Dynamical fluctuations and the $\surd$3 $times$ $\surd$3 $\leftrightarrow$ 3 $times$ 3 transition in $\alpha$-Sn/Ge(111) and Sn/Si(111)

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Abstract
A critical review of our understanding of the reversible phase transition × ↔ 3 × 3 for the α-phase of the Sn/Ge(111) and Sn/Si(111) interfaces is presented. Experimental data and theoretical calculations suggest the following. (a) Sn/Ge(111) presents a reversible phase transition associated with the freezing of a surface soft phonon. At high temperature the dynamical fluctuations of the underlying 3 × 3 structure give rise to the observed × symmetry. (b) Sn/Si(111) does not present the 3 × 3 phase although the × reconstruction should be understood within the dynamical fluctuations model found for Sn/Ge(111). (c) These properties are modulated, but not substantially modified, by either defects or by electron many-body effects. We discuss these two cases and analyse how they affect the surface geometry and the electronic properties of the system.