Final-state effect on X-ray photoelectron spectrum of n-doped SrTiO$_3$*

X-ray photoelectron spectroscopy (XPS) is a widely used technique to determine the oxidation states of chemical elements. In stoichiometric SrTiO$_3$, the Ti$^{4+}$ peak appears at a binding energy of about 459.0 eV for photoelectrons ejected from the Ti $2p$ core level. In lightly n-doped SrTiO$_3$, a weak shoulder at a binding energy of about 1.5 eV lower than the Ti$^{4+}$ peak appears in the XPS spectrum that has been conventionally interpreted as a Ti$^{3+}$ signal. By taking the final-state effect into account, i.e. by considering the response of the valence electrons in the presence of a core hole, we argue that such a Ti$^{3+}$ peak does not necessarily imply the existence of spatially localized Ti$^{3+}$ ions, and explicitly show that a spatially uniform Ti$^{(4-x)+}$ distribution also leads to the multi-peak structure. Spectra from metallic n-doped SrTiO$_3$ (e.g. La replacing Sr, Nb replacing Ti, or even oxygen vacancy doping) should be interpreted as the latter case. Several experiments based on this interpretation are discussed.