Interstitial or interstitialcy: effect of the cation size on the migration mechanism in NaSICON materials.

Sodium superionic conductors (NaSICONs) with general formula NaM2A3O12 have attracted significant attention as solid electrolytes for all solid-state batteries owing to their remarkable room temperature ionic conductivity in the order of 10-3 S cm-1. Their flexible structural framework, which allows the incorporation of various aliovalent cations, affects the Na+ ion transport. However, establishing a straightforward correlation between Na+ mobility and NaSICON composition proves challenging due to competing influences such as framework alteration and stoichiometric changes of the cation substituents and thus the mobile Na+ ions. Therefore, we systematically investigate the NaSICON system across various Na1+xM2SixP3-xO12 compositions. We unravel and examine independently two key aspects impacting the Na+ ion transport in NaSICONs: structural factors determined by introduced M4+ framework cations and the substitution level (x). By employing DFT calculations, we explore the interstitial- and interstitialcy-like migration mechanisms, revealing that these mechanisms and the associated migration energies are primarily influenced by metastable transient states traversed during the Na+ ion migration. The stability of these transient states, in turn, depends on the spatial arrangement of the Na+ ions, the size of the M4+ cations defining the structural framework, and x. This study enhances our fundamental understanding of Na+ ion migration within NaSICONs across a wide range of compositions. The findings offer valuable insights into the microscopic aspects of NaSICON materials and provide essential guidance for prospective studies in this field.

First principles approach for promising oxide ion conducting ABGa3O7 melilite structures.

Melilite type structures of the general composition A3+1+xB2+1-xGa3O7+x/2 provide high oxide ion conductivity for x > 0 due to the presence of mobile oxide interstitials. While the structure can accommodate a variety of A- and B-cations, compositions besides La3+/Sr2+ are rarely investigated and the literature is inconclusive. In this contribution, combinations of A-cations (Ce, La, Nd, Pr, Sm) and B-cations (Mg, Ca, Sr, Ba) are investigated by density functional theory calculations. Two criteria for high ionic conductivity are examined: The variation of the site energies for different configurations and the average migration barriers. Promising combinations of cations are suggested for further investigation.

The origin of high Na+ ion conductivity in Na1+xZr2SixP3-xO12 NASICON materials.

Due to the high sodium ion conductivity, sodium super ionic conductors (NASICONs) are among the most promising candidates as solid electrolytes in solid state batteries and have therefore gained enormous attention in recent years. Previous experimental and computational investigations show excellent sodium ion conductivity for Na1+xZr2SixP3-xO12 with x = 2-2.5. In order to elucidate the conductivity maximum at high substitution levels, we investigate the influence of the cation environment on the site energies of sodium ions and the correlated migration using density functional theory. The results reveal that the site energy strongly depends on electrostatic effects. In addition, an increasing fraction of sodium and silicon ions enhances the correlated migration due to the increasing coulombic repulsion of sodium ions and opening of the bottleneck along the migration path. Based on the results, we generate an energy model to predict configurational and migration energy in subsequent Kinetic Monte Carlo simulations. We show that sodium ions are trapped due to introduced silicon ions but percolate in the system for x≥2.0, which greatly increases the ionic conductivity.

The effect of defect interactions on the reduction of doped ceria.

Doped cerium oxide is known for its reduction properties that are utilized in catalytic applications as well as in thermochemical cycling to produce solar fuels. Upon reduction of the lattice, oxygen vacancies and polarons are formed leading to a highly concentrated solution of defects in which the interactions of defects cannot be neglected anymore. In this study, the effect of defect interactions on the free energy of reduction for doped ceria with composition Ce1-x-yRExZryO2-x/2-δ was investigated by large scale Metropolis Monte Carlo multi-stage sampling simulations based on first-principles calculations. The simulations allowed the prediction of the relation between oxygen partial pressure and non-stoichiometry for the highly interacting, non-ideal system. The results show that the non-ideality observed in experiments can be traced back to the interactions of defects and allow prediction of the reduction behavior for various dopant types, dopant fractions, temperatures, and non-stoichiometries.

The effect of ionic defect interactions on the hydration of yttrium-doped barium zirconate.

Hydrated acceptor-doped barium zirconate is a well-investigated proton conductor. In the analysis of most experimental studies, an ideal defect model is applied to fit the measured hydration data and obtain corresponding enthalpies and entropies. However, the data show a distinct deviation from ideal behaviour and thus defect interactions cannot be neglected. In the present contribution, the thermodynamics of water uptake into the yttrium-doped bulk material are investigated on the microscopic level with regards to ionic defect interactions. Metropolis Monte Carlo simulations using interaction models from first-principles energy calculations are applied to obtain an estimation of the free energy of interaction. The present results indicate that the ionic defect interactions are the primary reason for the non-ideality observed in experiments with varying yttrium fraction, proton fraction, and temperature. The activity coefficient quotients for the mass action law are obtained, which connect the ideal and real model and are of relevance to data evaluation and theoretical calculations.

MOCASSIN: Metropolis and kinetic Monte Carlo for solid electrolytes.

The combination of density functional theory and Monte Carlo simulations is a powerful approach for the atomistic modeling of defect transport in solid electrolytes. The present contribution introduces the MOCASSIN software (Monte Carlo for Solid State Ionics) for kinetic and Metropolis Monte Carlo simulations of crystalline materials. MOCASSIN combines model building, visualization, and simulation, aiming to provide accessible MC for end users. Developed for the investigation of solid electrolytes, MOCASSIN is ideal for screening common variation parameters, such as temperature and doping fraction. The input effort is minimized using space groups for processing symmetry. The graphical interface for model building allows complex model input, including multiple mobile species, multiple migration paths, small polaron hopping, vehicle movements, multiple complex migration mechanisms, and custom interaction clusters. The software is provided free of charge for noncommercial usage.

Publisher Correction: Nanoscale percolation in doped BaZrO3 for high proton mobility.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Nanoscale percolation in doped BaZrO3 for high proton mobility.

Acceptor-doped barium zirconate is a promising proton-conducting oxide for various applications, for example, electrolysers, fuel cells or methane-conversion cells. Despite many experimental and theoretical investigations there is, however, only a limited understanding as to how to connect the complex microscopic proton motion and the macroscopic proton conductivity for the full range of acceptor levels, from diluted acceptors to concentrated solid solutions. Here we show that a combination of density functional theory calculations and kinetic Monte Carlo simulations enables this connection. At low concentrations, acceptors trap protons, which results in a decrease of the average proton mobility. With increasing concentration, however, acceptors form nanoscale percolation pathways with low proton migration energies, which leads to a strong increase of the proton mobility and conductivity. Comparing our simulated proton conductivities with experimental values for yttrium-doped barium zirconate yields excellent agreement. We then predict that ordered dopant structures would not only strongly enhance the proton conductivities, but would also enable one- or two-dimensional proton conduction in barium zirconate. Finally, we show how the properties of other dopants influence the proton conductivity.

Defect formation and migration in Nasicon Li1+xAlxTi2-x(PO4)3.

NASICON-structured materials of the composition Li1+xAlxTi2-x(PO4)3 are regarded as solid electrolytes with high Li-ion conductivity applicable in all solid-state batteries. In this study we investigate the migration paths of constituting ions and monovalent charge carriers including K+, Na+, and, H+. The results proof that Li is the most mobile species in the investigated composition and that the formation of intrinsic defects is unlikely. In addition, we find surprisingly low migration energy for oxygen vacancies in the structure of the dedicated Li-ion conductor.

Influence of the lattice constant on defects in cerium oxide.

The lattice constant has a crucial effect on the defect chemistry and defect kinetics in solid state materials. However, within density functional theory, some functionals perform badly in reproducing the experimental lattice constant. In this study, energies of defect formation, interaction and migration in the model system ceria were calculated for different lattice constants to investigate the impact on the energies. The GGA+U functional in the PBE and PBEsol parametrization as well as the hybrid functional HSE06 were applied and results are compared among these three commonly applied functionals. The results suggest a strong influence of the lattice constant on the energies especially regarding oxygen ion migration. This influence has an impact on the accurate prediction of defect properties from first principles but can also be utilized for specific tailoring of material properties by chemo-mechanical design. In addition, the issue of the correct lattice constant, which should be used in the defect calculations, is discussed in this paper.

Understanding the ionic conductivity maximum in doped ceria: trapping and blocking.

Materials with high oxygen ion conductivity and low electronic conductivity are required for electrolytes in solid oxide fuel cells (SOFC) and high-temperature electrolysis (SOEC). A potential candidate for the electrolytes, which separate oxidation and reduction processes, is rare-earth doped ceria. The prediction of the ionic conductivity of the electrolytes and a better understanding of the underlying atomistic mechanisms provide an important contribution to the future of sustainable and efficient energy conversion and storage. The central aim of this paper is the detailed investigation of the relationship between defect interactions at the microscopic level and the macroscopic oxygen ion conductivity in the bulk of doped ceria. By combining ab initio density functional theory (DFT) with Kinetic Monte Carlo (KMC) simulations, the oxygen ion conductivity is predicted as a function of the doping concentration. Migration barriers are analyzed for energy contributions, which are caused by the interactions of dopants and vacancies with the migrating oxygen vacancy. We clearly distinguish between energy contributions that are either uniform for forward and backward jumps or favor one migration direction over the reverse direction. If the presence of a dopant changes the migration energy identically for forward and backward jumps, the resulting energy contribution is referred to as blocking. If the change in migration energy due to doping is different for forward and backward jumps of a specific ionic configuration, the resulting energy contributions are referred to as trapping. The influence of both effects on the ionic conductivity is analyzed: blocking determines the dopant fraction where the ionic conductivity exhibits the maximum. Trapping limits the maximum ionic conductivity value. In this way, a deeper understanding of the underlying mechanisms determining the influence of dopants on the ionic conductivity is obtained and the ionic conductivity is predicted more accurately. The detailed results and insights obtained here for doped ceria can be generalized and applied to other ion conductors that are important for SOFCs and SOECs as well as solid state batteries.

Ab initio calculation of the attempt frequency of oxygen diffusion in pure and samarium doped ceria.

The rate of oxygen ion jumps in a solid oxide depends not only on the activation energy but also on the pre-exponential factor of diffusion. In order to allow a fully ab initio prediction of the oxygen ion conductivity in pure and samarium doped ceria, we calculated the attempt frequency for an oxygen ion jump from first principles combining DFT+U, the NEB method, phonon calculations and the transition state theory. Different definitions of the jump attempt frequency are presented. The equivalence of the Eyring and the Vineyard method is shown without restriction to the Gamma point. Convergence checks of the phonon mesh reveal that the common reduction to the Gamma point is not sufficient to calculate the attempt frequency. Calculations of Sm doped ceria revealed an increase of the prefactor. The attempt frequency for the constant pressure case in quasi-harmonic approximation is larger than the attempt frequency at constant volume in harmonic approximation. The calculated electronic energies, enthalpies and entropies of migration are in agreement with the experimental diffusion coefficients and activation energies.

Association of defects in doped non-stoichiometric ceria from first principles.

We investigate the interaction and distribution of defects in doped non-stoichometric ceria Ce1-xRExO2-x/2-δ (with RE = Lu, Y, Gd, Sm, Nd, and La) by combining DFT+U calculations and Monte Carlo simulations. The concentrated solution of defects in ceria is described by the pair interactions of dopant ions, oxygen vacancies, and small polarons. The calculated interaction energies for polarons and oxygen vacancies are in agreement with experimental results and previously reported calculations. Simulations reveal that in thermodynamic equilibrium the configurational energy decreases with increasing non-stoichiometry as well as increasing dopant fraction similar to the observed behavior of the enthalpy of reduction in experiments. This effect is attributed to the attractive interaction of oxygen vacancies with polarons and dopant ions.

A combined DFT + U and Monte Carlo study on rare earth doped ceria.

We investigate the dopant distribution and its influence on the oxygen ion conductivity of ceria doped with rare earth oxides by combining density functional theory and Monte Carlo simulations. We calculate the association energies of dopant pairs, oxygen vacancy pairs and between dopant ions and oxygen vacancies by means of DFT + U including finite size corrections. The cation coordination numbers from ensuing Metropolis Monte Carlo simulations show remarkable agreement with experimental data. Combining Metropolis and Kinetic Monte Carlo simulations we find a distinct dependence of the ionic conductivity on the dopant distribution and predict long term degradation of electrolytes based on doped ceria.

Entropies of defect formation in ceria from first principles.

We calculate entropies of formation for fully charged point defects, including the small polaron Ce'(Ce), in undoped fluorite-structured ceria by means of density functional theory in the GGA + U approximation. We discuss the behaviour of the entropy for the constant volume and the constant pressure case. Our results for constant pressure (p = 0) suggest that the change in volume, due to the formation of defects, dominates the entropy of formation. From the individual entropies of formation the entropies of Frenkel, anti-Frenkel and Schottky disorder as well as the entropy of reduction of ceria are obtained. At temperatures of about 1000 K the entropic contributions to the Gibbs energy are up to 0.9 eV per defect and thus are no longer negligible. For our calculated entropy of reduction of about 17 kB we find a remarkable agreement with experimental data from the literature.