Electron transport via polaron hopping in bulk TiO2:: A density functional theory characterization

This work focuses on the intrinsic electron transport in stoichiometric TiO2. Electron hopping is described by a polaron model, whereby a negative polaron is localized at a Ti3+ site and hops to an adjacent Ti4+ site. Polaron hopping is described via Marcus theory formulated for polaronic systems and quasiequivalent to the Emin-Holstein-Austin-Mott theory. We obtain the relevant parameters in the theory (namely, the activation energy Delta G(*), the reorganization energy lambda, and the electronic coupling matrix elements V-AB) for selected crystallographic directions in rutile and anatase, using periodic density functional theory (DFT)+U and Hartree-Fock cluster calculations. The DFT+U method was required to correct the well-known electron self-interaction error in DFT for the calculation of polaronic wave functions. Our results give nonadiabatic activation energies of similar magnitude in rutile and anatase, all near similar to 0.3 eV. The electronic coupling matrix element V-AB was determined to be largest for polaron hopping parallel to the c direction in rutile and indicative of adiabatic transfer (thermal hopping mechanism) with a value of 0.20 eV, while the other directions investigated in both rutile and anatase gave V-AB values of about one order of magnitude smaller and indicative of diabatic transfer (tunneling mechanism) in anatase.