Abstract
Topological materials exhibit protected edge modes that have been proposed for applications in, for example, spintronics and quantum computation. Although a number of such systems exist^{1,2,3,4,5,6}, it would be desirable to be able to test theoretical proposals in an artificial system that allows precise control over the key parameters of the model^{7}. The essential physics of several topological systems can be captured by tightbinding models, which can also be implemented in artificial lattices^{6,7}. Here, we show that this method can be realized in a vacancy lattice in a chlorine monolayer on a Cu(100) surface. We use lowtemperature scanning tunnelling microscopy (STM) to fabricate such lattices with atomic precision and probe the resulting local density of states (LDOS) with scanning tunnelling spectroscopy (STS). We create analogues of two tightbinding models of fundamental importance: the polyacetylene (dimer) chain with topological domainwall states, and the Lieb lattice with a flat electron band. These results provide an important step forward in the ongoing effort to realize designer quantum materials with tailored properties.
Main
The topological theory of matter has its roots in the 70s and 80s studies of polyacetylene^{8}, the quantum Hall effect^{9} and superfluid ^{3}He (ref. 10). The interest in the field exploded after the 2006 discovery of topological insulators^{2,3}. Crystalline solids exhibit a spectrum of energy bands constrained by the material’s symmetries. Topological properties of the band structure give rise to protected boundary states that are the hallmark of topological materials. Wellknown examples include chiral and helical states in quantum (spin) Hall systems and Majorana modes in topological superconductors^{1,9,11,12,13,14}. A central goal of research has become to identify systems with a topological spectrum.
Theories of topological materials^{15} are often formulated using tightbinding models describing hopping between localized electronic orbitals. This means that, given sufficient control, one can implement these models by assembling the corresponding structure from individual constituents that are suitably coupled. It is hence possible to build ‘designer quantum materials’ based on specific Hamiltonians through atomic assemblies. Although there has been considerable success in this direction using ultracold atomic gases, STM can be used to fabricate and characterize structures at the atomic scale in the solid state^{7,16,17,18,19}. Recently, it was shown that vacancy defects in the c(2 × 2) chlorine superstructure on Cu(100) make an excellent system for largescale atomic assembly^{20}. We show that individual Cl vacancies host a welldefined vacancy state below the band edge of the chlorine layer which interact when sufficiently close to each other. We demonstrate that it is possible to construct coupled lattices and implement two Hamiltonians of general interest, the Su–Schrieffer–Heeger (SSH) dimer chain and the twodimensional Lieb lattice.
An overview image of the chlorine layer with several vacancies and typical dI/dV spectra of the vacancy states are shown in Fig. 1a (see Supplementary Information for additional spectroscopy). The surface is characterized by a slowly varying density of states around the Fermi energy and a prominent band edge at about 3.5 V (green line in Fig. 1a). The vacancy sites host an electronic state split off from this band edge (purple line).
We constructed vacancy dimers at different separations through lateral manipulation and investigated their properties. Figure 1c–f shows conductance spectra and maps acquired on two vacancy dimers separated by one or two chlorine sites. Coupling between the vacancies modifies the spectra considerably. Instead of a single resonance, we now find one component above and one component below 3.5 V. Conductance maps acquired at the respective energies, presented in Figs 1d and 1f, clearly show that the lowerenergy resonance is found in between the vacancy sites, while the higherenergy component is stronger on the outer edges (the highenergy component of the short dimer lies within the conduction band of the chlorine layer and thus gives poor contrast). These are clear signatures of the formation of bonding and antibonding combinations of the vacancy state wavefunctions^{21,22}. On the basis of our data, we estimate the following hopping amplitudes: t_{short} ≈ 0.14 eV, t_{long} ≈ 0.04 eV and t_{long}/t_{short} ≈ 0.28 (see Supplementary Information for details). These results establish our premise that the vacancy states may be used to construct interacting lattices.
One of the simplest models in which topological states emerge is the SSH dimer chain, originally developed to describe the polyacetylene molecule and the soliton states within it^{8}. The topologically protected domainwall states in the SSH model provide a condensedmatter realization of the domainwall states in the Jackiw–Rebbi model of the (1+1) dimensional Dirac equation with a mass kink^{23}. The SSH Hamiltonian is given by:
where the hopping parameter alternates between t_{i} = t_{0} ± δt for even/odd i, and c_{i}^{†} and c_{i} are electronic creation and annihilation operators, respectively, at site i. The system has a bandgap determined by δt. The chain exists in two phases depending on the sign of δt and distinguished by a topological index, the winding number. Physically, this means that the two phases are distinguished by the location of the strong bonds. Midgap states are expected on the boundary between these two phases where δt changes sign. The characteristic localization length of the domainwall states in units of lattice constant is ξ = t_{0} − δt/δt. In our experiment, we expect strongly localized states as δt ≈ 0.55t_{0}.
The structure and a set of spectra for a short dimer chain can be found in Fig. 2a, b. Two sets of states can be seen to extend throughout the chain above and below the vacancy state energy. In analogy to the SSH model, we refer to the region in between these two bands as the gap and to states within it as subgap states despite their location far above the Fermi energy. According to our estimation of the coupling constants, we expect a bandgap of
which agrees with our experimental findings. No subgap states exist within this structure.
We constructed a chain containing two domain walls (see Fig. 2c). The lower panel of Fig. 2c shows a constant height conductance map of this structure acquired at 3.53 V. The two domain walls are clearly visible at this bias. We confirmed the existence of midgap modes through point spectroscopy. Figure 2d shows a stacked contour plot of conductance spectra acquired along the chain. The bulk of the chain shows the same signatures as that shown in Fig. 2b. At the location of the domain walls, however, a single resonance centred around the midgap energy emerges. These are the topologically protected midgap states predicted by the SSH model. To show that the presence of topological states is independent of the chain form, we also studied a ringlike structure with two embedded phase boundaries. This structure can be found in Fig. 2e with the corresponding conductance map at the midgap energy. As expected, midgap states located on the domainwall sites are found.
To show that the concept of lattice engineering can be generalized to more complex systems, we constructed a Lieb lattice structure from chlorine vacancies. This is a linecentred square structure with a threeatom unit cell, resulting in a fermionic system with a lattice pseudospin of 1. The band structure in the infinite limit consists of a Dirac cone on the corners of the first Brillouin zone intersected by a flat band at the Dirac energy^{24}. Such flat bands are prone to electronic instabilities near halffilling and have been suggested to yield magnetic or superconducting order^{25,26,27,28}. The presence of flat bands could also enhance the properties of superconducting materials by increasing the critical temperature^{29,30}. Depending on the nature of the interactions in the lattice, topological states may also emerge^{26,31}. Although Lieb lattices have been studied in optical lattices and ultracold atomic gases^{32,33}, no realization using electronic states exist to our knowledge.
A small Lieb lattice can be seen in Fig. 3a with point spectra taken on the A, B and C sites of the threeatom unit cell presented in panel b. Figure 3 shows constant height conductance maps acquired at different bias voltages. At 2.85 V and 3.15 V, the maps reveal an extended electronic state with higher intensity on the A sites. This finding is reproduced well by our tightbinding simulation (see below). The signal distribution we observe here is characteristic of the dispersive bands in the Lieb lattice which converge to form the Dirac cone. The map taken at a bias of 3.5 V reveals a starkly different contrast: there now is nearly no signal on the A sublattice whereas the B and C sites show up more prominently. This is the hallmark of the flat electron band for which the Lieb lattice is known^{24}.
We performed tightbinding simulations of the Lieb lattice based on our experimental structure and estimates of the hopping amplitudes. As the nextnearest neighbour coupling (coupling between B and C sites) is nonzero, the flat band will be slightly distorted (see Supplementary Information). We estimate the nextnearest neighbour interactions by assuming that the coupling between vacancies depends exponentially on the separation distance to obtain a value of t_{NNN} ≈ 0.045 eV. The lower panel of Fig. 3 shows the results of the tightbinding simulation. A good agreement with the experiment is found using our coupling estimates of t_{NN} ≈ 0.14 eV and t_{NNN} ≈ 0.045 eV, giving a ratio of t_{NNN}/t_{NN} ≈ 0.33 (see Supplementary Information for details).
In summary, we have presented a general approach for producing tailormade band structures through atom manipulation using STM. We implemented two model systems with topological states and nearly flat electron bands in a precisely controlled environment. Our approach, combined with the possibilities of automating structure building at the atomic level^{20}, places a vast amount of relevant model systems within experimental reach. Additional customizability could come from adapting such assembly techniques to, for example, heavy metals with magnetic properties or significant spin–orbit coupling. The next major challenge is to identify systems in which the energies of the participating states are close to the Fermi level, or can be tuned through the relevant energy range, to observe effects such as the magnetic instability of the flat band or the fractional states of the SSH chain. Further progress in this direction may allow atomic assemblies to produce intriguing physics similar to those seen in ultracold atomic gases while working entirely with electronic states. The same concepts laid out here may be applied to mesoscopic building blocks such as quantum dots to produce quantum materials with tailored properties.
Methods
All sample preparations and experiments were carried out in an ultrahigh vacuum system with a base pressure of ∼10^{−10} mbar. The (100)terminated copper single crystal was cleaned by repeated cycles of Ne^{+} sputtering at 1.5 kV, annealing to 600 °C. To prepare the chloride structure, anhydrous CuCl_{2} was deposited from an effusion cell held at 300 °C onto the warm crystal (T ≈ 150–200 °C) for 180 s. The sample was held at the same temperature for 10 min followingthe deposition.
After the preparation, the sample was inserted into the lowtemperature STM (Unisoku USM1300) and all subsequent experiments were performed at T = 4.2 K. STM images were taken in the constant current mode. dI/dV spectra were recorded by standard lockin detection while sweeping the sample bias in an open feedback loop configuration, with a peaktopeak bias modulation of 20 mV at a frequency of 709 Hz. Line spectra were acquired in constant height; the feedback loop was not closed at any point between the acquisition of the first and last spectra. Manipulation of the chlorine vacancies was carried out using a procedure adapted from ref. 20. The tip was placed above a Cl atom adjacent to a vacancy site at 0.5 V bias voltage and the current was increased to 1 to 2 μA with the feedback circuit engaged. The tip was then dragged towards the vacancy site at a speed of up to 250 pm s^{−1} until a sharp jump in the zposition of the tip was observed. This procedure leads to the Cl atom and the vacancy site exchanging positions with high fidelity.
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
This research made use of the Aalto Nanomicroscopy Center (Aalto NMC) facilities and was supported by the European Research Council (ERC2011StG No. 278698 PRECISENANO), the Academy of Finland through its Centres of Excellence Program (projects no. 284594 and 284621) and Academy Research Fellow program (no. 256818), and the Aalto University Centre of Quantum Engineering.
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All authors jointly conceived and planned the experiment. R.D. performed the measurements. R.D. and P.L. analysed the STM data. T.O. proposed the SSH dimer chain structure and provided the description of its physics. A.H. proposed the Lieb lattice model and performed the tightbinding calculations for this structure. All authors jointly authored, commented, and corrected the manuscript.
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Drost, R., Ojanen, T., Harju, A. et al. Topological states in engineered atomic lattices. Nature Phys 13, 668–671 (2017). https://doi.org/10.1038/nphys4080
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DOI: https://doi.org/10.1038/nphys4080
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