Schottky Defect Formation Energy Research Articles

Numerical study of defects in protactinium carbide

Actinide carbides, like ThC and UC, have been widely studied in recent years, on the contrary, protactinium carbide has merited less attention. One issue of PaC that has not been approached, neither experimentally nor theoretically is the formation of defects and the incorporation of impurities. Therefore, we will focus on these topics through a numerically study. To accomplish this, we calculate formation energies of vacancy, tetrahedral interstitial, Frenkel pair, Schottky and antisite defects. We find that C vacancies are the energetically preferred defect. We analyze density of states, charge densities and distortion of lattices with these kind of defects. We also study the effect of incorporation of oxygen and helium in vacancy and interstitial defects. In view of the lack of both theoretical and experimental results in PaC, we compared with UC and ThC. As a measure of the efficacy of the protactinium pseudopotential, which we generate, we calculate structural parameters and phonon spectrum.

Molecular dynamics investigation of xenon, uranium, and oxygen diffusion in UO2 nuclear fuel

In this paper, atomistic simulations based on molecular dynamics (MD) are used to calculate the diffusion coefficients of xenon, uranium and oxygen in uranium dioxide containing different percentages of xenon atoms replaced by uranium from 2%Xe to 10%Xe at ten temperatures between 300 and 2500 K (covering three temperature regimes of athermal, intermediate and intrinsic). Since we did not take the irradiation conditions into account in the simulations, the calculated diffusion coefficients are “out-of-pile”. Initially, the validity of the interatomic potential was performed by calculating the formation energies of interstitial, vacancy, Frankel and Schottky defects by using molecular statics (MS) simulations. Then, the migration energies of Xe/U/O based on considering various mechanisms were obtained by employing the Nudge Elastic Band (NEB) method. The results were generally in good agreement with those reported in the literature and the potential was validated. The simulations of UO2 containing different percentages of xenon (percentage of replacement of uranium atoms with xenon atoms) from 2 to 10 showed that with increasing the xenon percentage, the number of xenon clusters, cluster size, and radius of gyration considerably increases, and bubble nucleation was considerable in the case of 10%Xe. Then, by determining the slope of the mean square displacements, diffusion coefficients of xenon, uranium and oxygen were obtained. The results showed that they are both temperature-dependent and also dependent on the percentage of xenon (which was previously assumed to be independent of concentration). In the end, three temperature and xenon percentage dependent equations were proposed for the diffusion coefficients of xenon, uranium and oxygen. Finally, by applying a reduction factor (constant) to the diffusion coefficients, they were compared with experimental data available in the literature and significant satisfaction was obtained.

Impact of defect migration on electrical and dielectric properties in molten salt synthesized CaCu3Ti4O12 and customizing the properties by compositional engineering with Mg doping

For the CaCu3Ti4O12 (CCTO) synthesized using the molten salt method, sintered at 973–1273 K, the effect of extrinsic and intrinsic point defects on conductivity and dielectric behavior in the temperature range 300–450 K and frequency range 20 Hz–1 MHz was studied. The crystal structure, microstructures, and ionic distribution were studied by XRD, SEM, and EDS. The energy-dispersive X-ray spectroscopy, EDS, revealed qualitatively that CCTO has a minor deficiency of O and Ca and a slight excess of Ti and Cu. To maintain electrical neutrality, this results in the formation of V●●O, e/, h●, Ti3+, and Cu1+, etc., as complimentary point defects leading to an increased loss. Dielectric loss (tan δ ∼ 6–7 at ∼300 K/20 Hz) in a low-temperature regime was observed due to extrinsic defects migration with activation energy. EaσdcMott's ∼0.028 eV through Mott's variable range hopping. For temperatures ≥350 K, long-range conduction occurs due to an avalanche of intrinsic defects accumulated at sample electrode interface having activation energy, EaArrhenius's ∼0.177 eV, culminating in loss, tanδ ∼40 at ∼450 K/20 Hz). To minimize excessive loss, compositional engineering was done in CCTO by substituting Mg at the Ti site. Judicious choice of Mg doping, having high Schottky defect formation energy (ΔHf ≈ 7.7 eV) at Ti site (CCTMgO) resulted in almost temperature-independent dielectric constant, εr ≈ 103 and a significant reduction in the loss at ∼ 300–370 K in the frequency range 20 Hz - 1 kHz. Phase formation, structural refinements, etc., were further investigated in CCTMgO using the Rietveld refinement technique.

Density Functional Theory Modeling of Cation Diffusion in Bulk Lanthanum Manganite and Tetragonal Zirconia for Solid Oxide Fuel Cell Applications

Cation diffusion in perovskites such as La1-xSrxMnO3±δ (LSM) and fluorites such as Yttria Stabilized Zirconia (YSZ) plays a key role in controlling performance and long-term stability of solid oxide fuel cells (SOFCs) and of the corresponding electrode/electrolyte interfaces. For point defects based cation migration mechanisms, discrepancies between experimental studies and atomistic modeling results are generally observed when trying to identify the apparent activation energies of the cation diffusivities. In particular, computational modeling of the simple point defect migration generally overestimates the apparent activation energies by several eVs [1-3] relative to experimental results. Despite the existence of few proposed defect cluster pathways in the SOFC literature, the corresponding energetics based on atomistic level modeling is quite limited. As a result, outstanding questions related to the influence of the local bonding environment of the material upon the activation energies of cation diffusion and upon the energetics of the defect cluster carriers are still open. In this presentation, density functional theory (DFT) calculations of the cation diffusion involving mechanisms beyond the isolated point defect migration mechanism are discussed in connection to SOFC applications. The barriers of the cations, cation vacancy clusters, and cation impurities migration as well as their corresponding saddle point configurations in bulk LaMnO3±δ (LMO) and tetragonal ZrO2 are investigated. For cation vacancy migration in both LaMnO3 and tetragonal ZrO2 systems, it was revealed that significant reduction in the barriers of 1~3 eV takes place when there exists an additional nearest neighbor vacancy or a nearest neighbor vacancy pair bound to the original cation vacancy transport carrier (i.e., partially or fully bound Schottky defects). Such a significant reduction in the migration barriers of isolated cation vacancy migration due to the presence of an additional nearest neighbor vacancy or a vacancy pair invokes the need to examine other possible defect cluster diffusion mechanisms that exhibit attractive or weakly repulsive interactions between point defects. For example, our recent studies [1,3] suggest that the presence of a nearest neighbor B-site cation vacancy in bulk LMO decreases the electrostatic repulsion and steric constraints to the migrating A-site cations in the transition state configurations, leading to a more active A-site cation diffusion with apparent activation energies in good agreement with the experimental measurements. An analogous scenario is also observed for the Lanthanide impurity migration (via a cation vacancy related mechanism) in tetragonal ZrO2, where the migration barriers of partial and full Schottky bound defects are about 1.5 and 2.4 eV lower than those of simple cation vacancy migration (about 3 eV). By considering the attractive interactions of -1.0 and -1.4 eV for the partial (VZr-VO) and full (VO-VZr-VO) bound Schottky defects along with the separated Schottky defect formation energies of 4~5 eV (in a 288-atom ZrO2 supercells), it follows that cation impurity diffusion via partially and fully bonded Schottky defects could be more active than the cation vacancy migration of the separated Schottky defects in tetragonal ZrO2. Finally, the trends in the migration barriers of the defect cluster mechanisms among several types of cation impurities relevant for SOFC applications in LMO and tetragonal ZrO2 will be also discussed.

The diffusion of point defects in uranium mononitride: Combination of DFT and atomistic simulation with novel potential

The properties of point defects in uranium mononitride (UN) are studied by ab initio calculations and molecular dynamics simulations with a new interatomic potential. Density functional theory (DFT) calculations are used for fitting of the parameters of the angular-dependent interatomic potential, as well as for evaluation of the defects formation and migration energies. Molecular dynamics (MD) simulations are applied to analyse what migration mechanisms are activated at finite temperatures and to calculate diffusion coefficients of point defects. It is shown that the U antisite defects play an important role in the U-rich UN1−x. During migration the interstitial uranium is able to knock-out nitrogen atom, and this act leads to formation of U in antisite and N interstitial. This effect results in dependence of the diffusivity of U-interstitials on the concentration of defects in the N sublattice. Another peculiarity of UN is the large athermal concentration of U-vacancies in the N-rich UN1+x. This is due to close formation energies of nitrogen Frenkel pairs and Schottky defects. In addition, the applicability of the new potential for description of various phase transitions in UN is discussed.

Efficient Explicitly Correlated Many-Electron Perturbation Theory for Solids: Application to the Schottky Defect in MgO.

We introduce a novel and efficient explicitly correlated implementation of second-order perturbation theory for solids. The required three-electron integrals are computed directly using a plane wave basis set. We parametrize the employed correlation factors using results previously obtained for a finite uniform electron gas simulation cell. We demonstrate for a range of solids that basis set converged correlation energies, equilibrium volumes, and bulk moduli can be obtained efficiently in this theory using a few ten orbitals per atom. To stretch the capabilities of this novel method we compute the Schottky defect formation energy in MgO, studying systems with 54 atoms in the supercell. We verify the accuracy of the calculated formation energies using the more accurate coupled cluster singles and doubles theory. Furthermore, we discuss other potential applications for the derived and implemented expressions such as an occupied orbital only correlation energy functional.

States of the Schottky defect in uranium dioxide and other fluorite type crystals: Molecular dynamics study

Mass transfer processes in fluorite-type systems are determined by the diffusion of cations via vacancies from Schottky defects. To predict diffusion coefficients of cations and other parameters based on it the Lidiard and Matzke approximation of the point defects model, namely the assumption of isolated vacancies, is widely used. States of the Schottky defect were studied with the high-speed molecular dynamics method in a wide temperature range, with six different interaction potentials. Schottky vacancies were dynamically detected during the simulation. It is shown that contrary to the Lidiard and Matzke model, the Schottky cation vacancy is always associated with anion vacancies.The degree of the Schottky defect association depends on the temperature, at high temperatures near the cation vacancy two or more anion vacancies are located. It is shown that the calculated formation energy of Schottky defects in the form of the trivacancy (5.8–7.4)eV for all potentials are close to the experimental value (6–7)eV, in contrast to the formation energy of Schottky defects in the form of isolated vacancies that exceeds 10eV.Point defects model of the simulated system in the presence of an artificially created Schottky defect was constructed and compared with calculation results. According to our study the point defects model is applicable only at low temperatures less than half of the melting temperature.It is shown that the presence of anion vacancies near the cation vacancy reduces the migration energy of cations. However, for systems with several Schottky defects the vacancy clusterization and the formation of voids are observed. This leads to an increase of the diffusion activation energy due to the contribution of the energy that needs to separate the single cation vacancy from the void.

Ab initio energetics of charge compensating point defects: A case study on MgO

Density functional theory (DFT) calculations using supercells have proven quite successful in predicting defect properties. Although forming defect groups, for example, the Schottky pair VO••+VMg″ in MgO, are usually energetically favorable in many ionic systems, it is useful to obtain the defect energies of such systems without any defect-defect interactions as reference energies. However, determining non-interacting energies through multi-defect supercell calculations is challenging due to interactions between the defects that are difficult to quantify and can only be avoided by using very large supercells. One solution to this problem is to build an effective multi-defect cell through separate isolated defect calculations, with each defect in their own supercell. However, this isolated defect approach requires that the charge compensation be introduced through charged supercells, and a careful treatment of the cell energetics and electron reference energy is required. In this paper we assess the use of an isolated defect approach for modeling charge-compensating defect groups using the test case of MgO. The appropriate asymptotic condition for the electron reference energy shift is formulated and a method to meet the condition is given. We also demonstrate the strong coupling effect between residual strain energy and electrostatic energy in charged cells, demonstrating that these two effects cannot generally be separated and treated in isolation. The key steps in an approach that yields accurate defect group energies from the isolated defect calculations are presented. The non-interacting Schottky defect formation energy in MgO is determined to be 6.1eV through calculation of separated isolated charged cells containing VO•• and VMg″, respectively, while the binding energy between the charged defects VO•• and VMg″ is determined to be 1.5eV. This approach may also be of value for accurate modeling of general isolated defects. The formation energy of an isolated neutral Mg vacancy is found to be lower than that of a Schottky pair, suggesting that it is possible to have significant thermal cation defect formation in MgO without forming charge compensating Schottky pairs.

Charge states of point defects in uranium oxide calculated with a local hybrid functional for correlated electrons

The formation energies and charge states of point defects in uranium dioxide are calculated from first principles, using the local hybrid functional for correlated electrons, which offers a nice alternative to LDA$+$U calculations. The possible occurrence of multiple minima in such calculations is carefully taken into account. Point defects in UO${}_{2}$ are generally found to be charged with a \ensuremath{-}4 charge for uranium vacancies, \ensuremath{-}2 for oxygen interstitials, and a charge ranging from 0 to $+$2 for oxygen vacancies depending on the position of the Fermi level. The calculated formation energies of noninteracting oxygen Frenkel pairs and Schottky defects made of the association of charged defects are in very good agreement with experimental values. A Brouwer diagram based on a point-defect model for stoichiometry variations in UO${}_{2+x}$ is built. It fails to predict the dominant concentration of oxygen interstitials in the overstoichiometric oxide but predicts that oxygen vacancies are in a $+$1 charge state in the hypostoichiometric oxide. This charge state, which differs from the one assumed in the traditional (fully ionic) picture of UO${}_{2}$, is confirmed by the obtained variation of the deviation from stoichiometry with oxygen pressure, which is consistent with experiments.

Defects induced melting in alkali halides

In the present paper we study the pressure dependence of melting of NaCl and CsCl crystals. A formulation has been presented for the pressure dependence of melting temperature on the basis of the vacancy model using the expression for the pressure dependence of the volume of Schottky defects from the Roy–Roy equation of state. Values of pressure derivatives of melting temperature have been calculated at elevated pressures to determine the rate of change of melting temperature with increase in pressures using the data of vacancy formation energy and effective volume of Schottky defects. The vacancy model revised in the present study takes into account the variation of bulk modulus with pressure, whereas in the Ksiazek and Gorecki model, it was treated constant. Results for pressure derivative of melting temperature are calculated for the solids under study. The melting curves have also been obtained and found to compare well with results based on molecular dynamics simulation and experimental data reported in recent literature.

First-principles study of the intrinsic defects inPbFCl

First-principles pseudopotential calculations have been performed to investigate intrinsic defects including vacancies, interstitials, antisite defects, as well as Schottky and Frenkel defects in $\mathrm{PbFCl}$ crystals. For the isolated vacancies and interstitials, their formation energies are critically dependent on the atomic chemical potentials and electron Fermi energy. The charged defects for vacancies and interstitials are stable in a wide range of Fermi level, suggesting a strong ionization tendency. The present calculations also reveal that the formation energies of Schottky defects are much lower than those of Frenkel defects. Therefore it can be concluded that the Schottky defects are dominant in $\mathrm{PbFCl}$ crystals. Furthermore, Schottky defects $({V}_{\mathrm{Pb}}+{V}_{\mathrm{F}}+{V}_{\mathrm{Cl}})$ exhibit the lowest formation energy of $2.21\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. Defect transition energy levels can occur between different charged states. The anion interstitials and the cation vacancies form acceptorlike defects with transition energies of 1.05, 0.57, and $(0.47,0.9)\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ for ${\mathrm{F}}_{i}$, ${\mathrm{Cl}}_{i}$ and ${V}_{\mathrm{Pb}}$, respectively, while ${V}_{\mathrm{F}}$, ${V}_{\mathrm{Cl}}$, and ${\mathrm{Pb}}_{i}$ have high transition energies of more than $3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. In addition, the calculated profiles of densities of states for the defective supercells show that some extra states may appear in the band gap.

Schottky defect formation energy in MgO calculated by diffusion Monte Carlo

The energetics of point defects in oxide materials plays a major role in determining their high-temperature properties, but experimental measurements are difficult, and calculations based on density functional theory (DFT) are not necessarily reliable. We report quantum Monte Carlo calculations of the formation energy ${E}_{\mathrm{S}}$ of Schottky defects in MgO, which demonstrate the feasibility of using this approach to overcome the deficiencies of DFT. In order to investigate system-size errors, we also report DFT calculations of ${E}_{\mathrm{S}}$ on repeating cells of up to $\ensuremath{\sim}1000$ atoms, which indicate that QMC calculations on systems of only 54 atoms should yield high precision. The DFT calculations also provide the relaxed structures used in the variational and diffusion Monte Carlo calculations. For MgO, we find ${E}_{\mathrm{S}}$ to be in close agreement with results from DFT and from model interaction potentials, and consistent with the scattered experimental values. The prospects for applying the same approach to transition metal oxides such as FeO are indicated.

Strain effects on point defects and chain-oxygen order-disorder transition in 123 cuprate compounds

The energetics of Schottky defects in 123 cuprate superconductor series, $\rm REBa_2Cu_3O_7$ (where RE = lanthandies) and $\rm YAE_2Cu_3O_7$ (AE = alkali-earths), were found to have unusual relations if one considers only the volumetric strain. Our calculations reveal the effect of non-uniform changes of interatomic distances within the RE-123 structures, introduced by doping homovalent elements, on the Schottky defect formation energy. The energy of formation of Frenkel Pair defects, which is an elementary disordering event, in 123 compounds can be substantially altered under both stress and chemical doping. Scaling the oxygen-oxygen short-range repulsive parameter using the calculated formation energy of Frenkel pair defects, the transition temperature between orthorhombic and tetragonal phases is computed by quasi-chemical approximations (QCA). The theoretical results illustrate the same trend as the experimental measurements in that the larger the ionic radius of RE, the lower the orthorhombic/tetragonal phase transition temperature. This study provides strong evidence of the strain effects on order-disorder transition due to oxygens in the CuO chain sites.

Energetics of Native Defects in Al<sub>2</sub>O<sub>3</sub> and SiO<sub>2</sub>

Formation energies of various defects in Al2O3and SiO2are calculated by using the plane-wave pseudopotential method. Also, the formation energies of Schottky defects and Frenkel defects are evaluated on the basis of these calculations. It is shown that formation energies of these defects are higher in SiO2than in Al2O3. In other words, less defects are formed in SiO2than in Al2O3. It is also found that the principal defect is the cation Frenkel defect in Al2O3but the anion Frenkel defect in SiO2. These results agree with the experimental results that Al ions diffuse preferably in Al2O3but oxygen ions diffuse in SiO2at high temperatures.

Systematic hardness measurements on single crystals and polycrystalline blanks of cesium halides

Vickers and knoop hardness measurements were carried out on CsBr and CsI single crystals. Polycrystalline blanks of CsCl,CsBr and CsI were prepared by melting and characterized by X-ray diffraction. Vickers hardness measurements were carried out on these blanks. The hardness values were correlated with the lattice constant and the Schottky defect formation energy.

Segregation in Yttrium Aluminum Garnet: II, Theoretical Calculations

Calculations based on ionic space‐charge models of doped yttrium aluminum garnet (YAG) have been compared to experimental measurements of surface segregation in crystals of various compositions. The comparison allows limits for vacancy‐formation energies to be set. The range for anion:cation formation‐energy ratios has been established to be 0.20‐0.23, based on the reasonable assumptions that the formation energy of the yttrium ion is 75% of that of the aluminum ion and the Schottky defect formation energy is 4.2 eV. The model explains the experimental observation of calcium at the surface regardless of net acceptor excess or net donor excess. The relationship between vacancy‐formation energy and dopant excess has been used to construct segregation maps for YAG, which are useful in materials design strategies.

Schottky defect energy of some binary metal hydride and deuteride crystals

Schottky defect energies of some binary metal hydrides and deutendes are discussed with reference to conductivity measurements and the recent computer simulation calculations. An empirical relation with Debye temperature is found to yield values of Schottky defect formation energies of Li(H,D) systems in reasonable agreement with experiments. The relationship is also utilized to obtain the formation energies for Na(H,D) systems for which experimental values are not available in the literature.

Defect energetics in oxide materials from first principles.

We have made fully ab initio calculations on the energetics of defects in the oxide materials MgO and ${\mathrm{Li}}_{2}$O using a parallel computing methodology. The calculations, based on density functional and pseudopotential theory, yield results for the formation and migration energies of Schottky defects in MgO and Frenkel defects in ${\mathrm{Li}}_{2}$O which are in excellent accord with experiment. The work gives new insight into the defect-induced redistribution of valence electrons.

Solid solutions of alkali halide compounds. II. Physical properties and vacancy contributions.

The advantages of describing solid solutions at each composition x and temperature T with effective interactions are illustrated by considering two issues in the present work: (1) Calculation of certain physical properties of interest at each x and T; and (2) handling of single impurities in a solid solution (the formation energy of Schottky defects has been estimated). Our results suggest that significant contributions to the total entropy of a solid solution can arise from configurational entropy due to vacancies.

Rigid-ion models for NiO and CoO: comparison between experimental and calculated thermal expansion

The authors developed rigid-ion interatomic potentials for NiO and CoO and assess their reliability by calculating Schottky and Frenkel defect formation energies. Then they studied the variation with temperature in the lattice parameter at zero pressure via constant-temperature constant-pressure molecular dynamics simulations. The agreement obtained with experimental results is remarkably good.