Electron-phonon coupling in $d$-electron solids: A temperature-dependent study of rutile $\mathrm{Ti}{\mathrm{O}}_2$ by first-principles theory and two-photon photoemission

Rutile $\mathrm{Ti}{\mathrm{O}}_2$ is a paradigmatic transition-metal oxide with applications in optics, electronics, photocatalysis, etc., that are subject to pervasive electron-phonon interaction. To understand how energies of its electronic bands, and in general semiconductors or metals where the frontier orbitals have a strong $d$-band character, depend on temperature, we perform a comprehensive theoretical and experimental study of the effects of electron-phonon $\left(e-p\right)$ interactions. In a two-photon photoemission (2PP) spectroscopy study we observe an unusual temperature dependence of electronic band energies within the conduction band of reduced rutile $\mathrm{Ti}{\mathrm{O}}_2$, which is contrary to the well-understood $sp$-band semiconductors and points to a so far unexplained dichotomy in how the $e-p$ interactions affect differently the materials where the frontier orbitals are derived from the $sp$- and $d$ orbitals. To develop a broadly applicable model, we employ state-of-the-art first-principles calculations that explain how phonons promote interactions between the $\mathrm{Ti}-3d$ orbitals of the conduction band within the octahedral crystal field. The characteristic difference in $e-p$ interactions experienced by the $\mathrm{Ti}-3d$ orbitals of rutile $\mathrm{Ti}{\mathrm{O}}_2$ crystal lattice are contrasted with the more familiar behavior of the $\mathrm{Si}-2s$ orbitals of stishovite $\mathrm{Si}{\mathrm{O}}_2$ polymorph, in which the frontier $2s$ orbital experiences a similar crystal field with the opposite effect. The findings of this analysis of how $e-p$ interactions affect the $d$- and $sp$-orbital derived bands can be generally applied to related materials in a crystal field. The calculated temperature dependence of $d$-orbital derived band energies agrees well with and explains the temperature-dependent inter-$d$-band transitions recorded in 2PP spectroscopy of $\mathrm{Ti}{\mathrm{O}}_2$. The general understanding of how $e-p$ interactions affect $d$-orbital derived bands is likely to impact the understanding of temperature-dependent properties of highly correlated materials.

Figure 1

2PP spectra of $\mathrm{Ti}{\mathrm{O}}_2$ taken with UV excitation between 370 and 330 nm with $p$-polarized light. The maximum intensity at 3.44–3.54 eV (350–360 nm) corresponds to the $t_{2g}-e_g$ resonance at 300 K for the sample aligned with the $\left[1\overline{1}0\right]$ axis in the optical plane.

Figure 2

Wavelength-dependent 2PP measurements for 600, 300, 100, and 31 K showing the position of the final photoelectron energy at which the $e_g$ state is observed vs the photon energy. The intermediate $e_g$ state energy decreases with temperature. A slope of 1 is fixed in fitting the photon energy dependence of the 2PP peak energies for all temperatures; the intercept of each plot gives the $e_g$-state energy relative to $E_F$. The data are obtained from $s$-polarized light for the [001] aligned sample.

Figure 3

Experimental change of the $e_g$ and $t_{2g}$ state energies with temperature from their reference positions at 300 K. The vertical arrows and dotted lines indicate the total shifts for each band in the measured temperature range.

Figure 4

(a) CBM orbitals of $\mathrm{Ti}{\mathrm{O}}_2$ and $\mathrm{Si}{\mathrm{O}}_2$ are plotted within their unit cells (red balls-O ions, gray balls-Ti ions, blue balls-Si ions, yellow-CBM orbitals). (b) Calculated VBM- and CBM-state energy shifts for $\mathrm{Ti}{\mathrm{O}}_2$ and $\mathrm{Si}{\mathrm{O}}_2$. The VBM shifts are caused by the O-$2p$ orbitals, and are similar for $\mathrm{Ti}{\mathrm{O}}_2$ and $\mathrm{Si}{\mathrm{O}}_2$; for CBMs, the Ti-$3d$ and Si-$2s$ orbitals shift oppositely. Each type of orbital has a characteristic temperature-dependent behavior. (c) The $e-p$ factor $f_{jq}$ for phonons of increasing frequencies for the CBM states of $\mathrm{Ti}{\mathrm{O}}_2$ (gray) and $\mathrm{Si}{\mathrm{O}}_2$ (blue), which cause their opposite shifts.

Figure 5

Calculated temperature dependence of the $e-p$ renormalization of the CBM ($t_{2g}$) and $e_g$ states.

Figure 6

Theoretical and experimental resonance energies for optical transitions from the $t_{2g}$ to $e_g$ states. The dotted line represents the theoretical prediction including only the $e-p$ renormalization. The solid line represents the total theoretically predicted shift, which also includes the thermal expansion (TE). By considering both contributions, the theoretical value comes into excellent agreement with the experimental data (red points).

Figure 7

(a) Electronic orbitals ($t_{2g}$ and $e_g$) of $\mathrm{Ti}{\mathrm{O}}_2$ involved in the optical transitions that contribute to the 2PP spectra at the $\mathrm{\Gamma }$ point. The $t_{2g}$ state has the $d_{xy}+d_{z^2}$ orbital character while the $e_g$ state has the $d_{xz}+d_{yz}$ orbital character [47, 81]. (b) The $e-p$ factor with respect to phonon frequencies for the $t_{2g}$ and $e_g$ states.