Designing an artificial Lieb lattice on a metal surface

Recently, several experiments [K. K. Gomes et al., Nature (London) 483, 306 (2012); S. Wang et al., Phys. Rev. Lett. 113, 196803 (2014)] have illustrated that metal surface electrons can be manipulated to form a two-dimensional (2D) lattice by depositing a designer molecule lattice on a metal surface. This offers a promising new technique to construct artificial 2D electron lattices. Here we theoretically propose a molecule lattice pattern to realize an artificial Lieb lattice on a metal surface, which shows a flat electronic band due to the lattice geometry. We show that the localization of electrons in the flat band may be understood from the viewpoint of electron interference, which may be probed by measuring the local density of states with scanning tunneling microscopy. Our proposal may be readily implemented in experiment and may offer an ideal solid state platform to investigate the novel flat band physics of the Lieb lattice.

Figure 1

(a) The muffin-tin potential proposed in this Rapid Communication to induce an artificial Lieb lattice with a nearly flat electron band. In the model calculation, the potential is $U_0$ inside the gray dots, which represent the absorbed molecules, and zero outside the dots. $d$ is the diameter of the potential dots and $a_0$ is the lattice constant of the molecule lattice. The red dashed lines indicate the unit cell of the 2D system in the presence of the muffin-tin potential. The blue circles represent the three sites ($A,B,C$) in a unit cell of the artificial Lieb lattice for the surface electrons. $t$ and $t^{\prime }$ are the NN and NNN hopping integrals, respectively. (b) The equivalent Lieb lattice.

Figure 2

Left panels: three lowest bands (blue lines) calculated within the plane wave (PW) method for the muffin-tin potential given in Fig. 1 and the fitting electron bands (red dotted lines) within the tight-binding (TB) model of the Lieb lattice in Eq. (1) for three sets of parameters in Fig. 1. $U_0=7$ eV, $a_0=0.95$ nm, and $d=0.5$ nm. The fitting parameters in TB are $t=0.11$ eV, $t^{\prime }=0.034$ eV, $\mathrm{\Delta }=0.012$ eV, and ${\epsilon }_0=1.27$ eV. (c) $U_0=9$ eV, $a_0=0.95$ nm, and $d=0.5$ nm. The fitting parameters in Eq. (1) are $t=0.11$ eV, $t^{\prime }=0.025$ eV, $\mathrm{\Delta }=0.014$ eV, and ${\epsilon }_0=1.35$ eV. (e) $U_0=7$ eV, $a_0=1.2$ nm, and $d=0.5$ nm. The fitting parameters in Eq. (1) are $t=0.069$ eV, $t^{\prime }=0.023$ eV, $\mathrm{\Delta }=0.006$ eV, and ${\epsilon }_0=0.71$ eV. Right panel: (b), (d), and (f) are the corresponding DOS for the cases of (a), (c), and (e), respectively. The black dotted line is the Fermi level.

Figure 3

The lowest three energy bands calculated by the plane wave method in the FBZ. (a) is for the case of Fig. 2. (b) is for the case of Fig. 2.

Figure 4

The LDOS pattern in real space with different energy. The potential parameters are the same as that used in Fig. 2. (a) $\epsilon =1.2$ eV. (b) $\epsilon =1.37$ eV. (c) $\epsilon =1.5$ eV. (d) $\epsilon =1.6$ eV.