Abstract
Geometry, whether on the atomic or nanoscale, is a key factor for the electronic band structure of materials. Some specific geometries give rise to novel and potentially useful electronic bands. For example, a honeycomb lattice leads to Diractype bands where the charge carriers behave as massless particles^{1}. Theoretical predictions are triggering the exploration of novel twodimensional (2D) geometries^{2,3,4,5,6,7,8,9,10}, such as graphynes and the kagomé and Lieb lattices. The Lieb lattice is the 2D analogue of the 3D lattice exhibited by perovskites^{2}; it is a squaredepleted lattice, which is characterized by a band structure featuring Dirac cones intersected by a flat band. Whereas photonic and coldatom Lieb lattices have been demonstrated^{11,12,13,14,15,16,17}, an electronic equivalent in 2D is difficult to realize in an existing material. Here, we report an electronic Lieb lattice formed by the surface state electrons of Cu(111) confined by an array of carbon monoxide molecules positioned with a scanning tunnelling microscope. Using scanning tunnelling microscopy, spectroscopy and wavefunction mapping, we confirm the predicted characteristic electronic structure of the Lieb lattice. The experimental findings are corroborated by muffintin and tightbinding calculations. At higher energies, secondorder electronic patterns are observed, which are equivalent to a superLieb lattice.
Main
The Lieb lattice is a squaredepleted lattice, described by three sites in a square unit cell, as illustrated in Fig. 1a. Two of the sites (indicated in red) are neighboured by two other sites. The third site in the unit cell (blue) has four neighbours. In the remainder of this article, these sites will be referred to as edge (red) and corner (blue) sites, respectively. This geometry results in an electronic band structure exhibiting two characteristic features: two dispersive bands, which form a Dirac cone at the M point in the first Brillouin zone, and a flat band crossing the Dirac point (Fig. 1b). It is well established that Dirac cones give rise to unusual behaviour, such as effectively massless fermions. Similarly, flat bands may potentially facilitate the realization of magnetic order^{18,19}, give rise to the fractional quantum (spin) Hall and the quantum anomalous Hall effect^{20,21}, and enhance the critical temperature of superconductors^{22,23}.
The electronic band structure of the Lieb lattice can be calculated from the following tightbinding Hamiltonian:
where ε_{i} and t (t′) indicate the onsite energy of site i and the (next)nearestneighbour hopping constants, respectively. Taking only nearestneighbour hopping into account and using the same onsite energy for the three sites results in the band structure shown in Fig. 1b. The flat band exclusively contains electronic states which are localized on edge sites. In contrast, all sites contribute to the dispersing bands converging to the Dirac cone. Hence, the local density of states (LDOS) exhibits a characteristic spatial variation, see Fig. 1c.
Thus far, a 2D electronic Lieb lattice has not been realized. In principle, lithography could be used to impose a Lieb pattern on a 2D electron gas^{7,24}. Alternatively, a strategy similar to the one employed for generating artificial graphene could be used—that is, assembling a molecular lattice on a substrate featuring a surface state to force the electrons into the desired geometry^{25}. In the following, we will choose the latter approach and describe how atomic scale manipulation of carbon monoxide molecules on Cu(111) with a scanning tunnelling microscope is used to generate and characterize an electronic Lieb lattice.
The design of the molecular Lieb lattice is not trivial for several reasons. First, the Lieb lattice has fourfold rotational symmetry, whereas substrates that exhibit a surface state close to the Fermi energy such as Cu(111) have hexagonal symmetry. Furthermore, the carbon monoxide (CO) molecules on Cu(111) act as repulsive scatterers, confining the electrons to the space between the CO molecules^{25,26,27,28}. This implies that the CO molecules should compose the antilattice of the electronic Lieb lattice. Our design consists of a CO square lattice, which defines the trivial antilattice of a square lattice, with one CO placed in the centre of four CO molecules to form the antilattice of a depleted square lattice (see Fig. 1d). This design was recently proposed independently by Qiu and colleagues^{29}. The size of the unit cell is chosen to be (≈ 2.66 nm × 2.56 nm), where a_{0} = 0.256 nm is the Cu(111) nearestneighbour distance. Two factors play a critical role in the design. First, this arrangement of CO molecules provides the best approximation to the perfect fourfold symmetry of the Lieb lattice on the hexagonal Cu(111) substrate. Furthermore, the size of the unit cell determines the position of the bands of the lattice with respect to the Fermi level of the Cu(111)^{25}. With the lattice constants described above, the flat band is close to the Fermi level (see below).
To establish whether the design described above confines the electrons into an electronic Lieb lattice, we performed calculations based on the nearlyfree electron model, in which the CO molecules are modelled by a muffintin potential. The band structure calculated using this approach is given by the black curve in Fig. 1e. These results can be reproduced well using a tightbinding model including orbital overlap and nextnearestneighbour interactions (t′/t = 0.6), see the grey curve in Fig. 1e. Hence, the arrangement of CO molecules on Cu(111) shown in Fig. 1d generates an electronic Lieb lattice. The large t′/t ratio shows that nextnearestneighbour hopping in this system is important. This can be rationalized by the fact that the distance between individual CO molecules is fairly large on the atomic scale. A detailed description of the correspondence between the nearlyfree electron and tightbinding calculations is given in the Supplementary Information.
A lattice of 5 × 5 unit cells was assembled in the way shown in Fig. 2a. To provide further evidence that any observed features are due to the Lieb lattice, a square lattice was created immediately next to the Lieb lattice. Differential conductance spectra were acquired above various positions of the lattice (indicated by the blue and red circles in Fig. 2a). The spectra were normalized by the average spectrum acquired on the clean Cu(111) surface, analogously to ref. 25. The resulting spectra above corner (blue) and edge sites (red) are shown in Fig. 2b. We first focus on the spectrum acquired above a corner site (blue). Two peaks are observed, one at V = −0.20 V and one at +0.18 V. These peaks can be assigned to the lowest and highestenergy bands in the nearestneighbour tightbinding model of the Lieb lattice. In between these two peaks, the LDOS reaches a minimum, which should correspond to the Dirac point. In contrast, the edgesite spectrum (red) exhibits a maximum, which is located at V = −0.07 V. This peak can be assigned to the flat band. The neighbouring peaks are again due to the lowest and highestenergy bands.
In principle, a flat band should give rise to an (infinitely) narrow feature in the LDOS. In contrast, the peak at V = −0.07 V observed above the edge sites is fairly broad. We attribute this broadening to the influence of nextnearestneighbour hopping, as well as to the limited lifetime of the electrons in the surface state.
The experimentally observed differential conductance spectra are reproduced very well when nextnearestneighbour hopping is included in tightbinding calculations of a finite lattice (Fig. 2b). Nextnearestneighbour hopping is essential to account for the observed asymmetry in the LDOS of the low and highenergy bands (blue spectrum, peaks at −0.20 V and +0.18 V), as well as for the peak at 0.09 V in the edgesite spectrum. A fit of the tightbinding result to the experimental data yields t = (89 ± 15) meV, which is in excellent agreement with earlier results^{25}. Using this hopping parameter, we calculate the Fermi velocity of the electrons in the Dirac cones to be v_{F} = (3.5 ± 0.6) × 10^{5} m s^{−1}.
To investigate the spatial distribution of the electronic states, we acquired differential conductance maps (see below), as well as 100 spectra along the line indicated in Fig. 2a. This line starts and ends at an edge site and passes four corner sites. The resulting contour plot is shown in Fig. 2c. The peaks described above can be clearly recognized for each site, demonstrating that the LDOS features are a property of the lattice.
For comparison, a differential conductance spectrum acquired over a site in the square lattice is shown in Fig. 2d, while a contour plot showing 125 spectra along a line is shown in Fig. 2e. The spectra along the line again demonstrate the similarity of the features for equivalent sites (Fig. 2e). Importantly, the spectra are qualitatively different from the spectra obtained over the Lieb lattice and display a good agreement with the LDOS calculated for the square lattice using the tightbinding model (using the same parameters as for the Lieb lattice) (see Fig. 2d). This further demonstrates that the features observed in the differential conductance spectra shown in Fig. 2b are due to the Lieb lattice.
Figure 3 shows several experimental and simulated constantheight differential conductance maps of the two lattices. For the square lattice, all equivalent sites appear identical at all three energies. In contrast, for the Lieb lattice at V = −0.20 V, both the edge and corner sites contribute significantly to the density of states. At the energy of the flat band (V = −0.05 V), the contribution of the edge sites to the density of states dominates. At V = +0.15 V, again both corner and edge sites contribute significantly, with the first being dominant. The simulated maps using the tightbinding model (Fig. 3d–f) and using the muffintin approach (Fig. 3g–i) reproduce the features observed experimentally.
A careful inspection of the contour plots shown in Fig. 2c and e indicates that for both the square and Lieb lattice there is structure in the spectra at higher energy (around V = +0.60 V). For both lattices, these highenergy states are localized in between different sites. To account for these states in the tightbinding calculations, additional basis functions need to be included. This can be done by adding sites in between the original sites. To first order, the simple square lattice is then described by a threesite quasiLieb model, with corner and edge sites having different onsite energies (Fig. 4a). Likewise, the Lieb lattice is described by a superLieb (Fig. 4b) geometry involving 11 sites per unit cell. Differential conductance maps of the highenergy states of the square and Lieb lattice with indicated unit cells are shown in Fig. 4c and d, respectively. Note that 3 and 11 sites are required to describe the unit cells, respectively. Using this model, we again simulated differential conductance maps. The experimental and simulated maps at higher energy are in good agreement.
The peak positions with respect to the Fermi energy can be shifted to lower energies by increasing the lattice constant^{25}. We make use of this effect to access states with even higher energy in the square lattice. Figure 4e–g shows differential conductance maps of a square lattice with a four times larger unit cell. For this large square lattice, the pseudoLieb character emerges at bias voltages as low as −0.30 V and −0.15 V for the bottom and flat bands, respectively. At higher bias voltages, a superLieb character appears, as depicted in Fig. 4g. The higher onsite energies of the ‘bridging sites’ results in a band gap between the lowerenergy bands (which retain their square/Lieb character) and the higherenergy bands where localization is more pronounced on the bridging sites.
The ability to generate electronic lattices using CO molecules on Cu(111) opens the path to the experimental realization and characterization of many 2D geometries for which nontrivial properties dictated by the lattice have been anticipated theoretically. Typical examples, which apply to the studied Lieb lattice geometry, are the quantum spin Hall effect, the superKlein tunnelling paradox, and the Hofstadter butterfly^{5,11,30,31}. The Cu(111)/CO system is an ideal model system, as it allows one to tune parameters that cannot be easily varied in a real solidstate material. In addition, one can create junctions and study the effects of disorder, which can be designed in a controlled manner. The inherent versatility and the direct access to structural and electronic characterization allow a reality check for advanced theory and a first step in the design of truly novel electronic materials.
Methods
Scanning tunnelling microscope (STM) experiments.
The experiments were performed in a Scienta Omicron LTSTM, operating at a temperature of 4.6 K and a pressure in the 10^{−10} mbar range. Prior to the experiments, a clean Cu(111) crystal surface was prepared by several cycles of sputtering and annealing. After cooling down in the STM microscope head, CO was deposited on the surface by leaking in this gas to P = 2 × 10^{−8} mbar for 3 min. For all measurements a Cucoated tungsten tip was used. Assisted by an inhouse developed program, atomic manipulations were performed following previously described procedures^{32,33}. STM images were acquired in constant current mode. dI/dV spectroscopy and mapping were performed in constantheight mode using a standard lockin amplifier modulating the sample bias with an amplitude of 10–20 mV r.m.s. at a frequency of 273 Hz. Various tips, characterized by differently shaped Cu(111) spectra, were used to corroborate the features arising from the Lieb lattice.
Tightbinding calculations.
Tightbinding calculations were performed for periodic and finitesized lattices. For dispersion and LDOS calculations, we utilized a grid of 50 × 50 kpoints in the first Brillouin zone, whereas n × n kpoints were used for calculating the differential conductance maps of the higherorder lattices. The used tightbinding parameters were t′/t = 0.6 and an orbital overlap of s = 0.15. The calculations on the experimentally realized geometry are Γpoint calculations with periodic boundary conditions, utilizing the same tightbinding parameters as the periodic lattice calculations. The local density of states was inferred directly from sites in the centre of both lattices, using a Lorentzian energy level broadening of Γ = 0.8t. Simulated differential conductance maps were obtained by taking again Γ = 0.8t and by expanding the wave functions by normalized Gaussians of width σ = 0.4a, where a is the lattice constant of the Lieb lattice.
Muffintin calculations.
The surface state electrons of Cu(111) can be considered a 2D electron gas. The CO molecules are modelled as disks (radius 0.3 nm), centred at a CO molecule, with a repulsive potential of 0.9 eV. See Supplementary Information for details.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Additional Information
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Acknowledgements
Financial support from the Foundation for Fundamental Research on Matter (FOM, grants 16PR3245 and DDC13), which is part of the Netherlands Organisation for Scientific Research (NWO), as well as the European Research Council (‘FIRSTSTEP’,692691) is gratefully acknowledged. We thank J. van der Lit and N. van der Heijden for fruitful discussions.
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M.R.S., T.S.G., P.H.J. and I.S. planned the experiment, including the proposal of the design of the CO lattice. M.R.S. and T.S.G. performed the experiments and analysed the data. P.H.J. carried out the tightbinding calculations and G.C.P.v.M. performed the muffintin model calculations. S.J.M.Z. developed a program that partially automates the lattice assembly. All authors contributed to the discussions and the manuscript.
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Slot, M., Gardenier, T., Jacobse, P. et al. Experimental realization and characterization of an electronic Lieb lattice. Nature Phys 13, 672–676 (2017). https://doi.org/10.1038/nphys4105
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DOI: https://doi.org/10.1038/nphys4105
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