X-ray absorption linear dichroism at the Ti K-edge of rutile (001) TiO2 single crystal

Linear dichroism at the Ti K-edge is investigated in rutile TiO2 single crystal. A complete assignment of the pre-edge is provided based on finite difference method calculations and spherical tensor analysis. The origin of weak peak A2 similar to anatase TiO2 is discussed from the presence of oxygen vacancies or from an intrinsic quadrupolar transition.

1 Sample synthesis and characterization 1

.1 Synthesis
(001)-oriented TiO 2 rutile thin films were prepared by Pulsed Laser Deposition. The vacuum chamber was evacuated to a base pressure of 10 −5 Pa. The samples were grown by ablating a commercially available TiO 2 target with a KrF excimer laser (Lambda Physik LPX 300, 25 ns pulses, λ = 248 nm). The energy density was set at 3.0 J/cm 2 with a spot size of about 1.5 mm 2 .
The films were grown under an O 2 partial pressure of 10 Pa. The target-substrate distance was fixed at 40 mm. The rutile films were grown on 10 × 10 × 0.5 mm double side polished, (1010)oriented sapphire substrates (CrysTec GmBH). The substrate temperature was set at 670 • C read out using a pyrometer directed onto a spot of Pt paste painted on the sample holder near the substrate edge. The thermal contact between substrate and heating stage was provided by Pt paste.

X-ray diffraction and rocking curve
The crystalline structure of the samples was determined by X-ray Diffraction (PANalytical X'pert Pro MPD with Cu K α1 radiation at 1.540Å). The X-ray diffraction pattern of the TiO 2 thin film is shown in Figure 1a which confirms the (001)-orientation of the film. The diffraction pattern shows two peaks at 62.9 • and 68.5 • corresponding to the (002) diffraction of r-TiO 2 and the (1010) diffraction of the M-sapphire substrate, respectively. Figure 1b shows a rocking curve measured at the rutile (002) diffraction peak. The full width at half maximum is 0.48 • associated to a mean crystallite size of about 3.4 nm. The in-plane lattice mismatch between the film (a rutile = 4.594Å) and the substrate (a sapphire = 4.785Å) is about -4.0 %. Despite this considerable value, the Mplane sapphire represents a suitable platform for an epitaxial growth of (001)-oriented r-TiO 2 .

Ellipsometry
The ellipsometry measurements were carried out at room temperature with a SE-2000 spectroscopic ellipsometer from Semilab and microspot optics in rotating attenuator mode. Prior to each measurement, the vertical position of the sample was optimized for maximal reflection intensity.
The absorption coefficient was measured by spectroscopic ellipsometry on the thin film of r-TiO 2 with L the sample thickness, E the energy of the resonance and n the refractive index, we estimate a thickness fluctuation of 3.7Å which is less than 1% of the total film thickness. and fitting (blue line).

X-ray reflectometry
The X-ray reflectometry was measured with a Bruker D8 Advance (Cu K α1 radiation at 1.540Å).
A vertical slit of 0.1 mm is used to limit the vertical extension of the X-rays at grazing incidence.
Soler slits are used to select the diffracted beam. The experimental X-ray reflectometry is shown in Figure 2b. The experimental data was fitted with the open source GenX software [3]. Effect of incident intensity and background intensity variation were included. A bulk alumina substrate was considered with a bulk r-TiO 2 layer on top (a = b = 4.593Å, c = 2.959Å, α = β = γ = 90 • ).
Surface roughness was included for both the r-TiO 2 layer and the alumina substrate. The r-TiO 2 layer thickness is fitted to 45.9 ± 0.5 nm.
2 Fitting of the pre-edge peaks The pre-edge in the experimental data was fitted in Matlab with a set of four gaussians while the pre-edge from the theory was fitted with a set of four pseudo-Voigt functions. Overlap between the spectra and the fits are shown in Figure 3. Fittings of experimental spectra at a given incidence angle are shown in Figure 4. The X-ray absorption spectrum (XAS) amplitude in the experiment and in the theory do not match because the experiment deals with a number of photons detected by an APD in a given emission cone while the calculations provide absorption cross-sections. The detected fluorescence yield per incident photon depends on the linear absorption coefficient µ in a non-linear fashion [4], where is the probability to emit a X-ray fluorescence photon, ∆Ω the solid angle of detection, µ f the linear absorption coefficient of the fluorescence through a path length d inside the sample. In the thin film approximation, (µ − aµ f )d → 0 such that the fluorescence yield becomes, The normalization at the isosbestic points predicted by the theory accounts for the changes in ∆Ω providing spectra proportional to µ. The linear absorption coefficient is linearly related to the absorption cross-section σ by, where N is the atomic density. Hence, a multiplicative coefficient needs to be applied to the experimental data after normalization to match the evolution of the theoretical absorption crosssection providing the experimental points depicted in Figure 4 of the main text.      there is a mismatch between the Euler angles in the site frame and in the crystal frame. The rotation to go from the crystal frame to the site frame is given by the matrix R, We would like to go from the site frame to the crystal frame which is given by the transformation and the following quadrupolar spherical tensors, Now, we can convert these site-symmetrized tensors from the site frame to the crystal frame with the rotation R −1 in matrix 5. This is performed with the formula, where g are the Euler angles associated to the symmetry operation to go from frame 1 to the frame 2. The Euler angles associated to R −1 are (−3π/4, −π/2, 3π/4). It provides the transformed sitesymmetrized cross-sections for the dipole cross-section, and for the quadrupole cross-section, where we have used the time-reversal symmetry relation σ(l, m) * = (−1) m σ(l, −m), the fact that σ Q (4, 2) and σ Q (2, 2) are real so that σ Q (4, 2) = σ Q (2, 2) = 0 and the fact that σ Q (4, 3) = σ Q (4, 1) = 0 [5]. Now that we have the site-symmetrized tensors in the crystal frame, we can calculate the crystal symmetrized tensors which enter equations (3) and (4) in the main text.

Derivation of the crystal-symmetrized spherical tensors of rutile in the crystal frame
The coset method is a powerful way to calculate the spherical tensors averaged over the crystal from the spherical tensor symmetrized over a single site which has been developed by Brouder and coworkers [6]. The crystal-symmetrized tensor σ(l, m) X of a given site is obtained from the site-symmetrized tensor σ(l, m) by the operation sites) whose representatives are (x, y, z) and (−y + 1 2 , x + 1 2 , z + 1 2 ). The symmetry operation which allows going from (x, y, z) to (−y + 1 2 , x + 1 2 , z + 1 2 ) is represented by the rotation matrix R (without translations), The Euler angles associated to R −1 required in equation 12 are (0, 0, −π/2) in ZYZ convention. We can now calculate the crystal-symmetrized tensors in the crystal frame and give an expression with respect to the site-symmetrized tensors in the site frame by taking the average over the equivalent sites. It provides for the dipole cross-section, and for the quadrupole cross-section, Introducing the values of the spherical tensors in Tables 1 and 2 we can now calculate the expected expression of the dipole and quadrupole cross-section assuming a given final state represented by a single orbital in the monoelectronic approximation. This provides the angular dependencies for the dipole and quadrupole cross-section given in Table 2 of the main text.