Imperfections are not 0 K: free energy of point defects in crystals.

Defects determine many important properties and applications of materials, ranging from doping in semiconductors, to conductivity in mixed ionic-electronic conductors used in batteries, to active sites in catalysts. The theoretical description of defect formation in crystals has evolved substantially over the past century. Advances in supercomputing hardware, and the integration of new computational techniques such as machine learning, provide an opportunity to model longer length and time-scales than previously possible. In this Tutorial Review, we cover the description of free energies for defect formation at finite temperatures, including configurational (structural, electronic, spin) and vibrational terms. We discuss challenges in accounting for metastable defect configurations, progress such as machine learning force fields and thermodynamic integration to directly access entropic contributions, and bottlenecks in going beyond the dilute limit of defect formation. Such developments are necessary to support a new era of accurate defect predictions in computational materials chemistry.

Chemical Bonding in Colossal Thermopower FeSb2.

FeSb2 exhibits a colossal Seebeck coefficient ( S ) and a record-breaking high thermoelectric power factor. It also has an atypical shift from diamagnetism to paramagnetism with increasing temperature, and the fine details of its electron correlation effects have been widely discussed. The extraordinary physical properties must be rooted in the nature of the chemical bonding, and indeed, the chemical bonding in this archetypical marcasite structure has been heavily debated on a theoretical basis since the 1960s. The two prevalent models for describing the bonding interactions in FeSb2 are based on either ligand-field stabilization of Fe or a network structure of Sb hosting Fe ions. However, neither model can account for the observed properties of FeSb2 . Herein, an experimental electron density study is reported, which is based on analysis of synchrotron X-ray diffraction data measured at 15 K on a minute single crystal to limit systematic errors. The analysis is supplemented with density functional theory calculations in the experimental geometry. The experimental data are at variance with both the additional single-electron Sb-Sb bond implied by the covalent model, and the large formal charge and expected d-orbital splitting advocated by the ionic model. The structure is best described as an extended covalent network in agreement with expectations based on electronegativity differences.

Quantification of the Magnetic Anisotropy of a Single-Molecule Magnet from the Experimental Electron Density.

Reported here is an entirely new application of experimental electron density (EED) in the study of magnetic anisotropy of single-molecule magnets (SMMs). Among those SMMs based on one single transition metal, tetrahedral CoII-complexes are prominent, and their large zero-field splitting arises exclusively from coupling between the d x 2 - y 2 and dxy orbitals. Using very low temperature single-crystal synchrotron X-ray diffraction data, an accurate electron density (ED) was obtained for a prototypical SMM, and the experimental d-orbital populations were used to quantify the dxy-d x 2 - y 2 coupling, which simultaneously provides the composition of the ground-state Kramers doublet wave function. Based on this experimentally determined wave function, an energy barrier for magnetic relaxation in the range 193-268 cm-1 was calculated, and is in full accordance with the previously published value of 230 cm-1 obtained from near-infrared spectroscopy. These results provide the first clear and direct link between ED and molecular magnetic properties.

X-ray electron density investigation of chemical bonding in van der Waals materials.

Van der Waals (vdW) solids have attracted great attention ever since the discovery of graphene, with the essential feature being the weak chemical bonding across the vdW gap. The nature of these weak interactions is decisive for many extraordinary properties, but it is a strong challenge for current theory to accurately model long-range electron correlations. Here we use synchrotron X-ray diffraction data to precisely determine the electron density in the archetypal vdW solid, TiS2, and compare the results with density functional theory calculations. Quantitative agreement is observed for the chemical bonding description in the covalent TiS2 slabs, but significant differences are identified for the interactions across the gap, with experiment revealing more electron deformation than theory. The present data provide an experimental benchmark for testing theoretical models of weak chemical bonding.

Electron Density Studies in Materials Research.

Rational material design requires a deep understanding about the relationship between the structure and properties of materials, which are both intimately related to their chemical bonding. Through the experimentally observable electron density, chemical bonding can be understood from experimental and theoretical points of view on an equal footing, and advances in accurate X-ray diffraction measurements and computational techniques over the past decades have provided access to electron density distributions in increasingly complex functional materials. In this Review, selected electron density studies from the literature on a wide range of materials classes are presented, including studies of thermoelectric materials, high pressure electrides, coordination polymers and non-linear optical materials. These studies demonstrate how detailed analysis of chemical bonding based on the electron density provides important understanding of materials beyond arguments based on structure and simple chemical concepts. In cases such as understanding the conducting properties of Zintl semiconductors or the effect of mutual electrical polarization in host-guest systems, it is clearly imperative to go beyond structure and examine the chemical bonding in detail. In the Review, the complementarity between theory and experiment is underlined, which allows for mutual validation of new chemical bonding concepts, and indeed experiment and theory may challenge each other based on the different strengths and weaknesses of each method.

Low-Barrier Hydrogen Bonds in Negative Thermal Expansion Material H3 [Co(CN)6 ].

The covalent nature of the low-barrier N-H-N hydrogen bonds in the negative thermal expansion material H3 [Co(CN)6 ] has been established by using a combination of X-ray and neutron diffraction electron density analysis and theoretical calculations. This finding explains why negative thermal expansion can occur in a material not commonly considered to be built from rigid linkers. The pertinent hydrogen atom is located symmetrically between two nitrogen atoms in a double-well potential with hydrogen above the barrier for proton transfer, thus forming a low-barrier hydrogen bond. Hydrogen is covalently bonded to the two nitrogen atoms, which is the first experimentally confirmed covalent hydrogen bond in a network structure. Source function calculations established that the present N-H-N hydrogen bond follows the trends observed for negatively charge-assisted hydrogen bonds and low-barrier hydrogen bonds previously established for O-H-O hydrogen bonds. The bonding between the cobalt and cyanide ligands was found to be a typical donor-acceptor bond involving a high-field ligand and a transition metal in a low-spin configuration.

Exploring the High-Temperature Stabilization of Cubic Zirconia from Anharmonic Lattice Dynamics.

Finite-temperature stability of crystals is of continuous relevance in solid-state chemistry with many important properties only emerging in high-temperature polymorphs. Currently, the discovery of new phases is largely serendipitous due to a lack of computational methods to predict crystal stability with temperature. Conventional methods use harmonic phonon theory, but this breaks down when imaginary phonon modes are present. Anharmonic phonon methods are required to describe dynamically stabilized phases. We investigate the high-temperature tetragonal-to-cubic phase transition of ZrO2 based on first-principles anharmonic lattice dynamics and molecular dynamics simulations as an archetypical example of a phase transition involving a soft phonon mode. Anharmonic lattice dynamics calculations and free energy analysis suggest that the stability of cubic zirconia cannot be attributed solely to anharmonic stabilization and is thus absent for the pristine crystal. Instead, an additional entropic stabilization is suggested to arise from spontaneous defect formation, which is also responsible for superionic conductivity at elevated temperatures.

Low-Cost Vibrational Free Energies in Solid Solutions with Machine Learning Force Fields.

The rational design of alloys and solid solutions relies on accurate computational predictions of phase diagrams. The cluster expansion method has proven to be a valuable tool for studying disordered crystals. However, the effects of vibrational entropy are commonly neglected due to the computational cost. Here, we devise a method for including the vibrational free energy in cluster expansions with a low computational cost by fitting a machine learning force field (MLFF) to the relaxation trajectories available from cluster expansion construction. We demonstrate our method for two (pseudo)binary systems, Na1-xKxCl and Ag1-xPdx, for which accurate phonon dispersions and vibrational free energies are derived from the MLFF. For both systems, the inclusion of vibrational effects results in significantly better agreement with miscibility gaps in experimental phase diagrams. This methodology can allow routine inclusion of vibrational effects in calculated phase diagrams and thus more accurate predictions of properties and stability for mixtures of materials.

Room-temperature stacking disorder in layered covalent-organic frameworks from machine-learning force fields.

The local structures of layered covalent-organic frameworks (COFs) deviate from the average crystal structures assigned from X-ray diffraction experiments. For two prototype COFs of Tp-Azo and DAAQ-TFP, density functional theory calculations have shown that the eclipsed structure is not an energy minimum and that the internal energy is lowered for an inclined stacking arrangement. Here we explore the structural disorder of these frameworks at 300 K through molecular dynamics (MD) simulations using an on-the-fly machine learning force field (MLFF). We find that an initially eclipsed stacking mode spontaneously distorts to form a zigzag configuration that lowers the free energy of the crystal. The simulated diffraction patterns show good agreement with experimental observations. The dynamic disorder from the MLFF MD trajectories is found to persist in mesoscale MD simulations of 155 thousand atoms, giving further confidence in our conclusions. Our simulations show that the stacking behaviour of layered COFs is more complicated than previously understood.

Towards pump-probe single-crystal XFEL refinements for small-unit-cell systems.

Serial femtosecond crystallography for small-unit-cell systems has so far seen very limited application despite obvious scientific possibilities. This is because reliable data reduction has not been available for these challenging systems. In particular, important intensity corrections such as the partiality correction critically rely on accurate determination of the crystal orientation, which is complicated by the low number of diffraction spots for small-unit-cell crystals. A data reduction pipeline capable of fully automated handling of all steps of data reduction from spot harvesting to merged structure factors has been developed. The pipeline utilizes sparse indexing based on known unit-cell parameters, seed-skewness integration, intensity corrections including an overlap-based combined Ewald sphere width and partiality correction, and a dynamically adjusted post-refinement routine. Using the pipeline, data measured on the compound K4[Pt2(P2O5H2)4]·2H2O have been successfully reduced and used to solve the structure to an R1 factor of ∼9.1%. It is expected that the pipeline will open up the field of small-unit-cell serial femtosecond crystallography experiments and allow investigations into, for example, excited states and reaction intermediate chemistry.

Mixology of MA1-x EA x PbI3 Hybrid Perovskites: Phase Transitions, Cation Dynamics, and Photoluminescence.

Mixing molecular cations in hybrid lead halide perovskites is a highly effective approach to enhance the stability and performance of optoelectronic devices based on these compounds. In this work, we prepare and study novel mixed 3D methylammonium (MA)-ethylammonium (EA) MA1-x EA x PbI3 (x < 0.4) hybrid perovskites. We use a suite of different techniques to determine the structural phase diagram, cation dynamics, and photoluminescence properties of these compounds. Upon introduction of EA, we observe a gradual lowering of the phase-transition temperatures, indicating stabilization of the cubic phase. For mixing levels higher than 30%, we obtain a complete suppression of the low-temperature phase transition and formation of a new tetragonal phase with a different symmetry. We use broad-band dielectric spectroscopy to study the dielectric response of the mixed compounds in an extensive frequency range, which allows us to distinguish and characterize three distinct dipolar relaxation processes related to the molecular cation dynamics. We observe that mixing increases the rotation barrier of the MA cations and tunes the dielectric permittivity values. For the highest mixing levels, we observe the signatures of the dipolar glass phase formation. Our findings are supported by density functional theory calculations. Our photoluminescence measurements reveal a small change of the band gap upon mixing, indicating the suitability of these compounds for optoelectronic applications.

Multipole electron densities and structural parameters from synchrotron powder X-ray diffraction data obtained with a MYTHEN detector system (OHGI).

Powder X-ray diffraction has some inherent advantages over traditional single-crystal X-ray diffraction in accurately determining electron densities and structural parameters due to the lower requirements for sample crystallinity, simpler corrections and measurement simultaneity. For some simple inorganic materials, it has been shown that these advantages can compensate for disadvantages such as peak overlap and error-prone background subtraction. Although it is challenging to extend powder X-ray diffraction-based electron-density studies to organic materials with significant peak overlap, previous results using a dedicated vacuum diffractometer with a large image-plate camera (AVID) demonstrated that it can be done. However, the vacuum setup with the off-line detector system was found to prohibit a widespread use. Fast microstrip detectors, which have been employed at a number of powder diffraction beamlines, have the potential to facilitate electron-density studies. Nevertheless, no electron-density studies even for materials with slight peak overlap have been performed with microstrip detectors. One of the most critical problems has been a difference in sensitivity between microstrip channels, which substantially defines the dynamic range of a detector. Recently, a robust approach to this problem has been developed and applied to a total scattering measurement system (OHGI) with 15 MYTHEN microstrip modules. In the present study, synchrotron powder X-ray diffraction data obtained with OHGI are evaulated in terms of multipole electron densities and structural parameters (atomic positions and displacement parameters). These results show that, even without a dedicated setup and perfect samples, electron-density modelling can be carried out on high-quality powder X-ray diffraction data. However, it was also found that the required prior information about the sample prohibits widespread use of the method. With the presently obtainable data quality, electron densities of molecular crystals in general are not reliably obtained from powder data, but it is an excellent, possibly superior, alternative to single-crystal measurements for small-unit-cell inorganic solids. If aspherical atomic scattering factors can be obtained from other means (multipole databases, theoretical calculations), then atomic positions (including for hydrogen) and anisotropic atomic displacement parameters (non-hydrogen atoms) of excellent accuracy can be refined from synchrotron powder X-ray diffraction data on organic crystals.

Direct observation of one-dimensional disordered diffusion channel in a chain-like thermoelectric with ultralow thermal conductivity.

Structural disorder, highly effective in reducing thermal conductivity, is important in technological applications such as thermal barrier coatings and thermoelectrics. In particular, interstitial, disordered, diffusive atoms are common in complex crystal structures with ultralow thermal conductivity, but are rarely found in simple crystalline solids. Combining single-crystal synchrotron X-ray diffraction, the maximum entropy method, diffuse scattering, and theoretical calculations, here we report the direct observation of one-dimensional disordered In1+ chains in a simple chain-like thermoelectric InTe, which contains a significant In1+ vacancy along with interstitial indium sites. Intriguingly, the disordered In1+ chains undergo a static-dynamic transition with increasing temperature to form a one-dimensional diffusion channel, which is attributed to a low In1+-ion migration energy barrier along the c direction, a general feature in many other TlSe-type compounds. Our work provides a basis towards understanding ultralow thermal conductivity with weak temperature dependence in TlSe-type chain-like materials.

Expression and interactions of stereochemically active lone pairs and their relation to structural distortions and thermal conductivity.

In chemistry, stereochemically active lone pairs are typically described as an important non-bonding effect, and recent interest has centred on understanding the derived effect of lone pair expression on physical properties such as thermal conductivity. To manipulate such properties, it is essential to understand the conditions that lead to lone pair expression and provide a quantitative chemical description of their identity to allow comparison between systems. Here, density functional theory calculations are used first to establish the presence of stereochemically active lone pairs on antimony in the archetypical chalcogenide MnSb2O4. The lone pairs are formed through a similar mechanism to those in binary post-transition metal compounds in an oxidation state of two less than their main group number [e.g. Pb(II) and Sb(III)], where the degree of orbital interaction (covalency) determines the expression of the lone pair. In MnSb2O4 the Sb lone pairs interact through a void space in the crystal structure, and their their mutual repulsion is minimized by introducing a deflection angle. This angle increases significantly with decreasing Sb-Sb distance introduced by simulating high pressure, thus showing the highly destabilizing nature of the lone pair interactions. Analysis of the chemical bonding in MnSb2O4 shows that it is dominated by polar covalent interactions with significant contributions both from charge accumulation in the bonding regions and from charge transfer. A database search of related ternary chalcogenide structures shows that, for structures with a lone pair (SbX 3 units), the degree of lone pair expression is largely determined by whether the antimony-chalcogen units are connected or not, suggesting a cooperative effect. Isolated SbX 3 units have larger X-Sb-X bond angles and therefore weaker lone pair expression than connected units. Since increased lone pair expression is equivalent to an increased orbital interaction (covalent bonding), which typically leads to increased heat conduction, this can explain the previously established correlation between larger bond angles and lower thermal conductivity. Thus, it appears that for these chalcogenides, lone pair expression and thermal conductivity may be related through the degree of covalency of the system.

Accurate high-resolution single-crystal diffraction data from a Pilatus3†X CdTe detector.

Hybrid photon-counting detectors are widely established at third-generation synchrotron facilities and the specifications of the Pilatus3 X CdTe were quickly recognized as highly promising in charge-density investigations. This is mainly attributable to the detection efficiency in the high-energy X-ray regime, in combination with a dynamic range and noise level that should overcome the perpetual problem of detecting strong and weak data simultaneously. These benefits, however, come at the expense of a persistent problem for high diffracted beam flux, which is particularly problematic in single-crystal diffraction of materials with strong scattering power and sharp diffraction peaks. Here, an in-depth examination of data collected on an inorganic material, FeSb2, and an organic semiconductor, rubrene, revealed systematic differences in strong intensities for different incoming beam fluxes, and the implemented detector intensity corrections were found to be inadequate. Only significant beam attenuation for the collection of strong reflections was able to circumvent this systematic error. All data were collected on a bending-magnet beamline at a third-generation synchrotron radiation facility, so undulator and wiggler beamlines and fourth-generation synchrotrons will be even more prone to this error. On the other hand, the low background now allows for an accurate measurement of very weak intensities, and it is shown that it is possible to extract structure factors of exceptional quality using standard crystallographic software for data processing (SAINT-Plus, SADABS and SORTAV), although special attention has to be paid to the estimation of the background. This study resulted in electron-density models of substantially higher accuracy and precision compared with a previous investigation, thus for the first time fulfilling the promise of photon-counting detectors for very accurate structure factor measurements.

Multipole electron densities and atomic displacement parameters in urea from accurate powder X-ray diffraction.

Electron density determination based on structure factors obtained through powder X-ray diffraction has so far been limited to high-symmetry inorganic solids. This limit is challenged by determining high-quality structure factors for crystalline urea using a bespoke vacuum diffractometer with imaging plates. This allows the collection of data of sufficient quality to model the electron density of a molecular system using the multipole method. The structure factors, refined parameters as well as chemical bonding features are compared with results from the high-quality synchrotron single-crystal study by Birkedal et al. [Acta Cryst. (2004), A60, 371-381] demonstrating that powder X-ray diffraction potentially provides a viable alternative for electron density determination in simple molecular crystals where high-quality single crystals are not available.

Chemical bonding origin of the unexpected isotropic physical properties in thermoelectric Mg3Sb2 and related materials.

The Mg3Sb2 structure is currently being intensely scrutinized due to its outstanding thermoelectric properties. Usually, it is described as a layered Zintl phase with a clear distinction between covalent [Mg2Sb2]2- layers and ionic Mg2+ layers. Based on the quantitative chemical bonding analysis, we unravel instead that Mg3Sb2 exhibits a nearly isotropic three-dimensional bonding network with the interlayer and intralayer bonds being mostly ionic and surprisingly similar, which results in the nearly isotropic structural and thermal properties. The isotropic three-dimensional bonding network is found to be broadly applicable to many Mg-containing compounds with the CaAl2Si2-type structure. Intriguingly, a parameter based on the electron density can be used as an indicator measuring the anisotropy of lattice thermal conductivity in Mg3Sb2-related structures. This work extends our understanding of structure and properties based on chemical bonding analysis, and it will guide the search for and design of materials with tailored anisotropic properties.

Accurate charge densities from powder X-ray diffraction - a new version of the Aarhus vacuum imaging-plate diffractometer.

In recent years powder X-ray diffraction has proven to be a valuable alternative to single-crystal X-ray diffraction for determining electron-density distributions in high-symmetry inorganic materials, including subtle deformation in the core electron density. This was made possible by performing diffraction measurements in vacuum using high-energy X-rays at a synchrotron-radiation facility. Here we present a new version of our custom-built in-vacuum powder diffractometer with the sample-to-detector distance increased by a factor of four. In practice this is found to give a reduction in instrumental peak broadening by approximately a factor of three and a large improvement in signal-to-background ratio compared to the previous instrument. Structure factors of silicon at room temperature are extracted using a combined multipole-Rietveld procedure and compared with ab initio calculations and the results from the previous diffractometer. Despite some remaining issues regarding peak asymmetry, the new diffractometer yields structure factors of comparable accuracy to the previous diffractometer at low angles and improved accuracy at high angles. The high quality of the structure factors is further assessed by modelling of core electron deformation with results in good agreement with previous investigations.