB for a +3% deformation to 1.15 μB at −3%. A more dramatic reduction in the spin moment is seen if ripples are allowed to form in the graphene layer. Figure 3B shows that the spin moment of the LS-rippled (i.e., low-spin and free to ripple) vacancy decreases sharply when the compression exceeds 0.5%. In fact, the ground state of the monovacancy becomes nonmagnetic for compressive strains in the range of 1.5−2%. The distance between defects shows up in our calculations as dependence with respect to the supercell size: larger sizes cause an increase in the rippling amplitude. This change leads to the killing of the magnetic moment at small compressions The changes in the spin moments of vacancies may be understood by analyzing their electronic structure. When

It seems that even for large supercells the main determinant for ripples is the applied strain. However, the structural patterns that occur in defective graphene define a preferential direction and break the up−down symmetry that exists perpendicular to the layer. It is this preferential direction and the breaks in the symmetry that are explained by the reconstructed vacancies. The vacancies break the hexagonal symmetry of graphene with their goggles-pentagon structure and thus play a key role in determining the shape and symmetry of the global deformation patterns of graphene. It is noteworthy that the vacancies locate at the saddle points of the ripples. We characterize the local curvature at the vacancies and the height of atom 2 above the local tangent-plane in terms of the bending angle θ12, as defined in Figure 1. In Figure 2B, we plot the bending angle for both defective and pristine graphene. Under moderate compression (1−2%), the local bending angle for defective graphene is about three times greater than that of pristine graphene and the graphene rippling is clearly made easier by the presence of the vacancies. The dependence of θ12 on the size of the supercell, as shown in the inset of Figure 2B, further demonstrates the coupling between the local geometry of the defect and the global deformation in that for 4 × 4 supercell a strain larger than 1.0% is required to obtain an increase in θ12 and allow the corrugation of the layer to begin. However, for larger supercells, much smaller strains cause appreciable deformations around the vacancies. Energetics. The range of applied strain used here is comparable to that used in experiments in which an appreciable corrugation of graphene was reported for supported layers,33,39,40 chemically functionalized graphene,13 or defective layers.14 As a result of the imposed periodicity and the finite sizes used in our calculations, long-wavelength deformations appear primarily to be related to the strain, i.e., vacancies do not create significant corrugation in the relaxed geometry at the equilibrium lattice constant. However, we observe that for the range of strain considered here the system is compressed and assumes a rippled configuration with an energy between two and three times lower for defective graphene than for the pristine layer. For a 10 × 10 supercell, the energy required to create ripples for a compression of 3% is reduced to almost one-half its value in the presence of vacancies, as seen in Figure 3A. This difference in energy is consistent with the proposed role of vacancies as a source of ripples in recent experiments.14 Magnetism versus Strain: Several Spin Solutions. We now focus on the influence of these structural changes on the 7604



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support from the CSIC JAE-predoc program, co-financed by the ESF, and from DIPC.

carbon atom 2 is restricted to remain in the plane, the spin moment is mainly associated with the sp2 dangling bond and the contribution from the pz states is smaller. The straininduced deformation of the vacancy and subsequent out-ofplane displacement of atom 2 gives rise to the hybridization between the out-of-plane pz states and the in-plane sp2 states. For flat graphene the spin-polarized impurity level associated with the vacancy therefore has a strong sp2 character and remains essentially decoupled from the delocalized electronic levels of graphene due to their different symmetries. For the rippled layer, these two types of electronic states are strongly hybridized. This hybridization results in delocalization of the defect levels which eliminate the magnetism and account for the lower spin moment of the LS-rippled configuration. Taking into account the change in three dimensions, the rippling transforms the hybridization of the C vacancy atoms in sp3, thereby removing the cause of local magnetism, as seen at the higher compression of the case of Figure 1B. Rippling of the graphene layer occurs in a range of cases. One such case is where graphene is deposited on substrates in which there is a significant mismatch in between lattice parameters. For example, ripples have been previously observed for graphene deposited on Ru(0001)33 as well as on other metallic substrates. Strain can also be applied and controlled by placing exfoliated graphene on flexible substrates, as reported recently.41,42 Under such conditions, it may be possible to create defects in regions of different curvature using a focused electron beam by applying a technique similar to that recently demonstrated by Rodriguez-Manzo and Banhart.43 The existence and spatial distribution of the spin moment associated with the vacancies and its variations with deformation could then in principle be monitored using a spin-polarized scanning tunneling microscope27,44 or spin-dependent transport.45,46 In summary, we have shown that the magnetic and structural properties of carbon vacancies depend strongly on the local curvature induced by the defects. Together with the global rippling pattern, this curvature can be controlled by application of a strain. By compressing graphene by up to 2%, the spin moment of the vacancy can be adjusted to be between 0 and 1.5 μB. At our high-compression limit, the rippling allows the defect C atoms to be arranged in a geometry that is different from the vacancy as generally observed. Such strains may be achieved experimentally41,42 and are comparable with those found when graphene is deposited on a range of different substrates. In fact, our results suggest that the magnetism that is induced in graphene by the presence of defects can be controlled using isotropic strain and other mechanical deformations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the support of the Basque Departamento de Educación and the UPV/EHU (Grant No. IT-366-07), the Spanish Ministerio de Innovación, Ciencia y Tecnologa (Grant No. FIS2007-66711-C02-02), and the ETORTEK research program funded by the Basque Departamento de Industria and ́ the Diputacion Foral de Guipuzcoa. EJGS acknowledges 7605



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Magnetism of Single Vacancies in Rippled Graphene - The Journal of

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