Structure determination of Pb/Ge(111)-$\beta \text{-}(\sqrt{3}\times \sqrt{3})\mathrm{R}30^{\circ} $ by dynamical low-energy electron diffraction analysis and first-principles calculation

We have determined the atomic structure of the Pb/Ge(111)- surface, which was shown to exhibit a large Rashba spin splitting in a metallic surface state by dynamical low-energy electron diffraction analysis. The Pb coverage for the optimized atomic structure is 4/3 with one Pb atom located at every third H3 site of the bulk-truncated Ge(111) surface and the other three near the T1 sites but slightly displaced towards the T4 sites. The determined atomic structure agrees well with the energetically optimized one obtained from the first-principles calculation. The calculation also revealed that the potential for the Pb atoms on the H3 sites is very soft along the surface normal, suggesting that their vertical position is distributed within a range of about 0.2–0.3 Å. The previously proposed phase transition associated with the surface melting is discussed.