  Charge trapping and formation of polarons is a pervasive phenomenon in
transition metal oxide compounds, in particular at the surface, affecting
fundamental physical properties and functionalities of the hosting materials.
Here we investigate via first-principle techniques the formation and dynamics
of small polarons on the reduced surface of titanium dioxide, an archetypal
system for polarons, for a wide range of oxygen vacancy concentrations. We
report how the essential features of polarons can be adequately accounted in
terms of few quantities: the local structural and chemical environment, the
attractive interaction between negatively charged polarons and positively
charged oxygen vacancies, and the spin-dependent polaron-polaron Coulomb
repulsion. We combined molecular dynamics simulations on realistic samples
derived from experimental observations with simplified static models, aiming to
disentangle the various variables at play. We find that depending on the
specific trapping site, different types of polarons can be formed, with
distinct orbital symmetries and different degree of localization and structural
distortion. The energetically most stable polaron site is the subsurface Ti
atom below the undercoordinated surface Ti atom, owing to a small energy cost
to distort the lattice and a suitable electrostatic potential. Polaron-polaron
repulsion and polaron-vacancy attraction determine the spatial distribution of
polarons as well as the energy of the polaronic in-gap state. In the range of
experimentally reachable oxygen vacancy concentrations the calculated data are
in excellent agreement with observations, thus validating the overall
interpretation.
