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

Single layer Pb on top of (111) surfaces group IV semiconductors hosts charge density wave and superconductivity depending the coverage substrate. These systems are normally considered to be experimental realizations single band Hubbard models their properties mostly investigated using lattice with frozen structural degrees freedom, although reliability this approximation is unclear. Here, we consider case Pb/Ge(111) at $1/3$ coverage, for which surface x-ray diffraction angle-resolved photoemission spectroscopy data available. By performing first-principles calculations, demonstrate that nonlocal exchange between substrate drives system into a $3\ifmmode\times\else\texttimes\fi{}3$ wave. The electronic structure ordered phase mainly determined by two effects: magnitude distortion large spin-orbit coupling. Finally, show effect applies also Pb/Si(111) where Pb-substrate interaction increases bandwidth more than factor $1.5$ respect functional theory $+\mathrm{U}$, in better agreement scanning tunneling data. delicate interplay substrate, structural, freedom invalidates widespread interpretation available literature considering these compounds as physical models.