Letter pubs.acs.org/NanoLett
Coexisting Honeycomb and Kagome Characteristics in the Electronic Band Structure of Molecular Graphene Sami Paavilainen,*,† Matti Ropo,†,‡ Jouko Nieminen,† Jaakko Akola,†,‡ and Esa Ras̈ an̈ en† †
Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland COMP Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
‡
S Supporting Information *
ABSTRACT: We uncover the electronic structure of molecular graphene produced by adsorbed CO molecules on a copper (111) surface by means of first-principles calculations. Our results show that the band structure is fundamentally different from that of conventional graphene, and the unique features of the electronic states arise from coexisting honeycomb and Kagome symmetries. Furthermore, the Dirac cone does not appear at the K-point but at the Γ-point in the reciprocal space and is accompanied by a third, almost flat band. Calculations of the surface structure with Kekulé distortion show a gap opening at the Dirac point in agreement with experiments. Simple tight-binding models are used to support the first-principles results and to explain the physical characteristics behind the electronic band structures. KEYWORDS: Molecular graphene, density functional theory, tight binding, Kekulé, Kagome cone, which itself has been relocated at the Γ-point in reciprocal space. A necessary condition for this band structure is a sufficiently low CO concentration with respect to surface copper atoms; as it is also required to produce the Kekulé distortion similarly to the experiments.3 Indeed, our calculations show the energy gap opening at the Γ-point due to additional CO molecules within the Kekulé arrangement. Our analysis of the Kohn−Sham states of the density-functional calculations, together with the accompanying tight-binding results, explain the observed features in a detailed level. Previously, some of the present authors have calculated the band electronic structure of molecular graphene with high CO surface coverages.18 Although (nonisolated) Dirac points were observed, and the role of the copper layer depth was explained, the intricate details of the band structure remained unresolved. Here we focus on much lower and more realistic CO concentrations and find that the extra space in between the CO molecules changes the band structure of Cu surface states dramatically. The low CO concentrations are also required to construct the Kekulé distortion. The studied systems of molecular graphene (MG), without and with the Kekulé distortion (KD), are visualized in Figure 1A and B, respectively. The primitive cell for the studied MG is based on a (111) surface configurations, and it contains 4 × 4 Cu atoms and a single CO molecule. The primitive cell has been further replicated 3 × 3 times to get a supercell with a size identical to smallest possible Kekulé distortion. Thus, both MG
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n the past few years, the research on graphene1 has extended to other structures with honeycomb arrangements.2−8 Some of those systems are called as “artificial graphene” as they provide controllability in terms of, e.g., the lattice constant.9 High-precision tunability of a graphene-like band structure has been demonstrated for molecular graphene.3 In this system, adsorbate CO molecules are positioned in a triangular configuration on a copper (111) surface to act as repulsive constraints for the Cu conduction electrons that are confined to move in an interfacial honeycomb geometry. Remarkably, the system also allows opening of a gap at the Dirac point through Kekulé distortion, which can be induced by modifying the CO adsorption pattern. Recently, also finite flakes of molecular graphene have been studied in external magnetic fields.10,11 Another example showing graphene-like features is the Kagome structure, which is composed of interlaced triangles. Kagome lattices have been the subject for various theoretical studies for decades, partly due to their specific magnetic properties.12−14 In these lattices, the Dirac cone is accompanied by a third, flat band with no dispersion. Recently, physical realizations for the Kagome conformations have been successful, e.g., with Pc molecules 15 or colloidal-sized structures.16 Kagome lattice has also been discovered on a Cl-covered Si(111) surface corresponding to “sd2-graphene”.17 In this Letter, we show by electronic structure calculations that the molecular graphene system displays both honeycomb and Kagome patterns at the interface between the CO molecules and the topmost Cu layer. The graphene-like features in the electronic structure arise from electronic states with coexisting honeycomb and Kagome symmetries. As a result, a flat energy band appears in the middle of the Dirac © XXXX American Chemical Society
Received: January 29, 2016 Revised: April 29, 2016
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DOI: 10.1021/acs.nanolett.6b00397 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 2. Band structure calculated with DFT for (A) molecular graphene and (B) Kekulé structure in the 4 × 4 construction. In (C,D), the states shown as bolded blue in (A,B) are plotted as a function of kx and ky, respectively. E−EF refers to the Kohn−Sham state energies with respect to the Fermi level. Figure 1. Visualization of (A) molecular graphene and (B) Kekulé distortion constructed with CO molecules (shown as red-turquoise licorice) on Cu(111) based on the 4 × 4 construction. Blue and green lines show the honeycomb and Kagome structures formed in the uppermost Cu layer, respectively. The lines are broken in the Kekulé formation, and these points are depicted with silver color.
recent results of photonic crystals.24 To better visualize the cone-like shape these three states for MG and KD are shown as a function of kx and ky close to Γ-point in Figure 2C,D. A closer inspection of the Dirac-cone-like states reveals that they are not folded to the Γ-point from the K-point as a consequence of the unit cell replication for the studied supercell (consisting of 3 × 3 unit cells). This is confirmed by the fact that the states are not degenerate as they should be in the case of folding. Furthermore, the same cone still exists at the Γ-point in the band structure for the unit cell, as shown in SI. Projection of the states to the atomic orbitals shows that CO has basically no contribution to these states, which thus originate from the copper states. The role of CO in the emergence of the Dirac cone is in preventing a part of the copper atoms from contributing to these surface states, as discussed in the following and in the SI. Next we focus on the structure and symmetry of the three states that coincide at the Γ-point for the MG unit cell (see Figure 2A). In particular, we analyze the local density of states (LDOS) for the corresponding Kohn−Sham states. Two of the three states have nearly identical LDOS, and one of them is shown in Figure 3A,B, where A is mapped 1.8 Å above and B just below the uppermost copper layer. The third state, which is different, is plotted in Figure 3C (above),D (below). Here we do not attribute the LDOS to certain bands in Figure 2. The symmetry in LDOS patterns in Figure 3 can be rationalized by arranging the uppermost Cu atoms of the unit cell into three groups: (1) two triangles with three Cu atoms each (enclosed by the white triangles in the figures), (2) intermediate Cu atoms between the triangles (enclosed by the circles), and (3) the Cu atom below CO with its six nearest neighbors (dark regions). Interestingly, the triangles form a honeycomb lattice, while the circles form a Kagome lattice. The atoms of the last group (3) and the CO molecule itself have only a minor contribution to any of the studied states at the Γpoint. The Cu atoms in the triangles in Figure 3 contribute to the first two Kohn−Sham states through the atomic d-orbitals. Also the Cu atoms enclosed by the circles have a large weight on these states, but their contribution arise mainly from the pzorbitals. Figure 3A is taken above the surface, and it shows that due to their slower decay the p-orbitals dominate the LDOS, and correspondingly, the contribution of the circled Cu atoms in LDOS will be more significant than that of triangle Cu atoms. Thus, these two states appear with Kagome symmetry above the surface. Interestingly, for the third Kohn−Sham state shown in Figure 3C,D, the roles of the atomic contributions are opposite
and KD have 144 Cu atoms in each layer in the supercell. Still, the Kekulé distortion has a larger CO coverage than that measured by Gomes et al.3 The density-functional theory (DFT) calculations were carried out using the projector augmented wave method19,20 as implemented in the VASP code.21 For the exchange and correlation energy functionals we applied the Perdew−Burke− Ernzerhof (PBE) functional22 within the generalized-gradient approximation (GGA). The surface Brillouin zone was sampled using 6 × 6 k-points, and the plane wave cutoff energy was 400 eV. For the simulated scanning tunneling microscopy (STM) images we applied the Tersoff−Hamann approximation based on the local density of states.23 The DFT calculations were carried out for a Cu slab consisting of three layers of metal atoms and covered by MG and KD surface arrangements of CO. The adsorbate molecules were positioned on the top sites of Cu atoms, each. The periodically repeated Cu slabs were separated by a 14 Å thick vacuum. For all systems, the atomic structure was relaxed until forces were smaller than 0.02 eV/Å. However, only the uppermost Cu layer atoms were allowed to relax, while the atoms in the two other layers were fixed to their bulk position. In addition, for the Kekulé distortion the CO molecules were not allowed to relax parallel to the surface in order to maintain the broken symmetry of the initial structure. We used only three layers of Cu due to computational limitations: even for a system of this size there are already 486 atoms in the supercell. Furthermore, the band structure is more clear with fewer layers (less atoms, less energy bands) enabling also a direct comparison with the tight-binding results. However, whenever feasible we have verified the results with more Cu layers. Furthermore, since PBE functional may give a small error in the CO−Cu bond strength, we verified the results with varying CO−Cu distances, as discussed in Supporting Information (SI). Figure 2A,B shows the band structures for the MG and KD systems, respectively. Interestingly, there is a Dirac-cone-like crossing at the Γ-point in MG, which opens up as a distinct energy gap in the KD system (see the bolded blue lines). There are no other bands near Fermi level where opening of a gap is visible. Instead, the size of the gap is 28 meV at the Γ-point for KD, which is in the same range as in experiments.3 In addition to the two states forming the Dirac cone there is a third state with a weak parabolic dispersion in close resemblance with B
DOI: 10.1021/acs.nanolett.6b00397 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 3. Local density of states of MG for the Kohn−Sham states forming the Dirac cone and the accompanied state extracted at Γ-point mapped at two different heights. Two of the states are nearly identical and only one of them is shown in (A,B), while the third one is different and is shown in (C,D). (A,C) correspond to cross sections taken 1.8 Å above the uppermost Cu layer, whereas (B,D) are taken at 0.2 Å below the layer. All figures show the 3 × 3 unit cells, and their size is 30.82 × 26.69 Å2. The triangles enclose Cu atoms forming the honeycomb lattice while the circles identify the Cu atoms forming the Kagome lattice. The color scale is the same in all panels.
set to zero and difference Δ = ϵα − ϵβ. The detailed derivations of the results for the model are carried out in the SI. Based on this model, we are able to generate eigenvalues ϵ as a function of the wave vector k shown scaled with interaction strength t in Figure 5 for the cases Δ = 0 and Δ = t. The TB
to the other two states: The main contribution from the triangled Cu atoms is of pz-type, while the circled Cu atoms display d-type characteristics. Thus, also the LDOS contributions are opposite above the surface as shown in Figure 3C, and this state appears to have a honeycomb symmetry above the surface. However, below the surface the combined contribution of both patterns is clearly visible as a path connecting the honeycomb and Kagome sites (Figure 3D). The combined symmetries of Figure 3A,C are reflected in the simulated STM images of constant LDOS surfaces for MG and KD, as shown in Figure 4C and D, respectively. The signal
Figure 5. Band structure for the five-atom TB Hamiltonian shown with solid blue lines for Δ = 0 and with dashed black lines for Δ = t. The energy of the state ϵ is scaled with the interaction strength t to obtain a unitless number.
Figure 4. (A,B) Top view of the atomic models and extended supercells used in the calculation of MG (A) and KD (B). The CO molecules are shown as red spheres. The formed honeycomb-like and Kagome-like structures are shown as light blue and green circles, respectively. α and β refer to atomic sites in tight-binding model. (C,D) Simulated small-bias STM images of MG and KD formations with corrugations of 0.38 and 0.58 Å, respectively, mapped at about 4 Å above the surface. The STM images are of the same size as the computational supercells.
model verifies our assumption that with the coexisting honeycomb and Kagome arrangements there is a Dirac cone at the Γ-point accompanied by a third, f lat band. The third band does not have any dispersion since the simple TB model does not include next-nearest neighbor interactions. The result is similar to the one obtained with TB for the bare Kagome structure with two important differences: the cone is at the Γ-point instead of the K-point and the flat band is not above the other bands but in between the bands forming the Dirac cone. For the Dirac cone, the TB model gives Fermi velocity of one-third compared to the TB model of the normal graphene. When the difference Δ between on-sites is not zero, a gap opens between the cones, as in the case of normal graphene. Interestingly the flat band is still attached to the upper Dirac cone band as in the DFT calculation of the Kekulé distortion shown in Figure 2B. The gap opens similarly also when the next-nearest neighbor hopping integrals are included in the TB model as nonzero. Our five-atom TB model is at first glance similar to the one used for graphyne25 where in between the honeycomb atoms there are always two atoms connected to each other with a triple bond. This approach also leads to a Dirac cone moving away from the K-point to the M−Γ-line where the exact spot can be adjusted by the choice of parameters.25 Graphyne is still fundamentally different from our result where the cone appears always at the Γ-point and is always accompanied by the Kagome flat band. The five-atom-basis TB model cannot explain all the features of the DFT calculation. For example, why do the triangle Cu
corrugation in the simulated images is 0.38 and 0.58 Å for MG and KD, respectively. The simulated results are very similar to the experimental STM images,3 which verify that the electronic structure obtained from the calculations is realistic. It is noteworthy that the symmetry of the simulated STM image of KD is broken exactly as in the experimental images. The salient features of the DFT band structure can be highlighted and explained with the help of a simple tightbinding (TB) model using one orbital per atom for a basis. We use the sites for honeycomb (referred as α) and Kagome (referred as β) configurations shown as the light blue and green circles in Figure 4A as the atomic sites corresponding to five atoms in the unit cell of MG. The model is directly comparable to the DFT calculations of MG by identifying all three triangled Cu atoms with a single α site and the circled Cu atom with a β site. The obtained 5 × 5 TB Hamiltonian matrix has elements Hα,β = t exp(i(k·aj)) where aj refers to unit cell lattice vectors and t is the strength of the hopping integral. The next-nearestneighbor matrix elements Hα,α and Hβ,β are zero. The on-sites 1 for Hamiltonian are ϵα and ϵβ with average ϵav = 2 (ϵα + ϵβ ) C
DOI: 10.1021/acs.nanolett.6b00397 Nano Lett. XXXX, XXX, XXX−XXX
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symmetry and not of an accidental degeneracy as in the case of photonic crystals. As a summary, we have shown that the molecular graphene produced by a triangular arrangement of CO molecules on Cu(111) surface confines the electronic surface states of copper to coexisting honeycomb and Kagome symmetries. The resulting electronic band structure has fundamentally different nature from graphene: The Dirac cone is resituated at the Γpoint and always accompanied by a third, nearly flat band energetically in between the Dirac cones. The existence of a Dirac cone accompanied by the weakly dispersive band at Γpoint is a robust feature appearing with a similar overall geometry, and as shown in SI for another case, it is valid at various low adsorption densities of CO on Cu(111). Kekulé distortion breaks the symmetry of molecular graphene, which is seen in calculations as an opening of a gap at the Dirac point. This finding manifests the corresponding experimental observations showing the change of the zero effective mass of an electron in molecular graphene into nonzero by the symmetry breakdown.
atoms constituting the honeycomb lattice contribute a different orbital component to the Dirac cone states than the Cu atoms constituting the Kagome arrangement? To tackle that question we formed a 9 × 9 atom TB model where the atoms (on-sites ϵα) forming the triangle (see Figure 3) are considered separately. They interact with each other via a hopping of strength t1. The Kagome sites are still treated with single atoms (on-sites ϵβ) interacting with triangle atoms via a hopping of strength t2. As shown in SI, also this slightly more complicated TB model leads to a Dirac cone with an accompanied third state. However, by choosing suitable parameters, ϵα, ϵβ, t1, and t2, one can produce two Dirac cones: one below and one above Fermi-level. Both cones can open up with a gap, which depends on the choice of the parameters. However, the gap is closed only if the triangle Cu atom orbitals are different from the Kagome Cu atom orbitals, i.e., they must have different on-sites. The nine-atom TB model casts new light on the DFT results since the orbitals contributing to the Dirac cone states have to be different at the Kagome and honeycomb sites. This condition is fulfilled for MG so that d- and p-orbitals at different sites contribute to these states. The different nature of the orbitals is further reflected on LDOS above the surface since the d-orbitals decay faster than p-orbitals. However, the character of different states cannot be distinguished in the STM simulations, and hence, the resulting pattern is a combination of both honeycomb and Kagome symmetries, as shown in Figure 4C. While the DFT calculations verify the experimental Dirac cone and its opening for the CO adsorption geometry with the Kekulé distortion, the energy of the Dirac point with respect to the Fermi-level is different from the experiments. This might be due to small number of Cu layers used in the calculations. However, increasing the number of layers to nine only worsens the comparison.18 Most likely, laterally larger supercells (enabling lower CO coverage) would be required to obtain better comparison with the experiments. This is, however, not yet feasible due to limited computational resources. It is worth noting that there has to be enough space in between the CO molecules for the coexistence of both the honeycomb and Kagome lattice. The 2 × 2 structure of the CO is too tight, and no similar Dirac cone was found at the Γpoint.18 Instead, as shown in the SI, the (2√3) × (2√3) structure meets this requirement and shows the Dirac cone at the Γ-point accordingly. The obtained grid has also a certain similarity with the Ni3C12S12 lattice.26 Thus, we propose that a configuration with a five-atom basis shown in Figure 4A could be implemented for organic molecules similarly as for Kagome structures.15,26 As a benefit, the flat band arising from the Kagome configuration would be directly at the Fermi level. Using carbon atoms as a basis for the coexisting honeycomb and Kagome structures does not lead to a feasible electronic structure (nor to a stable geometry) owing to changing hybridization of p-orbitals. However, one can manipulate graphene with adatoms placed in the Kagome arrangement to obtain similar states. Finally, there is a clear resemblance of the band structure of Figures 2A and 5 with the results obtained for photonic crystals, which also show a Dirac cone with an auxiliary flat band between the cones.24 Our case corresponds to “bosonic” conical intersection, i.e., integer pseudospin variation of the Dirac equation.27 However, for the molecular graphene the Dirac cone with the auxiliary flat band is a consequence of the
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b00397.
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DFT calculations for bare copper, 4 × 4, and 2 3 × 2 3 MG unit cells, and tight-binding models for MG with five-atom and nine-atom basis (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail: sami.paavilainen@tut.fi. Phone: +358-40-849 0366. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work has been supported by the Academy of Finland through its Centre of Excellence Program (project no. 284621) and through project no. 126205, COST Action CM1204 (XLIC), and the Nordic Innovation through its Top-Level Research Initiative (project no. P-13053). The computer resources of the Finnish IT Center for Science (CSC) and Finnish Grid Infrastructure (FGI) are acknowledged.
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REFERENCES
(1) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. Rev. Mod. Phys. 2009, 81, 109. (2) Reich, E. S. Nature 2013, 497, 422. (3) Gomes, K. K.; Mar, W.; Ko, W.; Guinea, F.; Manoharan, H. C. Nature 2012, 483, 306. (4) Tarruell, L.; Greif, D.; Uehlinger, T.; Jotzu, G.; Esslinger, T. Nature 2012, 483, 302. (5) Gibertini, M.; Singha, A.; Pellegrini, V.; Polini, M.; Vignale, G.; Pinczuk, A.; Pfeiffer, L. N.; West, K. W. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 241406. (6) Singha, A.; Gibertini, M.; Karmakar, B.; Yuan, S.; Polini, M.; Vignale, G.; Katsnelson, M. I.; Pinczuk, A.; Pfeiffer, L. N.; West, K. W.; Pellegrini, V. Science 2011, 332, 1176. (7) Räsänen, E.; Rozzi, C. A.; Pittalis, S.; Vignale, G. Phys. Rev. Lett. 2012, 108, 246803. (8) Kalesaki, E.; Delerue, C.; Morais Smith, C.; Beugeling, W.; Allan, G.; Vanmaekelbergh, D. Phys. Rev. X 2014, 4, 011010. D
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Nano Letters (9) Polini, M.; Guinea, F.; Lewenstein, M.; Manoharan, H. C.; Pellegrini, V. Nat. Nanotechnol. 2013, 8, 625. (10) Aichinger, M.; Janecek, S.; Kylänpäa,̈ I.; Räsänen, E. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 235433. (11) Kylänpäa,̈ I.; Aichinger, M.; Janecek, S.; Räsänen, E. J. Phys.: Condens. Matter 2015, 27, 42. (12) Atwood, J. L. Nat. Mater. 2002, 1, 91. (13) Helton, J. S.; Matan, K.; Shores, M. P.; Nytko, E. A.; Bartlett, B. M.; Yoshida, Y.; Takano, Y.; Suslov, A.; Qiu, Y.; Chung, J.-H.; Nocera, D. G.; Lee, Y. S. Phys. Rev. Lett. 2007, 98, 107204. (14) Yan, S.; Huse, D. A.; White, S. R. Science 2011, 332, 1173. (15) Mao, J.; Zhang, H.; Jiang, Y.; Pan, Y.; Gao, M.; Xiao, W.; Gao, H.-J. J. Am. Chem. Soc. 2009, 131, 14136. (16) Chen, Q.; Bae, S. C.; Granick, S. Nature 2011, 469, 381. (17) Zhou, M.; Liu, Z.; Ming, W.; Wang, Z.; Liu, F. Phys. Rev. Lett. 2014, 113, 236802. (18) Ropo, M.; Paavilainen, S.; Akola, J.; Räsänen, E. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 241401. (19) Blöchl, P. E. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 50, 17953. (20) Kresse, G.; Joubert, D. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758. (21) Kresse, G.; Furthmüller, J. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 11169. (22) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (23) Tersoff, J.; Hamann, D. R. Phys. Rev. B: Condens. Matter Mater. Phys. 1985, 31, 805. (24) Fang, A.; Zhang, Z. Q.; Louie, S. G.; Chan, C. T. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 035422. (25) Kim, B. G.; Choi, H. J. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 115435. (26) Wang, Z. F.; Su, N.; Liu, F. Nano Lett. 2013, 13, 2842. (27) Leykam, D.; Desyatnikov, A. S. Advances in Physics: X 2016, 1, 1.
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