Diffuse scattering from correlated electron systems

Abstract

The role of inhomegeneity in determining the properties of correlated electron systems is poorly understood because of the dearth of structural probes of disorder at the nanoscale. Advances in both neutron and x-ray scattering instrumentation now allow comprehensive measurements of diffuse scattering in single crystals over large volumes of reciprocal space, enabling structural correlations to be characterized over a range of length scales from 5 to 200 angstroms or more. When combined with new analysis tools, such as three-dimensional difference pair-distribution functions, these advanced capabilities have produced fresh insights into the interplay of structural fluctuations and electronic properties in a broad range of correlated electron materials. This review describes recent investigations that have demonstrated the importance of understanding structural inhomogeneity pertaining to phenomena as diverse as superconductivity, charge density wave modulations, metal-insulator transitions, and multipolar interactions.

Diffuse scattering reveals the importance of structural inhomogeneity in the properties of correlated electron systems.

INTRODUCTION

In the majority of solids, the effective interactions among the mobile electrons are weak and their behavior can be understood as if they were independent particles. In many interesting systems, however, strong interactions lead to pronounced correlations among electrons, which causes a breakdown of the independent particle picture and often leads to rich physics and unprecendented emergent phases (1). Strongly correlated electron materials are at the forefront of modern condensed matter research: They include prominent systems such as high-temperature superconductors (2) and low-dimensional quantum magnets (3), as well as materials with a wide range of ordering phenomena, such as metal-insulator transitions (MITs) (4) and various density wave–ordered phases (5, 6). Strongly correlated systems are thus of tremendous importance with regard to our fundamental understanding of the quantum many-body physics of electrons in solids. Moreover, these materials have vast potential for applications, such as in computing, sensing, as well as magnetic information and energy storage (7).

It has long been recognized that inhomogeneity profoundly affects the properties of correlated electron materials (8–11), and it is generally difficult to discern if inhomogeneity is driven by structural or electronic degrees of freedom. In principle, structural and electronic inhomogeneity can emerge together, but it is also possible that structural inhomogeneity is the underlying cause of electronic inhomogeneity or vice versa. On the one hand, structural disorder can have deleterious effects on long-range electronic order or the formation of spin-liquid ground states (12, 13), but it can also be pivotal to generating emergent electronic order, for example, in the case of substitutional doping to tune charge-carrier densities (14–16) or by inducing quantum fluctuations close to a quantum phase transition (17–19). On the other hand, classic examples of electronically driven inhomogeneity are systems with (incommensurate) density wave order. The mechanisms that connect structural inhomogeneity and electronic correlations are often poorly understood, and theoretical approaches tend to fall into two camps. Crystalline disorder is either treated as randomly substituted point defects within a rigid-band model or, conversely, as phase-separated into more homogeneous domains (14). These simplifications are often necessitated by limitations in the experimental methods available to probe more complex forms of structural inhomogeneity that fall between these two extremes (9) and involve translational symmetry breaking on the nanoscale (10).

Inhomogeneity is most frequently characterized by bulk local probes, such as nuclear magnetic resonance (NMR) (20), muon spin rotation (μSR) (21), or x-ray absorption fine structure (XAFS) (22), which provide only indirect information on the length scales of structural correlations. However, scanning tunneling microscopy (STM) measurements have shown that, at least on the surface, the interplay between electronic and structural inhomogeneity can occur on a range of length scales, which leads to complex short-range order resulting from phase competition and competing interactions (23). Pair distribution function (PDF) analysis of both neutron and x-ray powder diffraction data is a widely used bulk probe of crystalline disorder (24, 25), although some structural fluctuations, such as transverse displacements, are obscured by spherical averaging, and long-range correlations are hard to separate from the average structure. Nevertheless, the relative ease of measurement and the sophistication of PDF analysis software has made this a powerful and popular tool for both identifying and characterizing short-range order at length scales of a few tens of angstroms (26).

Single-crystal diffuse scattering, with both neutrons and x-rays, can overcome many of the limitations of these techniques (27, 28). The method is sensitive to three-dimensional (3D) structural correlations over length scales of 5 to 200 Å or more, and it provides information on both local atomic relaxations around point defects and the ways defects self-organize on the nanoscale into more complex short-range order. In the past, it was challenging to measure diffuse scattering because the signal is generally several orders of magnitude weaker than Bragg scattering and spread over many Brillouin zones. However, instrumental developments over the past 10 to 15 years now make possible the collection of large contiguous volumes of scattering in reciprocal space, encompassing hundreds and often thousands of Brillouin zones, on timescales ranging from a few minutes, e.g., with synchrotron x-rays at the Advanced Photon Source (29), to a few hours with neutrons, e.g., on the CORELLI diffractometer at the Spallation Neutron Source (30). These speeds enable diffuse scattering data to be collected as a function of temperature and composition, allowing the evolution of structural fluctuations to be tracked across entire phase diagrams in a matter of days.

Since such experiments have become increasingly routine, the challenge has shifted to modeling the large 3D data volumes. Software applications that perform atomistic simulations of diffuse scattering, such as DISCUS (31) or SCATTY (32), provide the necessary computational framework to calculate the structure factor S(Q), but optimizing the conditional interatomic vector probabilities that underlie these models is difficult to perform reliably. However, the datasets are now comprehensive enough to enable new modes of analysis, such as the 3D-ΔPDF method pioneered at ETH Zürich (33, 34), which are much simpler to interpret, even without large-box simulations. 3D-ΔPDF analysis converts broad reciprocal space intensity distributions into a discrete set of peaks in real space that represent only those interatomic vector probabilities that differ from the average crystalline structure (29). Even without atomistic modeling, 3D-ΔPDF maps can reveal the length scales over which these structural deviations are correlated, and therefore provide a novel method of extracting critical exponents in real space both above and below structural phase transitions (35).

In this review, we will discuss a number of recent examples in which comprehensive single-crystal diffuse scattering measurements have transformed our understanding of quantum materials. These examples cover a broad range of correlated electron behavior over large length-scale ranges. The work has provided unique insights into MITs, unconventional superconductivity, Goldstone mode fluctuations, the effect of disorder on charge density waves (CDWs) and quantum spin liquids, and many other phenomena. This review does not include diffuse magnetic neutron scattering, although it implicitly probes electronic fluctuations, because of space limitations.

Following this introduction, the review is divided into five sections. The next section briefly summarizes recent advances in diffuse scattering instrumentation before describing the computational approaches required to interpret the large datasets now produced routinely. The following section covers the interplay between electronic and structural fluctuations, as well as their respective length scales, at electronic phase transitions. We then describe experiments probing inhomogeneity in superconducting oxides, before a section covering recent experiments that probe the role of extended defects on both unconventional superconductivity and spin-liquid behavior. In the final section, we discuss the outlook for these investigations in the future.

RECENT ADVANCES IN DIFFUSE SCATTERING

In crystalline materials, diffuse scattering comprises all the contributions to the measured scattering cross section produced by deviations from the translationally invariant atomic structure, i.e., all the scattering apart from (and mostly between) the Bragg peaks (27, 28). Data in reciprocal space are frequently difficult to interpret in terms of real-space correlations, although a few general rules of thumb can be formulated. For example, 1D correlations lead to planar diffuse scattering, 2D correlations cause rod-like diffuse signals, and 3D correlations are relatively localized in reciprocal space. Beyond these guidelines, diffuse scattering displays a bewildering diversity of patterns, a point that is well illustrated by the examples we describe in this review. It is also much weaker than Bragg scattering, by several orders-of-magnitude, and spread broadly over large volumes of reciprocal space. The technical challenges of performing measurements and understanding the results has limited the use of this technique to a small number of specialists in the past. However, with both x-rays and neutrons, the situation is now changing because of recent developments in both instruments and detectors. We provide a brief summary of those advances here and refer to (36) and (37) for more comprehensive reviews of x-ray and neutron diffuse scattering techniques, respectively.

At x-ray synchrotron sources, a new generation of photon-counting area detectors, which combine fast readout times with a wide dynamic range and low backgrounds, allow the use of the classic rotation method to measure $S(\mathbf{Q})$ in a finely gridded mesh with minimal contamination from Bragg scattering artifacts. In most of the measurements described here, the sample is rotated continuously through 360^∘ in a monochromatic beam, while frames are collected on a fast area detector. The high energies available at third-generation sources, such as the Advanced Photon Source, minimize backgrounds and absorption corrections while expanding the Q range of the collected data. For example, with an incident energy of 87 keV, $S(\mathbf{Q})$ is measured with values of ∣Q∣ up to 15 Å⁻¹, covering thousands (and sometimes tens of thousands) of Brillouin zones (29), in under 20 min. Backgrounds are also reduced by using helium or nitrogen cryostreams to vary the temperature, rather than a helium flow cryostat or displex.

Diffuse scattering can also be measured on a number of diffractometers at both reactor and spallation neutron sources (37), but single-crystal time-of-flight neutron diffractometers are particularly efficient at measuring large reciprocal space volumes (38). Among these, CORELLI is unique in modulating the incident white beam with a statistical chopper, which allows the quasi-static contributions to diffuse scattering to be separated from phonon scattering (30). Phonon scattering, generally known as thermal diffuse scattering, cannot be distinguished from other diffuse contributions by high-energy x-rays, whose scattering is integrated over all frequencies, so it is usually identified by its temperature dependence. In cases where that identification is ambiguous, CORELLI can play an important role in separating different scattering contributions.

The instrumental advances that have enabled large volumes of $S(\mathbf{Q})$ to be collected on a routine basis require the development of more efficient computational tools to process the data and facilitate scientific interpretation. Even after data reduction, a single dataset can exceed 10 gigabyte in size, making conventional approaches to data analysis impractical or prone to selection bias. The problem is only exacerbated by the ever increasing speed of data collection, particularly at synchrotron x-ray sources, so that they are often repeated at many temperatures and for a range of sample compositions. It is clear that novel computational methods are necessary to ensure that more than a small fraction of the data is used.

Two approaches have been developed over the past decade to address this challenge, and most of the investigations described in this review have used one or both of them. The first is to use machine learning (ML) to identify important features in the data by their distinctive temperature dependences. The second approach is to transform entire datasets into real-space maps of interatomic vector probabilities, which are usually much easier to interpret than the original intensity distributions measured in reciprocal space. We briefly describe here both methods and then discuss in subsequent sections the science that they have enabled.

Unsupervised ML

The ability to measure a large number of data sets as a function of a parametric variable, such as temperature, allows for novel ways to interrogate the data. For example, unsupervised ML algorithms that group voxels within $S(\mathbf{Q})$ volumes into clusters that share similar temperature dependences have been implemented in a program called X-TEC (x-ray diffraction temperature clustering) (39). This is a particularly powerful way to search for a set of superlattice peaks that are absent above a structural phase transition but grow with decreasing temperature with common critical exponents (Fig. 1). This method is not only confined to identifying peaks associated with long-range order but can also be used to identify distinct diffuse scattering contributions. The temperature dependence of each cluster results from the physical origin of the scattering at the respective wavevectors, so thermal diffuse scattering will typically increase linearly with temperature, whereas structural diffuse scattering from quenched defects is often temperature independent. The temperature dependence of other diffuse scattering contributions may reflect the growth of emergent order or of low-frequency critical fluctuations (35, 39).

Fig. 1. Unsupervised machine learning using X-TEC.

X-TEC is a software application that analyzes a billion voxels or more measured in reciprocal space as a function of temperature and clusters them according to their distinctive temperature dependences (39). (A) In Sr₃Rh₄Sn₁₃, data at 30 K reveal superlattice peaks, at a relative wave vector of (¹/₂,¹/₂,0) and symmetric equivalents. (B) These superlattice peaks are absent at 150 K. (C) Temperature dependence of all the voxels above a threshold value rescaled to their mean intensities. (D) These voxels were assigned by X-TEC to two clusters (red and blue). The solid lines and shaded regions represent the mean intensities and variance, respectively, of each cluster, showing that X-TEC has separated superlattice peaks showing typical order-parameter behavior below T_c = 135 K from other Bragg peaks. (E) Q map showing the locations of voxels assigned to each cluster, i.e., which share a similar temperature dependence, confirming that the superlattice peaks (red) occupy (¹/₂,¹/₂,0) positions. (F) Temperature dependence of three CDW order parameters in CsV₃Sb₅ identified by X-TEC (40). (G) Momentum distribution in the out-of-plane (L) axis for each cluster.

This method was used to characterize the sequence of CDW transitions in the kagome metals CsV₃Sb₅ and ScV₆Sn₆ (40, 41), but it can also be effective in analyzing diffuse contributions, such as Goldstone modes and short-range CDW fluctuations above the transition (39, 41, 42), as will be discussed later in this review. Last, the method can be extended to provide statistical analyses of scattering associated with each cluster. For example, it was used to analyze the Q dependence of the CDW peak spread to provide evidence of Bragg glass correlations in Pd_xErTe₃ (42).

3d-ΔPDF maps of structural correlations

Another way to ensure that all the data collected in single-crystal diffuse scattering experiments are used is to transform the reciprocal space data to real space. When performed over a sufficiently large contiguous scattering volume, these transforms produce 3D-PDFs, i.e., maps of interatomic vector probabilities summed over all the atomic sites. These are equivalent to Patterson maps, which have long been used in crystallography as a tool for solving crystal structures (43). When both Bragg peaks and diffuse scattering are included in the transforms, the PDF maps contain peaks at interatomic vectors from both the average structure and local deviations from the average.

1D PDFs derived from powder diffraction data are regularly used to identify deviations from the average crystallographic structure. However, in 1D, it is not possible to separate contributions from the average crystal structure and local deviations without sophisticated modeling because they overlap when spherically averaged. However, Weber and colleagues (33) at the ETH Zürich realized that this is not the case for 3D-PDFs because the Bragg peaks are highly localized in reciprocal space. If the Bragg peaks are removed from $S(\mathbf{Q})$ and replaced by interpolations of the surrounding diffuse scattering, a procedure known as “punch-and-fill,” then the resulting PDFs only contain peaks at interatomic vectors whose probabilities differ from the average. These difference pair distribution functions, which are now known as 3D-ΔPDF maps, are playing an increasingly important role in the analysis of neutron and x-ray diffuse scattering (29, 33, 34, 44), as well as complementary techniques such as electron diffraction (45).

Representing the experimental results as 3D-ΔPDF maps produces an effective dimensional reduction of the raw data, with broad distributions of diffuse intensity transformed into peaks in real space. The peak positions correspond to different interatomic distance vectors, while the peak amplitudes correspond to the relative probabilities of finding a given pair of atoms at that particular distance, summed over all pairs. These probabilities can be positive or negative, depending on whether the corresponding interatomic vector is more or less probable than in the average crystalline structure, either because of local changes in site occupations, or local distortions, or a combination of the two. The measured probabilities are weighted by the product of the atomic scattering factors of the atoms in the pair, so a comparison of neutron and x-ray 3D-ΔPDF maps can be valuable in elucidating the role of different atomic species when the origin of the interatomic vectors is ambiguous (46).

As R. Welberry has pointed out, the 3D-ΔPDF peak intensities “are simply related to the Warren-Cowley short-range order parameters that have frequently been used to parametrize diffuse scattering models” (27). These Warren-Cowley parameters, which are the conditional probabilities for each set of interatomic pairs, effectively represent all that can be determined about local structural correlations from $S(\mathbf{Q})$. 3D-ΔPDF maps are often intuitive to interpret without computationally expensive modeling and, at the very least, provide a guide to the appropriate disorder models. For example, in sodium-intercalated V₂O₅, it was possible to determine that sodium ions on two-leg ladders only occupy next-nearest neighbor sites, forming a zig-zag pattern, without requiring any further modeling (Fig. 2) (29). Welberry and Weber (36) provide further examples of 3D-ΔPDF maps generated by different kinds of disorder.

Fig. 2. 3D-ΔPDF maps of Na_xV₂O₅.

(A) Intercalant sodium ions occupy approximately 50% of the sites on two-leg ladders. The figure illustrates a possible zig-zag configuration of occupied sites (dark green) interleaved with unoccupied sites (light green) (29). The axes are in lattice units (l.u.). (B) Evidence for this configuration is provided by 3D-ΔPDF maps, which consist of peaks at real-space interatomic vectors connecting sites that are occupied with a greater probability (red) or lesser probability (blue) than in the average structure. The correspondence between real space and 3D-ΔPDF maps are illustrated by three interatomic vectors labeled A, B, and C, which connect sites occupied with higher than 50% probability. Because the vector origin can be on sites on either leg, a two-leg ladder produces a three-leg ladder in the 3D-ΔPDF.

Locally correlated displacements can also be seen via 3D-ΔPDF. A good example of this is the correlated dipole system PbTe, whose simple rock salt crystal structure hosts local dipoles formed by off-centered Pb atoms (47, 48). These Pb displacements are locally correlated only over several unit cells, leaving the average rock salt structure unchanged while producing broad diffuse scattering rods. The 3D-ΔPDF of this diffuse scattering shows the distinct quadrupolar signature of a positive displacement correlation and the length scale of the correlation. 3D-ΔPDF analysis is also capable of imaging more complex displacement correlations, including those coupled to local occupation, as well-demonstrated in studies of bixbyite (49). While this system is better-known as a case study of diffuse magnetic scattering (50), it also hosts complex structural short-range order. Using both reciprocal space and 3D-ΔPDF maps, the authors were able to assemble a detailed model of stacking faults, intergrowths, and relaxations consistent with the long-range structure, definitively determining the local structure and showing the applicability of 3D-ΔPDF beyond the simplest crystal systems.

STRUCTURAL CORRELATIONS VERSUS ELECTRONIC CORRELATIONS

Changes in the electronic structure are usually accompanied by changes in the atomic structure and vice versa, but the respective length scales of electronic and structural correlations have historically been difficult to ascertain. In some cases, the coupling of the two order parameters is so strong that both undergo simultaneous, possibly first-order phase transitions, but in other cases, the interplay is more subtle. Diffuse scattering provides a direct probe of the structural response to changes in the electronic structure, whether short range or long range. We will describe examples in which structural correlations profoundly alter our understanding of the physics that underlies the electronic transitions observed in transport and spectroscopic measurements.

Metal-insulator transitions

MITs are often concomitant with structural phase transitions, whether they are driven by local Mott physics or by a Peierls distortion (4, 51). We will discuss the specific example of VO₂. On the one hand, there are reports of an apparent suppression of any structural response to the MIT in epitaxial films of VO₂ (52, 53). In contrast, in bulk VO₂, the strongly first-order MIT occurs at 340 K, with a transition from a high-temperature tetragonal rutile structure to the monoclinic M₁ phase, in which the vanadium ions dimerize along buckled c-axis chains (54). However, the substitution of other transition metals for vanadium can induce the related M₂ phase, in which only half the vanadium ions dimerize (55), which indicates that the M₁ phase consists of a superposition of the two equivalent M₂ phases along orthogonal [110] directions (56).

Molybdenum substitution suppresses the MIT from 340 K to about 150 K in V_1−xMo_xO₂ with x = 0.19 (57) and weakens the strength of the electronic transition. Nevertheless, the transition remains first order, so it was unexpected that diffuse scattering measurements showed a complete collapse of the structural phase transition (58). At x = 0.17, superlattice peaks in the $L={\displaystyle \frac{1}{2}}$ plane indicate the development of a long-range M₁ structure below the MIT. However, at slightly higher doping, the superlattice peaks are replaced by wavy rods that indicate purely 2D correlations persisting over length scales of 50 Å or less along 〈110〉 directions (Fig. 3). The waviness results from weak correlations transverse to these 〈110〉 directions.

Fig. 3. MITs in V_1−xMo_xO₂.

(A and B) Diffuse scattering below the first-order electronic phase transition with (A) x = 0.17 and (B) x = 0.19 in the $L={\displaystyle \frac{1}{2}}$ plane. At x = 0.17, superlattice peaks consistent with the M₁ phase appear at the MIT, but at higher x values, these are replaced by rods of scattering, indicating short-range 2D correlations characteristic of the M₂ phase. This is confirmed by the respective 3D-ΔPDF maps, which show (C) long-range M₁ order at x = 0.17, but (D) 2D correlations along (108) directions at x = 0.19 (58).

3D-ΔPDF maps confirm that the local distortions at x = 0.19 correspond to the M₂ structure, in which only half the vanadium ions form dimer pairs. This behavior is also seen in niobium-doped VO₂ (59). Structurally, the short-range character of the correlations results from a geometric frustration of the phase of the lattice buckling in neighboring planes. Electronically, this result demonstrates that only short-range lattice relaxations are required to stabilize long-range modifications to the electronic bands (51), and this is the likely explanation for the absence of structural changes at the MIT of thin films of VO₂ (52, 53).

Order-disorder transitions

For over 60 years, second-order structural phase transitions have been discussed in terms of two limiting categories, as either “displacive” or “order-disorder” transitions (60). In the former, the amplitude of the time-averaged local distortions that define the low-temperature phase fall to zero at T_c, whereas in the latter category, local distortions persist above T_c but only become phase-coherent at the transition. This distinction has important implications for the origin of the phase transition and the nature of the associated changes in electronic structure. If the transition is order-disorder, then the electronic excitations could become incoherent above T_c because of the persistence of quasi-static disorder in the high-temperature phase.

Diffuse scattering measurements on the quasi-skutterudite Sr₃Rh₄Sn₁₃ provide strong evidence of just such a scenario (Fig. 4) (35). This compound undergoes a structural phase transition at 135 K in which neighboring tin icosahedra counter-rotate, to a first approximation (Fig. 4A), with the emergence of superlattice peaks at q_s = (¹/₂,¹/₂,0) and equivalent wave vectors. The growth of these peaks with decreasing temperature could either be due to an increase in the distortion amplitude or an increase in phase coherence, but conventional analysis cannot distinguish between the two. However, 3D-ΔPDF transforms of the diffuse scattering data show that the distortion amplitudes are independent of temperature (Fig. 4B), from 30 K to at least 200 K, i.e., even above the transition, the distortions are undiminished. This is only possible because the 3D-ΔPDF transforms include both the superlattice peaks and the surrounding diffuse scattering, which allows the real-space variations of the PDF peak intensities to be used to measure the correlation lengths of structural fluctuations from 10 to 200 Å, both above and below T_c (Fig. 4C), and their respective critical exponents to be determined (Fig. 4D).

Fig. 4. Order-disorder transition in Sr₃Rh₄Sn₁₃.

(A) Neighboring tin icosahedra above and below the structural phase transition. (B) 3D-ΔPDF maps of the interatomic vector probabilities in the X-Y plane (Z = 0). Positive values (red) at ±X, ±Y = 0.35, and 0.45 show that the tin ion displacements from their high-temperature average positions (blue) at ±X and ±Y = 0.4 do not change with temperature, even above T_c. (a) 30 K, (b) 100 K, (c) 120 K, (d) 130 K, and (e) 150 K. (C) Fits of PDF peak intensities to an exponential decay showing the correlation lengths in real space. (D) Critical scaling curves for (Sr_1−xCa_x)₃Rh₄Sn₁₃ at x = 0, 0.1, 0.6, and 0.65 showing the temperature dependence of (a) the order parameter and (b) the inverse correlation lengths (35).

It had been suggested that Sr₃Rh₄Sn₁₃ and related skutterudites were unusual examples of systems with 3D CDW order (61), although the nature of the charge disproportionation was never established. However, the 3D-ΔPDF results call that into question. Instead, optical spectroscopy provides evidence of pseudogap behavior (62), which suggests a loss of electronic coherence caused by quasi-static structural disorder persisting well above the transition [see also (63)].

It is commonly thought that most transitions display a mixture of properties associated with both displacive and order-disorder character (60). However, the 3D-ΔPDF method, which allows distortion amplitudes to be tracked down to very low temperatures where thermal activation of structural disorder is minimized, can remove some of the ambiguities inherent in other classification methods (64).

Structural fluctuations in spin-orbit coupled systems

It has been theoretically predicted that metallic systems with strong spin-orbit coupling will exhibit a variety of unprecedented electronic phases, such as multipolar nematicity and metallic ferroelectricity (65), but the structural response in candidate systems is often anomalously small (66). One example is Cd₂Re₂O₇, which is a pyrochlore metal with two structural phase transitions at 113 and 200 K that have been attributed to parity-breaking nematic transitions (67, 68). The phase transitions produce very small lattice distortions, making conventional crystallographic analysis exceedingly challenging (69–71). Second-harmonic generation (SHG) optical spectroscopy, which is extremely sensitive to parity-breaking phases, seemed to indicate that the primary order parameter is inconsistent with earlier diffraction measurements (67), so synchrotron x-ray experiments initially sought to resolve this question. The use of X-TEC identified four classes of Bragg peaks (two of which are shown in Fig. 5A), whose temperature dependences showed distinct behavior at the two phase transitions. Selection rules derived from the Q dependence of these clusters were associated with the sensitivity of the structure factors to in-plane and out-of-plane cation distortions.

Fig. 5. Structural phase transitions in Cd₂Re₂O₇.

(A) Temperature dependences of two clusters of superlattice peaks (green and yellow) identified by unsupervised machine learning and (B) temperature dependences of diffuse scattering from Goldstone mode fluctuations. The red and blue diffuse clusters correspond to the yellow and green superlattice clusters, respectively (39).

The earlier structural investigations indicated that the upper transition involved a lowering of cubic symmetry (space group $\mathit{Fd}\overline{3}m$) to tetragonal symmetry (space group $I\overline{4}m2$), which is produced by a condensation of two-component E_u modes (68, 69). If these components are nearly degenerate, then there would be strong Goldstone mode fluctuations between the two, corresponding to a switch between in-plane and z-axis ionic distortions. Raman scattering had already found some evidence of the existence of these Goldstone modes, so X-TEC was used to determine their Q dependence by analyzing diffuse scattering around hundreds of Bragg peaks. Stronger fluctuations were observed around Bragg peaks that were particularly sensitive to in-plane distortions (Fig. 5B). This is consistent with a Landau theory analysis, which predicts fluctuations toward the second E_u component with I4₁22 symmetry, which then condense at the lower first-order transition (39). This is the first time that selection rules for diffuse scattering have been correlated with those for superlattice reflections, which provided detailed insights into the mechanisms that drive the two structural phase transitions.

Bragg glass correlations

Nearly 50 years ago, Imry and Ma (72) predicted that ordered phases with continuous symmetry would be unstable to arbitrarily weak random fields below the upper critical dimension of four. One example is the destruction of long-range order when fluctuations in the phase of an incommensurate CDW are pinned by random defects. The competition between the disorder potential and elastic strain was predicted to result in a vestigial nematic phase with a short-range correlation length (73). However, it was later pointed out that this prediction is modified when the periodicity of the phase is taken into account (74). Instead, strong phase fluctuations are predicted to result in an unusual form of quasi–long-range order, in which there is an algebraic decay of structural correlations (75, 76). This was termed a Bragg glass, a completely new phase that has been extremely difficult to observe because of the exacting resolution required to distinguish it from true long-range order.

STM studies of incommensurate CDW compounds have provided evidence of Bragg glass behavior through an analysis of topological defects in the presence of disorder (77, 78), but the first bulk-probe evidence was only recently provided by a novel ML analysis of diffuse x-ray scattering data described below (42). The suppression of CDW order in palladium-intercalated ErTe₃ has been extensively studied by scattering, transport, and STM measurements (78, 79). In pure ErTe₃, there are two CDW phases with transition temperatures of 260 and 135 K, resulting from orthogonal modulations of the tellurium square-planar nets (79). The lower transition is rapidly suppressed by an intercalation of less than 1% palladium, but evidence of CDW peaks from the upper transition persist up to 3% intercalation. However, the momentum resolution of x-ray diffraction is insufficient to determine if these peaks display the power-law tails characteristic of a Bragg glass.

To overcome this limitation, X-TEC was modified to investigate the linewidths of the broadened CDW peaks above the apparent CDW transitions (42). The theoretical basis of this analysis is that peak broadening due to phase fluctuations is Q independent, and so can be distinguished from the quadratic Q dependence of displacement fluctuations. By analyzing the peak spread of thousands of CDW peaks, it proved possible to establish that the correlation lengths associated with phase fluctuations diverge at a nonzero temperature, whereas a vestigial nematic should remain short range at all temperatures. The inferred phase diagram of the Bragg glass phases is in good agreement with the onset of in-plane anisotropy in transport measurements (79).

It is evident from this example that the unsupervised machine learning approach implemented by X-TEC can be adapted to a variety of different problems whenever there are sufficient data to generate robust statistical analyses of both the Q and temperature dependence of features in reciprocal space.

SUPERCONDUCTING OXIDES

Complex oxides that exhibit high-temperature superconductivity have been intensely studied for more than three decades, including a wide array of experiments sensitive to structural inhomogeneity. Yet, the extent and role of nanoscale structural correlations in the phenomenology of these materials remains heavily debated, often due to tremendous difficulties in resolving the many different kinds of disorder. The lamellar high-T_c cuprates have been most heavily investigated due to their extraordinary superconducting and normal-state properties (2), yet oxides such as the bismuthates (80) also show high-temperature superconductivity and other electronic ordering tendencies that are not well understood. It has been long known that complex oxides are prone to various types of inhomogeneity of both electronic and structural origin (9). The vast majority of superconducting oxides must be chemically doped to achieve superconductivity, which necessarily introduces point disorder. Moreover, the most prominent oxide superconductors have perovskite-derived structures, which generically exhibit structural instabilities due to atomic size mismatch. The latter can lead to both long-range lowering of the structural symmetry and short-range fluctuations embedded in a high-symmetry phase. In addition, a host of inhomogeneous phases of apparent electronic origin has been found, including spin and CDWs, with coherence lengths of only a few unit cells in some systems. Scattering has played an essential role in the discovery of long- and short-range density wave order in the cuprates, from the pioneering neutron work on La-based materials (81) to more recent x-ray scattering experiments in multiple cuprate families (82–87). The development of sophisticated resonant soft x-ray scattering techniques has largely been motivated by studies of CDWs in cuprates. Most of these experiments, however, have focused on limited reciprocal space volumes and have not provided systematic insight into different types of bulk inhomogeneity. With the recent development of high-throughput diffuse scattering instruments, 3D-ΔPDF analysis, and advanced numerical modeling, this important question is beginning to be addressed in several model systems.

We highlight two sets of results here: the finding of inversion-breaking atomic correlations in the prototypical superconducting bismuthate Ba_1−xK_xBiO₃ (BKBO) (88) and the detailed characterization of nanoscale structural correlations in the model cuprate HgBa₂CuO_4+δ (Hg1201) (89). These examples comprehensively showcase the strengths and possibilities of state-of-the-art diffuse scattering and associated analysis, indicate the presence of unexpected interactions between the local structure and electronic degrees of freedom, and provide the foundation for a broad range of further investigations.

Bismuthates

Superconductivity in the bismuthates was found nearly five decades ago (80), and the BKBO family shows a maximum T_c above 30 K, similar to the La-based cuprates (90). Yet, bismuthate research has been somewhat overshadowed by the cuprates, and major questions pertaining to the doping-temperature phase diagram and superconducting pairing mechanism remain open. Although their average structure is close to a simple cubic perovskite with Bi-O octahedra, the bismuthates display considerable structural and electronic complexity (80, 91). The stoichiometric parent compound, BaBiO₃, is an insulator with pronounced charge disproportionation: the total charge periodically changes from one Bi-O octahedron to the next, in what can be viewed as a commensurate CDW. Upon substitutional doping, either via Bi→Pb or Ba→K, the long-range CDW order disappears, and a metallic/superconducting phase emerges at sufficiently high doping levels. It has long been speculated that short-range CDW correlations survive deep into the metallic phase and play an important role in the superconducting pairing mechanism (92, 93). Alternatively, the bismuthates have been proposed to be conventional electron-phonon superconductors, with a large electronic coupling to optical phonon branches that involve oxygen (94). Because diffuse scattering is sensitive to short-range CDW correlations, this pivotal conundrum can be resolved through studies of the local structure. Notably, the Bi-Pb system is more complicated compared to Ba-K (BKBO) due to the presence of metastable structural variants (80). This leads to interesting mesoscale structures that might be easily tunable with strain, but it also makes this bismuthate family less suitable for systematic diffuse scattering studies. BKBO, in contrast, is nearly ideal: The simple average structure and small unit cell enable both fruitful 3D-ΔPDF analysis and quantitative modeling.

X-ray diffuse scattering measurements on BKBO have yielded two central results (Fig. 6) (88). First, no trace of short-range CDW correlations is observed in crystals without long-range CDW order. This includes both insulating and metallic/superconducting BKBO, and it implies that CDW correlations are likely not relevant for bismuthate superconductivity. The second important result is the unexpected finding of nanoscale structural correlations that break inversion symmetry (Fig. 6A). These polar octahedral distortions lead to characteristic diffuse scattering features (Fig. 6C) that are much stronger in metallic than in insulating samples. They are also visible in the 3D-ΔPDF, especially through the opposite signs of the Ba-O and Ba-Bi correlations (Fig. 6, D and E). Moreover, the polar distortions are seen in classical Monte Carlo (MC) simulations based on effective bond valence sums, similar to previous work on relaxor ferroelectrics (95). The simulations also provide insight into the origin of the correlations: An intrinsic tendency toward this octahedral deformation is amplified by the local charge inhomogeneity introduced by the Ba-K substitution. In metallic BKBO, electronic screening renders electrostatic interactions short-ranged and thus likely both enhances the correlations and sets their length scale.

Fig. 6. Local structural correlations in superconducting Ba_1−xK_xBiO₃.

(A and B) Two characteristic distortions of Bi-O octahedra: an inversion-breaking dislacement (A) and a breathing distortion (B). The latter is associated with a CDW phase in the parent compound BaBiO₃. (C) X-ray diffuse scattering data for a single crystal of Ba_0.6K_0.4BiO₃ (Exp), compared to classical Monte Carlo (MC) modeling. A cut with L = 0 is shown. The diffuse patterns predominantly originate from short-range inversion-breaking distortions of the type seen in (A). (D and E) 3D-ΔPDF in the z = 0 and z = 0.5 planes generated from x-ray scattering data, with the most important pair correlations labeled in each panel. No evidence of breathing distortions is found, and the opposite signs of Ba-O and Ba-Bi correlations are only consistent with an inversion-breaking distortion (88).

The presence of locally broken inversion symmetry opens up fresh perspectives in bismuthate physics. Effective Rashba interactions between conducting electrons and phonons become possible and might contribute to superconducting pairing. Moreover, because BKBO turns out to be a locally noncenstrosymmetric superconductor, the possibility for exotic superconducting order parameter symmetries arises (96). Without inversion, the usual classification into parity-even and parity-odd superconducting order parameters is no longer applicable, and mixed-parity states are allowed. In turn, these might exhibit broken time-reversal symmetry, which could be detected using complementary local probes such as muon spin rotation. Last, diffuse scattering studies of other bismuthate families, as well as related compounds such as antimonides (97), should provide further insight into their similarities and differences and might help to explain why BKBO shows the highest T_c values among the bismuthates.

Cuprates

One of the defining features of the lamellar high-T_c cuprates is the interplay between perovskite-derived copper-oxygen planes and the intervening ionic rock-salt layers that separate the planes, and this generic structure can host a wide variety of distortions. Structural and electronic inhomogeneity has been extensively investigated in the cuprates since the early days (98), with numerous prominent STM (99–101), NMR (102–104), x-ray (105, 106), and neutron scattering (81, 107) studies. Yet, no consensus has emerged on the common characteristics and importance of nanoscale correlations, as different cuprate families exhibit various specific forms of inhomogeneity, along with doping-related point disorder (108). As noted, the cuprates harbor short-range CDW order that has been extensively studied with scattering techniques, including detailed recent diffuse x-ray scattering work on La_2−xBa_xCuO₄ (Fig. 7, A to C) (109). The Bi-based cuprates, which have been especially favorable for investigations with surface-sensitive probes, display both long-range and short-range superstructures (110–112) that can be modified, e.g., by Pb codoping (113). Another important system, YBa₂Cu₃O_7−δ (YBCO), shows complex ordering patterns of oxygen interstitials that cause extensive diffuse scattering (114, 115). Body-centered systems such as the La-based cuprates La_2−xSr_xCuO₄ (LSCO) and La_2−xBa_xCuO₄ (LBCO) exhibit a series of symmetry-lowering structural transitions due to rigid rotations of the Cu-O octahedra (116), and extensive scattering investigations have shown that the associated short-range fluctuations are prominent across the temperature-doping phase diagram (98, 117, 118). Recent diffuse scattering work established a universal exponential temperature dependence of short-range orthorhombic fluctuations in LSCO and the Tl-based system Tl₂Ba₂CuO_6+δ (Tl2201) (119), which closely resembles the superconducting fluctuation behavior (Fig. 7, D to G) (120, 121). This unusual observation has been interpreted as a signature of rare ordered regions that appear well above the bulk phase transition temperatures due to some underlying doping- and family-independent correlated inhomogeneity. If such inhomogeneity is indeed present, it would have far-reaching consequences for our understanding of cuprate physics, and diffuse scattering is one of the most versatile tools to search for it.

Fig. 7. Diffuse scattering in lanthanum- and thallium-based cuprates.

(A) Diffuse scattering in La_1.875Ba_0.125CuO₄. Weak incommensurate CDW peaks are seen in multiple zones (109). (B) Measured CDW peak intensities are compared with (C) those calculated by a model of La and Cu modulations in both layers of the crystal structure. (D) 1D cuts through a half-integer Bragg position in the high-temperature tetragonal phase of La_2−xSr_xCuO₄ (LSCO) (119). Neutron scattering measured with CORELLI (red, quasi-elastic scattering; orange, energy-integrated scattering) shows a split diffuse peak, while only a single peak is seen in x-ray scattering. Because neutron scattering is sensitive to a considerably smaller energy range than x-ray scattering, this stark difference was ascribed to the presence of both (E) dynamic and (F) quasistatic orthorhombic fluctuations, with the latter showing an antiphase boundary that leads to the observed low-energy incommensurability. (G) Exponential scaling of the diffuse superstructure intensity above the tetragonal-to-orthorhombic transition temperature T_LTO for LSCO with several Sr concentrations, as well as optimally doped Tl2201, compared to measurements of superconducting fluctuations. a.u., arbitrary units. r.l.u., reciprocal lattice unit.

To this end, HgBa₂CuO_4+δ (Hg1201) was chosen as a model system for detailed diffuse scattering measurements. The main advantage of this compound is a simple tetragonal average structure and small unit cell, along with an absence of structural transitions. Moreover, Hg1201 is doped using interstitial oxygen, which resides relatively far from the quintessential Cu-O planes and perturbs the lattice less severely than substitutional doping. Early diffuse scattering work (122–124) uncovered a tendency for the interstitial oxygen atoms to form chain-like structures in samples with transition temperatures above about 80 K, yet at lower densities, the interstitials are essentially randomly distributed. Electronically, Hg1201 shows the highest T_c values of any cuprate with a single Cu-O plane per primitive cell, as well as quantum oscillations (125) and negligible residual resistivities (126), which demonstrates a weak influence of the interstitials on the Cu-O planes. All this indicates that Hg1201 is one of the most pristine cuprates and representative of the entire cuprate family.

A combined neutron and x-ray diffuse scattering study of samples with relatively low interstitial oxygen densities revealed extensive and highly structured reciprocal-space features in Hg1201, implying that nanoscale structural correlations are prominent in this material (89). It is immediately clear that there is little diffuse scattering within the H-K planes, which implies that the atomic correlations are predominantly in the out-of-plane direction (Fig. 8). Moreover, the elastic discrimination enabled by the CORELLI instrument at the Spallation Neutron Source shows that the diffuse features are mostly static, an observation that was further confirmed in a targeted inelastic neutron scattering experiment. Both the neutron and x-ray scattering data were of sufficient quality to generate 3D-ΔPDFs, which provide further insight. The characteristic length scales associated with the correlations turn out to be ∼10 unit cells within the Cu-O planes and ∼3 unit cells perpendicular to the planes. Both length scales are comparable to or larger than the superconducting coherence lengths. This implies that the pairing is affected by the inhomogeneity, which could explain the observation of the unusual exponential fluctuation regime (Fig. 7).

Fig. 8. Local structure of the model cuprate superconductor Hg1201.

(A to C) Combined neutron and x-ray scattering data in underdoped Hg1201 crystals that show similar lobe-like features in the HK plane, indicative of complex nanoscale atomic displacements perpendicular to the Cu-O planes. (D and E) Atomic pair correlation functions obtained from reverse MC fits to the reciprocal space data. Mercury and apical oxygen atoms show strong positive correlations (D), while copper and apical oxygen display negative short-range correlations (E). This is consistent either with the formation of local Hg-O dipoles in the ionic layer, or breathing distortions of the Cu-O octahedra (89).

Although Hg1201 has a relatively small unit cell, there is still significant overlap among different atomic pair vectors in the 3D-ΔPDF, which makes interpretation challenging. To reliably determine the real-space nature of the atomic displacements, reverse Monte Carlo (RMC) refinement was used on both the x-ray and neutron scattering data in reciprocal space. The large supercells that are produced numerically enable detailed analysis of atomic correlation functions and show that the most important pair correlations are between apical oxygen and mercury atoms, as well as apical oxygen and in-plane copper. Mercury and apical oxygen displacements are positively correlated, while the copper and apical oxygen displacements are anticorrelated, which points to two possible origins: the formation of local Hg-O dipoles in the ionic layer or a breathing distortion of the Cu-O octahedra. While further insights from theory or ab initio modeling might resolve this question, both effects are not specific to the simple-tetragonal Hg1201 structure and might thus be a generic property of the cuprates. Diffuse scattering measurements of other cuprate families will therefore be highly valuable. More broadly, the comprehensive neutron, x-ray, and numerical work on Hg1201 constitutes both a technical and a scientific benchmark for structural studies of other quantum materials.

EXTENDED DEFECTS

Defects that extend over many unit cells, such as dislocations and stacking faults, cause specific diffuse scattering signatures, and the new generation of high-sensitivity instruments enables unprecedented scientific opportunities to examine them. While extended defects are crucial in materials science and metallurgy, they have been much less studied in the context of quantum materials, although they can lead to marked effects. Dislocations are associated with enormous local lattice strains, which can cause qualitative changes in the electronic subsystem in their vicinity. In turn, stacking faults destroy the long-range periodicity of layered crystal structures and can thus substantially affect electronically ordered states. We discuss here two recent examples: self-organized dislocation structures induced by plastic deformation in the perovskite oxides SrTiO₃ (STO) and KTaO₃ (KTO), and stacking faults in the van der Waals spin-liquid candidate RuCl₃.

Plastic deformation in oxides

A straightforward way to introduce dislocations into a material is via irreversible, plastic deformation. Yet for this to be possible, cracking must not occur, i.e., the energy of dislocation formation and/or migration must be sufficiently low. Given that these processes are thermally activated, most materials become ductile at temperatures that are a sizable fraction of their melting points. However, some systems display anomalous ductility at much lower temperatures, with STO a prominent example (127). STO is also a well-known quantum material, with low-temperature electronic properties that have been the subject of debate for six decades (128, 129). Pristine STO is a band insulator and incipient ferroelectric: It is very close to a ferroelectric instability but does not show long-range order down to the lowest temperatures. Upon doping with electrons, the material becomes superconducting at record-low charge-carrier densities and shows puzzling normal-state transport properties. Both the superconductivity and ferroelectricity are extremely sensitive to lattice strain, which makes STO an ideal model system to study the effects of plastic deformation on the quantum properties of a material.

For a ceramic material, STO is strikingly ductile at ambient temperature, with plastic deformation up to 10% possible in compression. Moreover, the deformation process leads to a remarkable self-organization of dislocations into mesoscale structures, whose properties have recently been revealed using diffuse scattering (130). The most obvious effect in deformed single crystals is an elongation of the usual Bragg peaks into arcs (Fig. 9A), which are known as asterisms from early studies of plastically deformed metals (131). The presence of asterisms implies that the sample breaks up into nearly unstrained tilted domains, with internal strain concentrated around the domain boundaries (Fig. 9, B and C). The local structure of the domain walls can be determined from weak diffuse streaks that are observed away from the asterisms, which are fully consistent with scattering from a periodic array of dislocations (Fig. 9, D to G). The dislocations thus self-organize to form long-range periodic domain walls, with a strain field that decays quickly away from the walls. This structural information was essential to obtain a deeper understanding of the unexpected effects of plastic deformation on the electronic properties of STO, including the appearance of quantum-critical ferroelectric fluctuations, a large boost of the superconducting T_c (130), and emergent magnetism and multiferroicity (132).

Fig. 9. The structure of plastically deformed strontium titanate.

(A) Neutron scattering data obtained for a STO crystal deformed to ε = 4.2% in compression along 〈001〉 (130). Upon deformation, sharp Bragg peaks transform into arcs (asterisms) due to the creation of tilted domains, as shown schematically in (B). The dislocations self-organize into walls [red lines in (B)], where the internal strain is highly concentrated. (C) Dependence of the asterism angular spread on the strain level for STO and KTO crystals (133). The observed behavior implies that the dislocation density within the walls increases with strain, which provides a simple method to tune this important property. (D to G) Diffuse streaks that originate from the long-range dislocation correlations within the walls. Panels (D) and (F) show neutron scattering measurements in two Brillouin zones, whereas panels (E) and (G) are the corresponding scattering intensities calculated for strain fields generated by periodic dislocation arrays (130, 133).

The initial work on deformed STO has introduced the use of plastic deformation to tune the electronic properties of quantum materials, opening a new avenue in the field. The perovskite KTaO₃ (KTO) was very recently shown to be ductile at ambient temperature as well (133, 134), with signatures of similar structural and electronic features. Moreover, recently developed high-force uniaxial strain cells have enabled pioneering x-ray diffuse scattering experiments with in situ applied stress, which have provided detailed insights into the formation of asterisms in STO with increasing strain (133). Given the importance of uniaxial stress as a tuning knob for the properties of quantum materials, these devices will be useful for diffuse scattering measurements on a wide range of interesting material systems.

Stacking faults in layered materials

Many important materials exhibit irregularities in the stacking sequence of their crystallographic planes. Such stacking faults are a specific type of planar defect and can play a prominent role in determining electronic properties. Strongly anisotropic, layered materials with weak van der Waals bonds between the layers are particularly susceptible due to the low energies needed to create stacking faults. Given their planar nature, these defects typically lead to rod-like diffuse features, which can be used to determine both their structure and concentration. Perhaps the most prominent recent scattering work on stacking faults in quantum materials has been in the context of magnetic systems such as RuCl₃ (Fig. 10) (135, 136). This material has been the subject of tremendous attention (137, 138) because it is a candidate to host the elusive Kitaev quantum spin-liquid state, which is expected to exhibit exotic excitations and holds promise for quantum computation (139). Yet due to residual interactions between van der Waals–bonded hexagonal RuCl₃ layers, the system orders magnetically below about 10 K and does not show a spin-liquid ground state, at least in the absence of an applied magnetic field. The long-range magnetic order is exceedingly sensitive to the stacking sequence (137, 140), which has motivated efforts to grow and characterize crystals with ever smaller stacking fault concentrations. Neutron and x-ray diffuse scattering have been indispensable in the efforts to quantify the stacking fault density and uncover their interplay with other structural features.

Fig. 10. Stacking faults in the Kitaev spin-liquid candidate material RuCl₃.

(A) X-ray diffuse scattering in a RuCl₃ crystal cycled through a structural transition that occurs between 100 and 200 K and rearranges the stacking sequence (135). Stacking faults are absent in the as-grown sample (a), while each cycle induces more defects both above and below the transition (b) to (d). (B) Neighboring ruthenium planes in the high-temperature C2/m structure are displaced by different vectors than the $R\overline{3}$ structure prevalent at low temperature. (C) 3D-ΔPDF generated from the x-ray scattering data measured at 30 K. Cuts for two values of the out-of-plane coordinate z are shown, z = 0 and z = 1/3; the latter corresponds to the distance between two adjacent Ru-Cl planes. The 3D-ΔPDF maps show that the low-temperature structure contains a mixture of the two stacking sequences in (B).

The stacking fault density in the highest-quality RuCl₃ crystals is negligibly low at room temperature; however, the material undergoes a first-order structural transition between 100 and 200 K, which involves a rearrangement of the planar stacking (135, 136). Neutron diffuse scattering measurements with CORELLI have shown that the sharpness of the structural transition strongly depends on the sample quality and is correlated with the intensity of the diffuse rods associated with the stacking faults (136). Electronic properties such as the magnetic ordering transitions and spin thermal transport are also sensitive to the stacking fault concentration. High-sensitivity x-ray scattering experiments have demonstrated that even the best crystals acquire stacking faults below the structural transition (Fig. 9A) due to an incomplete rearrangement of the layers and extremely low defect activation energies (135). Moreover, 3D-ΔPDF analysis was successfully used to provide insight into different stacking sequences and their relative weights (Fig. 9C) (135), which, to our knowledge, is the first application of the method in studies of extended defects. This work has provided clarity on the structural complexities of RuCl₃ that is essential to understand the interplay between structure and magnetism. The methodology is also relevant for a broad range of interesting layered materials from systems displaying magnetic or CDW order to exotic superconductors.

OUTLOOK

The previous sections have shown insights into correlated electron systems that existing single-crystal diffuse scattering capabilities can provide. In this concluding section, we briefly describe developments in instrumentation, detectors, sample environments, and data analysis, which will expand the scope of future scientific investigations. Most of the work reviewed here represents the initial steps in studies of broad classes of physical phenomena where diffuse scattering can provide unique insights. Some of the major questions that we expect will prominently feature in future investigations include the complex real-space nature of MITs; low-temperature intrinsic inhomogeneity associated with quantum phase transitions; quantification of extended defect densities and correlations in a wide range of systems, from layered van der Waals materials to complex oxide superconductors such as cuprates and ruthenates; and the influence of external stimuli—strain, pressure, magnetic and electric fields and light—on short-range correlations. Rapid technical advances will make many of these investigations possible in the near future.

While current x-ray diffuse scattering measurements over large volumes of reciprocal space are limited to an approximate temperature range of 15 to 700 K, many properties of interest in quantum and strongly correlated materials require measurements to much lower temperatures, if possible into the millikelvin range. At the same time, extending the range to higher temperatures will also enable experiments that probe the creation and evolution of short-range correlations at temperatures approaching the melting point, as well as open enhanced capabilities for in situ monitoring of high-temperature plastic deformation and dislocation engineering.

The challenge for developing such capabilities will be to keep background scattering to a minimum while still enabling measurements over large sample rotation angles and, in the case of x-rays, to avoid beam heating of the sample at low temperatures. Furthermore, the ability to simultaneously apply pressure and magnetic or electric fields would enable detailed studies of correlated disorder across multidimensional phase diagrams and result in valuable insights into quantum phase transitions and other correlated electron phenomena. As discussed earlier, uniaxial strain is another important parameter that can tune or qualitatively modify the properties of quantum materials in the elastic and plastic regimes. First tests of a dedicated, specially designed uniaxial strain cell have been performed at the Advanced Photon Source (133), and a device that is compatible with CORELLI is under development as well.

Low- and high-temperature sample environments and magnetic fields are in principle all available for diffuse neutron scattering measurements, although the required sample sizes may impact crystal quality. Even with larger samples, measurement times are still an order of magnitude longer than for x-ray experiments, limiting investigations to a few points in the phase diagram. New instrumentation at future facilities, such as the time-of-flight Laue instrument PIONEER (141) proposed to be built at the Second Target Station of the Spallation Neutron Source, will be optimized for small samples with linear dimensions in the 0.1 to 1 mm in range and a large solid-angle detector coverage of about 4 sr.

Recent developments in x-ray detector technologies will enable higher frame rates and higher dynamic range. Both are important to reduce artifacts due to the spray from strong Bragg peaks within the detector sensor layer. These are currently removed by elaborate procedures, which are difficult to optimize (27, 29, 142). Updates to synchrotron facilities are producing smaller beam sizes and higher brilliance, which, when combined with higher detector frame rates, open up the possibility of time-resolved and spatially resolved measurements. This could enable, e.g., scanning-probe diffuse scattering measurements of inhomogeneities at the micron scale. Stroboscopic measurements would allow us to probe correlated disorder in electric field–driven states.

Last, optimizing the extraction of all the information contained in complete datasets requires further developments of tools for automated feature detection and physically interpretable models of correlated disorder embedded in a long-range ordered crystalline lattice. Traditional methods are based on parametrizing the diffuse scattering with Warren-Cowley parameters (27), performing MC simulations of effective Hamiltonians (143, 144), or using RMC simulations to generate real-space structures (145, 146). The MC method has the advantage that it generally only involves relatively few interatomic interaction parameters and provides direct physical insights. However, it requires a specific model to be tailored for each system with parameters based on known physical and chemical principles. In the RMC method, on the other hand, the adjustable parameters are the positions of all the atoms in a box, which does not provide direct physical insight without further statistical analysis to obtain, e.g., conditional probabilities. 3D-ΔPDF has recently been incorporated into RMC approaches either to build a starting model for the simulations, e.g., in the program YELL (147), or to help validate and interpret RMC results (148).

The importance of machine learning in solving models of structural disorder will inevitably grow. Autoencoders have been successfully trained to optimize spin Hamiltonians from magnetic diffuse scattering (149, 150). However, it will be challenging to do the same with structural diffuse scattering because the interaction parameters are not as well defined, although they can sometimes be effectively approximated by mean field methods (44, 151, 152). An alternative approach would be to use ML to model short-range order directly from measured 3D-ΔPDF maps, as has already been achieved for long-range ordered structures (153). The problem can be simplified by using tools such as symmetry-mode analysis to constrain allowed distortions derived from higher symmetry phases (154). Ultimately, large supervised ML models incorporating these approaches in real, Patterson, and reciprocal space to obtain physical models could lead to a much more user friendly analysis of single-crystal diffuse scattering, on a par with standard PDF analysis. This would almost certainly lead to the more widespread adoption of the techniques described here in future investigations of the role of inhomogeneity in the properties of correlated electron materials.