arXiv subjects

Chris Wolverton

Publications and source records attributed to Chris Wolverton.

At least 19 recordsLinked to original sources

arXiv2026

Autonomous Optimization of Complex Oxides for Thermochemical Fuel Production

Two-step thermochemical fuel production, including H2O and CO2 splitting, offers a promising route to sustainable fuel manufacturing, with performance governed by redox-active oxides that enable cyclic reduction-oxidation reactions. Maximizing thermal-to-fuel conversion efficiency demands materials that simultaneously satisfy multiple stringent thermodynamic and kinetic targets. Addressing these requirements has increasingly driven materials design toward complex, multi-cation oxides, such as mixed-cation fluorites, perovskites, and high-entropy oxides, wherein composition, defect chemistry, phase stability, and morphology should be co-optimized. This creates a challenging materials optimization problem that is poorly suited to traditional trial-and-error approaches. In this review, we argue that thermochemical fuel production provides a compelling frontier for autonomous materials design and optimization. We first examine why redox-active complex oxides are difficult to develop, owing to multidimensional phase spaces, harsh operating conditions, and competing functional targets. We then discuss how high-throughput computation, automated synthesis, characterization and testing, and machine learning can be integrated into closed-loop workflows to address these challenges. Building on broader oxide materials research, we organize recent progress into a capability roadmap for complex-oxide optimization, spanning compositionally diverse synthesis, operando characterization, robotic testing, operation-condition computation, and multi-objective optimization. Finally, we outline key experimental, computational, and data challenges for building self-improving materials development platforms for materials development in thermochemical fuel production.

cond-mat.mtrl-sci↗

arXiv2026

Mechanism-Dependent Descriptors Enable Predictive Design of Oxygen Capacity in Perovskite Oxides

Perovskite oxides can reversibly accommodate substantial changes in oxygen stoichiometry, making them attractive for clean-energy technologies including chemical looping and oxygen storage. Despite extensive efforts to optimize their redox properties, predictive descriptors capable of assessing oxygen capacity across diverse compositions remain under development. Here, we combine experiments and first-principles calculations to establish composition and oxygen-capacity relationships in the model perovskite series LnxSr1-xCoO3. We confirm that increasing Sr2+ content promotes the formation of high-valence Co4+, expanding the cationic redox reservoir available during oxygen release and thereby enhancing oxygen capacity. In this regime, oxygen-vacancy formation energy captures the observed trend because oxygen release is primarily compensated by Co4+/Co3+/Co2+ redox. Across the rare-earth series, however, oxygen capacity decreases from La to Lu despite progressively lower oxygen-vacancy formation energies. We reveal that this counterintuitive behavior originates from an alternative charge-compensation pathway, in which lattice oxygen is partially oxidized to O1- -like species during oxygen removal. Heavy rare-earth compositions (Tb-Lu) preferentially stabilize these oxygen-hole species through distinct local bonding environments, with charge compensation involving both oxidized lattice oxygen and reduced rare-earth and cobalt cations, thereby suppressing net oxygen release despite favorable vacancy thermodynamics. We further identify average metal-oxygen bond strength, quantified by integrated crystal orbital Hamilton population, as a physically meaningful descriptor for oxygen capacity when anionic redox becomes dominant.

cond-mat.mtrl-sci↗

arXiv2026

Long- and Short-Range Anion Order in SrTiO$_{3-x}$H$_x$ Perovskite Oxyhydrides: DFT+$U$ Sensitivity and HSE06 Cluster Expansion

Anion ordering in perovskite oxyhydrides can remain significant even in disordered states, particularly at non-dilute hydrogen concentrations. Nevertheless, hydride substitution poses challenges for accurate simulations based on density functional theory due to configurational complexity and Ti 3$d$ reduction. Here, we develop a cluster expansion (CE) framework for SrTiO$_{3-x}$H$_x$ incorporating HSE06 hybrid-DFT energetics. We first demonstrate that calculated mixing energies and ordering stability are highly sensitive to the choice of DFT+$U$, with maximum variations on the order of 100 meV/anion. We benchmark ordering energetics against HSE06 calculations and identify $U$ = 3.3 eV as an HSE06 proxy, which enables extensive configurational exploration while limiting costly HSE06 calculations to key configurations for efficient learning of ordering energetics. Together, ground-state orderings, correlations between octahedral configurations and structural stability, and MC sampling of CE models all support a strong preference for the O$_4$H$_2$ cis configuration in SrTiO$_{3-x}$H$_x$, in which two hydride ions occupy first-nearest-neighbor anion sites. This cis-type preference was overlooked in previous ATiO$_{3-x}$H$_x$ studies, despite its sizable stabilization of ~200 meV per hydride comparable to reported anion-migration and polaron-formation energies. This study addresses both the previously underexplored sensitivity of CE-based ordering analyses to DFT+$U$ and anion-ordering in perovskite oxyhydrides.

cond-mat.mtrl-sci↗

arXiv2026

Building a physics-aware AI ecosystem for solid-state hydrogen storage materials

Hydrogen storage remains a central bottleneck for scalable hydrogen energy systems due to the multiscale and coupled nature of the thermodynamics, kinetics, and microstructural evolution of hydrogen storage materials (HSMs). Although artificial intelligence (AI) has accelerated materials discovery, current approaches remain constrained by fragmented data, limited physical consistency, and weak integration with experimental validation. Here, we propose a unified framework that integrates coherent data infrastructure, physics-grounded modeling, and AI-driven inverse design within a closed-loop discovery paradigm. By embedding physical constraints and experimental feedback, this approach enables adaptive, physically consistent optimization, thereby establishing a pathway toward autonomous, digital-twin-enabled discovery of HSMs.

cond-mat.mtrl-sci↗

arXiv2025

Lattice-to-Total Thermal Conductivity Ratio: A Phonon-Glass Electron-Crystal Descriptor for Data-Driven Thermoelectric Design

Thermoelectrics (TEs) are promising candidates for energy harvesting with performance quantified by figure of merit, $ZT$. To accelerate the discovery of high-$ZT$ materials, efforts have focused on identifying compounds with low thermal conductivity $\kappa$. Using a curated dataset of 71,913 entries, we show that high-$ZT$ materials reside not only in the low-$\kappa$ regime but also cluster near a lattice-to-total thermal conductivity ratio ($\kappa_\mathrm{L}/\kappa$) of approximately 0.5. This optimal ratio provides a quantitative descriptor for the well-known phonon-glass electron-crystal (PGEC) design concept. Building on this insight, we construct a framework consisting of two machine learning models for the lattice and electronic components of thermal conductivity that jointly provide both $\kappa$ and $\kappa_\mathrm{L}/\kappa$ for screening and guiding the optimization of TE materials. By applying this framework to 104,567 inorganic compounds, we identify 2,522 ultralow-$\kappa$ candidates while simultaneously evaluating their proximity to the optimal PGEC regime. A follow-up case study on chemical doping demonstrates how the framework can qualitatively provide optimization strategies that shift pristine materials toward the ideal $\kappa_\mathrm{L}/\kappa$ $\approx$ 0.5 target. Ultimately, by integrating rapid screening with PGEC-guided optimization, our data-driven framework takes a critical step towards closing the gap between materials discovery and performance enhancement.

cond-mat.mtrl-sci↗

arXiv2025

Interpretable Nanoporous Materials Design with Symmetry-Aware Networks

Reticular frameworks hold promise for diverse sustainable applications, yet their immense chemical space limits efficient and systematic design. While machine learning provides a compelling pathway to accelerate exploration, existing approaches often lack either interpretability or fidelity in linking crystal geometry to emergent properties. Here, we introduce a site-resolved equivariant learning framework based on three-dimensional periodic space sampling, which decomposes reticular structures into local geometric environments for simultaneous property prediction and site-wise contribution analysis. Trained on a combination of constructed and retrieved datasets, the model achieves state-of-the-art accuracy and data efficiency across gas storage, gas separation, and electronic-property prediction tasks. Importantly, the framework reveals interpretable local structure-property relationships by identifying transferable high-contribution sites across diverse frameworks. Leveraging these learned motifs, we further demonstrate inverse design of new metal-organic frameworks exhibiting record-high N2 storage, strong CO2/N2 separation performance, and near-zero electronic band gaps, validated by physics-based simulations.

cond-mat.mtrl-sci↗

arXiv2025

Origin of Glass-like Thermal Conductivity in Crystalline TlAgTe

Ordered crystalline compounds exhibiting ultralow and glass-like thermal conductivity are both fundamentally and technologically important, where phonon quasi-particles dominate their heat transport. Understanding the microscopic mechanisms that govern such unusual transport behavior is necessary to unravel the complex interplay of crystal structure, phonons, and collective excitations of these quasi-particles. Here, we use state-of-the-art first-principles calculations based on quantum density functional theory to investigate the origin of experimentally measured unusually low and glassy thermal conductivity in semiconducting TlAgTe. Utilizing a unifying framework of anharmonic lattice dynamics theory that combine phonon self-energy induced frequency renormalization, particle-like Peierls ($\kappa_{l}^{P}$) and wave-like coherent ($\kappa_{l}^{C}$) thermal transport contributions including three and four-phonon scattering channels, we successfully explain the experimental results both in terms of magnitude and temperature dependence. Our analysis reveals that TlAgTe exhibits several localized phonon modes arising from concerted rattling-like vibration of Tl atoms, which show strong temperature dependence and enhanced four-phonon scattering rates that are dominated by Umklapp processes, suppressing $\kappa_{l}^{P}$ to ultralow values. The ensuing strong anharmonicity induced by local structural distortions, lone-pair electrons, and rattling-like vibrations of the heavy cations lead to a transition from particle-like behavior to wave-like tunneling characteristics of the phonon modes above 40 cm$^{-1}$, contributing significantly to $\kappa_{l}^{C}$ which increases with temperature. Our analysis uncovers important structure-property relationship, which may be used in designing of novel materials with tunable thermal conductivity.

cond-mat.mtrl-sci↗

arXiv2025

Short-Range Order and Li$_x$TM$_{4-x}$ Probability Maps for Disordered Rocksalt Cathodes

Short-range order (SRO) in the cation-disordered state is a controlling factor influencing the probability of finding Li$_{4}$ tetrahedron clusters in disordered rocksalt (DRX) cathode materials. However, the prevalent Li$_4$ probability below the random limit across reported DRX compositions has not been systematically investigated, active strategies to surpass the random limit of Li$_4$ probability are lacking, and the fundamental ordering behavior on the face-centered cubic (FCC) lattice remains insufficiently explored. This research quantitatively examines pair SRO parameters and Li$_x$TM$_{4-x}$ probabilities via exhaustive Monte Carlo mapping across a simplified subset of the parameter space. The results indicate that, in the disordered state, the Li$_4$ probability is governed by the nearest neighbor (NN) pair-wise SRO parameter, and that these quantities do not necessarily represent a simple attenuation of their corresponding low-temperature long-range order, particularly for the important cases of Layered and Spinel-like orderings. Strategies are proposed to mitigate or even reverse the lithium and transition metals mixing tendency of NN pair SRO to achieve Li$_4$ probabilities that exceed the random limit. This study advances the fundamental thermodynamic understanding of ordering behaviors, which can be generalized to any FCC system.

cond-mat.mtrl-sci↗

arXiv2025

Data-driven Discovery of Novel High-performance Quaternary Chalcogenide Photovoltaics

Photovoltaic materials facilitate the conversion of sunlight into electricity by harnessing the interaction between light and matter, offering an eco-friendly and cost-efficient energy solution. Combining data-driven approaches with static and time-dependent density functional theories and nonadiabatic molecular dynamics simulations, we predict 14 high-performance photoabsorber materials from a family of known quaternary semiconductors. Among these, we investigate four compounds - SrCuGdSe3, SrCuDyTe3, BaCuLaSe3, and BaCuLaTe3 in greater detail. Hybrid density functional theory calculations including spin-orbit coupling reveal that SrCuGdSe3, SrCuDyTe3, BaCuLaSe3 and BaCuLaTe3 possess direct band gaps of 1.65, 1.79, 1.05, and 1.01 eV, respectively. These band gap values lie close to an optimal range ideal for visible-light absorption. Consequently, the calculated optical absorption coefficient and spectroscopic limited maximum efficiency for these compounds become comparable or larger than crystalline silicon, GaAs, and methylammonium lead iodide. Calculated exciton binding energies for these compounds are relatively small (30-32 meV), signifying easy separation of the electron-hole pairs, and hence enhanced power conversion efficiencies. Investigations of photoexcited carrier dynamics reveal a relatively long carrier lifetime (~ 30-40 ns), suggesting suppressed nonradiative recombination and enhanced photo-conversion efficiencies. We further determined the defect formation energies in these compounds, which showed that despite the likely formation of cation vacancies and interstitial defects, midgap states remain absent making these defects non-detrimental to carrier recombination. Our theoretical predictions invite experimental verification and encourage further investigations of these and similar compounds in this quaternary semiconductor family.

cond-mat.mtrl-sci↗

arXiv2025

High-throughput computational framework for lattice dynamics and thermal transport including high-order anharmonicity: an application to cubic and tetragonal inorganic compounds

Accurately predicting lattice thermal conductivity (kL) from first principles remains a challenge in identifying materials with extreme thermal behavior. While modern lattice dynamics methods enable routine predictions of kL within the harmonic approximation and three-phonon scattering framework (HA+3ph), reliable results, especially for low-kL compounds, require higher-order anharmonic effects, including self-consistent phonon renormalization, four-phonon scattering, and off-diagonal heat flux (SCPH+3,4ph+OD). We present a high-throughput workflow integrating these effects into a unified framework. Using this, we compute kL for 773 cubic and tetragonal inorganic compounds across diverse chemistries and structures. From 562 dynamically stable compounds, we assess the hierarchical effects of higher-order anharmonicity. For about 60% of materials, HA+3ph predictions closely match those from SCPH+3,4ph+OD. However, SCPH corrections often increase kL, sometimes by over 8 times, while four-phonon scattering universally reduces it, occasionally to 15% of the HA+3ph value. Off-diagonal contributions are minor in high-kL systems but can be comparable to the diagonal ones in highly anharmonic, low-kL compounds. We highlight four cases-Rb2TlAlH6, Cu3VS4, CuBr, and KTlCl4-exhibiting distinct anharmonic behaviors. This work delivers not only a robust workflow for high-fidelity kL dataset but also a quantitative framework to determine when higher-order effects are essential. The hierarchy of kL results, from the HA+3ph to SCPH+3,4ph+OD level, offers a scalable, interpretable route to discovering next-generation extreme thermal materials.

cond-mat.mtrl-sci↗

arXiv2025

Strong acoustic phonon suppression leads to ultralow thermal conductivity and enhanced thermoelectric performance in BaCuGdTe$_3$

Excitations and scatterings among the quantized lattice vibrations, i.e., phonons, govern the lattice thermal conductivity ($\kappa_l$) in crystalline solids. Therefore, effective modulation of $\kappa_l$ can be achieved through selective manipulation of phonon modes that strongly participate in the heat transport mechanisms. Here, combining accurate first-principles density functional theory calculations and Boltzmann transport theory, we report a layered quaternary chalcogenide semiconductor, BaCuGdTe$_3$, which exhibits unusually low $\kappa_l$ ($\sim$ 0.14 W/mK at room temperature) despite its ordered crystalline structure. Our analysis reveals that the ultralow $\kappa_l$ arises mainly from a strong suppression of acoustic phonon modes induced by local distortion, shear vibrations among the layers, and large acoustic-optical avoided-crossing between phonons, which collectively enhances the phonon-scattering rates. Further calculations of the electrical transport properties with explicit consideration of electron-phonon interactions reveal a high thermoelectric figure of merit exceeding unity for this compound at moderate temperature (400-700 K) and carrier concentration $(1\text{--}5 \times 10^{19}\ \text{cm}^{-3})$ ranges. Our theoretical predictions warrant experimental investigations of the intriguing phonon dynamics, thermal transport mechanisms, and thermoelectric properties in this compound. Moreover, insights from our analysis can be used to design and engineer compounds with ultralow $\kappa_l$.

cond-mat.mtrl-sci↗

arXiv2025

Anomalous Reversal of Stability in Mo-containing Oxides: A Difficult Case Exhibiting Sensitivity to DFT+U and Distortion

Accurate predictions of the properties of transition metal oxides using density functional theory (DFT) calculations are essential for the computational design of energy materials. In this work, we investigate the anomalous reversal of the stability of structural distortions (where distorted structures go from being energetically favorable to sharply unfavorable relative to undistorted ones) induced by DFT+U on Mo d-orbitals in layered AMoO$_2$ (A = Li, Na, K) and rutile-like MoO$_2$. We highlight the significant impact of varying U$_{\text{eff}}$ values on the structural stability, convex hull, and thermodynamic stability predictions, noting that deviations can reach up to the order of 100 meV/atom across these energetic quantities. We find the transitions in stability are coincident with changes in the electron localization and magnetic behavior. The anomalous reversal persists across PBE, r$^2$SCAN functionals, and also with vdW-dispersion energy corrections (PBE+D3). In Mo-containing oxide systems, high U$_{\text{eff}}$ leads to inaccurate descriptions of physical quantities and structural relaxations under artificial symmetry constraints, as demonstrated by the phonon band structures, the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional results, and comparisons with experimental structural data. We conclude that high U$_{\text{eff}}$ values (around 4 eV and above, depending on the specific structures and compositions) might be unsuitable for energetic predictions in A-Mo-O chemical spaces. Our results suggest that the common practice of applying DFT+U to convex hull constructions, especially with high U$_{\text{eff}}$ values derived from fittings, should be carefully evaluated to ensure that ground states are correctly reproduced, with careful consideration of dynamic stability and possible energetically favorable distortions.

cond-mat.mtrl-sci↗

arXiv2024

Accelerating High-Throughput Phonon Calculations via Machine Learning Universal Potentials

Phonons play a critical role in determining various material properties, but conventional methods for phonon calculations are computationally intensive, limiting their broad applicability. In this study, we present an approach to accelerate high-throughput harmonic phonon calculations using machine learning universal potentials. We train a state-of-the-art machine learning interatomic potential, based on multi-atomic cluster expansion (MACE), on a comprehensive dataset of 2,738 crystal structures with 77 elements, totaling 15,670 supercell structures, computed using high-fidelity density functional theory (DFT) calculations. Our approach significantly reduces the number of required supercells for phonon calculations while maintaining high accuracy in predicting harmonic phonon properties across diverse materials. The trained model is validated against phonon calculations for a held-out subset of 384 materials, achieving a mean absolute error (MAE) of 0.18 THz for vibrational frequencies from full phonon dispersions, 2.19 meV/atom for Helmholtz vibrational free energies at 300K, as well as a classification accuracy of 86.2% for dynamical stability of materials. A thermodynamic analysis of polymorphic stability in 126 systems demonstrates good agreement with DFT results at 300 K and 1000 K. In addition, the diverse and extensive high-quality DFT dataset curated in this study serves as a valuable resource for researchers to train and improve other machine learning interatomic potential models.

cond-mat.mtrl-sci↗

arXiv2024

CSPBench: a benchmark and critical evaluation of Crystal Structure Prediction

Crystal structure prediction (CSP) is now increasingly used in discovering novel materials with applications in diverse industries. However, despite decades of developments and significant progress in this area, there lacks a set of well-defined benchmark dataset, quantitative performance metrics, and studies that evaluate the status of the field. We aim to fill this gap by introducing a CSP benchmark suite with 180 test structures along with our recently implemented CSP performance metric set. We benchmark a collection of 13 state-of-the-art (SOTA) CSP algorithms including template-based CSP algorithms, conventional CSP algorithms based on DFT calculations and global search such as CALYPSO, CSP algorithms based on machine learning (ML) potentials and global search, and distance matrix based CSP algorithms. Our results demonstrate that the performance of the current CSP algorithms is far from being satisfactory. Most algorithms cannot even identify the structures with the correct space groups except for the template-based algorithms when applied to test structures with similar templates. We also find that the ML potential based CSP algorithms are now able to achieve competitive performances compared to the DFT-based algorithms. These CSP algorithms' performance is strongly determined by the quality of the neural potentials as well as the global optimization algorithms. Our benchmark suite comes with a comprehensive open-source codebase and 180 well-selected benchmark crystal structures, making it convenient to evaluate the advantages and disadvantages of CSP algorithms from future studies. All the code and benchmark data are available at https://github.com/usccolumbia/cspbenchmark

cond-mat.mtrl-sci↗

arXiv2024

Thermal stabilities Landscape of A$_2$BB$^{\prime}$O$_6$ compounds

Perovskite oxides have been extensively studied for their wide range of compositions and structures, as well as their valuable properties for various applications. Expanding from single perovskite ABO$_3$ to double perovskite $A_2BB^{\prime}$O$_6$ significantly enhances the ability to tailor specific physical and chemical properties. However, the vast number of potential compositions of $A_2BB^{\prime}$O$_6$ makes it impractical to explore them all experimentally. In this study, we conducted high-throughput calculations to systematically investigate the structures and stabilities of 4,900 $A_2BB^{\prime}$O$_6$ compositions (with $A$ = Ca, Sr, Ba, and La; $B$ and $B^{\prime}$ representing metal elements) through over 42,000 density functional theory (DFT) calculations. Our analysis lead to the discovery of more than 1,500 new synthesizable $A_2BB^{\prime}$O$_6$ compounds, with over 1,100 of them exhibiting double perovskite structures, predominantly in the $P2_1/c$ space group. By leveraging the high-throughput dataset, we developed machine learning models that achieved mean absolute errors of 0.0444 and 0.0330 eV/atom for formation energy and decomposition energy, respectively. Using these models, we identified 803 stable or metastable compositions beyond the chemical space covered in our initial calculations, with 612 of them having DFT-validated decomposition energies below 0.1 eV/atom, resulting in a success rate of 76.2 \%. This study delineates the stability landscape of $A_2BB^{\prime}$O$_6$ compounds and offers new insights for the exploration of these materials.

cond-mat.mtrl-sci↗

arXiv2024

Developments and applications of the OPTIMADE API for materials discovery, design, and data exchange

The Open Databases Integration for Materials Design (OPTIMADE) application programming interface (API) empowers users with holistic access to a growing federation of databases, enhancing the accessibility and discoverability of materials and chemical data. Since the first release of the OPTIMADE specification (v1.0), the API has undergone significant development, leading to the upcoming v1.2 release, and has underpinned multiple scientific studies. In this work, we highlight the latest features of the API format, accompanying software tools, and provide an update on the implementation of OPTIMADE in contributing materials databases. We end by providing several use cases that demonstrate the utility of the OPTIMADE API in materials research that continue to drive its ongoing development.

cond-mat.mtrl-sci↗

arXiv2023

Oxygen Vacancy Formation Energy in Metal Oxides: High Throughput Computational Studies and Machine Learning Predictions

The oxygen vacancy formation energy ($\Delta E_{vf}$) governs defect dynamics and is a useful metric to perform materials selection for a variety of applications. However, density functional theory (DFT) calculations of $\Delta E_{vf}$ come at a greater computational cost than the typical bulk calculations available in materials databases due to the involvement of multiple vacancy-containing supercells. As a result, available repositories of direct calculations of $\Delta E_{vf}$ remain relatively scarce, and the development of machine learning models capable of delivering accurate predictions is of interest. In the present, work we address both such points. We first report the results of new high-throughput DFT calculations of oxygen vacancy formation energies of the different unique oxygen sites in over 1000 different oxide materials, which together form the largest dataset of directly computed oxygen vacancy formation energies to date, to our knowledge. We then utilize the resulting dataset of $\sim$2500 $\Delta E_{vf}$ values to train random forest models with different sets of features, examining both novel features introduced in this work and ones previously employed in the literature. We demonstrate the benefits of including features that contain information specific to the vacancy site and account for both cation identity and oxidation state, and achieve a mean absolute error upon prediction of $\sim$0.3 eV/O, which is comparable to the accuracy observed upon comparison of DFT computations of oxygen vacancy formation energy and experimental results. Finally, we demonstrate the predictive power of the developed models in the search for new compounds for solar-thermochemical water-splitting applications, finding over 250 new AA$^{\prime}$BB$^{\prime}$O$_6$ double perovskite candidates.

cond-mat.mtrl-sci↗

arXiv2023

Phase Stability of Lead Phosphate Apatite Pb$_{10-x}$Cu$_{x}$(PO$_{4}$)$_{6}$O, Pb$_{10-x}$Cu$_{x}$(PO$_{4}$)$_{6}$(OH)$_{2}$, and Pb$_{8}$Cu$_{2}$(PO$_{4}$)$_{6}$

Recently, Cu-substituted lead apatite LK-99 was reported to have room-temperature ambient-pressure superconductivity. Here we utilize density functional theory (DFT) total energy and harmonic phonon calculations to investigate the thermodynamic and dynamic stability of two lead phosphate apatites in their pure and Cu-substituted structures. Though Pb$_{10}$(PO$_4$)$_6$O and Pb$_{10}$(PO$_4$)$_6$(OH)$_2$ are found to be thermodynamically stable (i.e., on the T=0K ground state convex hull), their Cu-substituted counterparts are above the convex hull. Harmonic phonon calculations reveal dynamic instabilities in all four of these structures. Oxygen vacancy formation energies demonstrate that the addition of Cu dopant substituting for Pb increases the likelihood of the formation of oxygen vacancies on the anion site. We propose a new possible phase in this system, Pb$_8$Cu$_2$(PO$_4$)$_6$, where two monovalent Cu atoms are substituted for two Pb(1) atoms and the anion oxygen is removed. We also propose several reaction pathways for Pb$_9$Cu(PO$_4$)$_6$O and Pb$_8$Cu$_2$(PO$_4$)$_6$, and found that both of these two structures are likely to be synthesized under a 1:1 ratio of reactants Pb$_2$SO$_5$ and Cu$_3$P. Our work provides a thorough foundation for the thermodynamic and dynamic stabilities of LK-99 related compounds and we propose several possible novel synthesis reaction pathways and a new predicted structure for future studies.

cond-mat.mtrl-sci↗