Correlation-Driven Charge Order in a Frustrated Two-Dimensional Atom Lattice

We thoroughly examine the ground state of the triangular lattice of Pb on Si(111) using scanning tunneling microscopy and spectroscopy. We detect electronic charge order, and disentangle this contribution from the atomic configuration which we find to be 1-down—2-up, contrary to previous predictions from density functional theory. Applying an extended variational cluster approach we map out the phase diagram as a function of local and nonlocal Coulomb interactions. Comparing the experimental data with the theoretical modeling leads us to conclude that electron correlations are the driving force of the charge-ordered state in $\mathrm{Pb}/\mathrm{Si}(111)$. These results resolve the discussion about the origin of the well-known $3\times 3$ reconstruction. By exploiting the tunability of correlation strength, hopping parameters, and band filling, this material class represents a promising platform to search for exotic states of matter, in particular, for chiral topological superconductivity.

Figure 1

(a) Structure model of a $\sqrt{3}\times \sqrt{3}$ adatom lattice on a Si(111) substrate. The Wigner-Seitz unit cells of the substrate surface (gray), the adatom lattice (blue), and the charge-ordered state (red) are depicted. (b) Illustration of the involved parameters governing the different ground states: hopping integrals $t_1$, $t_2$, $t_3$, local Coulomb interaction $U_0$, and nearest-neighbor Coulomb interaction $U_1$. (c) Model representation of $\mathrm{Pb}/\mathrm{Si}(111)$ in the $3\times 3$ charge-ordered state. Red and blue charge clouds reflect the excess and reduced charge density at the respective adatom sites. (d) Experimental STM data of $\mathrm{Pb}/\mathrm{Si}(111)$ showing the charge-ordered state.

Figure 2

Topography (left) and the corresponding LDOS maps (right) of $\mathrm{Pb}/\mathrm{Si}(111)$ for bias voltages (a) $V=-100$, (b) $V=-125$, and (c) $V=-150\mathrm{mV}$. A $3\times 3$ Wigner-Seitz unit cell is marked in each panel. While the LDOS maps show the CO state with a qualitatively unchanged appearance, the contrast in the topographic maps switches due to the interplay of LDOS effects from the CO state with the concomitant topographic buckling. The modulation voltage used to record the LDOS maps was 10 mV at a frequency of 653 Hz.

Figure 3

(a) Quasiparticle interference pattern obtained from Fourier-transforming a $28\times 28{\mathrm{nm}}^2$ $dI/dV$ map of $\mathrm{Pb}/\mathrm{Si}(111)$ measured at $V=+10\mathrm{mV}$ (modulation voltage 2.5 mV, image symmetrized with respect to the threefold symmetry, three-point Gaussian smoothing). The seemingly complex pattern can be explained by a single scattering channel as indicated by the red circles around selected Bragg spots. (b) Theoretically calculated $k$-resolved spectral function of $\mathrm{Pb}/\mathrm{Si}(111)$ for parameters $(U_0/t_1,U_1/t_1)=(5,3)$. The Fermi vector $k_f\approx 0.22{\AA }^{-1}$ is in good agreement with the scattering vector observed in experiment. The enlargement on the right side of the panel includes further scattering vectors obtained from QPI at different sample areas or bias voltages (red circles), underlining the consistency of experimental data and theoretical modeling.

Figure 4

Quantitative phase diagram for the extended Hubbard Hamiltonian [Eq. (1)] as obtained within XVCA. For dominant on-site repulsion $U_0$ we find an antiferromagnetic insulator with row-wise order, for sufficiently strong nearest-neighbor repulsion $U_1$ a charge-ordered insulator. For weaker interactions also metallic regimes are present. Used parameters are $t_2/t_1=-0.383$, $t_3/t_1=0.125$ with $(U_0/t_1,U_1/t_1)=(5,3)$ for PbSi and (12,4) for SnSi. The energy scale for PbSi (SnSi) is set by $t_1=58.5\mathrm{meV}$ ($t_1=52.7\mathrm{meV}$).