Polaron-mediated surface reconstruction in the reduced Rutile TiO$_2$ (110) surface.

The role of polarons is of key importance for the understanding of the fundamental properties and functionalities of TiO$_2$. We use density functional theory with an on-site Coulomb interaction and molecular dynamics to study the formation and dynamics of small polarons in the reduced rutile (110) surface. We show that excess electrons donated by oxygen-vacancies (V$_{\rm{O}}$) form mobile small polarons that hop easily in subsurface and surface Ti-sites. The polaron formation becomes more favorable by increasing the V$_{\rm{O}}$ concentration level (up to ~20\%) due to the progressively lower energy cost needed to distort the lattice. However, at higher V$_{\rm{O}}$ concentration the shortening of the averaged polaron-polaron distance leads to an increased Coulomb repulsion among the trapped charges at the Ti-sites, which weakens this trend. This instability is overtaken by means of a structural $1\times2$ surface reconstruction, characterized by a distinctively more favorable polaron distribution. The calculations are validated by a direct comparison with experimental AFM and STM data. Our study identifies a fundamentally novel mechanism to drive surface reconstructions and resolves a long standing issue on the origin of the reconstruction in rutile (110) surface.