Applicability of coherent x-ray diffractive imaging to ferroelectric, ferromagnetic, and phase change materials

A basic understanding of fundamental magnetic phenomena can inspire novel designs and facilitate numerous applications in advanced magnetic technologies.¹ In general, the ground state of a ferromagnetic system is multidomain, resulting from the competitive interactions of exchange, anisotropy, magnetostatic, and Zeeman energies. Neighboring domains² with non-collinear spin configuration are typically connected by a transition region, commonly known as the domain wall (DW).³ Mathematically, the width of a DW $Δ$ is defined by $Δ≈πJ/KU$⁠, with J being the exchange energy and $KU$ the anisotropy constant. $Δ$ can range from a few nanometers for hard magnets, such as $Nd2Fe14B$⁠,⁴ to tens of nanometers for soft magnetic materials, e.g., Permalloy $(Fe20Ni80)$⁴ or CoFe/Pd multilayers.⁵ The magnetic DWs typically exist in a state of complex non-collinear spin textures, and the magnetic DWs are particularly interesting due to their complex spin textures with specific symmetry, chirality, and structural/magnetic heterogeneities. Typically, the magnetic textures of the systems can be tailored according to the external magnetic field vs temperature phase diagrams, with many other conjugate fields (also known as order parameters) relevant to the systems applied onto the system of interest. Magnetic properties can be manipulated by applying several conjugate fields onto the systems, not limited to, such as spin current, varying exchange interactions, bias fields, controlling temperatures, modifying the chemical compositions by doping, etc.

Ferroelectricity is greatly influenced by microscopic properties, such as interfaces, the crystallinity of the systems, granular structures, ferroelectric domains heterogeneities, ferroelectric DWs, piezoelectric coefficients, etc. Like its magnetic counterpart, the polarization in ferroelectric materials also forms three-dimensional (3D) non-collinear structures, subject to influences of the relevant external conjugate fields, such as external electric field, temperature, and chemical compositions in the associated phase diagrams of the system. DWs and ferroelectric vortice formations and evolution can be driven by several conjugate fields, giving rise to structural/electronic phase transitions in the system. Ultimately, this mechanism can be implemented for remote sensing, quantum computing, and energy storage applications. Traditionally, DWs of the ferroelectric materials are considerably narrower than that of ferromagnetic systems, typically of the average size of only a few nm or lower, making them harder to image with conventional tools. Nevertheless, one could still tune/modify the ferroelectric properties for the investigation of structural dynamics of the systems to undergo a phase transformation, with relatively good spatial resolution and local structural sensitivities. A recent study suggests the potential applications of unit-cell-thick domain in the free-standing quasi-two-dimensional (2D) ferroelectric material,⁶ and this opens a new avenue for potential applications that require unit-cell-thick domain size. To maximize the ferroelectric systems’ performance and electronic properties, it is crucial to understand the fundamental behaviors of the converse piezoelectric coefficient, as well as the flexoelectric coefficient, as a function of an external electric field, temperature, and strain/strain gradient in the systems.

To illustrate the complex interrelation in complex materials, among the most significant external forces (electric field, magnetic field, and stress), the relevant materials’ conjugate fields (electric polarization, magnetization, and strain), and their coupling coefficients (electric susceptibility, magnetic susceptibility, and compliance tensor), Fig. 1 shows their general correlation in informative diagrams.⁷ This review paper focuses on the fundamental principles and the recent developments in phase changes that relate to these complex correlations in materials of interest.

Recent advancement of multiferroics⁸ has attracted enormous attention in the condensed matter physics community due to its wide applications in magnetostrictive/electrostrictive nano-devices, remote sensing, nano-actuators, semiconductor devices, quantum computation, etc. A great endeavor for searching emergent ferroelectric and ferromagnetic materials has been fruitful in the recent decade, fostering solutions for the dilemma of so-called “the contra-indication” between ferroelectricity and ferromagnetism due to its contradictory nature of the formation mechanisms. To enable the formation of ferroelectricity, the minimization of short-range Coulomb repulsion is preferred, with the ions being packed in a symmetric and non-ferroelectric mechanism. Furthermore, a ferroelectric system is formed when the ionic distance between the neighboring is minimized, obeying the second-order Jahn–Teller mechanism. To generate a magnetic moment in a ferromagnetic system, an unpaired electron is required but this is unpreferred for ferroelectricity as it prefers empty valence d-electron manifolds, resulting in zero unpaired electrons in the state. This competing mechanism generally hinders the formation of intrinsic multiferroic systems. However, some exceptional candidates satisfy the criteria, such as bismuth ferrite and $BiFeO3$⁠,^8,9 which exhibit both ferroelectricity and antiferromagnetic ordering when applying an external electric field. Other notable examples of single-phase intrinsic multiferroics are hexagonal manganites $RMnO3$ (R = Y, Ho, Lu),^10,11 where improper ferroelectrics is the resultant of structural distortion, and magnetite $Fe3O4$ that is both ferrimagnetic and ferroelectric¹² in the orthorhombic crystallographic structural phase. Figure 2 shows a summary of phase diagrams of several hexagonal $RMnO3$ from a number of studies.¹¹ The formation of extrinsic multiferroics usually has been devoted to the creation and design of multiferroic stacks with alternating ferromagnetic and ferroelectric layers so that artificial multiferroics systems are achieved with possessing both ferroelectric and ferromagnetic orderings.

Understanding phase transitions in complex materials is key for optimizing and manipulating the functional properties of ferroelectric or ferromagnetic devices. The first-order phase transitions (FOPTs) are of forms of discontinuous processes, occurring in $BaTiO3$⁠.¹³ Some other single crystals manifest the mechanism that the first derivatives of Gibbs free energy (G) with respect to relevant conjugate fields for the systems (such as temperature, electric field, and pressure) are discontinuous. On the other hand, second-order phase transitions (SOPTs) represent continuous first derivatives of G with respect to the conjugate fields, such as normal metal to superconducting transition, where the materials behave like quantum fluids in the superconducting state. Another example of SOPTs is ferromagnetic to the paramagnetic phase transition, where the thermodynamic properties of the phase transition can be calculated by using Landau theory.^14–17 Both FOPTs and SOPTs are typically accompanied by crystallographic structural transitions, such as cubic, tetragonal, orthorhombic to rhombohedral crystallographic structures, from the highest to the lowest temperatures in $BaTiO3$⁠. One example of FOPTs is the structural phase transition of the monoclinic phase at a lower temperature to the rutile phase at a higher temperature in vanadium dioxide,^18,19 and which is accompanied simultaneously by the SOPT electronic insulator-to-metal transition (IMT).²⁰ Figure 3 shows a detailed binary phase diagram of $VO2$ with changing stoichiometry and temperature.²⁰ Understanding the microscopic process of the phase transition could help reveal the fundamental drive force behind. Thus, it is desirable to characterize the crystallographic lattice strains’ evolution during the phase transformations, with high spatial resolution and sensitivity to local lattice displacements. The ultimate goal is to control and manipulate the phase transitions in ferroelectric, ferromagnetic, and other important complex systems to apply phase transition to tune the materials properties,^21–25 and to design their potential applications by demand.

X-ray imaging on complex materials has been under active development as an alternative to high-resolution electron microscopy imaging. While electron-based imaging techniques provide ultra-high spatial resolution with the ability to perform direct real-spacing mapping of the structural heterogeneities, the thickness of the specimens is typically limited to $∼$100 nm to avoid dynamical scattering and ensure Born approximation still applies to the systems. On the contrary, x-ray imaging can usually investigate bulk sample properties, with the attenuation depth of typical x-ray energy $∼$9 keV reaching tens of micrometers for common hard-condensed materials. Furthermore, x-ray-based imaging techniques benefit from their chemical and elemental sensitivities. When phase-contrast is implemented, refraction-sensitivity can be realized in a variety of systems of scientific interest. Direct x-ray imaging techniques include Transmission X-ray Microscopy (TXM) and Scanning Transmission X-ray Microscopy (STXM), both with $∼$20 nm best achievable spatial resolution, depending on the experimental setups and the properties of the specimens. Reciprocal-space based imaging techniques include x-ray holography,^26,27 and most recently, coherent x-ray diffraction imaging (CXDI).²⁸ Figure 4 illustrates a typical Bragg coherent x-ray diffraction imaging experiment at a synchrotron radiation facility.²⁹ CXDI is the first lensless x-ray imaging tool that uses coherent diffraction intensities in the reciprocal-space to retrieve the real-space complex sample wave functions using iterative algorithms. The iterative algorithms iterate between real-space and reciprocal-space by imposing various constraints in both spaces so that converged solutions are expected to reach for retrieval of sample complex wave function in real-space. The constraints in real-space are typically the so-called support constraint that estimates sample sizes and its known/estimated electronic densities. In the reciprocal-space, the Fourier modulus constraint is usually applied to replace the calculated amplitude of the sample exit wave function with the square-root values of the measured coherent diffraction intensities. To ensure successful CXDI data reconstructions, a stringent oversampling criterion³⁰ is a prerequisite. Bragg CXDI^31–33 was invented specifically for imaging crystalline specimens^34–36 with high sensitivity to lattice displacements and local structural heterogeneities.

The sophisticated amalgamation of STXM with CXDI, namely, ptychography,^37–39 can address the issue of non-uniqueness in the reconstructed solutions in CXDI occasionally, and the technique can study extended specimens with the simultaneous retrieval of the complex illumination function. The original overlap constraint⁴⁰ was proposed to retrieve phase information from the overlapping Bragg reflections in electron diffraction experiments with a convergent beam. In ptychography, the real-space support constraint is replaced with the overlap constraint^37,41,42 that is moved to real-space when coherent diffraction intensities are recorded with overlapping illumination on the specimens. Figure 5 illustrates a typical Bragg projection ptychography (BPP) experiment at a synchrotron radiation facility.⁴³ 

Spatial resolution in direct x-ray imaging techniques depends heavily on the quality and perfection of the x-ray lenses, which are technologically challenging due to the refraction indices of x rays. Therefore, the characteristics of the lenses are determined by the best achievable lithographic capabilities available. Both CXDI and ptychography circumvent the strict requirements for lithography, and the best achievable spatial resolution is, in principle, only diffraction limited.

X-ray ptychography and Bragg CXDI are ideal imaging techniques for studies of structural/electronic/magnetic dynamical changes, phase transitions, and structural/electronic/magnetic excitations driven by relevant external stimuli due to several factors. First, both techniques could be implemented in conjunction with other well-established x-ray methodologies, such as computed tomography,^44–48 near-edge x-ray absorption/refraction fine-structure microscopy,^49–51 and x-ray fluorescence imaging.^52,53 Second, minimal sample preparation effort is required to perform measurements comparing to high-resolution electron-based imaging techniques. Thus, the native state of the specimens can be better preserved, resulting in more practical in situ and operando studies on potentially real-world operating devices.

A lot of progress has been made for a better understanding of phase transformations in ferroelectric materials using Bragg CXDI, notably, ferroelectric to paraelectric phase transition,⁵⁴ antiferroelectrics structural dynamics in epitaxial monodomain $BiFeO3$⁠,⁵⁵ structural phase transitions in single-crystalline systems. The phase changes in the $ABO3$ systems can be shown in generalized diagrams as a function of various order parameters of the associated materials. Figure 6 illustrates some typical physics of phase diagrams in the general systems (⁠$ABO3$⁠, A = Ba, Pb; B = Ti, Zr), for example, and this applies to BaTi$O3$ nanocrystals. The order parameters referred here may include external electric field, magnetic field, temperature, stress, pressure, and so on.

A recent breakthrough in imaging structural phase transition driven by temperature in situ has been reported, and the structural transition has resulted in dislocations associated with the phase transformation process.⁵⁴ Figure 7 illustrates the evolution of a $BaTiO3$ nanoparticle upon crossing through its tetragonal-cubic phase transition.⁵⁴ The structural phase transition is accompanied simultaneously with the ferroelectric to paraelectric phase transition at a temperature of around 395 K. Additionally, the phase transformation appears to be reversible. Although, some hysteresis effects were observed due to a number of reasons, such as some experimental limitations due to the relatively long time required for the sample to be stabilized at the thermal equilibrium of the system.

Similarly, a recent study imaged ferroelectric vortices with an external electric field in situ in $BaTiO3$ with ferroelectric vortices dynamics as a function of external electric field.⁵⁶ Figure 8 ⁵⁶ shows the correlations between Bragg coherent diffraction measurements and phase-field simulations on a $BaTiO3$ nanoparticle. The study suggests that the transformation of the vortices in the system occurs simultaneously with the structural phase transition from coexisting tetragonal (T) and monoclinic (M) to a predominant M structural phase. The 3D morphology of the vortex core was successfully reconstructed, and the movement and the transformation of the vortex core were observed. Furthermore, the 3D toroidal moment of the polarization as the order parameter of the applied electric field was reconstructed as the curl of the 3D lattice displacements of the nanocrystal.

Studies on a $V2O3$ nanocrystal reveal mixed and partial dislocations present in the particle with high local structural sensitivity.⁵⁷ The study investigated the types of the reconstructed dislocations within the $V2O3$ nanocrystal by fitting the lattice displacements by particular Burgers vectors at several radii so that the type of the dislocation in the nanocrystal was correctly identified. The study concludes that the identification of the mixed dislocation present is partly due to a stacking fault near the base of the nanocrystal. Figure 9 displays Bragg CXDI results⁵⁷ of the outline of the dislocation cores in the base of the $V2O3$ particle, based on the phase of the reconstructed crystal, with dislocation lines being approximately along [001]. Further understanding of the role partial and mixed dislocations plays in crystal growth is helpful for a better understanding of structural properties in the growth of nanocrystals.

A recent endeavor has demonstrated exciting results on multiferroic materials $BaFe12O19$⁠, revealing 3D structural dynamics of topological defects, strain, and improper ferroelectric ordering driven by external electric field in situ.⁵⁸ The results demonstrate that Bragg CXDI can study topological defects that resemble cosmic strings, which is a commonly known hypothesis in cosmology. The study combines the Bragg CXDI reconstructed results with the Landau phase-field modeling to reveal reconstructions of multiple topological loops in a nanoparticle of around 500 nm size. Figure 10 ⁵⁸ shows the evolution of the vorticity of electric polarization and depth profiling for a 1D topological string in multiferroic materials $BaFe12O19$ nanocrystal. Future studies can be extended to the ultrafast field of research to investigate the associated symmetry changes in the systems upon selected external stimuli.

Bragg CXDI studies on the effects of high pressure onto a silver/gold nanocrystal have been successfully demonstrated with exciting structural dynamics as a function of external pressure in situ.^59,60 The study used water as the pressure transmitting medium, and once the water becomes solidified, the specimen movement in both translational and rotational dimensions becomes constrained to enable reliable 3D reconstructions. In the report, four pressure conditions were investigated. With the highest pressure level (2.9 GPa), the nanocrystal became stabilized although drastically deformed, which resulted in reduced interference fringes recorded. The study claims that the momentum transfer induced by the Bragg diffraction has significant effects on the silver nanocubes. Also, strain and morphology studies on the same crystal were performed as a function of time elapsed. The Bragg CXDI reconstructions identified a slip deformation. Figure 11 ⁵⁹ shows strain and morphology evolution of a silver nanocube under 2.1 GPa pressure in four sequential measurements, using Bragg CXDI measured along the measured [111] direction.

In Bragg ptychography, studies were conducted on structural dynamics,^61,62 structural heterogeneities,⁶³ polarization mapping in ferroelectric domains,⁶⁴ phase-shifted domain structural analysis on a single-crystalline metallic thin film,⁴³ and so on. In the study of a single 400 nm-thick InGaN/GaN coreshell nanowire,⁶³ x-ray Bragg ptychography with a nanofocused beam was used to investigate 3D strain distribution, with a supporting model of finite element method (FEM). The lattice mismatch between the GaN core and InGaN shell was quantitatively retrieved, the strain distribution and a variation in misfit strain relaxation were observed. The strain distribution/relaxation reconstructed by Bragg ptychography has shown asymmetry, and the results were independently verified by the scanning x-ray diffraction microscopy (SXDM) method. Figure 12 ⁶³ displays reconstructed amplitudes and phases of 2D x-ray Bragg ptychography (XBP) on a single 400 nm-thick InGaN/GaN coreshell nanowire.

Recently, focused-beam Bragg projection ptychography (BPP) was implemented to image $180°$ strip domains in a ferroelectric $PbTiO3$ single-crystalline thin film. The reconstruction results recover quantitative mapping of out-of-plane electric polarization with sub-10-nm spatial resolution. The BPP is particularly sensitive to the strip domains that are orientated relatively parallel to the incident beam direction and less sensitive to that orientated away from the scattering plane due to the 2D projection nature. This limitation can be circumvented by applying 3D Bragg ptychography.^62,65 Figure 13 displays the reconstructed amplitudes and phases on $PbTiO3$ thin film using Bragg projection ptychography (BPP).⁶⁴ The results have shown a correlation to the out-of-plane polarization distribution quantitatively.

There have been some fruitful results for using Bragg CXDI to probe magnetic/structural heterogeneities in ferromagnetic nanocrystals.^66–68 A late study on the 3D Bragg electronic density and strain distribution within ferromagnetic nickel nanoparticles⁶⁶ demonstrated the feasibility of using Bragg CXDI for imaging local strain heterogeneities in single ferromagnetic crystals, where dislocations and twin domain defects were successfully reconstructed. The reconstructed nanoparticles have specific rough interfaces with a twin domain defect. The quantitative results were encoded in the phase difference between the two disconnected domains, and the phase offset represents the width of twin defects. Figure 14 shows the ferromagnetic nickel nanoparticle shape and the atomic displacement projected onto the surface, with some highlights of the three-dimensional dislocations.

Last but not least, there has been growing interest in systems, such as single-crystalline magnetostriction,⁶⁹ using Bragg CXDI^69,70 with external magnetic fields in situ. By recording the Bragg reflections of (002) and (111) of single-crystal nickel nanowires, with the external magnetic fields of 200 and 600 Oe, the nucleation of local 3D strain fields was observed with high spatial resolution, leading to the enhancement of the magnetostriction coefficients in the systems. The mechanisms responsible for the enhanced magnetostriction coefficients include strain-induced anisotropy, shape, and magnetic anisotropies. Figure 15 shows the coherent diffraction intensities of the nickel nanowires for the external magnetic fields of 0, 200, and 600 Oe respectively. Also, a lattice displacement component of $u111$ projected on an isosurface of the single-crystal nickel nanowire without the external magnetic field is shown in Fig. 15.

Active research on ferroelectric, ferromagnetic, and multiferroics has been performed with enormous efforts in the past decade, aiming at a better comprehension of their structural dynamics, underlying mechanisms of the ferroelectric and magnetic orderings, etc. Despite the fact that significant progress has been made for unraveling fundamental principles and implementing device applications in recent years, many problems remain to be investigated, both theoretically and experimentally, to further advance our understanding of the material systems.

Particularly, for FOPTs and SOPTs in complex electronic systems, the following questions are to be explored:

1. How the structural phase transition couples to the electronic phase transition in ferroelectric/piezoelectric single-crystal systems such as $BaTiO3$ and $PbTiO3$?

2. What roles do lattice strains play in the magnetoelastic phase transition in the relevant systems?

3. What is the mechanism that accounts for topological defects/dislocations formations during FOPTs?

Characterization of structural dynamics of the hard-condensed matter when undergoing phase transformation is of paramount importance for the design and manufacturing of next-generation high-performance nanoscale functional devices, with potential applications in the area ranging from ferroelectricity, magnetism, multiferroics, electrostriction, magnetostriction, magnetoelasticity, and so on. The breathtaking advancement in the active research field of complex electronic materials, particularly in the viewpoint of the advent of high-performance complex oxides-based multiferroics, high-temperature superconductivity, drives the invention/improvisation of high-resolution characterization tools that are structural, chemical, and magnetic sensitive. Characterization tools such as imaging techniques that are sensitive to the change in structural heterogeneities, electronic (dis-)order, lattice strain, chemical elements/compounds, are particularly desirable. The exemplary candidates are diffraction imaging techniques CXDI and its scanning version, ptychography, particularly in Bragg geometry. Studies on in situ and operando experiments are on the rise in the past few years; thus, dynamical mechanisms and functional properties of the materials systems in real-life operating conditions could be explored.

X-ray-based imaging tools have overwhelming advantages in bulk-sensitivities, high penetration depth, elemental specificities, and high intensity. CXDI and ptychography eliminate the requirements for x-ray lenses, promising diffraction-limited spatial resolution with a minimal level of sample preparation. They are both desirable because of their large scattering cross section, multi-scale capabilities with chemical, elemental, and refraction sensitivities. The tunability of x-ray energies enables measurements using resonant x-ray absorption of soft or hard x rays, especially helpful for studies of elemental mapping and characterization of magnetic and structural heterogeneous textures. Although Bragg CXDI and Bragg ptychography have fascinating potentials in structural and phase characterization, both techniques have certain limitations in sample size. In Bragg CXDI, the size of a single-crystal particle is limited by extinction length, which could be well below 1 $μ$m because of the kinematic approximation of the mathematical bases of the technique and the iterative algorithms. Nevertheless, recent developments of advanced iterative algorithms that account for dynamical diffraction⁷¹ have further advanced the field with theoretical analyses. Also, the corrections for the refraction and absorption of diffraction effects in Bragg CXDI data of single crystals are proposed with analytical simulations. For Bragg ptychography, while the sample dimension can be extended in 2D, the third dimension is still limited to 10s–100s of nanometers. This is because the phase retrieval algorithm assumes that the wavefront of the probe is not substantially perturbed while propagating through the sample, to satisfy the Fourier slice theorem in the projection geometry.

Due to the kinematic approximation in diffraction to satisfy the Born approximation, both the x-ray and electron-based imaging techniques have certain limitations in sample size. Bragg CXDI and Bragg ptychography are relatively advantageous in this regard because of the relatively larger extinction depths of x rays in hard-condensed matter specimens in general when compared to electron-based imaging techniques. Additionally, Bragg CXDI and Bragg ptychography have promised diffraction-limited spatial resolution, primarily determined by the coherent flux density. The spatial resolution is also highly dependent on the highest obtainable reciprocal-space Q-vectors of good signal-to-noise ratio, thus enabling the highest measurable numerical apertures in the experiments.

The atomic spatial resolution of the electron microscopy is higher than its x-ray counterparts, with the best achievable spatial resolution in Bragg CXDI being around 4–9 nm.⁷² On the other hand, high-resolution electron microscopy routinely achieves atomic spatial resolution due to the highly focused electron beam using well-developed electromagnetic lenses. As a result, electron and x-ray-based imaging techniques are complementary methods, which are equally important and valuable for materials characterizations.

Future studies should continue working toward a better understanding of the fundamental properties of the materials regarding their ferroelectric, magnetic, multiferroic, magnetostrictive, and electrostrictive nature, contributing to potential real-life technological applications in the research areas. Research and development of complex oxide-based nanoelectronics, their applications, as well as optimization of their operating conditions and maximization of the devices’ functional performance are important goals to achieve in the coming years and decades for scientific breakthroughs in the coming era.

To develop CXDI imaging techniques, optimization of experimental instrumentations and capabilities should be prioritized to address and accommodate challenging experiments with multiple external stimuli applied for structural dynamics and phase changes. CXDI and ptychography, in both transmission and Bragg geometry, are on the rise to be one of the most promising imaging tools owing to their versatility and applicability to a wide range of specimens. Currently, due to the limited coherent fluxes at the existing endstations in the existing synchrotron facilities, some of the photon-deficient measurements are extremely challenging to be carried out. With the ongoing upgrades to diffraction-limited fourth generation light sources worldwide with enhanced coherent beams in the next decade, probing structural dynamics and their associated fluctuations at time scales ranging from nanoseconds to femtoseconds is to become a reality.

Furthermore, intense development in other areas, such as commissioning and upgrading new/existing endstations with the state-of-art sample environments, optimization of the existing iterative reconstruction algorithms to account for sample instability/fluctuation, and data inaccuracy, is particularly important to further advance the experimental and methodological capabilities. Technological improvement in x-ray detectors is also crucial to better harness the best available coherent x-ray flux. Last but not least, x-ray free-electron laser sources that are operating worldwide offer an order of magnitude higher coherent flux with up to femtoseconds x-ray pulses, facilitating x-ray characterization of next-generation nanoscale electronics and materials sciences in general. All of the above proposed outlooks aim to build a bridge for fruitful inter-disciplinary research in the coming decades and beyond.

We envisage that future developments in the Bragg CXDI and Bragg ptychography will be primarily focused on developing multiple-detector measurement systems with high throughput and efficiency. With the incrementing coherence for the fourth generation diffraction-limited synchrotron facilities worldwide, imaging single dislocations could be feasible with potential technological applications in various research fields. For example, studies on hidden transitions in photosynthesis, resolving the coherent scattering of individual electron/atom, could be realized when spatial resolution is being pushed to the limits. Exciting advancements to date, such as single-shot crystallography implemented by the “diffract-and-destroy” experiments⁷³ with an x-ray free-electron laser, can also benefit from the diffraction-limited next-generation light sources available shortly.

This work was supported by the U.S. Department of Defense, Air Force Office of Scientific Research under Award No. FA9550-14-1-0363 (Program Manager: Dr. Ali Sayir) and funds from Rensselaer Polytechnic Institute. J.S. acknowledges the Air Force Office of Scientific Research under Award No. FA9550-18-1-0116. E.F. and J.S. acknowledge the National Science Foundation (NSF) under Award No. 2024972.

The authors have no conflict of interest to disclose.

Raw data were measured at the synchrotron radiation facilities worldwide and are permanently deposited there. The data that support the findings of this study are available from the corresponding author upon reasonable request.