First-Principles Modeling of Polaron Formation in TiO₂ Polymorphs

We present a computationally efficient and predictive methodology for modeling the formation and properties of electron and hole polarons in solids. Through a nonempirical and self-consistent optimization of the fraction of Hartree-Fock exchange (α) in a hybrid functional, we ensure the generalized Koopmans' condition is satisfied and self-interaction error is minimized. The approach is applied to model polaron formation in known stable and metastable phases of TiO2 including anatase, rutile, brookite, TiO2(H), TiO2(R), and TiO2(B). Electron polarons are predicted to form in rutile, TiO2(H), and TiO2(R) (with trapping energies ranging from -0.02 eV to -0.35 eV). In rutile the electron localizes on a single Ti ion, whereas in TiO2(H) and TiO2(R) the electron is distributed across two neighboring Ti sites. Hole polarons are predicted to form in anatase, brookite, TiO2(H), TiO2(R), and TiO2(B) (with trapping energies ranging from -0.16 eV to -0.52 eV). In anatase, brookite, and TiO2(B) holes localize on a single O ion, whereas in TiO2(H) and TiO2(R) holes can also be distributed across two O sites. We find that the optimized α has a degree of transferability across the phases, with α = 0.115 describing all phases well. We also note the approach yields accurate band gaps, with anatase, rutile, and brookite within six percent of experimental values. We conclude our study with a comparison of the alignment of polaron charge transition levels across the different phases. Since the approach we describe is only two to three times more expensive than a standard density functional theory calculation, it is ideally suited to model charge trapping at complex defects (such as surfaces and interfaces) in a range of materials relevant for technological applications but previously inaccessible to predictive modeling.