1. Introduction

Titanium dioxide (TiO₂) is a wide-band-gap insulator which exhibits superior properties in a wide range of applications (Liu & Aydil, 2009; Miyoshi et al., 2018; Gao et al., 2019). Its natural polymorph in bulk crystals is rutile TiO₂ (r-TiO₂) in which TiO₆ octahedra with D_2h point group symmetry are connected to ten neighbouring octahedra via a corner or an edge (Fig. 1). Anatase is another polymorph of TiO₂ (a-TiO₂) which becomes the most abundant phase in nanomaterials. It is also made of TiO₆ octahedra but in a different spatial arrangement which generates an indirect band gap, contrary to rutile. Although a-TiO₂ exhibits more appealing properties in a lot of applications such as photocatalysis (Linsebigler et al., 1995), the two polymorphs exhibit similar performances in photovoltaics (Park et al., 2000).

Figure 1 Conventional unit cell of rutile TiO2. Two equivalent Ti sites are present in the unit cell at the corners and in the centre which correspond to different orientations of the TiO6 octahedra.

Optoelectronic device performances are in part governed by the mobility of photogenerated charge carriers. The mobility can be reduced by coupling with the lattice modes, generating heavier quasiparticles which can localize in a few unit cells and slow down their recombination (Okamoto et al., 2010, 2011). Anatase TiO₂ is prone to lattice deformation which generates self-trapped excitons under band gap excitation (Tang et al., 1995) while free excitons and bound excitons are observed in the photoluminescence spectra of rutile (Amtout & Leonelli, 1992). Recent progress in the description of the bound excitons spatial extension (Baldini et al., 2017a,b) has shown that experimental techniques which are more sensitive to the local character of photogenerated quasiparticles are required to obtain the fine details of their geometrical and electronic structure.

XAS at the transition metal K-edge is an element-specific spectroscopy technique which can access both the electronic structure and local geometry with subatomic spatial resolution (Rehr & Albers, 2000). It is usually decomposed into three spectral regions around the absorption edge which correspond to the absorption of the X-ray photon by an electron in the 1s orbital: (i) the extended X-ray absorption fine structure (EXAFS) at >50 eV above the edge corresponds to single scattering events of the photoemitted electron by surrounding atoms – it provides information about the bond distances and is straightforward to interpret (Hollas, 2004); (ii) the X-ray absorption near-edge structure (XANES) at the absorption edge up to ∼50 eV above it corresponds to multiple scattering (MS) events of the photoemitted electrons which provide information about bond angles and distances. XANES is usually more complicated to interpret because of the contribution of transitions to the continuum and below to diffuse 4p orbitals, as well as to MS; (iii) the pre-edge exhibits weak features below the absorption edge which are due to transitions to bound states above the Fermi level. These transitions can have quadrupolar and/or dipolar character depending on the selection rules and the local symmetry around the atom absorbing the X-radiation. Since the pioneering work of Brümmer and Dräger, it is established that the dipole and quadrupolar matrix elements of the transitions can be separated with linear dichroism (LD) XAS experiments in which different crystal orientations are probed by the electric field and wavevectors of the X-rays (Brümmer et al., 1971; Dräger et al., 1988). More recent methods such as resonant X-ray emission spectroscopy (Szlachetko et al., 2014) or resonant inelastic X-ray scattering (Bagger et al., 2017) provide the same information since they involve the XAS matrix elements. Quadrupolar transitions are usually 10–100 times weaker than dipolar ones which makes them difficult to study experimentally with standard LD XAS techniques as used in this work. In this respect, the Borrmann effect can be used to enhance the amplitude of the quadrupole transitions (Pettifer et al., 2008; Tolkiehn et al., 2011).

The XAS of r-TiO₂ at the K-edge exhibits three pre-edge peaks labelled A1, A3 and B in this work.¹ Previous XAS measurements have shown a pronounced LD of the pre-edge features especially for peak A1 (Brouder et al., 1990) which undergoes a 90°-periodic 30% amplitude variation when the sample is rotated in the ([100], [010]) plane indicating a quadrupolar component. A pronounced LD is also observed in the XANES (Poumellec et al., 1991) when the sample is rotated in the ([100], [001]) plane where components along the ordinary and extraordinary axes are exchanged (Poumellec et al., 1995). The LD of peaks A1 and A3 has been confirmed more recently as well as the quadrupolar nature of peak A1 by angular-resolved resonant Auger spectroscopy (Le Fèvre et al., 2005). We have shown how the analysis of the LD can assign final states to the bound transitions in the pre-edge of a-TiO₂ (Rossi et al., 2019). In particular, for distorted octahedral systems such as a-TiO₂ and r-TiO₂, the usual crystal field splitting into t_2g and e_g final states is inappropriate and does not account for the experimental evolution of the peak amplitudes with incident electric field and wavevector (Ruiz-Lopez & Munoz-Paez, 1991). In this paper, we analyze the LD of r-TiO₂ (001) single crystal which provides an assignment of the final states involved in each pre-edge transition in the monoelectronic approximation. The novelty of the approach resides in the combination of ab initio and analytical theoretical techniques to infer the presence of given orbitals in the final states involved in the pre-edge transitions.

The assignment of the final states involved in the pre-edge goes hand in hand with theoretical modelling. Muffin-tin MS calculations are insufficient to describe the XAS of anisotropic materials which require full potential methods (Joly et al., 1999; Bunău & Joly, 2009; Kravtsova et al., 2010). The finite difference method (FDM) developed by Y. Joly (Joly et al., 1999; Joly, 2001) has shown that, in r-TiO₂, A1 is a quadrupolar transition to t_2g orbitals, A3 is essentially dipolar with a weak e_g quadrupolar contribution, and B is purely dipolar (Cabaret et al., 1999). The spatial extension of the final state can be used to address the dipolar or quadrupolar character of the transition. We have shown that the quadrupole transitions in a-TiO₂ populate a final state localized in the TiO₆ octahedron where the Ti atom absorbs the X-radiation (Rossi et al., 2019). Final states which result from on-site or off-site p–d orbital hybridization can have various spatial extensions sensitive to the first coordinating shell of TiO₆ octahedra or even further away. In this work, we investigate the spatial extension of the final states in detail using ab initio FDM calculations.

The pre-edge of a-TiO₂ and r-TiO₂ differs by the presence of a pronounced A2 peak on the low-energy side of peak A3 in a-TiO₂ which is absent in r-TiO₂. Empirical studies have shown that the amplitude of A2 is related to the amount of oxygen vacancies in a-TiO₂ while no difference is observed in the r-TiO₂ case (Luca et al., 1998; Hanley et al., 2002; Luca, 2009). Recent theoretical modelings have shown that the local structural disorder also plays an important role in the amplitude of A2 in a-TiO₂ (Zhang et al., 2008; Triana et al., 2016). Our recent LD study has shown that peak A2 is present in a-TiO₂ single crystals and is due to an intrinsic quadrupolar transition which exhibits LD (Rossi et al., 2019). It is the result of the weak p–d hybridization and p-density of states (DOS). Ab initio calculations including the explicit treatment of the core-hole interaction in the Bethe–Salpeter equation are in agreement with this assignment (Vorwerk et al., 2017) and suggest that peak A2 may be seen in r-TiO₂ on the low-energy side of peak A3 under azimuthal rotation (Shirley, 2004). The final states of peaks A1 and A2 in a-TiO₂ have orthogonal p-DOS contributions which indicate a difference in the binding energy with the core-hole. The energy splitting between these bound states has a similar origin to the bound excitonic states found in a-TiO₂ in the optical range (Baldini et al., 2017a; Chiodo et al., 2010). It is thus surprising a priori that the A2 peak is not observed in the r-TiO₂ polymorph either due to an intrinsic quadrupolar transition in the bulk or because of penta-coordinated Ti atoms due to oxygen vacancies. This is clarified in this work as we show that the degree of p–d hybridization is key to the amplitude of peak A2 in both cases.

In regard to the discrepancy of orbital assignment and spatial extension of the final state in previous studies as well as the absence of discussion about the absence of peak A2 in r-TiO₂, a combined experimental and theoretical LD XAS study is valuable to provide an accurate description of the orbital content in the pre-edge. In this work, we combine the measured XAS at different incidence angles at the surface of (001) r-TiO₂ with FDM calculations and spherical harmonic analysis (Brouder et al., 2008) to provide an unambiguous assignment beyond the usual octahedral crystal field splitting approximation which is inappropriate for r-TiO₂ (Ruiz-Lopez & Munoz-Paez, 1991). Previous assignments of the pre-edge peaks are provided in Table 1 together with the ones established in this work.

Table 1 Summary of the previous assignments of the pre-edge transitions in the XAS of r-TiO2 at the Ti K-edge Off-site transitions are shown in bold. E1 stands for dipole transitions and E2 for quadrupolar transitions. A1 A3 B Reference   3d(t2g) 3d(eg) (Fischer, 1972) E2: 3dxz     (Brouder et al., 1990) E2: 3dxz E1,E2: 3dxy,yz E1,E2: 3dxy,yz (Poumellec et al., 1991) 3dx2-y2 3d(t2g) 3d(eg) (Uozumi et al., 1992) 3d–4p 3d–4p 4p–4s (Wu et al., 1997) Core exciton 3d(t2g) 3d(eg) (Beaurepaire et al., 1993) E2: 3d(t2g) E1,E2: 3d(eg) E1: 3d–4p (Cabaret et al., 1999)   3d(t2g) 3d(eg) (Shirley, 2004) E2: 3d(t2g)     (Le Fèvre et al., 2005) E2: 3d(t2g) E1: pz–3d(t2g), E2: 3d(eg) E1: pz–3d(eg) (Cabaret et al., 2010) () E2: 3d(t2g) E1: px, y–3d(t2g) E1: px, y–3d(eg) (Cabaret et al., 2010) () E2: 3dz2,3dxy,3dx2-y2,E1: 4px,y E1: 4pz,3dxz,yz E1: 4pz (This work)

2. Experimental

The LD measurements were performed at the MicroXAS beamline of the SLS in Villigen, Switzerland, using a double Si(311) crystal monochromator to optimize the energy resolution. Energy calibration is performed from the first derivative of the XAS of a thin Ti foil. We used a moderately unfocused rectangular-shaped X-ray beam of 20 µm (horizontal) × 200 µm (vertical). The XAS is obtained in total fluorescence yield with a Ketek Axas detector system with Vitus H30 SFF and ultralow capacitance Cube-Asic preamplifier (Ketek GmbH).

The sample consists of a (001)-oriented crystalline r-TiO₂ thin film [thickness 46 nm; see §S1 of the supporting information (SI) for details on the sample growth and characterizations] mounted on a set of translation and rotation stages allowing a fixed centre of rotation at the sample surface. By convention, a set of Euler angles (θ, ϕ, ψ) orients the electric field and wavevector with respect to the sample crystallographic axes (Fig. 2). θ measures the angle between and the [001] crystal direction ( axis of the sample frame) orthogonal to the surface. ϕ measures the angle between and the sample rotation axis . In principle, a third angle ψ is necessary to fix the position of the wavevector in the orthogonal plane to the electric field but here ψ = 0°. The θ angles reported in the experimental datasets are with a maximum systematic offset of ±0.2° which comes from the precision setting up the θ = 0° reference from the sample half-clipping of the X-ray beam at grazing incidence. The precision of the rotation stage of ±0.01° is negligible with respect to this angular offset.

Figure 2 Linear dichroism experimental geometry and angular conventions with (a) side view and (b) top view. The sample surface is in grey while the incident X-ray beam is in pink. A set of Euler angles (θ, ϕ, ψ) is used to orient the electric field and wavevector of the incident X-ray beam with respect to the sample.

LD is usually studied with the sample rotated in the plane orthogonal to the incident X-ray beam (ϕ-rotation) (Brouder et al., 1990). Here, the sample is rotated around (θ-rotation) where the changes in the XAS are the most pronounced. Under this rotation, the X-ray footprint onto the sample surface changes with θ. However, we clearly show that this procedure does not introduce spectral distortions because the effective penetration depth of the X-rays through the material [between ∼85 nm and ∼540 nm across the absorption K-edge of r-TiO₂ for the largest footprint at θ = 1° used here (Henke et al., 1993)] is kept constant as the sample is thinner than the attenuation length at the Ti K-edge. Instead, the total amount of material probed by the X-rays changes due to the larger X-ray footprint when θ increases and a renormalization over the number of detected X-ray fluorescence photons is required. This is done with the support of finite difference method near-edge structure (FDMNES) calculations (vide infra) since a few energy points have θ-independent cross-sections as previously reported on other systems (Frétigny et al., 1986; Stizza et al., 1986; Oyanagi et al., 1987, 1989; George et al., 1989; Loupias et al., 1990; Pettifer et al., 1990; Rossi et al., 2019).² With this renormalization procedure performed at a single energy point (4987.3 eV), we could obtain a set of experimental points with θ-independent cross-sections at the energies predicted by the theory confirming the reliability of the method (vide infra). Hence, crystalline thin films with suitable thicknesses with respect to the X-ray penetration depth offer more possibilities to study LD effects than single crystals and prevent the usual self-absorption distortion of bulk materials using total fluorescence yield detection (Carboni et al., 2005).

2.2. Spherical tensor analysis of the dipole and quadrupole cross-sections

Analytical expressions of the dipole and quadrupole XAS cross-sections [ and , respectively] are obtained from their expansion into spherical harmonic components (Brouder, 1990; Brouder et al., 1990). The expressions of and depend on the crystal point group which is D_2h (2/mmm) for r-TiO₂. For the dipole cross-section we obtain

and, for the quadrupole cross-section,

where (θ, ϕ, ψ) are Euler angles in the Ti point group frame. σ^X(l, m) with X = D, Q is the spherical tensor with rank l and projection m. The Euler angles (θ, ϕ, ψ) in the experiment are referenced to the crystal frame which is rotated with respect to the Euler angles in the Ti site frame. Consequently, the angles in equations (1) and (2) differ from the angles defined in Fig. 2 by rotation around θ and ϕ. For instance, the axis of the site frame is rotated by an angle π/2 around a bisecting axis in the plane of the site frame with respect to the crystal frame. Consequently, the formulas (1) and (2) need to be modified to match the incidence angle θ used in the experiment [details in §S7 (SI)]. The dipole cross-section is now given by

and the quadrupole cross-section by

Although some terms of and may be negative, the total dipolar and quadrupolar cross-sections must be positive putting constraints on the values of σ^D(l, m) and σ^Q(l, m). The electric field and wavevector coordinates in the basis of Fig. 2 are given by

Hence the detail of the XAS cross-section angular dependence requires an estimate of the spherical tensors σ^D(l, m) and σ^Q(l, m) as performed elsewhere (Brouder et al., 2008). The XAS measured experimentally is an average over equivalent Ti atoms under the symmetry operations of the crystal point group. The analytical formula representing this average requires the site symmetrization and crystal symmetrization of the spherical tensors, which is provided in §S7 and §S7.2 (SI). In the case of θ-rotation at ϕ = 0° described in this paper, the two equivalent Ti sites have the same XAS spectra. We provide spectra calculated at the two equivalent Ti sites of r-TiO₂ under θ [Fig. S8 (SI)] and ϕ rotation [Fig. S9 (SI)] showing this effect. Assuming pure 3d and 4p final states in the monoelectronic approximation, analytical expressions are provided for and whose angular dependence with θ are given in Table 2. The full expressions of the cross-sections are provided in §S7.4 (SI). In this paper, we analyze the angular dependence of the pre-edge peak intensities with θ and assign them to specific final states corresponding to Ti-3d and/or 4p orbitals with the support of both FDM and spherical tensor analysis.

Table 2 Angular evolution of the dominant term in the dipole and quadrupole cross-section with Euler angle θ as defined in the main text for a given final state in the monoelectronic approximation Final state or θ-dependence px,py pz dz2 dx2-y2 dxy dxz dyz

3. Results

The experimental evolution of the normalized XAS with angle θ is depicted in Fig. 3(b) with an emphasis on the pre-edge given in Fig. 3(a). The results are similar to previous published spectra under similar rotation between [001] and (Aïfa et al., 1997). In the pre-edge, the amplitudes of peaks A3 and B are notably affected by the sample orientation with a red shift of the maximum for peak A3 and a blue shift for peak B with θ. Strong changes are also observed in the post-edge region showing that the LD remains well above the edge. Calculated spectra using FDMNES are shown in Figs. 3(c) and 3(d) with the same polarization conditions as in the experiment. The evolution of the peak B amplitude and energy shift is nicely reproduced while a red shift with θ appears for peak A3 with an underestimated amplitude change. The ratio between A3 and B amplitudes changes between the experiment and the theory [Figs. 3(a) and 3(c)] similarly to previous calculations with the same computational method (Joly et al., 1999) but also with full multiple scattering (Zhang et al., 2008). The evolution of peak A1 is weak in accord with the experiment and with previous results which assigned it to a purely quadrupolar transition (Uozumi et al., 1992; Parlebas, 1993; Parlebas et al., 1995). An excellent reproduction of the experimental data is also obtained above the edge where the isosbestic points are found in agreement, marked by black arrows in Fig. 3. This shows the absence of self-absorption effects in the experimental data which would distort the spectra (Carboni et al., 2005).

Figure 3 (a), (b) Experimental and (c), (d) theoretical evolution of the XAS at the Ti K-edge of r-TiO2 for different angles of incidence θ. (a), (c) Pre-edge region. (b), (d) XANES and EXAFS. Isosbestic points are shown with black arrows. Thick lines in (c) are spectra calculated with dipole and quadrupole matrix elements; thin lines show the quadrupole matrix elements contribution only.

We have fitted the pre-edge peaks with Gaussians for the experimental data and with pseudo-Voigt functions for the theoretical data in order to extract the evolution of the amplitude with the incidence angle θ. The fitted amplitudes of the three experimental pre-edge edge peaks appear as black circles with error bars in Fig. 4. The vertical scale is modified to match the theoretical evolution of the XAS cross-section according to a procedure described in §S2 (SI). Fittings to individual spectra are given in Fig. S4 (SI). Peak A1 undergoes an almost pure quadrupolar evolution with 90°-periodicity, reaching its maximum amplitude for θ = 45° [Fig. 4(a)]. However, we notice a slight deviation to this behaviour with the cross-section being larger when θ → 90° than when θ → 0°. This behaviour is in excellent agreement with the theory [blue circles in Fig. 4(a)] which indicates a weak dipolar contribution underlying the overwhelming quadrupolar one. Experimentally, both peaks A3 and B have a monotonic decrease with θ which does not match the theory for peak A3 in which an increase in amplitude appears from θ ≃ 60° [Figs. 4(b) and 4(c)]. This is likely due to the underestimated amplitude of peak A3 and its evolution which competes with peak broadening changes with the incidence angle.

Figure 4 Evolution of the experimental (black circles with error bars) and theoretical (coloured circles) amplitudes of the pre-edge peaks (a) A1, (b) A3 and (c) B in r-TiO2 going from an electric field along [001] (θ = 0°) to [100] (θ = 90°). The error bars represent 95% confidence interval in the fitted experimental amplitude. The sum of dipole and quadrupole components is fitted in the theoretical data. Details about the fitting procedure and the fitting results are given in §S2 and §S3 (SI).

In order to assign the 3d and 4p orbitals contributing to the final state of the pre-edge transitions, we look at the DOS in the pre-edge provided by FDMNES [Fig. S7 (SI)] as well as the expected evolution from the spherical tensor analysis (Table 2).

In the region of peak A1, the p-DOS is negligible [Fig. S7(b) (SI)] while the DOS of all possible d-states is present except d_yz [Fig. S7(c) (SI)]. This is in agreement with the dominant quadrupolar evolution observed experimentally [Fig. 4(a)], compatible with 3d orbitals contributing to the final state. Similar results were obtained by others under ϕ-rotation (Parlebas, 1993; Parlebas et al., 1995). The d_z² and d_xy-DOS is ∼500 times larger than the d_xz, d_yz-DOS [Fig. S7(c) (SI)]. The spherical tensor analysis of the cross-section involving d_z² and d_xy orbitals in the final state gives a maximum for θ = 45° [Fig. S6(b) (SI)], in agreement with the experiment [Fig. 4(a)] and the calculated DOS [Fig. S7(c) (SI)]. The deviation from the ideal quadrupolar evolution goes in the direction of a weak hybridization with 4p_x,y states which reach their maximum amplitude for θ = 90° according to the spherical tensor analysis [Fig. S6(a) (SI)].

The experimental amplitude of peak A3 decreases [Fig. 4(b)], reflecting the dominant contribution of the p_z-DOS to the transition according to the spherical tensor analysis [Fig. S6(a) (SI)] in agreement with the calculated DOS [Fig. S7(b) (SI)] and conclusions from previous studies (Parlebas et al., 1995). However, the calculated p_z-DOS is only slightly larger than the p_x,y-DOS [Fig. S7(b) (SI)] giving less than 10% amplitude variation with respect to the maximum in the theoretical spectra [Fig. 4(b)]. The fitted theoretical evolution of the peak A3 amplitude shows a non-monotonic behaviour indicating the overlapping contributions of a dipolar and quadrupolar transitions. The calculation of the quadrupolar cross-section indicates a contribution in the region of peak A3, which reaches its maximum for θ = 0°[90°] [Fig. 3(c)], in agreement with the fitted behaviour which shows that peak A3 originates from the superposition of a dipolar and quadrupolar transition. A quadrupolar evolution with maximum cross-section at θ = 0[90]° is only compatible with the presence of d_xz, d_yz states according to the spherical tensor analysis [Fig. S6(b) (SI)]. It corresponds to the largest d-DOS found in the region of peak A3 [Fig. S7c) (SI)]. The quadrupolar contributions of peaks A1 and A3 found in this work are in agreement with temperature-dependent XAS studies in which only quadrupolar transitions undergo a change in cross-section at the metal K-edge (Durmeyer et al., 2010; Brouder et al., 2010).

At the peak B, the p_z-DOS dominates over p_x,y [Fig. S7(b) (SI)] while all d-states have similar contributions except d_xy [Fig. S7(c) SI)]. Experimentally, peak B shows a monotonic decrease in amplitude with θ both experimentally and theoretically [Fig. 4(c)] which is compatible with the dominant p_z-DOS identified in our calculations [Fig. S7(b) (SI)] and the spherical tensor analysis which predicts a monotonic decay when p_z-DOS is dominant [Fig. S6(a) (SI)]. Peaks A3 and B are also observed by bremsstrahlung isochromat spectroscopy (BIS) which shows that they originate from essentially empty p-DOS which is due to the strong 3d–4p hybridization for peak A3 and the 4p-DOS for peak B (Beaurepaire et al., 1993), in agreement with our results. The absence of peak A1 in BIS is due to the absence of core-hole effects (Hüfner, 2013).

Hence, using a combination of ab initio FDM calculations and spherical tensor analysis with our experimental LD, the assignment of the dominant final states contributing to the pre-edge transitions at the Ti K-edge of r-TiO₂ is established. The assignments are given in Table 1.

4. Discussion

4.1. Spatial extension of the final states in the pre-edge

The spatial extension of the final states involved in the pre-edge transitions can be approached from the calculations of XAS for various cluster sizes involving more and more distant coordination shells around the absorbing titanium atom. Fig. 5 shows calculated spectra for increasing cluster sizes from the smallest TiO₆ octahedron up to a cluster with 5.5 Å radius involving the second nearest-neighbour shell of Ti atoms. Peak A1 can be reproduced with a simple TiO₆ octahedron in agreement with its quadrupolar nature involving 3d final states centred on the absorbing Ti atom (red curve). Peaks A3 and B are absent for the TiO₆ octahedron showing that they involve final states which are delocalized away from the central TiO₆ octahedron. The former appears when the nearest coordinating shell of TiO₆ octahedra is included for a cluster size of 4 Å (green curve) while the latter involves the next nearest neighbour shell which appears for a cluster size of 5.5 Å (purple curve). These results shine a light on the sensitivity of pre-edge peaks to different length scales around the atoms absorbing the radiation. It is clear that the p–d hybridization occurring in the final states of peak A3 and the pure p-DOS component in peak B contribute to their increased sensitivity to coordinating shells. These results are similar to the results obtained on a-TiO₂ for the peaks A3 and B (Rossi et al., 2019).

Figure 5 Influence of cluster size on the calculated XAS cross-section (sum of dipole and quadrupole components) (a) for θ = 90° and (b) for θ = 0°.

4.2. Appearance of transitions to pentacoordinated titanium atoms in the pre-edge

A remarkable difference between the XAS of a-TiO₂ and r-TiO₂ is the absence of a peak A2 in the latter. Previous assignments of this peak in a-TiO₂ include undercoordinated titanium atoms due to oxygen vacancies (Farges et al., 1997; Luca et al., 1998; Luca, 2009) and structural disorder (Zhang et al., 2008; Triana et al., 2016). At high defect concentrations, both phases converge to an amorphous phase where the A2 peak becomes dominant. EXAFS studies have shown that r-TiO₂ retains a crystalline structure when nanoparticle materials are synthesized by sol-gel (Manzini et al., 1995) or calcined at different temperatures (Luca, 2009), which may explain the absence of peak A2. Wu and co-workers have reported changes in the pre-edge of r-TiO₂ at the Ti K-edge from bulk to nanosized materials with a red shift of peak A3 and a red shift and decrease of the amplitude of peak B, while peak A1 is unaffected (Wu et al., 2002). The most pronounced differences are located in the XANES. Lemercier and co-workers have shown that, under UV irradiation, peak A3 undergoes a red shift which is compatible with the formation of Ti³⁺ centres at the surface of r-TiO₂ (LeMercier et al., 1995). Recently, a combined experimental and theoretical study at the Ti L_2, 3-edge of r-TiO₂ nanoparticles with oxygen vacancies has shown blue-shifted peak replicas from the t_2g and e_g derived final states which are due to V_O ²⁺ vacancies (Vásquez et al., 2016). Similar peaks are observed in a-TiO₂ nanoparticles at the same edge (Thakur et al., 2011; Krüger et al., 2017). It is thus surprising that defect-related peaks have been observed in only a few studies at the Ti K-edge of r-TiO₂ while several studies at the O K-edge and Ti L_2,3-edge report well defined resonances due to oxygen vacancies (Thomas et al., 2007; Thakur et al., 2011; Chen et al., 2015; Tian et al., 2015; Vásquez et al., 2016). They often appear as extra peaks on the high-energy side of the bulk resonances. These resonances become more pronounced with surface-sensitive techniques such as electron energy-loss spectroscopy (EELS) (Göpel et al., 1984; Eriksen & Egdell, 1987; Henderson et al., 2003), which shows the importance of polaron states in the trapping of charges at oxygen vacancies (Eriksen & Egdell, 1987). These results support our approach of using a relaxed lattice structure around V_O ²⁺ oxygen vacancies (O_vac) to find the expected pre-edge peaks due to pentacoordinated titanium atoms at the Ti K-edge.

A qualitative simulation of the Ti K-edge XAS of undercoordinated Ti atoms with O_vac in r-TiO₂ can help in understanding the expected changes from the bulk structure. For these simulations with the full potential FDM, we have used the relaxed local structure from density functional theory calculations with hybrid functionals for the exchange and the correlation terms reported in an earlier work (Janotti et al., 2010). A bulk 4 × 4 × 4 supercell of r-TiO₂ is considered first, which contains 739 atoms. A doubly ionized oxygen vacancy (V_O ²⁺) close to the centre of the supercell is simulated by removing an oxygen atom and displacing the nearest Ti atoms by the bond distances reported by Janotti et al. (2010) along the broken Ti—O bonds due to the O_vac. The XAS are calculated separately at the Ti K-edge of the atoms having the O_vac in equatorial or in apical position. The results are depicted in Fig. 6 (blue and red thick lines). Similarly to our previous results on a-TiO₂ (Rossi et al., 2019), we find that, for the equatorial O_vac, peak A1 is absent while a peak appears on the low-energy side of peak A3 (red curve). The XAS of the apical O_vac shows mainly one peak centred at the position of peak A3. A similar trend has been previously obtained by others with a smaller cluster size in the multiple scattering approach (Jeanne-Rose et al., 1997). This additional peak is consistent with DOS appearing at the Ti L_2,3-edge upon formation of oxygen vacancies on the blue side of t_2g and e_g-DOS (Vásquez et al., 2016). These results show that a defect-related resonance due to pentacoordinated Ti atoms may be expected on the low-energy side of peak A3, where the peak A2 is present in a-TiO₂, which originates from the relaxed projected DOS of Ti atoms in the vicinity of an O_vac. We noticed that some experimental studies at the Ti K-edge of r-TiO₂ seem to show a peak A3 asymmetry on the red side, suggesting the presence of peak A2 (Manzini et al., 1995; Finkelstein et al., 2002). The spectrum measured by Finkelstein and co-workers is shown in Fig. 6 (blue circles) in which the shoulder at ∼4958 eV is in good agreement with the position of the maximum in our computed spectra with O_vac. We suspect that peak A2 can only be observed with a substantial amount of V_O ²⁺ and a monochromator with high energy resolution. The low amount of such vacancies in our crystalline r-TiO₂ (001) thin films partially explains the absence of peak A2 since the other possible contribution to the formation of this peak is a quadrupolar transition from the d-DOS which is discussed in Section 4.3.

Figure 6 Evolution of the pre-edge of r-TiO2 upon formation of VO 2+ at the Ti K-edge. The spectra are calculated for a Ti atom in the bulk structure supercell (black curve), with VO 2+ at the apical (blue curve) or the equatorial position (red curve). The calculations are without polarization of the X-ray field (angular average). A linear combination of the apical and equatorial Ovac spectra gives a spectrum representative of the titanium atoms surrounding the Ovac (green curve). Experimental data from r-TiO2 nanoparticles are reproduced from Finkelstein et al. (2002) (blue circles). The spectral trace 100 ps after the excitation of r-TiO2 (black line with circle markers) is reproduced from Budarz et al. (2017). The horizontal dashed line represents the zero signal level of the X-ray absorption transient.

Picosecond XAS of photoexcited TiO₂ nanoparticles shows an enhancement of the spectral signature of trapped charges around metal centres at defects (Rittmann-Frank et al., 2014). Our picosecond Ti K-edge study of r-TiO₂ nanoparticles (Budarz et al., 2017) shows the reduction of pentacoordinated Ti centres in the defect-rich shell of the nanoparticles. The transient spectrum at 100 ps after excitation shows an increase in amplitude on the red side of peak A3 and a decrease in peak B amplitude (black line with circle markers in Fig. 6). To model the XAS of Ti³⁺ centres with a trapped electron without the polarization information, we have taken a linear combination of the calculated apical and equatorial O_vac XAS (one-third of the first and two-thirds of the second in agreement with the relative weight of these Ti atoms around the vacancy) which is shown as a green curve in Fig. 6. A net positive signal appears on the red side of peak A3 and a decrease at peak A1 and B with respect to the bulk spectrum (black curve) in agreement with the transient spectrum (Budarz et al., 2017). The good qualitative agreement between our simulations and picosecond XAS is evidence of the presence of peak A2 buried under the red side of peak A3 in the XAS of r-TiO₂ with oxygen vacancies such as in nanoparticles.

We have recently shown that peak A2 is not only due to oxygen vacancies but can also originate from an intrinsic quadrupolar transition in a-TiO₂ which becomes dominant in single crystals (Rossi et al., 2019). We investigate the presence of such transition in the pre-edge of r-TiO₂.

5. Conclusion

In conclusion, we have experimentally investigated in detail the linear dichroism of a rutile TiO₂ (001) thin film and analyzed it with the support of FDMNES ab initio calculations and spherical tensor analysis. An unambiguous assignment of the pre-edge transitions is provided which completes the work for the most common rutile and anatase (Rossi et al., 2019) polymorphs of TiO₂. The effect of an oxygen vacancy on the XAS of a pentacoordinated Ti atom is investigated for the first time and compared with similar calculations on anatase TiO₂. The results show that peak A2 is only due to the presence of pentacoordinated titanium atoms at the Ti K-edge of r-TiO₂, in agreement with picosecond XAS (Budarz et al., 2017). The quadrupolar transition which may be expected from the d-DOS in the region of peak A2 cannot be observed because of the limited p–d hybridization in rutile contrary to anatase TiO₂ (Rossi et al., 2019).

6. Related literature

The following references, not cited in the main body of the paper, have been cited in the supporting information: Mo & Ching (1995); DeVore (1951); Björck & Andersson (2007); Als-Nielsen & McMorrow (2011).