The crystallography of correlated disorder

Manipulating disorder within cathodes of alkali-ion batteries

The fact that ordered materials are rarely perfectly crystalline is widely acknowledged among materials scientists, but its impact is often overlooked or underestimated when studying how structure relates to properties. Various investigations demonstrate that intrinsic and extrinsic defects, and disorder generated by physicochemical reactions, are responsible for unexpectedly detrimental or beneficial functionalities. The task remains to modulate the disorder to produce desired properties in materials. As disorder is often correlated with local interactions, it is controllable. In this Review, we explore the structural disorder in cathode materials as a novel approach for improving their electrochemical performance. We revisit cathode materials for alkali-ion batteries and outline the origins and beneficial consequences of disorder. Focusing on layered, cubic rocksalt and other metal oxides, we discuss how disorder improves electrochemical properties of cathode materials and which interactions generate the disorder. We also present the potential pitfalls of disorder that must be considered. We conclude with perspectives for enhancing the electrochemical performance of cathode materials by using disorder.

Heterogeneous field response of hierarchical polar laminates in relaxor ferroelectrics

Understanding the microscopic origin of the superior electromechanical response in relaxor ferroelectrics requires knowledge not only of the atomic-scale formation of polar nanodomains (PNDs) but also the rules governing the arrangements and stimulated response of PNDs over longer distances. Using x-ray coherent nanodiffraction, we show the staggered self-assembly of PNDs into unidirectional mesostructures that we refer to as polar laminates in the relaxor ferroelectric 0.68PbMg1/3Nb2/3O3-0.32PbTiO3 (PMN-0.32PT). We reveal the highly heterogeneous electric-field-driven responses of intra- and interlaminate PNDs and establish their correlation with the local strain and the nature of the PND walls. Our observations highlight the critical role of hierarchical lattice organizations on macroscopic material properties and provide guiding principles for the understanding and design of relaxors and a wide range of quantum and functional materials.

Quantifying chemical short-range order in metallic alloys

Metallic alloys often form phases-known as solid solutions-in which chemical elements are spread out on the same crystal lattice in an almost random manner. The tendency of certain chemical motifs to be more common than others is known as chemical short-range order (SRO), and it has received substantial consideration in alloys with multiple chemical elements present in large concentrations due to their extreme configurational complexity (e.g., high-entropy alloys). SRO renders solid solutions "slightly less random than completely random," which is a physically intuitive picture, but not easily quantifiable due to the sheer number of possible chemical motifs and their subtle spatial distribution on the lattice. Here, we present a multiscale method to predict and quantify the SRO state of an alloy with atomic resolution, incorporating machine learning techniques to bridge the gap between electronic-structure calculations and the characteristic length scale of SRO. The result is an approach capable of predicting SRO length scale in agreement with experimental measurements while comprehensively correlating SRO with fundamental quantities such as local lattice distortions. This work advances the quantitative understanding of solid-solution phases, paving the way for the rigorous incorporation of SRO length scales into predictive mechanical and thermodynamic models.

Defect-enabling zirconium-based metal-organic frameworks for energy and environmental remediation applications

This comprehensive review explores the diverse applications of defective zirconium-based metal-organic frameworks (Zr-MOFs) in energy and environmental remediation. Zr-MOFs have gained significant attention due to their unique properties, and deliberate introduction of defects further enhances their functionality. The review encompasses several areas where defective Zr-MOFs exhibit promise, including environmental remediation, detoxification of chemical warfare agents, photocatalytic energy conversions, and electrochemical applications. Defects play a pivotal role by creating open sites within the framework, facilitating effective adsorption and remediation of pollutants. They also contribute to the catalytic activity of Zr-MOFs, enabling efficient energy conversion processes such as hydrogen production and CO2 reduction. The review underscores the importance of defect manipulation, including control over their distribution and type, to optimize the performance of Zr-MOFs. Through tailored defect engineering and precise selection of functional groups, researchers can enhance the selectivity and efficiency of Zr-MOFs for specific applications. Additionally, pore size manipulation influences the adsorption capacity and transport properties of Zr-MOFs, further expanding their potential in environmental remediation and energy conversion. Defective Zr-MOFs exhibit remarkable stability and synthetic versatility, making them suitable for diverse environmental conditions and allowing for the introduction of missing linkers, cluster defects, or post-synthetic modifications to precisely tailor their properties. Overall, this review highlights the promising prospects of defective Zr-MOFs in addressing energy and environmental challenges, positioning them as versatile tools for sustainable solutions and paving the way for advancements in various sectors toward a cleaner and more sustainable future.

Resolving length-scale-dependent transient disorder through an ultrafast phase transition

Material functionality can be strongly determined by structure extending only over nanoscale distances. The pair distribution function presents an opportunity for structural studies beyond idealized crystal models and to investigate structure over varying length scales. Applying this method with ultrafast time resolution has the potential to similarly disrupt the study of structural dynamics and phase transitions. Here we demonstrate such a measurement of CuIr2S4 optically pumped from its low-temperature Ir-dimerized phase. Dimers are optically suppressed without spatial correlation, generating a structure whose level of disorder strongly depends on the length scale. The redevelopment of structural ordering over tens of picoseconds is directly tracked over both space and time as a transient state is approached. This measurement demonstrates the crucial role of local structure and disorder in non-equilibrium processes as well as the feasibility of accessing this information with state-of-the-art XFEL facilities.

Giant Modulation of Refractive Index from Picoscale Atomic Displacements

It is shown that structural disorder-in the form of anisotropic, picoscale atomic displacements-modulates the refractive index tensor and results in the giant optical anisotropy observed in BaTiS3, a quasi-1D hexagonal chalcogenide. Single-crystal X-ray diffraction studies reveal the presence of antipolar displacements of Ti atoms within adjacent TiS6 chains along the c-axis, and threefold degenerate Ti displacements in the a-b plane. 47/49Ti solid-state NMR provides additional evidence for those Ti displacements in the form of a three-horned NMR lineshape resulting from a low symmetry local environment around Ti atoms. Scanning transmission electron microscopy is used to directly observe the globally disordered Ti a-b plane displacements and find them to be ordered locally over a few unit cells. First-principles calculations show that the Ti a-b plane displacements selectively reduce the refractive index along the ab-plane, while having minimal impact on the refractive index along the chain direction, thus resulting in a giant enhancement in the optical anisotropy. By showing a strong connection between structural disorder with picoscale displacements and the optical response in BaTiS3, this study opens a pathway for designing optical materials with high refractive index and functionalities such as large optical anisotropy and nonlinearity.

Configurational Disorder, Strong Anharmonicity, and Coupled Host Dynamics Lead to Superionic Transport in Li3YCl6 (LYC)

In superionic crystals, liquid-like ionic diffusivities often come hand-in-hand with ultra-low thermal conductivity and soft vibrational dynamics. However, generalized relationships between ion transport and vibrational dynamics remain elusive due to the diversity of superionic materials and complex underlying mechanisms. Here, the links between vibrational dynamics and ion transport in close-packed lithium halide ion conductor Li3YCl6 (LYC) are examined using a suite of atomistic first-principles methods. It is shown that configurational disorder, lattice anharmonicity, and coupled host-mobile ion vibrational dynamics together induce a transition to the superionic state. Statistical correlations between ionic hops and activation of the distribution of vibrational modes are found. However, typical phenomena associated with superionic conductors such as selective breakdown of zone-boundary soft phonons, or long wavelength transverse acoustic modes as in the 'phonon-liquid-electron crystal' concept, are not present. Instead, anharmonic zone-boundary modes aiding Li diffusion are found to broaden and soften selectively but persist across the superionic transition. These anharmonic modes couple Li ion motion with the vibrations of the flexible close-packed anion framework, which remains stable and facilitates ionic hopping. The results provide insights into how configurational disorder and soft-yet-resilient vibrational modes enable ionic hopping, particularly in 3D close-packed crystals.

Impact of Correlated Disorder on Surface Superconductivity: Revealing the Robustness of the Surface Ordering Effect

Surface superconductivity, wherein electron pairing occurs at material surfaces or interfaces, has attracted a remarkable amount of attention since its discovery. Recent theoretical predictions have unveiled increased critical temperatures, especially at the surfaces of certain compounds and/or structures. The notion of "surface ordering" has been advanced to elucidate this phenomenon. Employing the framework of self-consistent Bogoliubov-de Gennes equations and a model incorporating correlated disorder, our study demonstrates the persistence of the surface ordering effect in the presence of weak to moderate bulk disorder. Intriguingly, our findings indicate that under moderate disorder conditions the surface critical temperature can be further increased, depending on the intensity and correlation of the disorder.

Exploring disorder correlations in superconducting systems: spectroscopic insights and matrix element effects

Understanding the intricate interplay between disorder and superconductivity has become a key area of research in condensed matter physics, with profound implications for materials science. Recent studies have shown that spatial correlations of disorder potential can improve superconductivity, prompting a re-evaluation of some theoretical models. This paper explores the influence of disorder correlations on the fundamental properties of superconducting systems, going beyond the traditional assumption of spatially uncorrelated disorder. In particular, we investigate the influence of disorder correlations on key spectroscopic superconductor properties, including the density of states, as well as on the matrix elements of the superconducting coupling constant and their impact on the localization length. Our findings offer valuable insights into the role of disorder correlations in shaping the behavior of superconducting materials.

Correlating Structural Disorder in Metal (Oxy)hydroxides and Catalytic Activity in Electrocatalytic Oxygen Evolution

Understanding the correlation between the structural evolution of electrocatalysts and their catalytic activity is both essential and challenging. In this study, we investigate this correlation in the context of the oxygen evolution reaction (OER) by examining the influence of structural disorder during and after dynamic structural evolution on the OER activity of Fe-Ni (oxy)hydroxide catalysts using operando X-ray absorption spectroscopy, alongside other experiments and theoretical calculations. The Debye-Waller factors obtained from extended X-ray absorption fine structure analyses reflect the degree of structural disorder and exhibit a robust correlation with the intrinsic OER activities of the electrocatalysts. The enhanced OER activity of in situ-generated metal (oxy)hydroxides derived from different pre-catalysts is linked to increased structural disorder, offering a promising approach for designing efficient OER electrocatalysts. This strategy may inspire similar investigations in related electrocatalytic energy-conversion systems.

POMFinder: identifying polyoxometallate cluster structures from pair distribution function data using explainable machine learning

Characterization of a material structure with pair distribution function (PDF) analysis typically involves refining a structure model against an experimental data set, but finding or constructing a suitable atomic model for PDF modelling can be an extremely labour-intensive task, requiring carefully browsing through large numbers of possible models. Presented here is POMFinder, a machine learning (ML) classifier that rapidly screens a database of structures, here polyoxometallate (POM) clusters, to identify candidate structures for PDF data modelling. The approach is shown to identify suitable POMs from experimental data, including in situ data collected with fast acquisition times. This automated approach has significant potential for identifying suitable models for structure refinement to extract quantitative structural parameters in materials chemistry research. POMFinder is open source and user friendly, making it accessible to those without prior ML knowledge. It is also demonstrated that POMFinder offers a promising modelling framework for combined modelling of multiple scattering techniques.

How Amorphous Nanomaterials Enhanced Electrocatalytic, SERS, and Mechanical Properties

There is ever-growing research interest in nanomaterials because of the unique properties that emerge on the nanometer scale. While crystalline nanomaterials have received a surge of attention for exhibiting state-of-the-art properties in various fields, their amorphous counterparts have also attracted attention in recent years owing to their unique structural features that crystalline materials lack. In short, amorphous nanomaterials only have short-range order at the atomic scale, and their atomic packing lacks long-range periodic arrangement, in which the coordinatively unsaturated environment, isotropic atomic structure, and modulated electron state all contribute to their outstanding performance in various applications. Given their intriguing characteristics, we herein present a series of representative works to elaborate on the structural advantages of amorphous nanomaterials as well as their enhanced electrocatalytic, surface-enhanced Raman scattering (SERS), and mechanical properties, thereby elucidating the underlying structure-function relationship. We hope that this proposed relationship will be universally applicable, thus encouraging future work in the design of amorphous materials that show promising performance in a wide range of fields.

Quantitative three-dimensional local order analysis of nanomaterials through electron diffraction

Structure-property relationships in ordered materials have long been a core principle in materials design. However, the introduction of disorder into materials provides structural flexibility and thus access to material properties that are not attainable in conventional, ordered materials. To understand disorder-property relationships, the disorder - i.e., the local ordering principles - must be quantified. Local order can be probed experimentally by diffuse scattering. The analysis is notoriously difficult, especially if only powder samples are available. Here, we combine the advantages of three-dimensional electron diffraction - a method that allows single crystal diffraction measurements on sub-micron sized crystals - and three-dimensional difference pair distribution function analysis (3D-ΔPDF) to address this problem. In this work, we compare the 3D-ΔPDF from electron diffraction data with those obtained from neutron and x-ray experiments of yttria-stabilized zirconia (Zr0.82Y0.18O1.91) and demonstrate the reliability of the proposed approach.

Large Perpendicular Magnetic Anisotropy Induced by an Intersite Charge Transfer in Strained EuVO2H Films

Perovskite oxides ABO3 continue to be a major focus in materials science. Of particular interest is the interplay between A and B cations as exemplified by intersite charge transfer (ICT), which causes novel phenomena including negative thermal expansion and metal-insulator transition. However, the ICT properties were achieved and optimized by cationic substitution or ordering. Here we demonstrate an anionic approach to induce ICT using an oxyhydride perovskite, EuVO2H, which has alternating layers of EuH and VO2. A bulk EuVO2H behaves as a ferromagnetic insulator with a relatively high transition temperature (TC) of 10 K. However, the application of external pressure to the EuIIVIIIO2H bulk or compressive strain from the substrate in the thin films induces ICT from the EuIIH layer to the VIIIO2 layer due to the extended empty V dxy orbital. The ICT phenomenon causes the VO2 layer to become conductive, leading to an increase in TC that is dependent on the number of carriers in the dxy orbitals (up to a factor of 4 for 10 nm thin films). In addition, a large perpendicular magnetic anisotropy appears with the ICT for the films of <100 nm, which is unprecedented in materials with orbital-free Eu2+, opening new perspectives for applications. The present results provide opportunities for the acquisition of novel functions by alternating transition metal/rare earth layers with heteroanions.

Thiocyanate-Stabilized Pseudo-cubic Perovskite CH(NH2)2PbI3 from Coincident Columnar Defect Lattices

α-FAPbI3 (FA+ = CH(NH2)2+) with a cubic perovskite structure is promising for photophysical applications. However, α-FAPbI3 is metastable at room temperature, and it transforms to the δ-phase at a certain period of time at room temperature. Herein, we report a thiocyanate-stabilized pseudo-cubic perovskite FAPbI3 with ordered columnar defects (α'-phase). This compound has a √5ap × √5ap × ap tetragonal unit cell (ap: cell parameter of primitive perovskite cell) with a band gap of 1.91 eV. It is stable at room temperature in a dry atmosphere. Furthermore, the presence of the α'-phase in a mixed sample with the δ-phase drastically reduces the δ-to-α transition temperature measured on heating, suggesting the reduction of the nucleation energy of the α-phase or thermodynamic stabilization of the α-phase through epitaxy. The defect-ordered pattern in the α'-phase forms a coincidence-site lattice at the twinned boundary of the single crystals, thus hinting at an epitaxy- or strain-based mechanism of α-phase formation and/or stabilization. In this study, we developed a new strategy to control defects in halide perovskites and provided new insight into the stabilization of α-FAPbI3 by pseudo-halide and grain boundary engineering.

Size dependence of the surface spin disorder and surface anisotropy constant in ferrite nanoparticles

The magnetic properties of nanoscale magnets are greatly influenced by surface anisotropy. So far, its quantification is based on the examination of the blocking temperature shift within a series of nanoparticles of varying sizes. In this scenario, the surface anisotropy is assumed to be a particle size-independent quantity. However, there is no solid experimental proof to support this simplified picture. On the contrary, our work unravels the size-dependent magnetic morphology and surface anisotropy in highly uniform magnetic nanoparticles using small-angle polarized neutron scattering. We observed that the surface anisotropy constant does not depend on the nanoparticle's size in the range of 3-9 nm. Furthermore, our results demonstrate that the surface spins are less prone to polarization with increasing nanoparticle size. Our study thus proves the size dependence of the surface spin disorder and the surface anisotropy constant in fine nanomagnets. These findings open new routes in materials based on a controlled surface spin disorder, which is essential for future applications of nanomagnets in biomedicine and magnonics.

As predicted and more: modulated channel occupation in YZn5+x

Like many complex intermetallic phases, the crystal structures of REZn5+x compounds (RE = lanthanide or Group 3 element) based on the EuMg5 type have gradually unfolded. The original reports described a complex hexagonal structure with an unusual combination of tetrahedrally close-packed regions and open spaces, as well as observations of superstructure reflections. More recently, we reinvestigated the structure of YZn5, reclassifying it as the EuMg5+x-type compound YZn5+x (x ≃ 0.2), in which disordered channels run along c through the spaces formerly considered open. In addition, DFT-chemical pressure (DFT-CP) analysis of ordered models of YZn5+x highlighted paths for communication between neighboring channels setting the stage for superstructure formation. Herein, the experimental elucidation of this effect is presented with the synthesis and structure determination of a modulated form of YZn5+x. By slow-cooling samples of YZn5+x from the annealing temperature, crystals were obtained that exhibit satellite reflections with the modulation wavevector q = {1\over 3}a* + {1\over 3}b* + 0.3041c*. Structure solution and refinement using a (3+1)D model in superspace group P31c({1\over 3}\,\!{1\over 3}σ3)00s reveals incommensurate order in the structure's channels. Here, two Zn sites associated with the channels are present, each with discontinuous atomic domains that are slanted in the x3x4 plane. Their slanting corresponds to adjustments along the c axis for the presence or absence of close neighbors along that axis, while the occupation patterns of neighboring channels are shifted by {1\over 3} of the modulation period. These features follow earlier predictions from CP analysis, highlighting how this approach can be used predictively in search of new phenomena.

Tuning electronic and phononic states with hidden order in disordered crystals

Disorder in crystals is rarely random, and instead involves local correlations whose presence and nature are hidden from conventional crystallographic probes. This hidden order can sometimes be controlled, but its importance for physical properties of materials is not well understood. Using simple models for electronic and interatomic interactions, we show how crystals with identical average structures but different types of hidden order can have very different electronic and phononic band structures. Increasing the strength of local correlations within hidden-order states can open band gaps and tune mode (de)localisation-both mechanisms allowing for fundamental changes in physical properties without long-range symmetry breaking. Taken together, our results demonstrate how control over hidden order offers a new mechanism for tuning material properties, orthogonal to the conventional principles of (ordered) structure/property relationships.

Breaking Rayleigh's Law with Spatially Correlated Disorder to Control Phonon Transport

Controlling thermal transport in insulators and semiconductors is crucial for many technological fields such as thermoelectrics and thermal insulation, for which a low thermal conductivity (κ) is desirable. A major obstacle for realizing low κ materials is Rayleigh's law, which implies that acoustic phonons, which carry most of the heat, are insensitive to scattering by point defects at low energy. We demonstrate, with large scale simulations on tens of millions of atoms, that isotropic long-range spatial correlations in the defect distribution can dramatically reduce phonon lifetimes of important low-frequency heat-carrying modes, leading to a large reduction of κ-potentially an order of magnitude at room temperature. We propose a general and quantitative framework for controlling thermal transport in complex functional materials through structural spatial correlations, and we establish the optimal functional form of spatial correlations that minimize κ. We end by briefly discussing experimental realizations of various correlated structures.

Anion-polarisation-directed short-range-order in antiperovskite Li2FeSO

Short-range ordering in cation-disordered cathodes can have a significant effect on their electrochemical properties. Here, we characterise the cation short-range order in the antiperovskite cathode material Li₂FeSO, using density functional theory, Monte Carlo simulations, and synchrotron X-ray pair-distribution-function data. We predict partial short-range cation-ordering, characterised by favourable OLi₄Fe₂ oxygen coordination with a preference for polar cis-OLi₄Fe₂ over non-polar trans-OLi₄Fe₂ configurations. This preference for polar cation configurations produces long-range disorder, in agreement with experimental data. The predicted short-range-order preference contrasts with that for a simple point-charge model, which instead predicts preferential trans-OLi₄Fe₂ oxygen coordination and corresponding long-range crystallographic order. The absence of long-range order in Li₂FeSO can therefore be attributed to the relative stability of cis-OLi₄Fe₂ and other non-OLi₄Fe₂ oxygen-coordination motifs. We show that this effect is associated with the polarisation of oxide and sulfide anions in polar coordination environments, which stabilises these polar short-range cation orderings. We propose that similar anion-polarisation-directed short-range-ordering may be present in other heterocationic materials that contain cations with different formal charges. Our analysis illustrates the limitations of using simple point-charge models to predict the structure of cation-disordered materials, where other factors, such as anion polarisation, may play a critical role in directing both short- and long-range structural correlations.

Hidden chemical order in disordered Ba7Nb4MoO20 revealed by resonant X-ray diffraction and solid-state NMR

The chemical order and disorder of solids have a decisive influence on the material properties. There are numerous materials exhibiting chemical order/disorder of atoms with similar X-ray atomic scattering factors and similar neutron scattering lengths. It is difficult to investigate such order/disorder hidden in the data obtained from conventional diffraction methods. Herein, we quantitatively determined the Mo/Nb order in the high ion conductor Ba₇Nb₄MoO₂₀ by a technique combining resonant X-ray diffraction, solid-state nuclear magnetic resonance (NMR) and first-principle calculations. NMR provided direct evidence that Mo atoms occupy only the M2 site near the intrinsically oxygen-deficient ion-conducting layer. Resonant X-ray diffraction determined the occupancy factors of Mo atoms at the M2 and other sites to be 0.50 and 0.00, respectively. These findings provide a basis for the development of ion conductors. This combined technique would open a new avenue for in-depth investigation of the hidden chemical order/disorder in materials.

Understanding the Role of Entropy in High Entropy Oxides

The field of high entropy oxides (HEOs) flips traditional materials science paradigms on their head by seeking to understand what properties arise in the presence of profound configurational disorder. This disorder, which originates from multiple elements sharing a single lattice site, can take on a kaleidoscopic character due to the vast numbers of possible elemental combinations. High configurational disorder appears to imbue some HEOs with functional properties that far surpass their nondisordered analogs. While experimental discoveries abound, efforts to characterize the true magnitude of the configurational entropy and understand its role in stabilizing new phases and generating superior functional properties have lagged behind. Understanding the role of configurational disorder in existing HEOs is the crucial link to unlocking the rational design of new HEOs with targeted properties. In this Perspective, we attempt to establish a framework for articulating and beginning to address these questions in pursuit of a deeper understanding of the true role of entropy in HEOs.

Truchet-tile structure of a topologically aperiodic metal-organic framework

When tiles decorated to lower their symmetry are joined together, they can form aperiodic and labyrinthine patterns. Such Truchet tilings offer an efficient mechanism of visual data storage related to that used in barcodes and QR codes. We show that the crystalline metal-organic framework [OZn₄][1,3-benzenedicarboxylate]₃ (TRUMOF-1) is an atomic-scale realization of a complex three-dimensional Truchet tiling. Its crystal structure consists of a periodically arranged assembly of identical zinc-containing clusters connected uniformly in a well-defined but disordered fashion to give a topologically aperiodic microporous network. We suggest that this unusual structure emerges as a consequence of geometric frustration in the chemical building units from which it is assembled.

Direct Visualization of Atomic Structure in Multivariate Metal-Organic Frameworks (MOFs) for Guiding Electrocatalysts Design

The direct utilization of metal-organic frameworks (MOFs) for electrocatalytic oxygen evolution reaction (OER) has attracted increasing interests. Herein, we employ the low-dose integrated differential phase contrast-scanning transmission electron microscopy (iDPC-STEM) technique to visualize the atomic structure of multivariate MOFs (MTV-MOFs) for guiding the structural design of bulk MOFs for efficient OER. The iDPC-STEM images revealed that incorporating Fe³⁺ or 2-aminoterephthalate (ATA) into Ni-BDC (BDC: benzenedicarboxylate) can introduce inhomogeneous lattice strain that weaken the coordination bonds, which can be selectively cleaved via a mild heat treatment to simultaneously generate coordinatively unsaturated metal sites, conductive Ni@C and hierarchical porous structure. Thus, excellent OER activity with current densities of 10 and 100 mA cm^-2 are achieved over the defective MOFs at small overpotentials of 286 mV and 365 mV, respectively, which is superior to the commercial RuO₂ catalyst and most of the bulk MOFs.

Eutectic CsHSO4-Coordination Polymer Glasses with Superprotonic Conductivity

Superprotonic phase transition in CsHSO₄ allows fast protonic conduction, but only at temperatures above the transition temperature of 141 °C (T_c). Here, we preserve the superprotonic conductivity of CsHSO₄ by forming a binary CsHSO₄-coordination polymer glass system, showing eutectic melting. Their anhydrous proton conductivities below T_c are at least 3 orders of magnitude higher than CsHSO₄ without compromising conductivity at higher temperatures or the need for humidification, reaching 6.3 mS cm^-1 at 180 °C. The glass also introduces processability to the conductor, as its viscosity below 10³ Pa·s can be achieved at 65 °C. Solid-state NMR and X-ray pair distribution functions reveal the oxyanion exchanges and the origin of the preserved conductivity. Finally, we demonstrate the preparation of a micrometer-scale thin-film proton conductor showing low resistivity with high transparency (transmittance >85% between 380-800 nm).

Hidden Local Symmetry Breaking in a Kagome-Lattice Magnetic Weyl Semimetal

Exploring the relationship between intriguing physical properties and structural complexity is a central topic in studying modern functional materials. Co₃Sn₂S₂, a newly discovered kagome-lattice magnetic Weyl semimetal, has triggered intense interest owing to the intimate coupling between topological semimetallic states and peculiar magnetic properties. However, the origins of the magnetic phase separation and spin glass state below T_C in this ordered compound are two unresolved yet important puzzles in understanding its magnetism. Here, we report the discovery of local symmetry breaking surprisingly co-emerges with the onset of ferromagnetic order in Co₃Sn₂S₂, by a combined use of neutron total scattering and half-polarized neutron diffraction. An anisotropic distortion of the cobalt kagome lattice at the atomic/nano level is also found, with distinct distortion directions among the two Co1 and four Co2 atoms. The mismatch of local and average symmetries occurs below T_C, indicating that Co₃Sn₂S₂ evolves to an intrinsically lattice disordered system when the ferromagnetic order is established. The local symmetry breaking with intrinsic lattice disorder provides new understanding of the puzzling magnetic properties. Our density functional theory (DFT) calculation indicates that the local symmetry breaking is expected to reorient local ferromagnetic moments, unveiling the existence of the ferromagnetic instability associated with the lattice instability. Furthermore, DFT calculation unveils that the local symmetry breaking could affect the Weyl property by breaking the mirror plane. Our findings highlight the fundamentally important role that the local symmetry breaking plays in advancing our understanding on the magnetic and topological properties in Co₃Sn₂S₂, which may draw attention to explore the overlooked local symmetry breaking in Co₃Sn₂S₂, its derivatives and more broadly in other topological Dirac/Weyl semimetals and kagome-lattice magnets.

Exploration of glassy state in Prussian blue analogues

Prussian blue analogues (PBAs) are archetypes of microporous coordination polymers/metal–organic frameworks whose versatile composition allows for diverse functionalities. However, developments in PBAs have centred solely on their crystalline state, and the glassy state of PBAs has not been explored. Here we describe the preparation of the glassy state of PBAs via a mechanically induced crystal-to-glass transformation and explore their properties. The preservation of short-range metal–ligand–metal connectivity is confirmed, enabling the framework-based functionality and semiconductivity in the glass. The transformation also generates unconventional CN⁻ vacancies, followed by the reduction of metal sites. This leads to significant porosity enhancement in recrystallised PBA, enabled by further accessibility of isolated micropores. Finally, mechanical stability under stress for successful vitrification is correlated to defect contents and interstitial water. Our results demonstrate how mechanochemistry provides opportunities to explore glassy states of molecular framework materials in which the stable liquid state is absent.

Developments in Prussian blue analogues (PBAs) have centred solely on their crystalline state. Here, the authors describe the preparation of the glassy state of PBAs via a mechanically induced crystal-to-glass transformation and explore their properties.

The pressure induced phase diagram of double-layer ice under confinement: a first-principles study

Here, we present double-layer ice confined within various carbon nanotubes (CNTs) using state-of-the-art pressure induced (-5 GPa to 5 GPa) dispersion corrected density functional theory (DFT) calculations. We find that the double-layer ice exhibits remarkably rich and diverse phase behaviors as a function of pressure with varying CNT diameters. The lattice cohesive energies for various pure double layer ice phases follow the order of hexagonal > pentagonal > square tube > hexagonal-close-packed (HCP) > square > buckled-rhombic (b-RH). The confinement width was found to play a crucial role in the square and square tube phases in the intermediate pressure range of about 0-1 GPa. Unlike the phase transition in pure bilayer ice structures, the relative enthalpies demonstrate that the pentagonal phase, rather than the hexagonal structure, is the most stable ice polymorph at ambient pressure as well as in a deep negative pressure region, whereas the b-RH phase dominates under high pressure. The relatively short O⋯O distance of b-RH demonstrates the presence of a strong hydrogen bonding network, which is responsible for stabilizing the system.

Crystal structures

A personal view is offered on various solved and open problems related to crystal structures: the present state of reconstructing the crystal electron density from X-ray diffraction data; characterization of atomic and molecular motion from a combination of atomic displacement parameters and quantum chemical calculations; Bragg diffraction and diffuse scattering: twins, but different; models of real (as opposed to ideal) crystal structures from diffuse scattering; exploiting unexplored neighbourhoods of crystallography to mathematics, physics and chemistry.

Atomic structure of an FeCrMoCBY metallic glass revealed by high energy x-ray diffraction

Amorphous bulk metallic glasses with the composition Fe₄₈Cr₁₅Mo₁₄C₁₅B₆Y₂have been of interest due to their special mechanical and electronic properties, including corrosion resistance, high yield-strength, large elasticity, catalytic performance, and soft ferromagnetism. Here, we apply a reverse Monte Carlo technique to unravel the atomic structure of these glasses. The pair-distribution functions for various atomic pairs are computed based on the high-energy x-ray diffraction data we have taken from an amorphous sample. Monte Carlo cycles are used to move the atomic positions until the model reproduces the experimental pair-distribution function. The resulting fitted model is consistent with ourab initiosimulations of the metallic glass. Our study contributes to the understanding of functional properties of Fe-based bulk metallic glasses driven by disorder effects.

Strategies and Considerations for Least-Squares Analysis of Total Scattering Data

The process of least-squares analysis has been applied for decades in the field of crystallography. Here, we discuss the application of this process to total scattering data, primarily in the combination of least-squares Rietveld refinements and fitting of the atomic pair distribution function (PDF). While these two approaches use the same framework, the interpretation of results from least-squares fitting of PDF data should be done with caution through carefully constructed analysis approaches. We provide strategies and considerations for applying least-squares analysis to total scattering data, combining both crystallographic Rietveld and fitting of PDF data, given in context with recent examples from the literature. This perspective is aimed to be an accessible document for those new to the total scattering approach, as well as a reflective framework for the total scattering expert.

Multivariate analysis of disorder in metal-organic frameworks

The rational design of disordered frameworks is an appealing route to target functional materials. However, intentional realisation of such materials relies on our ability to readily characterise and quantify structural disorder. Here, we use multivariate analysis of pair distribution functions to fingerprint and quantify the disorder within a series of compositionally identical metal–organic frameworks, possessing different crystalline, disordered, and amorphous structures. We find this approach can provide powerful insight into the kinetics and mechanism of structural collapse that links these materials. Our methodology is also extended to a very different system, namely the melting of a zeolitic imidazolate framework, to demonstrate the potential generality of this approach across many areas of disordered structural chemistry.

Structural disorder in materials is challenging to characterise. Here, the authors use multivariate analysis of atomic pair distribution functions to study structural collapse and melting of metal–organic frameworks, revealing powerful mechanistic and kinetic insight.

Local Atomic Configuration in Pristine and A-Site Doped Silver Niobate Perovskite Antiferroelectrics

Antiferroelectrics have attracted increasing research interests in recent years due to both their great potential in energy storage applications and intriguing structural characteristics. However, the links between the electrical properties and structural characteristics of distorted perovskite antiferroelectrics are yet to be fully deciphered. Here, we adopt local-structure methods to elucidate the nanoscale atomic structure of AgNbO₃-based antiferroelectrics and their structural evolution upon La doping. The local structural features including interatomic distance distributions and atomic displacements have been analyzed using neutron small-box pair distribution function (PDF) refinement in conjunction with large-box Reverse Monte Carlo modelling. Our results highlight the correlation of cation displacements in AgNbO₃ and its disruption by the incorporation of La, apparently in corroboration with the observed anomalous dielectric properties. Spatial ordering of cation vacancies is observed in La-doped AgNbO₃ samples, which coordinates with oxygen octahedral tilting to relieve lattice strain. These results provide renewed insights into the atomic structure and antiferroelectric phase instabilities of AgNbO₃ and relevant perovskite materials, further lending versatile opportunities for enhancing their functionalities.

Controlling disorder in self-assembled colloidal monolayers via evaporative processes

Monolayers of assembled nano-objects with a controlled degree of disorder hold interest in many optical applications, including photovoltaics, light emission, sensing, and structural coloration. Controlled disorder can be achieved through either top-down or bottom-up approaches, but the latter is more suited to large-scale, low-cost fabrication. Disordered colloidal monolayers can be assembled through evaporatively driven convective assembly, a bottom-up process with a wide range of parameters impacting particle placement. Motivated by the photonic applications of such monolayers, in this review we discuss the quantification of monolayer disorder, and the assembly methods that have been used to produce them. We review the impact of particle and solvent properties, as well as the use of substrate patterning, to create the desired spatial distributions of particles.

Physics makes the difference: Bayesian optimization and active learning via augmented Gaussian process

Both experimental and computational methods for the exploration of structure, functionality, and properties of materials often necessitate the search across broad parameter spaces to discover optimal experimental conditions and regions of interest in the image space or parameter space of computational models. The direct grid search of the parameter space tends to be extremely time-consuming, leading to the development of strategies balancing exploration of unknown parameter spaces and exploitation towards required performance metrics. However, classical Bayesian optimization (BO) strategies based on the Gaussian process (GP) do not readily allow for the incorporation of the known physical behaviors or past knowledge. Here we explore a hybrid optimization/exploration algorithm created by augmenting the standard GP with a structured probabilistic model of the expected system’s behavior. This approach balances the flexibility of the non-parametric GP approach with a rigid structure of physical knowledge encoded into the parametric model. The fully Bayesian treatment of the latter allows additional control over the optimization via the selection of priors for the model parameters. The method is demonstrated for a noisy version of a standard univariate test function used to evaluate optimization algorithms and further extended to physical lattice models. This methodology is expected to be universally suitable for injecting prior knowledge in the form of physical models and past data in the BO framework.

Efficient fitting of single-crystal diffuse scattering in interaction space: a mean-field approach

The diffraction patterns of crystalline materials with strongly correlated disorder are characterized by the presence of structured diffuse scattering. Conventional analysis approaches generally seek to interpret this scattering either atomistically or in terms of pairwise (Warren-Cowley) correlation parameters. Here it is demonstrated how a mean-field methodology allows efficient fitting of diffuse scattering directly in terms of a microscopic interaction model. In this way the approach gives as its output the underlying physics responsible for correlated disorder. Moreover, the use of a very small number of parameters during fitting renders the approach surprisingly robust to data incompleteness, a particular advantage when seeking to interpret single-crystal diffuse scattering measured in complex sample environments. As the basis of this proof-of-concept study, a toy model is used based on strongly correlated disorder in diammine mercury(II) halides.

rmc-discord: reverse Monte Carlo refinement of diffuse scattering and correlated disorder from single crystals

A user-friendly Python-based program has been developed to analyze diffuse scattering from single crystals with the reverse Monte Carlo method. The approach allows for refinement of correlated disorder from atomistic supercells with magnetic or structural (occupational and/or displacive) disorder.

A user-friendly program has been developed to analyze diffuse scattering from single crystals with the reverse Monte Carlo method. The approach allows for refinement of correlated disorder from atomistic supercells with magnetic or structural (occupational and/or displacive) disorder. The program is written in Python and optimized for performance and efficiency. Refinements of two user cases obtained with legacy neutron-scattering data demonstrate the effectiveness of the approach and the developed program. It is shown with bixbyite, a naturally occurring magnetic mineral, that the calculated three-dimensional spin-pair correlations are resolved with finer real-space resolution compared with the pair distribution function calculated directly from the reciprocal-space pattern. With the triangular lattice Ba₃Co₂O₆(CO₃)_0.7, refinements of occupational and displacive disorder are combined to extract the one-dimensional intra-chain correlations of carbonate molecules that move toward neighboring vacant sites to accommodate strain induced by electrostatic interactions. The program is packaged with a graphical user interface and extensible to serve the needs of single-crystal diffractometer instruments that collect diffuse-scattering data.

Paradigms of frustration in superionic solid electrolytes

Superionic solid electrolytes have widespread use in energy devices, but the fundamental motivations for fast ion conduction are often elusive. In this Perspective, we draw upon atomistic simulations of a wide range of superionic conductors to illustrate some ways frustration can lower diffusion cation barriers in solids. Based on our studies of halides, oxides, sulfides and hydroborates and a survey of published reports, we classify three types of frustration that create competition between different local atomic preferences, thereby flattening the diffusive energy landscape. These include chemical frustration, which derives from competing factors in the anion-cation interaction; structural frustration, which arises from lattice arrangements that induce site distortion or prevent cation ordering; and dynamical frustration, which is associated with temporary fluctuations in the energy landscape due to anion reorientation or cation reconfiguration. For each class of frustration, we provide detailed simulation analyses of various materials to show how ion mobility is facilitated, resulting in stabilizing factors that are both entropic and enthalpic in origin. We propose the use of these categories as a general construct for classifying frustration in superionic conductors and discuss implications for future development of suitable descriptors and improvement strategies. This article is part of the Theo Murphy meeting issue 'Understanding fast-ion conduction in solid electrolytes'.

Paradigms of frustration in superionic solid electrolytes

Superionic solid electrolytes have widespread use in energy devices, but the fundamental motivations for fast ion conduction are often elusive. In this Perspective, we draw upon atomistic simulations of a wide range of superionic conductors to illustrate some ways frustration can lower diffusion cation barriers in solids. Based on our studies of halides, oxides, sulfides and hydroborates and a survey of published reports, we classify three types of frustration that create competition between different local atomic preferences, thereby flattening the diffusive energy landscape. These include chemical frustration, which derives from competing factors in the anion-cation interaction; structural frustration, which arises from lattice arrangements that induce site distortion or prevent cation ordering; and dynamical frustration, which is associated with temporary fluctuations in the energy landscape due to anion reorientation or cation reconfiguration. For each class of frustration, we provide detailed simulation analyses of various materials to show how ion mobility is facilitated, resulting in stabilizing factors that are both entropic and enthalpic in origin. We propose the use of these categories as a general construct for classifying frustration in superionic conductors and discuss implications for future development of suitable descriptors and improvement strategies. This article is part of the Theo Murphy meeting issue 'Understanding fast-ion conduction in solid electrolytes'.

Probing Multiscale Disorder in Pyrochlore and Related Complex Oxides in the Transmission Electron Microscope: A Review

Transmission electron microscopy (TEM), and its counterpart, scanning TEM (STEM), are powerful materials characterization tools capable of probing crystal structure, composition, charge distribution, electronic structure, and bonding down to the atomic scale. Recent (S)TEM instrumentation developments such as electron beam aberration-correction as well as faster and more efficient signal detection systems have given rise to new and more powerful experimental methods, some of which (e.g., 4D-STEM, spectrum-imaging, in situ/operando (S)TEM)) facilitate the capture of high-dimensional datasets that contain spatially-resolved structural, spectroscopic, time- and/or stimulus-dependent information across the sub-angstrom to several micrometer length scale. Thus, through the variety of analysis methods available in the modern (S)TEM and its continual development towards high-dimensional data capture, it is well-suited to the challenge of characterizing isometric mixed-metal oxides such as pyrochlores, fluorites, and other complex oxides that reside on a continuum of chemical and spatial ordering. In this review, we present a suite of imaging and diffraction (S)TEM techniques that are uniquely suited to probe the many types, length-scales, and degrees of disorder in complex oxides, with a focus on disorder common to pyrochlores, fluorites and the expansive library of intermediate structures they may adopt. The application of these techniques to various complex oxides will be reviewed to demonstrate their capabilities and limitations in resolving the continuum of structural and chemical ordering in these systems.

The mesoscale order of nacreous pearls

A pearl's distinguished beauty and toughness are attributable to the periodic stacking of aragonite tablets known as nacre. Nacre has naturally occurring mesoscale periodicity that remarkably arises in the absence of discrete translational symmetry. Gleaning the inspiring biomineral design of a pearl requires quantifying its structural coherence and understanding the stochastic processes that influence formation. By characterizing the entire structure of pearls (∼3 mm) in a cross-section at high resolution, we show that nacre has medium-range mesoscale periodicity. Self-correcting growth mechanisms actively remedy disorder and topological defects of the tablets and act as a countervailing process to long-range disorder. Nacre has a correlation length of roughly 16 tablets (∼5.5 µm) despite persistent fluctuations and topological defects. For longer distances (>25 tablets , ∼8.5 µm), the frequency spectrum of nacre tablets follows [Formula: see text] behavior, suggesting that growth is coupled to external stochastic processes-a universality found across disparate natural phenomena, which now includes pearls.

Crystal Engineering of Supramolecular 1,4-Benzene Bisamides by Side-Chain Modification - Towards Tuneable Anisotropic Morphologies and Surfaces

Benzene bisamides are promising building blocks for supramolecular nano‐objects. Their functionality depends on morphology and surface properties. However, a direct link between surface properties and molecular structure itself is missing for this material class. Here, we investigate this interplay for two series of 1,4‐benzene bisamides with symmetric and asymmetric peripheral substitution. We elucidated the crystal structures, determined the nano‐object morphologies and derived the wetting behaviour of the preferentially exposed surfaces. The crystal structures were solved by combining single‐crystal and powder X‐ray diffraction, solid‐state NMR spectroscopy and computational modelling. Bulky side groups, here t‐butyl groups, serve as a structure‐directing motif into a packing pattern, which favours the formation of thin platelets. The use of slim peripheral groups on both sides, in our case linear perfluorinated, alkyl chains, self‐assemble the benzene bisamides into a second packing pattern which leads to ribbon‐like nano‐objects. For both packing types, the preferentially exposed surfaces consist of the ends of the peripheral groups. Asymmetric substitution with bulky and slim groups leads to an ordered alternating arrangement of the groups exposed to the surface. This allows the hydrophobicity of the surfaces to be gradually altered. We thus identified two leitmotifs for molecular packings of benzene bisamides providing the missing link between the molecular structure, the anisotropic morphologies and adjustable surface properties of the supramolecular nano‐objects.

Atomic arrangements in an amorphous CoFeB ribbon extracted via an analysis of radial distribution functions

We discuss the atomic structure of amorphous ferromagnetic FeCoB alloys, which are used widely in spintronics applications. Specifically, we obtain the pair-distribution functions for various atomic pairs based on high-energy x-ray diffraction data taken from an amorphous Co₂₀Fe₆₁B₁₉specimen. We start our reverse Monte Carlo cycles to determine the disordered structure with a two-phase model in which a small amount of cobalt is mixed with Fe₂₃B₆as a second phase. The structure of the alloy is found to be heterogeneous, where the boron atoms drive disorder through the random occupation of the atomic network. Our analysis also indicates the presence of small cobalt clusters that are embedded in the iron matrix and percolating the latter throughout the structure. This morphology can explain the enhanced spin polarization observed in amorphous magnetic materials.

Tuneable local order in thermoelectric crystals

Although crystalline solids are characterized by their periodic structures, some are only periodic on average and deviate on a local scale. Such disordered crystals with distinct local structures have unique properties arising from both collective and localized behaviour. Different local orderings can exist with identical average structures, making their differences hidden to Bragg diffraction methods. Using high-quality single-crystal X-ray diffuse scattering the local order in thermoelectric half-Heusler Nb1-x CoSb is investigated, for which different local orderings are observed. It is shown that the vacancy distribution follows a vacancy repulsion model and the crystal composition is found always to be close to x = 1/6 irrespective of nominal sample composition. However, the specific synthesis method controls the local order and thereby the thermoelectric properties thus providing a new frontier for tuning material properties.

Moving beyond the constraints of chemistry via crystal structure discovery with isotropic multiwell pair potentials

The rigid constraints of chemistry-dictated by quantum mechanics and the discrete nature of the atom-limit the set of observable atomic crystal structures. What structures are possible in the absence of these constraints? Here, we systematically crystallize one-component systems of particles interacting with isotropic multiwell pair potentials. We investigate two tunable families of pairwise interaction potentials. Our simulations self-assemble a multitude of crystal structures ranging from basic lattices to complex networks. Sixteen of the structures have natural analogs spanning all coordination numbers found in inorganic chemistry. Fifteen more are hitherto unknown and occupy the space between covalent and metallic coordination environments. The discovered crystal structures constitute targets for self-assembly and expand our understanding of what a crystal structure can look like.

Emergent electrochemical functions and future opportunities of hierarchically constructed metal-organic frameworks and covalent organic frameworks

Designing spatial and architectural features across from the molecular to bulk scale is one of the most important topics in materials science which has received a lot of attention in recent years. Looking back to the past research, findings on the influences of spatial features denoted as porous structures on the applications related to mass transport phenomena have been widely studied in traditional inorganic materials, such as ceramics over the past two decades. However, due to the difficulties in precise control of the porous structures at the molecular level in this class of materials, the mechanistic understanding of the effects of spatial and architectural features across from the molecular level to meso-/macroscopic scale is still lacking, especially in electrochemical reactions. Further understanding of fundamental electrochemical functions in well-defined architectures is indispensable for the further advancement of key next-generation energy devices. Furthermore, creating periodic porosity in reticular structures is starting to be recognized as an emerging approach to control the electronic structure of materials. In this review, we focus on the investigations on preparing well-defined molecular-level crystalline porous materials known as metal-organic frameworks (MOFs) and covalent organic frameworks (COFs) into hierarchically constructed architectures from molecular structures lower than the reticular frameworks to meso-/macroscopic scale structures. By connecting well-defined nanosized porous structures in MOFs/COFs and additional length-scale space or shapes, emergent electrochemical functions towards emerging devices, such as beyond Li-ion batteries including all-solid-state rechargeable batteries, are expected to be obtained. By summarizing recent advancements in synthetic strategies of hierarchically constructed MOF/COF based materials and fundamental investigation of their structural effect in a wide spectrum of electrochemical applications, we highlight the importance and future direction of this developing field of hierarchically constructed MOFs/COFs, while emphasizing the required chemical stability of the MOFs/COFs which meet the use in the game-changing electrochemical devices.

Embracing Defects and Disorder in Magnetic Nanoparticles

Iron oxide nanoparticles have tremendous scientific and technological potential in a broad range of technologies, from energy applications to biomedicine. To improve their performance, single-crystalline and defect-free nanoparticles have thus far been aspired. However, in several recent studies, defect-rich nanoparticles outperform their defect-free counterparts in magnetic hyperthermia and magnetic particle imaging (MPI). Here, an overview on the state-of-the-art of design and characterization of defects and resulting spin disorder in magnetic nanoparticles is presented with a focus on iron oxide nanoparticles. The beneficial impact of defects and disorder on intracellular magnetic hyperthermia performance of magnetic nanoparticles for drug delivery and cancer therapy is emphasized. Defect-engineering in iron oxide nanoparticles emerges to become an alternative approach to tailor their magnetic properties for biomedicine, as it is already common practice in established systems such as semiconductors and emerging fields including perovskite solar cells. Finally, perspectives and thoughts are given on how to deliberately induce defects in iron oxide nanoparticles and their potential implications for magnetic tracers to monitor cell therapy and immunotherapy by MPI.