Electron-Phonon Coupling in the Bulk of Anatase ${\mathrm{TiO}}_2$ Measured by Resonant Inelastic X-Ray Spectroscopy

We investigate the polaronic ground state of anatase ${\mathrm{TiO}}_2$ by bulk-sensitive resonant inelastic x-ray spectroscopy (RIXS) at the Ti $L_3$ edge. We find that the formation of the polaron cloud involves a single 95 meV phonon along the $c$ axis, in addition to the 108 meV $ab$-plane mode previously identified by photoemission. The coupling strength to both modes is the same within error bars, and it is unaffected by the carrier density. These data establish RIXS as a directional bulk-sensitive probe of electron-phonon coupling in solids.

Figure 1

(Left) Measured (red line) and calculated (black line) Ti $L_3$ XAS spectrum of anatase. (Right) RIXS data obtained at different photon energies. The middle inset depicts the experimental geometry with respect to the (100)-projected bulk Brillouin zone. The green arrow represents the transferred momentum $\mathit{q}$.

Figure 2

(a) Ti $L_3$ RIXS via a $t_{2g}$ intermediate state. The core hole is well screened by the excited electron. (b) RIXS via an $e_g$ intermediate state. The excited electron is partially delocalized. The poorly screened core hole charge gives rise to a lattice distortion. In the deexcitation process, phonons are emitted.

Figure 3

(a) RIXS data ($h\nu =459.6\mathrm{eV}$; $\pi$ polarization; $\alpha =68°$) with (blue dashed lines) and without (red dashed lines) oxygen partial pressure. The phonon contributions (solid lines) are compared after subtraction of the elastic peak (cf). A similar comparison is shown for $\alpha =45°$. (b) Phonon line shapes for $\sigma$ and $\pi$ polarization are compared to a model calculation with $\hslash {\omega }_0=95\mathrm{meV}$ and $\mathrm{\Gamma }=375\mathrm{meV}$ [18]. (c) Cartoon of locally distorted anatase. Schematic displacement patterns for (d) the $A_{2u}$ $c$-axis phonon, and (e) the $E_u$ in-plane phonon.

Figure 4

RIXS data taken with $h\nu =459.6\mathrm{eV}$ and $\pi$-polarized light with (blue lines) and without (red lines) an oxygen partial pressure. In contrast to the charge transfer (CT) excitations and the phonon shoulder (inset), oxidation leads to a larger elastic line but a lower $dd$ peak.