  The cubic perovsike strontium titanate SrTiO$_3$ (STO) is one of the most
studied, polarizable transition metal oxides. When excess charge is introduced
to this material, e.g., through doping or atomic defects, STO tends to host
polarons: Quasi-particles formed by excess charge carriers coupling with the
crystal phonon field. Their presence alters the materials properties, and is a
key for many applications. Considering that polarons form preferentially on or
near surfaces, we study small polaron formation at the TiO$_2$ termination of
the STO(001) surface via density functional theory calculations. We model
several supercell slabs of Nb-doped and undoped STO(001) surfaces with
increasing size, also considering the recently observed as-cleaved TiO$_2$
terminated surface hosting Sr-adatoms. Our findings suggest that small polarons
become less stable at low concentrations of Nb-doping, in analogy with polarons
localized in the bulk. Further, we inspect the stability of different polaron
configurations with respect to Nb- and Sr-impurities, and discuss their
spectroscopic properties.
