Phase Transitions in Two Dimensions: The Case of Sn Adsorbed on Ge(111) Surfaces

Accurate atomic coordinates of the room-temperature ( $\sqrt{3}\times \sqrt{3}$) $R30\mathrm{}°$ and low-temperature ( $3\times 3$) phases of $1/3$ monolayer Sn on Ge(111) have been established by grazing-incidence x-ray diffraction with synchrotron radiation. The Sn atoms are located solely at $T_4$ sites in the ( $\sqrt{3}\times \sqrt{3}$) $R30\mathrm{}°$ structure. In the low-temperature phase one of the three Sn atoms per ( $3\times 3$) unit cell is displaced outwards by $0.26\pm 0.04Å$ relative to the other two. This displacement is accompanied by an increase in the first to second double-layer spacing in the Ge substrate.