Electron transport via polaron hopping in bulkTiO2: 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 $\Delta G^{*}$, 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 $\left(\mathrm{DFT}\right)+U$ 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 $\sim 0.3\mathrm{eV}$. 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 $0.20\mathrm{eV}$, 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.