Surface soft phonon and the root3 x root3 <--> 3 x 3 phase transition in Sn/Ge(111) and Sn/Si(111)

Density Functional Theory (DFT) calculations show that the reversible Sn/Ge(111) $\sqrt{3}\times\sqrt{3} \leftrightarrow 3\times3$ phase transition can be described in terms of a surface soft phonon. The isovalent Sn/Si(111) case does not display this transition since the $\sqrt{3}\times\sqrt{3}$ phase is the stable structure at low temperature, although it presents a partial softening of the $3\times3$ surface phonon. The rather flat energy surfaces for the atomic motion associated with this phonon mode in both cases explain the experimental similarities found at room temperature between these systems. The driving force underlying the $\sqrt{3}\times\sqrt{3} \leftrightarrow 3\times3$ phase transition is shown to be associated with the electronic energy gain due to the Sn dangling bond rehybridization.