Electron transport via polaron hopping in bulk Ti O 2 : A density functional theory characterization

This work focuses on the intrinsic electron transport in stoichiometric $\mathrm{Ti}{\mathrm{O}}_{2}$. Electron hopping is described by a polaron model, whereby a negative polaron is localized at a ${\mathrm{Ti}}^{3+}$ site and hops to an adjacent ${\mathrm{Ti}}^{4+}$ 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 $\ensuremath{\Delta}{G}^{*}$, the reorganization energy $\ensuremath{\lambda}$, and the electronic coupling matrix elements ${V}_{AB}$) for selected crystallographic directions in rutile and anatase, using periodic density functional theory $(\mathrm{DFT})+U$ and Hartree-Fock cluster calculations. The $\mathrm{DFT}+U$ 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 $\ensuremath{\sim}0.3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. 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 $0.20\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, 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.