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Single layer Pb on top of (111) surfaces of group IV semiconductors hostscharge density wave and superconductivity depending on the coverage and on thesubstrate. These systems are normally considered to be experimentalrealizations of single band Hubbard models and their properties are mostlyinvestigated using lattice models with frozen structural degrees of freedom,although the reliability of this approximation is unclear. Here, we considerthe case of Pb/Ge(111) at 1/3 coverage, for which surface X-ray diffraction andARPES data are available. By performing first principles calculations, wedemonstrate that the non-local exchange between Pb and the substrate drives thesystem into a $3\times 3$ charge density wave. The electronic structure of thischarge ordered phase is mainly determined by two effects: the magnitude of thePb distortion and the large spin-orbit coupling. Finally, we show that theeffect applies also to the $3\times 3$ phase of Pb/Si(111) where thePb-substrate exchange interaction increases the bandwidth by more than a factor1.5 with respect to DFT+U, in better agreement with STS data. The delicateinterplay between substrate, structural and electronic degrees of freedominvalidates the widespread interpretation available in literature consideringthese compounds as physical realizations of single band Hubbard models.

Charge density wave in single-layer Pb/Ge(111) driven by Pb-substrate exchange interaction