Experimental realization and characterization of an electronic Lieb lattice

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 Dirac-type bands where the charge carriers behave as massless particles [1]. Theoretical predictions are triggering the exploration of novel 2D geometries [2-10], such as graphynes, Kagomé and the Lieb lattice. The latter is the 2D analogue of the 3D lattice exhibited by perovskites [2]; it is a square-depleted lattice, which is characterised by a band structure featuring Dirac cones intersected by a flat band. Whereas photonic and cold-atom Lieb lattices have been demonstrated [11-17], 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 CO molecules positioned with a scanning tunneling microscope (STM). Using scanning tunneling microscopy, spectroscopy and wave-function mapping, we confirm the predicted characteristic electronic structure of the Lieb lattice. The experimental findings are corroborated by muffin-tin and tight-binding calculations. At higher energies, second-order electronic patterns are observed, which are equivalent to a super-Lieb lattice.

Figure 1. Designing an electronic Lieb lattice.

a, Geometric structure of the Lieb lattice. The unit cell (black dashed line) contains two edge sites and one corner site, indicated in red and blue, respectively. b, Band structure of the Lieb lattice, only taking into account nearest-neighbor hopping. c, Calculated local density of states at edge (red) and corner (blue) sites. d, Geometric arrangement of CO molecules (black) on a Cu(111) surface to generate an electronic Lieb lattice. Red and blue circles correspond to the edge and corner sites in a. e, Band structure from muffin-tin (black) calculations along the high-symmetry lines of the Brillouin zone, overlaid with the tight-binding result using parameters that provide the best agreement with the muffin-tin simulations (gray).

Figure 2. Electronic structure of a Lieb lattice.

a, STM image of a 5x5 Lieb (top) and square (bottom) lattice. Two edge sites and one corner site of the Lieb lattice are indicated in red and blue, respectively. The green circle indicates a site of the square lattice. Imaging parameters: V = 50 mV, I = 1 nA. Scale bar: 5 nm. b, Normalized differential conductance spectra acquired above edge (red squares) and corner (blue circles) sites and local density of states at these sites calculated using the tight-binding method (solid lines). c, Contour plot of 100 spectra taken along the line indicated in a. The features observed in the spectra shown in b can be clearly recognized (see arrows). d,e Same as b,c, but for a square lattice. Note that the spectrum on the square lattice is qualitatively different from the spectra acquired on the Lieb lattice.

Figure 3. Wave function mapping.

a-c, Differential conductance maps acquired above a Lieb (top) and square (bottom) lattice at −0.200 V, −0.050 V, and +0.150 V, respectively. Scale bars: 5 nm. d-f, Differential conductance maps for the Lieb lattice at these energies simulated using tight-binding. Black circles representing the CO molecules have been added manually to the tight-binding maps. g-i, Same as d-f, but calculated using the muffin-tin model.

Figure 4. Higher-order effects.

a-b, Schematic picture to show extra sites, resulting in a quasi-Lieb and quasi-super-Lieb lattice, respectively. c-d, Experimental differential conductance maps acquired at 0.550 V above a square and a Lieb lattice, respectively. At these energies 3 and 11 sites per unit cell are required to provide an adequate description of the wave function localization. e-g, Experimental differential conductance maps acquired above a square lattice at −0.300 V, −0.150 V, and 0.575 V, respectively. In each of these maps, the unit cell is indicated by a red dashed line. Note that each unit cell still only contains one CO molecule (at the bottom right of the unit cell). All scale bars denote 5 nm.