Chapter 3.34. Diffraction anomalous fine structure: experiment and data analysis

Diffraction anomalous fine-structure spectroscopy (DAFS) provides information about the empty electronic orbitals and the local atomic environment of resonant atoms which are selected by the diffraction condition, thus combining the chemical selectivity of X-ray absorption fine-structure (XAFS) spectroscopy with the space- or site-selectivity of diffraction.

At the turn of the 21th century, increasing interest in structural studies of 0D to 2D epitaxial nanostructures and heterostructures, thin films and nanomaterials more generally led to the development and application of multiwavelength anomalous diffraction (MAD) and DAFS, in combination with XAFS and ab initio calculations, to determine strain, atomic composition and atomic long-range and short-range order as well as crystallographic polarity. MAD, DAFS, XAFS and ab initio calculations constitute a very efficient and powerful toolset for semiconductor heterostructures and nano­structures (Coraux et al., 2007; Katcho et al., 2011). The reader will find extensive information about DAFS spectroscopy (including DAFS in grazing-incidence geometry) in several review articles (Sorensen et al., 1994; Hodeau et al., 2001; Favre-Nicolin et al., 2012; Renevier & Proietti, 2013).

MAD is commonly used to phase structure factors in molecular crystallography and to determine cation occupancy in long-range ordered crystallographic sites. As shown in Renevier & Proietti (2024), MAD enables mapping of the modulus of the partial structure factors F_A and F_N and the phase difference φ_N − φ_A from reciprocal-space maps measured at several energies close to the absorption edge of resonant atoms A (N refers to nonresonant atoms).

In the case of homogeneous binary alloys, φ_N − φ_A = 0, where A and N atoms occupy the same crystallographic site, the ratio |F_A|/|F_N| readily gives the A-atom average content in the isostrain region picked up by the diffraction condition. However, the method applies as long as the isostrain scattering hypothesis is valid, i.e. when the heterostructures can be decomposed into isocell lattice-parameter regions which do not interfere, i.e. in the presence of large objects and strain gradients (Kegel et al., 2001). In the most general case, the crystallographic structure must be used to refine the occupation factor of the resonant atoms (Renevier et al., 1997).

MAD is an efficient method to disentangle strain and composition. In the following, we mainly focus on energy-scan measurements, data reduction and extended DAFS refinement.

A DAFS experiment consists of measuring the scattered/diffracted intensity at a fixed point in reciprocal space and Q = 2sinθ/λ value (where 2θ is the scattering angle) as a function of the X-ray beam energy E = hc/λ, spanning a typical energy range of 1000 eV across an absorption edge. Q is the scattering vector, Q = k′ − k, where k and k′ are the incoming and outgoing X-ray beam wavevectors, respectively, and E is the X-ray beam energy. A DAFS spectrum cusp, measured over a large energy range (of the order of 1 keV) and a small step size (0.5–5 eV), corresponds to oversampled MAD data which provide very precise values of |F_A|/|F_N| and φ_N − φ_A (or and φ_T − φ_A).

If one wants to quantitatively analyze the oscillatory fine structure of the DAFS spectrum, the latter must not present distortions and the monitor-corrected signal-to-noise ratio must be as high as 1000 or greater, i.e. comparable requirements to those for obtaining high-quality extended XAFS oscillations. This requires a fixed-exit diffraction beamline with a scanning monochromator coupled to a diffractometer, both with high-precision movements and monochromator dynamical tuning. The stability and footprint of the beam at the sample position are of major importance. To avoid distortions, high precision and reproducibility of the monochromator–diffractometer coupling are essential. For instance, 1 eV resolution at 10 keV with a Si(111) crystal monochromator requires a monochromator angle precision of about 2 × 10⁻⁵ rad and therefore the coupling precision must be of the same order (Renevier et al., 2003).

Another source of distortion is multiple Bragg diffraction in the sample or the substrate, which diverts intensity away from the sample. In this case the only way to try to improve the data is to rotate the sample about the azimuthal ψ angle. When using an orientation matrix, its accuracy in tracking the Bragg peak when scanning the energy must be verified. As for XAFS experiments, the incident beam must be carefully monitored, for instance, by measuring the fluorescence signal emitted from a high-purity metal foil of a few micrometres in thickness mounted in vacuum (Renevier et al., 2003).

With regard to detection, 2D pixel detectors have definite advantages in comparison with point or linear detectors. They cope with high counting rates with a low background intensity, allowing the collection of high-quality 3D reciprocal-space maps close to Bragg peaks on counting timescales of the order of tens of seconds to minutes.

One drawback of 2D detectors is the lack of a sufficient energy resolution to discriminate the fluorescence background. However, most often the 2D detectors are large enough to simultaneously record both the DAFS spectra and the fluorescence background in different regions of interest.

In the Born approximation, the intensity of the DAFS spectrum is related to the square modulus of the structure factor according to the formulawhere I is corrected for the background intensity (mainly fluorescence yield), K is a scale factor, D is the detector efficiency, L and P are the Lorentz and polarization factors for Thomson scattering, respectively, A is the absorption correction and F(Q, E) is the structure factor (Als-Nielsen & McMorrow, 2001; Prince, 2006). D takes into account the whole detection setup, comprising the detector efficiency and absorption all the way from the monitor to the diffraction detector. The energy dependence of D is often linear within the energy range of interest (about 1 keV), so that D may be fitted to the DAFS spectrum by a straight line [D = m(ΔE + 1)], where m is the only adjustable parameter, ΔE = E − E₀ and E₀ is the edge energy. Care must be taken to measure the DAFS spectrum far enough below and above the absorption edge up to the point where resonant scattering is negligible, otherwise the m parameter is correlated to the crystallo­graphic phase φ_T − φ_A (Proietti et al., 1999).

For a rotation scan, i.e. with the sample rotation axis perpendicular to the plane of incidence containing k and k′, L = λ³/sin2θ (Als-Nielsen & McMorrow, 2001; Prince, 2006). The polarization correction for the Thomson scattering is given by the dot product squared of the polarization vectors of the incoming and outgoing beams.

Together with technical improvements, data analysis has also improved, relying on one hand on the solid support of the established MAD data treatment and on the other hand on the EXAFS approach (Stragier et al., 1992) with available and well-known codes for ab initio calculations, data simulation and analysis. A DAFS spectrum contains contributions from both the real and imaginary parts of the complex resonant scattering factors, whereas XAFS is only proportional to the imaginary part. In the simplest case, where the resonant atoms are located on one crystallographic site only, one can use equation (3) in Renevier & Proietti (2024) to perform an iterative Kramers–Kronig analysis and recover and (Cross et al., 1998). In the case of a non­centrosymmetric structure with several resonant crystallo­graphic sites, the iterative Kramers–Kronig procedure cannot be applied due the presence of f′f′′ cross terms. Alternatively, the extended-region DAFS (EDAFS) above the edge can be analyzed according to a first-order approximation in χ_Q (see Section 3.1 of Renevier & Proietti, 2024), such as EXAFS oscillations (Proietti et al., 1999). Basically, the analysis consists of the following.

(i) Fitting equation (1) to the DAFS spectrum cusp (corrected for fluorescence) gives the scale factor S_D (see equation 7 in Renevier & Proietti, 2024) and the crystallo­graphic phase φ₀ − φ_A (see equation 8 in Renevier & Proietti, 2024).

(ii) χ_Q is obtained by multiplying S_D by [(I − I₀)/I₀](Q, E) (where I₀ is the smooth diffracted intensity, i.e. without oscillations). The phase difference φ₀ − φ_A is added to the XAFS theoretical phase provided, for example, by the FEFF code (Ankudinov et al., 2002; Kas et al., 2024).

(iii) The new theoretical XAFS phases are used to analyze χ_Q, for instance with the IFEFFIT code (Newville, 2001; Newville & Ravel, 2024) implemented in the ARTEMIS package (Ravel & Newville, 2005, 2024).

Note that the path-by-path multiple-scattering approach is no longer valid in the energy region close to the absorption edge (Rehr & Albers, 2000). However, as in the case of X-ray absorption near-edge structure (XANES), the anomalous diffraction near-edge structure (DANES) can be calculated with a dedicated program, such as for instance FDMNES (see Joly, 2001; Bunău et al., 2024).

In this chapter we give a brief state-of-the-art survey of the DAFS technique, describing its most important experimental and data-reduction aspects. We provide a basic bibliography to approach the use of DAFS. DAFS combines the capabilities of X-ray absorption and diffraction techniques and is particularly suited to the study of the structural properties of low-dimensionality systems such as thin films and nanostructures.