Co-existing honeycomb and Kagome characteristics. in the electronic band structure of molecular. graphene: Supporting Information

Transcription

1 Co-existing honeycomb and Kagome characteristics in the electronic band structure of molecular graphene: Supporting Information Sami Paavilainen,, Matti Ropo,, Jouko Nieminen, Jaakko Akola,, and Esa Räsänen Department of Physics, Tampere University of Technology, P.O. Box 69, FI Tampere, Finland, and COMP Centre of Excellence, Department of Applied Physics, Aalto University, FI Aalto, Finland To whom correspondence should be addressed Department of Physics, Tampere University of Technology, P.O. Box 69, FI Tampere, Finland COMP Centre of Excellence, Department of Applied Physics, Aalto University, FI Aalto, Finland 1

2 In this Supporting Information we carry out more detailed analysis of molecular graphene produced by Gomes et al. [Nature 483, 306 (01)] with CO molecules on a copper surface. First, we study in detail how density functional theory calculations show the Dirac cone for the unit cell and study the real-space interpretation of the corresponding electronic states with coexisting honeycomb and Kagome structures. We also extend our studies to ( 3) ( 3) unit cell. Secondly, we develop a simple tight-binding model which explains the features in density functional calculations. All calculations have been carried out with the same parameters as the results described in the main paper. Electronic structure of MG in 4 4 unit cell First, we make a comparison between a bare copper slab and molecular graphene (MG) 1 to reveal the appearance of the Dirac cone from a set of copper states. The sphere model of MG with 4 4 unit cell is shown in Figure 4A of the main paper. In Figure S1A,B are shown the density functional theory (DFT) based bands for MG unit cell (consists of 4 4 copper atoms and a single CO molecule adsorbed on top of one of the Cu atoms) and bare copper slab (consists of 4 4 copper atoms). The crossing formed at the Γ-point (surrounded with dashed ellipse) appears to extend to lesser k-point range than for the super cell with 3 3 CO unit cells (shown in Figure A in the main text) but this is due to the larger Γ K distance for the smaller super cell. Comparing the crossing to the bare copper case it can be seen to arise from five Cu bands which are split when CO molecule is added in the system. Two of the five original bands are found then at lower energy and three form the Dirac cone and the additional parabolic band. Note that the crossing at the K-point is already present for the bare copper. It should be noted that while the PBE functional is not the best approach to evaluate the interaction between CO and the Cu surface accurately, the obtained Dirac cone in the band structure is not too sensitive on this interaction. We verified this by forcing CO to be further away or closer to the surface. This does not affect the shape of the Dirac cone in the band structure. However,

3 1 A B 0.8 E E F (ev) Figure S1: Band structure for (A) MG and (B) bare copper with 16 atoms in the unit cell obtained from density functional calculations. ε refers the Kohn-Sham state energies with respect to the Fermi level. The red dashed ellipses are shown to mark up the bands which form the Dirac cone. pushing CO from 1.86Å to 1.67Å from surface makes the separation of the Dirac cone states to increase from 1.6 mev to 3.7 mev in the MG formation. Withdrawal of CO to.05å diminishes the separation below 0.1 mev. We also checked the effect of van der Waals dispersion correction on the adsorption geometry with optb86b-vdw functional 4 7 and found that the Cu-C distance is less than 0.01Å different from the PBE result. Electronic structure of MG in ( 3) ( 3) unit cell Sphere model for the ( 3) ( 3) structure of MG is shown in Fig. SA including 3 replicas of unit cell in both x- and y-directions. There are 1 Cu atoms in each layer of the unit cell. In contrast to 4 4 structure, here the Cu atom sites at the uppermost layer coincide both with the honeycomb and Kagome lattices as indicated with a number of spots marked with blue spheres for the honeycomb and green spheres for the Kagome. The band structure for the ( 3) ( 3) structure shown in Figure SB obtained from the 3

4 A 1 B 0.8 E E F (ev) α β 0. 0 Figure S: (A) Sphere model for the MG with ( 3) ( 3) structure. CO molecules appear as red spheres. The formed honeycomb-like and Kagome-like structures are shown as dashed light blue and solid green circles, respectively. Click on the icons on the left to toggle visibility of the honeycomb and Kagome structures (with PDF readers supporting layer information). (B) Band structure for molecular graphene with ( 3) ( 3) structure. The Dirac cone at the Γ-point with energy of about 0.8 ev above Fermi-level has a small energy gap of size 19 mev. E E F refers the Kohn-Sham state energies with respect to the Fermi level. DFT calculations is rather similar to the corresponding band structure of 4 4 structure with a few distinctions. First, the Dirac cone appears still at the Γ-point as expected, but there is a small energy gap of 19 mev already present for the bare MG. The cone is also at higher energy compared to Fermi-level and the third state associated with the cone is now attached to the lower part of the cone. The appearance of the gap might be related to the higher CO concentration compared to the 4 4 structure. This aspect is discussed later in this SI in the context of tight-binding approximations. The local densities of states shown in Figure S3 for the three Kohn-Sham states forming the cone are again similar to 4 4 structure. However, as the honeycomb sites are now formed from single atoms, the state from this atom has to reflect the correct symmetry by itself. So in the first two states the honeycomb sites appear as sp -hybridized while Kagome sites show symmetric s- type contributions. For the third state, honeycomb sites are symmetric s-type sites and combine with p-type Kagome sites. Due to the gap between the cones, the states in Figure S3A,C now have lower energy than the state in Figure S3E which now can be attributed to the upper of the Dirac cone bands in Figure SB. 4

5 A B C D E F Figure S3: Local density of states of MG for the Kohn-Sham states forming the Dirac and cone the accompanied state extracted at Γ-point mapped at two different heights for the ( 3) ( 3) structure. Two of the states have nearly identical density of states shown in (A,B) and (C,D) while the third is different and is shown in (E,F). Left column figures (A,C,E) correspond to crosssections taken 1.8 Å above and right column (B,D,F) 0. Å below the uppermost copper layer. All figures show 3x3 unit cells and their size is Å. The triangles enclose the single atoms forming honeycomb lattice while the circles identify Kagome lattice atoms. Tight-binding model Next we develop tight-binding (TB) models to explain the features found in the DFT calculations. In our simpler TB model, which is quite similar to one used for graphyne, we use a Hamiltonian with a basis of five single-orbital atoms as described by the green and blue spheres shown in Fig. S4. In this model, the blue atoms α form the honeycomb structure and the green β atoms form the Kagome structure. This model can be used to analyze both the 4 4 structure discussed in the 5

6 main article and the ( 3) ( 3) discussed earlier in this SI. For the 4 4 structure the α sites correspond to the triangles constructed out of three Cu atoms and β are the middle Cu atoms. For ( 3) ( 3) also α corresponds to single Cu atoms as visualized in Figure SB in this SI. α β Figure S4: Model for the five-atom basis Hamiltonian. Blue spheres α refer to atoms forming honeycomb structure. They are connected with green β atoms which form a Kagome structure. The d unit cell corresponding to unit cell of MG in the DFT calculation is shown as solid line and its replicas with dashed line. Since both honeycomb and Kagome lattice are present, what might be expected is that the formed electronic bands will have both Kagome-kind of flat band and a Dirac cone. However, the double structure makes the output more interesting. The Hamiltonian matrix has hopping integrals H α,β = t exp(i(k a j )), where α refers to honeycomb sites, β to Kagome sites and t is the strength of interaction between these sites. The next-nearest-atom hopping integrals H α,α and H β,β are considered zero. The exponential term is included only if the hopping occurs through cell boundaries in lattice direction j. The on-sites for Hamiltonian are ε α and ε β with average ε av = 1 (ε α + ε β ) and difference = ε α ε β. The eigenvalues of the Hamiltonian can be calculated analytically. Four of the states have the form of: ε(k) = ε av ± + 3 ± e iφ +iφ 1 + e iφ iφ 1 + e iφ 1 + e iφ + e iφ 1 + e iφ + 3 (1) 6

7 where φ j = k a j and = /t is the scaled difference between on-sites and ε refer to similarly scaled energies. The fifth eigenstate corresponds to the Kagome flat band 3 and is identically equal to ε av. 3 A 3 B 1 1 ε/t 0 ε/t Figure S5: Band structure for a five-atom tight-binding Hamiltonian shown with solid blue lines for = 0 (A) and = t (B). The red dashed lines in (A) show the Dirac cone of Equation (3). We will consider first the case where the on-sites are equal, i.e., = 0 and plot the states as blue solid lines in Figure S5A for ε av = 0. In addition to the flat band there are two more eigenstates which are zero when k = 0, i.e., at Γ-point. These two states correspond to ±-sign inside the root to be negative in Eq. (1). The energies of these states close to Γ-point can be approximated by writing the real parts of the exponential as cosine and taking the two first terms from its Taylor series: cos(x) 1 x /. One obtains for the inner root e iφ +iφ 1 + e iφ iφ 1 + e iφ 1 + e iφ + e iφ 1 + e iφ + 3 = 9 (φ 1 + φ + φ 1 φ ). Substituting this into Eq. (1) leads to scaled eigenenergies ε(k) = ε av ± (φ 1 + φ + φ 1φ ) ε av ± (φ 1 + φ + φ 1φ ) () for the two interesting states when φ j are small. While a 1 = a ˆx and a = 1 a ˆx + 3 aŷ the last term can be written: φ 1 + φ + φ 1φ = 3 4 ( k x + k x) a. Inserting this into Eq. gives for = 0 the 7

8 Dirac cone ε(k) = ε av ± 1 a k. (3) This gives Fermi velocity of one third compared to normal graphene. The Dirac cone close to Γ-point is plotted as dashed red lines in Fig. S5A. It is worth noting that the flat Kagome band is now not above the conduction band, but energetically in the middle of the two bands right. Its energy is at the Fermi level if all atoms contribute one electron to the electronic system. When is not zero, a gap opens between the cones, as in the conventional graphene case. The corresponding bands are shown in Figure S5B. Interestingly the flat band is still attached to the upper Dirac cone band similarly to the DFT calculation of the Kekulé structure. The gap opens similarly also when next nearest neighbor hopping integrals H α,α and H β,β are included as non-zero parameters in the TB model. This may be the reason why for the ( 3) ( 3) unit cell a gap opens without any Kekulé distortion; For the higher CO concentration the distance between lattice sites is smaller leading to larger (clearly non-zero) direct hopping integrals between orbitals at Kagome sites or between orbitals at honeycomb sites. It is worth noting that the appearance of the Dirac cone at the Γ-point can not result from the two- nor three-atom base Hamiltonian but it requires a non-direct interaction between the atoms at the honeycomb sites. While one can include the middle atom interactions as self-energies in the two-atom Hamiltonian, this solution still differs greatly from the normal two-atom case as the on-sites will depend on the hopping integrals. For more detailed consideration of the 4 4 structure we construct a bit more complicated TB model where the atoms (on-sites ε α ) forming the triangle at the honeycomb sites are considered separately. They interact with each other with strength of t 1. The middle atoms at the Kagome sites (on-sites ε β ) still mediate the interaction between the triangle atoms, but the hopping integral H α,β = t exp(i(k a j )) is non-zero only to the closest triangle atoms. Solving numerically the scaled eigenvalues of the corresponding 9 9 Hamiltonian while taking account k-dependence due to interaction through cell boundaries one obtains bands shown as blue lines in Fig. S6 for the case where t = t 1 and ε α = ε β = 0. 8

9 5 4 3 /t Figure S6: Bands for the TB model with nine-atom basis. The solid blue lines correspond to the case with t = t 1 and ε α = ε β = 0. The dashed black lines correspond to case with ε α ε β = (7/5)t 1 and t = 9/10t 1. The band structure shows two different energies which has an opened up Dirac cone. Both gaps can be closed by setting the ratio between the interaction strengths to match difference in on-site energies. For the upper gap the condition can be written as = ( ) t, (4) 3 t 1 where is again scaled difference between on-sites: = (ε α ε β )/t 1. The other cross-section lower in energy can be closed as a Dirac cone with condition = 8 ( ) t 1. (5) 3 t 1 Combining the above conditions one notices that both gaps close with exact condition of = 7/5 and t = 9/10t 1 which case is shown in Fig. S6 with black dashed lines. It is worth noting that the phase velocities related to these different cones are not identical: For the upper case the phase velocity is the same as for 5-atom-basis case while for the lower it is only half of that. Neither cone is at the Fermi-level if all atoms contribute one electron to the electron system. The above conditions imply that for the Dirac cone to exist, the atoms forming the triangle and the middle atoms must be of different species, or that different orbitals must contribute to the 9

10 states. This finding is again in agreement with DFT results for the 4 4 structure, which show that the triangle states are of d-type and middle atom states of p-type, or vice versa. References (1) Gomes, K. K.; Mar, W.; Ko, W.; Guinea, F.; Manoharan, H. C. Nature 01, 483, 306. () Kim, B. G.; Choi, H. J. Phys. Rev. B 01, 86, (3) Zhou, M.; Liu, Z.; Ming, W.; Wang, Z.; Liu, F. Phys. Rev. Lett. 014, 113, (4) Klime s, J.; Bowler, D. R.; Michaelides, A. Phys. Rev. B 011, 83, (5) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. Phys. Rev. Lett. 004, 9, (6) Román-Pérez, G.; Soler, J. M. Phys. Rev. Lett. 009, 103, (7) Klime s, J.; Bowler, D. R.; Michaelides, A. J. Phys.: Cond. Matt. 010,,