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We report measurements of anisotropic triple-$q$ charge density wave (CDW) fluctuations in the transition metal dichalcogenide 1$T$-TiSe$_2$ over a large volume of reciprocal space with X-ray diffuse scattering. Above the transition temperature, $T_{\text{CDW}}$, the out-of-plane diffuse scattering is characterized by rod-like structures which indicate that the CDW fluctuations in neighboring layers are largely decoupled. In addition, the in-plane diffuse scattering is marked by ellipses which reveal that the in-plane fluctuations are anisotropic. Our analysis of the diffuse scattering line shapes and orientations suggests that the three charge density wave components contain independent phase fluctuations. At $T_{\text{CDW}}$, long range coherence is established in both the in-plane and out-of-plane directions, consistent with the large observed value of the CDW gap compared to $T_{\text{CDW}}$, and the predicted presence of a hierarchy of energy scales.

We explore, through a combination of x-ray diffraction and elastoresistivity measurements, the effect of disorder on the strain-tuned charge density wave and associated multicriticality in Pd$_x$ErTe$_3$ (x = 0, 0.01, 0.02 and 0.026). We focus particularly on the behavior near the strain-tuned bicritical point that occurs in pristine ErTe$_3$ (x=0). Our study reveals that while Pd intercalation somewhat broadens the signatures of the CDW phase transitions, the line of first-order transitions at which the CDW reorients as a function of applied strain persists in the presence of disorder and still seemingly terminates at a critical point. The critical point occurs at a lower temperature and a lower strain compared to pristine ErTe$_3$. Similarly, the nematic elastoresistance of Pd$_x$ErTe$_3$, though suppressed in magnitude and broadened relative to that of ErTe$_3$, has a markedly more symmetric response around the critical point. These observations point to disorder driving a reduction in the system's electronic orthorhombicity even while the material remains irrevocably orthorhombic due to the presence of a glide plane in the crystal structure. Disorder, it would appear, reinforces the emergence of a "pseudo-tetragonal" electronic response in this fundamentally orthorhombic material.

The role of inhomegeneity in determining the properties of correlated electron systems is poorly understood because of the dearth of structural probes of disorder at the nanoscale. Advances in both neutron and x-ray scattering instrumentation now allow comprehensive measurements of diffuse scattering in single crystals over large volumes of reciprocal space, enabling structural correlations to be characterized over a range of length scales from 10~{\AA} to 200~{\AA} or more. When combined with new analysis tools, such as three-dimensional difference pair-distribution functions, these advanced capabilities have produced novel insights into the interplay of structural fluctuations and electronic properties in a broad range of correlated electron materials. This review describes recent investigations that have demonstrated the importance of understanding structural inhomogeneity pertaining to phenomena as diverse as superconductivity, charge-density-wave modulations, metal-insulator transitions, and multipolar interactions., Comment: 50 pages, 10 figures

Though the effects of metal dopants on the electrostructural transition of rutile VO$_2$ have been studied for many decades, there is still no consensus explanation for the observed trends. A major challenge has been to separate the impact of a dopant's size from other factors such as its electronic configuration, which stems from the difficulty in directly probing the local bonding environment around a dopant atom. This work addresses the special case of larger dopant ions by combining X-ray total scattering experiments on V$_{0.83}$Mo$_{0.17}$O$_2$ and V$_{0.89}$Nb$_{0.11}$O$_2$ single crystals with multiple Monte Carlo method models to simulate local size effects in the high-temperature tetragonal phase (R). We find that sufficiently long apical metal-oxygen bonds (M$-$O$_{ap}$) induce a strain field in the neighboring chains that locally resembles the metal-metal dimer formation present in the low-temperature distorted structure of VO$_2$ (M1). The dimer mode in the M1 structure is antisymmetric along M$-$O$_{ap}$, however, while the strain-induced pseudodimer motif is symmetric. The implied direct competition between motifs is verified experimentally. This finding provides a new mechanistic parameter toward understanding the phase transition. More generally, the work highlights how local strain fields around dopants can lead to complex distortions that are ordinarily attributed to electronic origins., Comment: Article: 19 pages, 9 figures; Appendix: 4 pages, 2 figures, 3 tables

The long-sought pair density wave (PDW) is an exotic phase of matter in which charge density wave (CDW) order is intertwined with the amplitude or phase of coexisting, superconducting order \cite{Berg2009,Berg2009b}. Originally predicted to exist in copper-oxides, circumstantial evidence for PDW order now exists in a variety of materials. Recently, scanning tunneling microscopy (STM) studies have reported evidence for a three-component charge density wave (CDW) at the surface of the heavy-fermion superconductor, UTe$_2$, persisting below its superconducting transition temperature. Here, we use hard x-ray diffraction measurements on crystals of UTe$_2$ at $T = 1.9$ K and $12$ K to search for a bulk signature of this CDW. Using STM measurements as a constraint, we calculate the expected locations of CDW superlattice peaks, and sweep a large volume of reciprocal space in search of a signature. We failed to find any evidence for a CDW near any of the expected superlattice positions in many Brillouin zones. We estimate an upper bound on the CDW lattice distortion of $u_{max} \lesssim 4 \times 10^{-3} \mathrm{\AA}$. Our results suggest that the CDW observed in STM is either purely electronic, somehow lacking a signature in the structural lattice, or is restricted to the material surface.

Dislocation engineering has the potential to open new avenues toward the exploration and modification of the properties of quantum materials. Strontium titanate (SrTiO3, STO) and potassium tantalate (KTaO3, KTO) are incipient ferroelectrics that show metallization and superconductivity at extremely low charge carrier concentrations, and have been the subject of resurgent interest. These materials also exhibit remarkable ambient-temperature ductility, and thus represent exceptional platforms for studies of the effects of deformation-induced dislocation structures on electronic properties. Recent work on plastically deformed STO revealed an enhancement of the superconducting transition temperature and the emergence of local ferroelectricity and magnetism near self-organized dislocation walls. Here we present a comprehensive structural analysis of plastically deformed STO and KTO, employing specially designed strain cells, diffuse neutron and x-ray scattering, Raman scattering, and nuclear magnetic resonance (NMR). Diffuse scattering and NMR provide insight into the dislocation configurations and densities and their dependence on strain. As in the prior work on STO, Raman scattering reveals evidence for local ferroelectric order near dislocation walls in plastically deformed KTO. Our findings provide valuable information about the self-organized defect structures in both materials, and they position KTO as a second model system with which to explore the associated emergent physics.

Understanding the extent and role of inhomogeneity is a pivotal challenge in the physics of cuprate superconductors. While it is known that structural and electronic inhomogeneity is prevalent in the cuprates, it has proven difficult to disentangle compound-specific features from universally relevant effects. Here we combine advanced neutron and x-ray diffuse scattering with numerical modeling to obtain insight into bulk structural correlations in HgBa$_2$CuO$_{4+\delta}$. This cuprate exhibits a high optimal transition temperature of nearly 100 K, pristine charge-transport behavior, and a simple average crystal structure without long-range structural instabilities, and is therefore uniquely suited for investigations of intrinsic inhomogeneity. We uncover diffuse reciprocal-space patterns that correspond to prominent nanoscale correlations of atomic displacements perpendicular to the CuO$_2$ planes. The real-space nature of the correlations is revealed through three-dimensional pair distribution function analysis and complementary numerical refinement. We find that relative displacements of ionic and CuO$_2$ layers play a crucial role, and that the structural inhomogeneity is not directly caused by the presence of conventional point defects. The observed correlations are therefore intrinsic to HgBa$_2$CuO$_{4+\delta}$, and thus likely important for the physics of cuprates more broadly. It is possible that the structural correlations are closely related to the unusual superconducting correlations and Mott-localization in these complex oxides. As advances in scattering techniques yield increasingly comprehensive data, the experimental and analysis tools developed here for large volumes of diffuse scattering data can be expected to aid future investigations of a wide range of materials., Comment: 14 pages, 5 figures. V2 changes: exchanged Fig. 4 and 5.; added neutron panels to Fig. 4.; fixed incorrect neutron panels in Fig. 5 and updated related text; added 3D-DeltaPDF resolution paragraph: clarified text about correlation lengths, temperature control, absence of occupational disorder and CuO2 plane buckling; removed appendix; typo fixes, small changes to phrasing, etc

We investigate a spin-$\frac{1}{2}$ antiferromagnet, CuBr$_2$, which has quasi-one-dimensional structural motifs. The system has previously been observed to exhibit unusual Raman modes possibly due to a locally deformed crystal structure driven by the low-dimensional magnetism. Using hard X-ray scattering and neutron total scattering, here we aim to verify a specific form of tetramerized lattice deformation proposed in the previous study. Apart from diffuse scattering signals which we can reproduce by performing a thorough modeling of the lattice's thermal vibrations, we do not observe evidence for a tetramerized lattice structure within our detection sensitivity. As a result, it is more likely that the unusual Raman modes in CuBr$_2$ arise from classical magnon-phonon hybridization, rather than from quantum spin-singlet-driven lattice deformation., Comment: 8 pages, 6 figures

The quasi-one-dimensional chiral compound (TaSe$_4$)$_2$I has been extensively studied as a prime example of a topological Weyl semimetal. Upon crossing its phase transition temperature $T_\textrm{CDW}$ $\approx$ 263 K, (TaSe$_4$)$_2$I exhibits incommensurate charge density wave (CDW) modulations described by the well-defined propagation vector $\sim$(0.05, 0.05, 0.11), oblique to the TaSe$_4$ chains. Although optical and transport properties greatly depend on chirality, there is no systematic report about chiral domain size for (TaSe$_4$)$_2$I. In this study, our single-crystal scattering refinements reveal a bulk iodine deficiency, and Flack parameter measurements on multiple crystals demonstrate that separate (TaSe$_4$)$_2$I crystals have uniform handedness, supported by direct imaging and helicity dependent THz emission spectroscopy. Our single-crystal X-ray scattering and calculated diffraction patterns identify multiple diffuse features and create a real-space picture of the temperature-dependent (TaSe$_4$)$_2$I crystal structure. The short-range diffuse features are present at room temperature and decrease in intensity as the CDW modulation develops. These transverse displacements, along with electron pinning from the iodine deficiency, help explain why (TaSe$_4$)$_2$I behaves as an electronic semiconductor at temperatures above and below $T_\textrm{CDW}$, despite a metallic band structure calculated from density functional theory of the ideal structure., Comment: 25 pages, 20 figures, 5 tables

Unconventional superconductivity has long been believed to arise from a lab-grown correlated electronic system. Here we report compelling evidence of unconventional nodal superconductivity in a mineral superconductor \rhs. We investigated the temperature-dependent London penetration depth $\Delta\lambda(T)$ and disorder evolution of the critical temperature $T_c$ and upper critical field $H_{c2}(T)$ in synthetic miassite \rhs. We found a power-law behavior of $\Delta\lambda(T)\sim T^n$ with $n\approx 1.1$ at low temperatures below $0.3T_c$ ($T_c$ = 5.4 K), which is consistent with the presence of lines of the node in the superconducting gap of \rhs. The nodal character of the superconducting state in \rhs~was supported by the observed pairbreaking effect in $T_c$ and $H_{c2}(T)$ in samples with the controlled disorder that was introduced by low-temperature electron irradiation. We propose a nodal sign-changing superconducting gap in the $A_{1g}$ irreducible representation, which preserves the cubic symmetry of the crystal and is in excellent agreement with the superfluid density, $\lambda^2(0)/\lambda^2(T)$.

Hybrid lead halide perovskites (LHPs) are a class of semiconductor with novel properties that are distinctively governed by structural fluctuations. Diffraction experiments sensitive to average, long-range order reveal a cubic structure in the device-relevant, high-temperature phase. Local probes find additional short-range order with lower symmetry that may govern the structure-function relationships of LHPs. However, the dimensionality, participating atoms, and dynamics of this short-range order are unresolved, impeding our understanding of technologically relevant properties including long carrier lifetimes and facile halide migration. Here, we determine the true structure of two prototypical hybrid LHPs, CH$_3$NH$_3$PbI$_3$ and CH$_3$NH$_3$PbBr$_3$, using a combination of single-crystal X-ray and neutron diffuse scattering, neutron inelastic spectroscopy, and molecular dynamics simulations. The remarkable collective dynamics we found are not suggested by previous studies and consist of a network of local two-dimensional, circular pancake-like regions of dynamically tilting lead halide octahedra (lower symmetry) that induce longer range intermolecular correlations within the CH$_3$NH$_3^+$ sublattice. The dynamic local structure can introduce transient ferroelectric or antiferroelectric domains that increase charge carrier lifetimes, and strongly affect the halide migration, a poorly understood degradation mechanism. Our approach of co-analyzing single-crystal X-ray and neutron diffuse scattering data with MD simulations will provide unparalleled insights into the structure of hybrid materials and materials with engineered disorder.

AMn6Bi5 materials (A = Na, K, Rb and Cs) consisting of unique Mn-cluster chains emerge as a new family of superconductors with the suppression of their antiferromagnetic (AFM) order under high pressures. Here, we report transverse incommensurate spin density waves (SDWs) for the Mn atoms with a propagating direction along the chain axes as a ground state for KMn6Bi5 by single crystal neutron diffraction. The SDWs have a refined amplitude of ~2.46 Bohr magnetons for the Mn atoms in the pentagons and ~0.29 Bohr magnetons with a large standard deviation for Mn atoms in the center between the pentagons. AFM dominate both the nearest-neighbor Mn-Mn interactions within the pentagon and next-nearest-neighbor Mn-Mn interactions out of the pentagon (along the propagating wave). The SDWs exhibit both local and itinerant characteristics probably formed by a cooperative interaction between local magnetic exchange and conduction electrons. A significant magnetoelastic effect during the AFM transition, especially along the chain direction, has been demonstrated by temperature-dependent x-ray powder diffraction. Single crystal x-ray diffraction below the AFM transition revealed satellite peaks originating from charge density waves along the chain direction with a q-vector twice as large as the SDW one, pointing to a strong real space coupling between them. Our work not only manifests a fascinating interplay among spin, charge, lattice and one dimensionality to trigger intertwined orders in KMn6Bi5 but also provides important piece of information for the magnetic structure of the parent compound to understand the mechanism of superconductivity in this new family.

The classification of structural phase transitions as displacive or order-disorder in character is usually based on spectroscopic data above the transition. We use single crystal x-ray diffraction to investigate structural correlations in the quasi-skutterudites, (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$, which have a quantum phase transition at $x\sim0.9$. Three-dimensional pair distribution functions show that the amplitudes of local atomic displacements are temperature-independent below the transition and persist to well above the transition, a signature of order-disorder behavior. The implications for the associated electronic transitions are discussed., Comment: 5 pages, 4 figures

Low-valent nickelates R$_{n+1}$Ni$_n$O$_{2n+2}$ (R = rare earth) containing Ni$^{1+}$ (d$^{9}$) with a quasi-two-dimensional (quasi-2D) square planar coordination geometry possess structural and electronic properties that are similar to those of high-Tc cuprates, including superconductivity itself in the doped infinite layer ($n = \infty$) RNiO$_2$ system. Within this R$_{n+1}$Ni$_n$O$_{2n+2}$ nickelate family, the crystallographic isomorphs Pr$_4$Ni$_3$O$_8$ and La$_4$Ni$_3$O$_8$ exhibit singularly different ground states: Pr$_4$Ni$_3$O$_8$ is metallic and La$_4$Ni$_3$O$_8$ is a charge- and spin-stripe ordered insulator. To explore and understand the ground state evolution from metallic Pr$_4$Ni$_3$O$_8$ to stripe-ordered La$_4$Ni$_3$O$_8$ in the R$_4$Ni$_3$O$_8$ family, we have grown a series of isovalent-substituted single crystals (Pr$_{1-x}$La$_x$)$_4$Ni$_3$O$_8$. Combining thermodynamic, transport, magnetic, and synchrotron X-ray single crystal diffraction measurements, we reveal a transition between metallic and stripe-insulator phase regions, with a putative quantum phase transition at x = 0.4. We propose two possible models for (Pr$_{1-x}$La$_x$)$_4$Ni$_3$O$_8$: an electronically inhomogeneous system that could serve as a candidate for exploring quantum Griffiths phase physics and a homogeneous system with a putative quantum critical point at the phase boundary.

Metal halides perovskites (MHPs) are attracting considerable interest for optoelectronic applications, with Cs$_2$AgBiBr$_6$ one of the main contenders among lead-free systems. Cs$_2$AgBiBr$_6$ crystallizes in a nominally double-perovskite structure, but exhibits a soft lattice with large atomic fluctuations characteristic of MHPs. While crucial to understand electron-phonon and phonon-phonon couplings, the spatio-temporal correlations of these fluctuations remain largely unknown. Here, we reveal these correlations using comprehensive neutron and x-ray scattering measurements on Cs$_2$AgBiBr$_6$ single-crystals, complemented with first-principles simulations augmented with machine-learned neural-network potentials. We report the discovery of an unexpected complex modulated ground state structure containing several hundred atoms, arising from a soft-phonon instability of the low-temperature tetragonal phase. Further, our experiments and simulations both reveal extensive correlated 2D fluctuations of Br octahedra at finite temperature, arising from soft anharmonic optic phonons, reflecting very shallow potential wells. These results provide new insights into the atomic structure and fluctuations in MHPs, critical to understand and control their thermal and optoelectronic properties., Comment: 12 pages, 4 figures

Neutron diffraction on the centrosymmetric square-net magnet EuGa$_2$Al$_2$ reveals multiple incommensurate magnetic states (AFM1,2,3) in zero field. In applied field, a new magnetic phase (A) is identified from magnetization and transport measurements, bounded by two of the $\mu_0H$~=~0 incommensurate magnetic phases (AFM1,helical and AFM3, cycloidal) with different moment orientations. Moreover, magneto-transport measurements indicate the presence of a topological Hall effect, with maximum values centered in the A phase. Together, these results render EuGa$_2$Al$_2$ a material with non-coplanar or topological spin texture in applied field. X-ray diffraction reveals an out-of-plane (OOP) charge density wave (CDW) below $T_{CDW} \sim$ 50 K while the magnetic propagation vector lies in plane below $T_N$ = 19.5 K. Together these data point to a new route to realizing in-plane non-collinear spin textures through an OOP CDW. In turn, these non-collinear spin textures may be unstable against the formation of topological spin textures in an applied field.

Recently discovered alongside its sister compounds KV$_3$Sb$_5$ and RbV$_3$Sb$_5$, CsV$_3$Sb$_5$ crystallizes with an ideal kagome network of vanadium and antimonene layers separated by alkali metal ions. This work presents the electronic properties of CsV$_3$Sb$_5$, demonstrating bulk superconductivity in single crystals with a T$_{c} = 2.5$K. The normal state electronic structure is studied via angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT), which categorize CsV$_3$Sb$_5$ as a $\mathbb{Z}_2$ topological metal. Multiple protected Dirac crossings are predicted in close proximity to the Fermi level ($E_F$), and signatures of normal state correlation effects are also suggested by a high temperature charge density wave-like instability. The implications for the formation of unconventional superconductivity in this material are discussed., Comment: 6 pages, 4 figures

The information content of crystalline materials becomes astronomical when collective electronic behavior and their fluctuations are taken into account. In the past decade, improvements in source brightness and detector technology at modern x-ray facilities have allowed a dramatically increased fraction of this information to be captured. Now, the primary challenge is to understand and discover scientific principles from big data sets when a comprehensive analysis is beyond human reach. We report the development of a novel unsupervised machine learning approach, XRD Temperature Clustering (X-TEC), that can automatically extract charge density wave (CDW) order parameters and detect intra-unit cell (IUC) ordering and its fluctuations from a series of high-volume X-ray diffraction (XRD) measurements taken at multiple temperatures. We apply X-TEC to XRD data on a quasi-skutterudite family of materials, (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$, where a quantum critical point arising from charge order is observed as a function of Ca concentration. We further apply X-TEC to XRD data on the pyrochlore metal, Cd$_2$Re$_2$O$_7$, to investigate its two much debated structural phase transitions and uncover the Goldstone mode accompanying them. We demonstrate how unprecedented atomic scale knowledge can be gained when human researchers connect the X-TEC results to physical principles. Specifically, we extract from the X-TEC-revealed selection rule that the Cd and Re displacements are approximately equal in amplitude, but out of phase. This discovery reveals a previously unknown involvement of $5d^2$ Re, supporting the idea of an electronic origin to the structural order. Our approach can radically transform XRD experiments by allowing in-operando data analysis and enabling researchers to refine experiments by discovering interesting regions of phase space on-the-fly.

The metal-to-insulator transition (MIT) in rutile VO$_2$ has proven uniquely difficult to characterize because of the complex interplay between electron correlations and atomic structure. Here we report the discovery of the sudden collapse of three-dimensional order in the low-temperature phase of V$_{1-x}$Mo$_x$O$_2$ at $x=0.17$ and the emergence of a novel frustrated two-dimensional order at $x=0.19$, with only a slight change in electronic properties. Single crystal diffuse x-ray scattering reveals that this transition from the 3D M1 phase to a 2D variant of the M2 phase results in long-range structural correlations along symmetry-equivalent (11L) planes of the tetragonal rutile structure, yet extremely short-range correlations transverse to these planes. These findings suggest that this two-dimensionality results from a novel form of geometric frustration that is essentially structural in origin., Comment: 6 pages, 5 figures

Single crystals of PrNiO3 were grown under an oxygen pressure of 295 bar using a unique high-pressure optical-image floating zone furnace. The crystals, with volume in excess of 1 mm3, were characterized structurally using single crystal and powder X-ray diffraction. Resistivity, specific heat, and magnetic susceptibility were measured, all of which evidenced an abrupt, first order metal-insulator transition (MIT) at ~130 K, in agreement with previous literature reports on polycrystalline specimens. Temperature-dependent single crystal diffraction was performed to investigate changes through the MIT. Our study demonstrates the opportunity space for high fugacity, reactive environments for single crystal growth specifically of perovskite nickelates but more generally to correlated electron oxides., Comment: 19 pages, 6 figures, Supporting Information included

The intercalation of alkali ions into layered materials has played an essential role in battery technology since the development of the first lithium-ion electrodes. Coulomb repulsion between the intercalants leads to ordering of the intercalant sublattice, which hinders ionic diffusion and impacts battery performance. While conventional diffraction can identify the long-range order that can occur at discrete intercalant concentrations during the charging cycle, it cannot determine short-range order at other concentrations that also disrupt ionic mobility. In this article, we show that the use of real-space transforms of single crystal diffuse scattering, measured with high-energy synchrotron x-rays, allows a model-independent measurement of the temperature dependence of the length scale of ionic correlations along each of the crystallographic axes in a sodium-intercalated V$_2$O$_5$. The techniques described here provide a new way of probing the evolution of structural ordering in crystalline materials., Comment: 16 pages, 17 figures

We report a study of the Ca$_{0.73}$La$_{0.27}$FeAs$_2$ single crystals. We unravel a monoclinic to triclinic phase transition at 58 K, and a paramagnetic to stripe antiferromagnetic (AFM) phase transition at 54 K, below which spins order 45$^\circ$ away from the stripe direction. Furthermore, we demonstrate this material is substantially structurally untwinned at ambient pressure with the formation of spin rotation walls (S-walls). Finally, in addition to the central-hole and corner-electron Fermi pockets usually appearing in Fe pnictide superconductors, angle-resolved photoemission (ARPES) measurements resolve a Fermiology where an extra electron pocket of mainly As chain character exists at the Brillouin zone edge., Comment: 5 pages, 3 figures