Investigating Polaron Formation in Anatase and Brookite\nTiO₂ by Density Functional Theory with Hybrid-Functional\nand DFT + <i>U</i> Methods

Abstract

Anatase and brookite\nare robust materials with enhanced photocatalytic\nproperties. In this study, we used density functional theory (DFT)\nwith a hybrid functional and the Hubbard on-site potential methods\nto determine electron- and hole-polaron geometries for anatase and\nbrookite and their energetics. Localized electron and hole polarons\nwere predicted not to form in anatase using DFT with hybrid functionals.\nIn contrast, brookite formed both electron and hole polarons. The\nbrookite electron-polaronic solution exhibits coexisting localized\nand delocalized states, with hole polarons mainly dispersed on two-coordinated\noxygen ions. Hubbard on-site potential testing over the wide 4.0–10\neV range revealed that brookite polarons are formed at <i>U</i> = 6 eV, while anatase polarons are formed at <i>U</i> =\n8 eV. The brookite electron polaron was always localized on a single\ntitanium ion under the Hubbard model, whereas the hole polaron was\ndispersed over four oxygen atoms, consistent with the hybrid DFT studies.\nThe anatase electron polarons were dispersed at lower on-site potentials\nbut were more localized at higher potentials. Both methods predict\nthat brookite has a higher driving force for the formation of polarons\nthan anatase.