Valence-selective local atomic structures in inorganic materials by X-ray fluorescence holography

Holographic oscillation data were obtained by correcting a geometric background of the sample from fluorescent X-ray intensities and normalizing them to incident X-ray intensities usually measured using an ion chamber. Extensions of the holographic data to the 4π sphere were usually carried out using crystal symmetry of the sample and measured X-ray standing wave lines.¹³⁾

A hologram is an intensity modulation of fluorescent X-rays with varying incident angles, and is expressed as

$$\begin{eqnarray}&&\chi ({\boldsymbol{k}})=-\int \rho ({\boldsymbol{r}})\displaystyle \frac{\cos ({kr}-{\boldsymbol{k}}\cdot {\boldsymbol{r}})}{r}\,d{\boldsymbol{r}},\end{eqnarray} \tag{ 1 }$$

where $\chi ({\boldsymbol{k}})$ and $\rho ({\boldsymbol{r}})$ indicate a hologram pattern and an electron density function, respectively, and the origin of ${\boldsymbol{r}}$ is set at the fluorescence emitter atom. In Eq. (1), a polarization effect is not included to avoid a complex formula depending on crystal angles.

Barton's method¹⁴⁾ is based on Fourier transforms, and is typically used for the data analysis. It is given by

$$\begin{eqnarray}&&u({\boldsymbol{r}})=-r\int \chi ({\boldsymbol{k}})\cos ({kr}-{\boldsymbol{k}}\cdot {\boldsymbol{r}})d{\boldsymbol{k}},\end{eqnarray} \tag{ 2 }$$

where $u({\boldsymbol{r}})$ is a reconstructed atomic image. It should be noted that for reconstructing a perfect 3D image, a 3D hologram in the whole ${\boldsymbol{k}}$ space is, in principle, necessary by changing k or the incident X-ray energy E ($k=2\pi E/{hc}$ with Planck constant h and speed of light c) from 0 to $\infty$, which is experimentally impossible. Thus, Barton's method employs a sum of a limited number of holograms with different k values, i.e., multi-wavelength XFH, in place of the integral by k in Eq. (2).

For valence-sensitive XFH measurements, however, the incident X-ray energy is the only one around an absorption edge, and thus, the required atomic images are reconstructed from much less experimental information. In fact, a number of artifacts including intrinsic twin images always appear in reconstructed images by Fourier transforms as shown in Refs. 3 and 4. Thus, this is a typical example of an equation for an underdetermined system, i.e. too little experimental data is input for the requested unknown parameters.

To solve this problem of an equation for an underdetermined system, a sophisticated algorithm is urgently required for obtaining reliable 3D atomic images. Matsushita recently developed a new algorithm, named SPEA-L1 (scattering pattern matrix extraction algorithm using L₁ regularized linear regression),¹⁵⁾ which is based on inverse problem and sparse modeling.¹⁶⁾

When an atomic form factor, f, is introduced, Eq. (1) is expressed as

$$\begin{eqnarray}&&\chi ({\boldsymbol{k}})=-\displaystyle \sum _{i}\displaystyle \frac{f({\theta }_{{{\boldsymbol{a}}}_{i},{\boldsymbol{k}}})\cos ({{ka}}_{i}-{\boldsymbol{k}}\cdot {{\boldsymbol{a}}}_{i})}{{a}_{i}},\end{eqnarray} \tag{ 3 }$$

where ${{\boldsymbol{a}}}_{i}$ is the position of the ith atom and ${\theta }_{{{\boldsymbol{a}}}_{i},{\boldsymbol{k}}}$ is an angle between ${{\boldsymbol{a}}}_{i}$ and ${\boldsymbol{k}}$. This equation is extended by introducing an atomic distribution function, $g({\boldsymbol{r}})$, as follows:

$$\begin{eqnarray}&&\chi ({\boldsymbol{k}})=-\int g({\boldsymbol{r}})f({\theta }_{{\boldsymbol{r}},{\boldsymbol{k}}})\cos ({kr}-{\boldsymbol{k}}\cdot {\boldsymbol{r}})d{\boldsymbol{r}}.\end{eqnarray} \tag{ 4 }$$

Here, the element of neighboring atoms cannot be identified in advance. A standard f function is thus assumed, which is atomic number Z normalized f function averaged over constituent elements, since spectral forms of f/Z are quite similar to one another. By using small voxels for describing $g({\boldsymbol{r}})$,¹⁷⁾ Eq. (4) is modified as

$$\begin{eqnarray}&&\chi ({{\boldsymbol{k}}}_{j})=-\displaystyle \sum _{i}g({{\boldsymbol{r}}}_{i})f({\theta }_{{{\boldsymbol{r}}}_{i},{{\boldsymbol{k}}}_{j}})\cos ({k}_{j}{r}_{i}-{{\boldsymbol{k}}}_{j}\cdot {{\boldsymbol{r}}}_{i}).\end{eqnarray} \tag{ 5 }$$

Since the $g({{\boldsymbol{r}}}_{i})$ function is very sparse, i.e. atomic numbers are very limited in space, an L₁ regularized linear regression¹⁶⁾ is applicable. For this, an evaluation function is given by

$$\begin{eqnarray}&&E=\displaystyle \sum _{j}| \chi ({{\boldsymbol{k}}}_{j})-\hat{\chi }({{\boldsymbol{k}}}_{j}){| }^{2}+\lambda \displaystyle \sum _{i}| g({{\boldsymbol{r}}}_{i})| ,\end{eqnarray} \tag{ 6 }$$

where $\hat{\chi }({{\boldsymbol{k}}}_{j})$ is an experimental hologram and λ is a penalty parameter to decrease $g({{\boldsymbol{r}}}_{i})$. To obtain $g({{\boldsymbol{r}}}_{i})$, a minimization of E is carried out by an iterative calculation of

$$\begin{eqnarray}&&{g}^{(n+1)}({{\boldsymbol{r}}}_{i})={g}^{(n)}({{\boldsymbol{r}}}_{i})-\alpha \displaystyle \frac{\partial {E}^{(n)}}{\partial g({{\boldsymbol{r}}}_{i})}\end{eqnarray} \tag{ 7 }$$

where n is an index for iteration. The parameter α is optimized using a gradient method. Note that a non-negative constraint is applied to the voxel value $g({{\boldsymbol{r}}}_{i})$.

The estimation of λ is very important. When λ is increased, the amount of zeros in the voxels is increased, and the system becomes more sparse. When λ = 0, on the other hand, Eq. (9) goes to the usual least squares method with many artifacts in the atomic images. In these studies, λ was determined by using

$$\begin{eqnarray}&&\lambda =\beta \max \left(-\displaystyle \frac{\partial {E}_{1}^{(n)}}{\partial g({{\boldsymbol{r}}}_{i})}\right),\end{eqnarray} \tag{ 8 }$$

with

$$\begin{eqnarray}&&{E}_{1}=\displaystyle \sum _{j}| \chi ({{\boldsymbol{k}}}_{j})-\hat{\chi }({{\boldsymbol{k}}}_{j}){| }^{2},\end{eqnarray} \tag{ 9 }$$

and β is usually set between 0.60 and 0.95, where the fit error comes to the experimental error, and λ typically reaches 2 × 10⁻⁵ at the final conditions of fits.

The voxel size is usually set to be 0.01 × 0.01 × 0.01 nm³ with a distance range of 0.6 nm for each direction from the central emitter atom. The iteration of the L₁ regularization is started from $g({{\boldsymbol{r}}}_{i})=0$. Examples of excellent improvements in atomic images by using L₁ regularized linear regression are given elsewhere.^4,18)

Pure yttrium (III) oxide, Y₂O₃, with a valence state of ${{\rm{Y}}}^{3+}$, is a well-known transparent and insulating rare-earth oxide. The crystal structure of Y₂O₃ is a distorted CaF₂ structure, having a ${Ia}\bar{3}$ crystal symmetry with lattice constant a = 0.918 68  nm in standardized form,²¹⁾ where one of the Y sites is displaced by 0.0347 nm from the ideal positions into a different direction.

On the other hand, yttrium monoxide, YO, with a valence state of ${{\rm{Y}}}^{2+}$, has recently been manufactured by pulsed laser deposition as an epitaxial thin film,²²⁾ which is dark-brown colored and a narrow gap semiconductor. In thin films, YO forms a tetragonal structure, a very slightly distorted rock salt structure, with lattice constants of a = 0.493 6 and c = 0.497 7  nm.²²⁾ The surface of YO films is easily oxidized to Y₂O₃.

Our first attempt at valence-selective XFH measurement was focused on separating the structural information of Y₂O₃ and YO in the same thin film. Although Y atoms with different vacancies are macroscopically located at different positions from each other, this sample is suitable to examine the potential of XFH as a method of valence-selective characterization of atomic structures. The obtained result can be compared with results using other techniques, such as surface XRD, reflection high-energy electron diffraction, or transmission electron microscopy (TEM).

The YO sample was deposited on a CaF₂ substrate to grow as an epitaxial layer with a thickness of about 180 nm. The top 30 nm was oxidized to become Y₂O₃, and the sample film was capped with a 10 nm AlO_x layer. Two different structures of YO and Y₂O₃ were observed by TEM, and the interface layer between YO and Y₂O₃ was as small as 2 nm.

To select appropriate energies for valence-selective XFH measurements, Y K XANES spectra were measured on the mixed Y₂O₃/YO sample and pure Y₂O₃ reference film at room temperature in fluorescence mode at BL13XU of the SPring-8, Sayo, Japan, and the results are shown in Fig. 4

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As seen in the figure, a slight shift is observed toward higher energies in the absorption edge when increasing the valence value from ${{\rm{Y}}}^{3+}$/Y²⁺ to ${{\rm{Y}}}^{3+}$. The inset shows the ratio of two curves on mixed Y₂O₃/YO and Y₂O₃. The ratio shows a peak at 17.041 keV with a height of about 1.30. Thus, ${{\rm{Y}}}^{2+}$ contributions are enhanced when the incident X-rays are chosen at this energy.

XFH measurements were carried out at BL13XU of the SPring-8 at incident X-ray energies of 17.041 and 17.054 keV as shown by the arrows in Fig. 4. The former was chosen to excite a sufficiently large portion of ${{\rm{Y}}}^{2+}$, and the latter was selected to excite both ${{\rm{Y}}}^{2+}$ and ${{\rm{Y}}}^{3+}$.

The obtained holograms of the mixed Y₂O₃/YO sample are shown in Fig. 5 measured at (a) 17.041 and (b) 17.054 keV as orthographic projections. They are centered at θ = 0°, and the radial and angular directions indicate θ and ϕ, respectively. At a glance, they look very different from each other, although the difference in the incident X-ray energy is only 13 eV.

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These holograms can be decomposed into contributions of pure Y₂O₃ and YO. The ratio is calculated from the X-ray intensities absorbed by two layers, which depend on θ and the layer thicknesses, and which are assumed to be proportional to the fluorescent X-ray intensities. Owing to the high absorption at 17.054 keV, most of the incident X-rays are absorbed at this energy in the top Y₂O₃ layer. At 17.041 keV, however, a significant part of the incident X-rays are transmitted through the top layer due to the lower absorption. The ratio of Y₂O₃/YO is about 1:3 at 17.041 keV and about 3:1 at 17.054 keV at θ = 0, and the contributions of the top Y₂O₃ layer increase with increasing θ.

Using the estimated fractions for the holograms at two different incident X-ray energies, the pure holograms of Y₂O₃ and YO were evaluated as shown in Figs. 5(a) and 5(b) of Ref. 19, and the atomic images were obtained using the SPEA-L1 algorithm as shown in Fig. 6 for Y₂O₃ on the (a) (001) and (b) (004) planes and for YO on the (c) (001) and (d) (002) planes.

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In Figs. 6(a) and 6(b) for Y₂O₃, the dashed lines indicate an ideal CaF₂ lattice and the atomic positions of Y₂O₃ are largely shifted to ovals expected for ${{\rm{Y}}}^{3+}$ atoms by XRD on bulk Y₂O₃,²¹⁾ where two different ${{\rm{Y}}}^{3+}$ sites in the unit cell are superimposed. Since the incident X-ray energy is only one, many artifacts are found outside of the expected ovals. Nevertheless, the crystal structure of Y₂O₃ can be recognized. The prominent images for near-neighboring atoms do not separate into two sites in the radial direction, and it is still an open question whether the sites are composed of two sites or not.

In Figs. 6(c) and 6(d) for YO, the dashed lines indicate an ideal lattice, which is mostly a rock salt-type fcc structure. In contrast to the images of Y₂O₃, ${{\rm{Y}}}^{2+}$ atoms are clearly observed near the ideal positions. Moreover, even the first and third ${{\rm{O}}}^{2-}$ atoms are reconstructed in the figures.

In summary, the crystal structures of YO and Y₂O₃ in the mixed sample could be distinguished by a valence-selective XFH measurement. For this sample, the two structures are macroscopically separated. This XFH method employing the incident X-ray energy in the XANES region can, in principle, be applied to microscopically valence-mixed crystals.

YbInCu₄ is well known as a valence transition material. An abrupt change was reported by Felner and Nowik²³⁾ in 1986 in the T dependence of the magnetic susceptibility of Yb_xIn_1−xCu₂ (x ∼ 0.3 − 0.6). A simple valence fluctuation model was proposed, by which a first-order Yb³⁺ to Yb²⁺ phase transition was predicted with simply increasing T. These compounds exhibit the sharpest T-dependent valence phase transition in any metallic systems. When decreasing T through the transition temperature, T_v, at about 42 K, the lattice constant of a cubic C15b structure with the space group of ${Fd}\bar{3}m$ shows a sudden increase by about 0.15%. In addition, ¹⁷⁰Yb Mössbauer spectroscopy,²³⁾ electrical resistivity,^23,24) specific heat,²³⁾ and ¹¹⁵In Knight shift²⁴⁾ measurements revealed magnetic, electronic, and thermal anomalies at T_v.

Concerning the $4f$ electron occupancy across T_v, the XANES analysis gave a change of $2.98\to 2.85$.²⁵⁾ A bulk-sensitive hard X-ray photoelectron spectroscopy showed a slightly larger change of $2.90\to 2.74$.²⁶⁾ The valency change was also observed by a resonant X-ray scattering spectroscopy at the Yb L_III edge, and a change of approximately $2.94\to 2.80$ was reported.²⁷⁾

The atomic radius of Yb²⁺ ions is much larger than that of Yb³⁺ ions by about 17%, even depending on the coordination number around the Yb ions.²⁸⁾ So, an increase in the averaged atomic radii of Yb ions can be estimated to be 2.2–2.7% on the valence transition, which is much larger than the actual increase of the lattice constant by 0.15% at T_v. Thus, the Yb²⁺ ions with large atomic size should squeeze into a rigid crystal lattice below T_v, and large lattice distortions are expected around the Yb²⁺ ions.

The features in the Yb L_III XANES spectra of YbInCu₄ were already shown in Fig. 2 at 7 and 300 K. At 7 K, a shoulder is observed at 8.939 keV, which is characterized by the Yb²⁺ ions. When the incident X-rays are employed at this energy, only Yb $2{p}_{3/2}$ electrons in Yb²⁺ ions can excite and emit fluorescent X-rays, while those in Yb³⁺ do not. Therefore, the obtained hologram measured at 8.939 keV composes valence-selective structural information around mainly Yb²⁺ ions.

Single-crystal YbInCu₄ was grown by a flux growth method. Constituent elements with stoichiometric ratios in a InCu flux were put in an alumina crucible, and sealed in an evacuated quartz ampoule. The sample was then heated to 1100 °C and cooled slowly down to 800 °C. After remaining at 800 °C for 20 h, the flux was removed. The crystal sample was cleaved so as to have a flat (001) surface with an area larger than 1 × 1 mm². The crystallinity of the sample was examined by taking a Laue photograph, and the concentration and homogeneity over the sample were confirmed to be within the experimental errors by EPMA.

Yb Lα (7.414 keV) XFH measurements were carried out at 7 and 300 K by using a cryostat designed solely for XFH experiments (Pretech Co. Ltd., type XFME-RR4K) at BL39XU of the SPring-8. The XFH signals were recorded at incident X-ray energies of 8.939 and 8.947 keV as indicated by the arrows in Fig. 2. The former excites the Yb²⁺ ions and the latter both the Yb²⁺ and Yb³⁺ ions.

Figure 7(a) shows the obtained 3D atomic images around Yb atoms in YbInCu₄ on the (001) plane measured at 8.947 keV at 300 K, where only Yb³⁺ ions are distributed. The dashed lines indicate an ideal lattice of the C15b structure obtained by XRD measurement.²⁴⁾ As clearly seen in the figure, the prominent atomic images are located at the ideal fcc positions, although weak artifacts are observed.

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Figure 7(b) shows the result at 7 K measured at the same incident energy. Owing to decreases of positional fluctuations at low T, the image becomes much clearer compared with (a). However, only the images at the first fcc positions in the $\langle 110\rangle$ directions get weaker, which may be due to the mixture of local structures around the Yb³⁺ and Yb²⁺ ions.

Figure 7(c) shows the result measured at the incident X-ray energy of 8.939 keV at 7 K, by which only the Yb²⁺ ions emit fluorescent X-rays. Although the energy difference is only 8 eV, the atomic images are very different. Firstly, the first neighboring Yb ions in the (001) plane are again very weak, indicating that the positional fluctuations of these atoms are very large. Secondly, the atomic images beyond the second neighboring Yb ions in the (001) plane are always of cross (+) form, and the more distant parts in the crosses are stronger than the closer parts in the crosses.

To explain the obtained atomic images, it is hard to imagine that every neighboring atom shifts toward the $\langle 100\rangle$ and $\langle 010\rangle$ directions. Instead, it is likely that the central Yb²⁺ ion shifts from the lattice point toward the $\langle 100\rangle$ and $\langle 010\rangle$ directions as shown in Fig. 8. The directions of the shifts are determined to avoid the first neighboring atoms in the $\langle 110\rangle$ directions. Owing to the positional shifts, all of the atomic images have the form of crosses. Due to the much larger atomic radii, the central Yb²⁺ atom pushes the first and second neighboring atoms, and thus, the distant sides of these images become prominent.

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In summary, we have carried out valence-selective XFH measurements on an YbInCu₄ valence transition material, from which it is clearly revealed that the local structures around Yb²⁺ and Yb³⁺ are very different from each other, and that there are large positional shifts of Yb²⁺ ions from the lattice points.

Fe₃O₄ is well known as a mixed valence material, which has plural valencies of Fe²⁺ (fraction: 1/3) and Fe³⁺ (2/3) in a crystal. Figure 9 shows the crystal structure of Fe₃O₄ magnetite.²⁹⁾ It is believed that half of Fe³⁺ ions occupy A-sites to form tetrahedral local structures with four O atoms, and the remaining half of Fe³⁺ ions and all Fe²⁺ ions enter B-sites to build octahedral local atomic arrangements with six O atoms. The fractions of the A- and B-sites in the crystal are 1/3 and 2/3, respectively. Note that the Fe–O bond length in the A- and B-sites are 0.1887 and 0.2042 nm, respectively.²⁹⁾

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Figure 10 shows Fe K XANES spectra of Fe₃O₄, FeO, and Fe₂O₃ indicated by the solid, dotted, and dashed curves, respectively.³⁰⁾ An energy shift in the absorption edges is observed between the divalent FeO and trivalent Fe₂O₃ of about 3 eV. In addition, a prepeak or pre-shoulder is observed at about 7.114 keV for Fe₂O₃ and Fe₃O₄, which originates from Fe³⁺.

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For the valence-selective XFH experiments, incident X-ray energies were chosen at 7.114 and 7.120 keV, as shown by the dashed lines in the figure, with which the trivalent Fe³⁺ and divalent Fe²⁺ contributions were highly enhanced for the central atom, respectively. The XFH experiments were carried out at BL39XU of the SPring-8 and BL6C of the PF-KEK.

Figure 11 shows the Fe Kα holograms obtained at incident X-ray energies of (a) 7.114 and (b) 7.120 keV.²⁰⁾ Although the energy difference is only 6 eV, the holographic features are quite different from each other. In particular, the holographic oscillations are more distinct at 7.114 keV, indicating that the local structures around the Fe³⁺ ions are simple, and small positional fluctuations are expected.

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Figure 12 represents the 3D atomic images around Fe atoms in Fe₃O₄ on the (001) plane obtained at incident X-ray energies of (a) 7.114 and (b) 7.120 keV.²⁰⁾ The large and small circles in the figures indicate the ideal positions of Fe ions around the A- and B-sites for the central Fe atoms, respectively. The most prominent atomic images in (a) are observed at the ideal positions of the A-site, where only the Fe³⁺ ions are located.

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This result looks rather inconsistent with the local structures obtained from XRD mentioned above. The local structures around the Fe³⁺ atoms are a mixture of tetrahedral A-sites and octahedral B-sites, which is much more complicated than those around the Fe²⁺ atoms. However, the present experimental results gave the opposite conclusion.

To solve this inconsistency, the spin-state natures of the atomic radii may be important. According to Shannon,²⁸⁾ atomic radii, d, are highly dependent on the spin states of high and low spin (HS and LS) and coordination numbers, N, around the Fe ions. In fact, the d values of HS Fe³⁺ ions in the A-site (N = 4) and B-site (N = 6) are obtained to be 0.049 and 0.0645 nm, respectively, and those of LS Fe³⁺ are larger by 0.01 nm. As mentioned above, the space for Fe atoms in the A-sites is much smaller than that in the B-sites, and thus, the HS Fe³⁺ ions are likely to enter the A-sites and the LS ones may prefer the B-sites.

If the prepeak in the Fe K XANES spectrum of Fe₃O₄ is concerning the HS states of the Fe³⁺ ions, the obtained holographic data measured at the incident X-ray energy of 7.114 keV selectively show the local structures of the A-site, which is consistent with the experimental result. On the other hand, the atomic images measured at 7.120 keV show weak indication of the B-site position with shifted positions. The images are composed of complex atomic arrangements, i.e., a mixture of central Fe ions with different atomic radii depending on the spin states and valencies.

In summary, valence-selective XFH experiments were carried out on mixed valence Fe₃O₄. Very different atomic images were obtained with a small difference in the incident X-ray energy by 6 eV near the Fe K absorption edge. For the further investigations, however, the obtained atomic images had many artifacts, and most of the prominent images were outside the ideal positions, owing to the small amplitudes of the experimental holograms of about 0.04%. Detailed experiments with better statistics are now in progress.