Intrinsic Hole Migration Rates in TiO2from Density Functional Theory

Formation and migration of hole polarons in bulk rutile and anatase TiO 2 were modeled using density functional theory (DFT) combined with the Marcus/Holstein theory of electron/polaron transfer. We previously applied a similar methodology to model electron polarons and extended the approach to hole polarons. Holes were formed by removal of an O(2p) valence electron, and the quantum mechanical characterization of hole hopping (reorganization energy and electronic coupling) was carried out with the DFT+U method, a method that corrects for self-interaction errors and facilitates charge localization, combined with Hartree−Fock cluster calculations. Several elementary hole transfer processes along various directions were investigated for both rutile and anatase. Hopping along most directions in rutile and along one direction in anatase was found to be adiabatic in character, i.e., thermal processes coupled to phonons. The activation energies for these processes were found to be about twice as large as the activation energy for electron hopping in rutile Deskins and Dupuis, Phys. Rev. B 2007, 75, 195212, in agreement with experiment that indicates electron diffusivity in TiO 2 to be faster than hole diffusivity. Comparison of our calculated rutile hole mobility with experiment shows good agreement (theoretical value of 0.16 cm 2 /V·s at 1300 K). Lattice distortions around hole polarons were found to be larger than around electron polarons. Our results showed also that holes are thermodynamically more stable in the rutile phase, while electrons are more stable in the anatase phase. A hole trapping site with hemibond structure (i.e., a “molecular” polaron with the hole shared between two nonbonded oxygen sites) was also identified in anatase. We also characterized the formation of hole and electron polarons at the (110) surface. The energy barriers for hole and electron hopping at the surface are larger than in the bulk, by ∼0.07 and ∼0.11 eV, respectively. These studies form the basis for further development of models to describe polaron transport in TiO 2 structures such as surfaces, interfaces, or nonperfect crystals.