Thermodynamics Of Point Defects Research Articles

Defect energy formalism for CALPHAD thermodynamics of dilute point defects: Theory

Defect energy formalism for CALPHAD thermodynamics of dilute point defects: Theory

A Single Model for the Thermodynamics and Kinetics of Metal Exsolution from Perovskite Oxides.

Exsolution has emerged as an outstanding route for producing oxide-supported metal nanoparticles. For ABO3-perovskite oxides, various late transition-metal cations can be substituted into the lattice under oxidizing conditions and exsolved as metal nanoparticles after reduction. A consistent and comprehensive description of the point-defect thermodynamics and kinetics of this phenomenon is lacking, however. Herein, supported by hybrid density-functional-theory calculations, we propose a single model that explains diverse experimental observations, such as why substituent transition-metal cations (but not host cations) exsolve from perovskite oxides upon reduction; why different substituent transition-metal cations exsolve under different conditions; why the metal nanoparticles are embedded in the surface; why exsolution occurs surprisingly rapidly at relatively low temperatures; and why the reincorporation of exsolved species involves far longer times and much higher temperatures. Our model's foundation is that the substituent transition-metal cations are reduced to neutral species within the perovskite lattice as the Fermi level is shifted upward within the bandgap upon sample reduction. The calculations also indicate unconventional influences of oxygen vacancies and A-site vacancies. Our model thus provides a fundamental basis for improving existing, and creating new, exsolution-generated catalysts.

Point-Defect Segregation and Space-Charge Potentials at the Σ5(310)[001] Grain Boundary in Ceria

The atomistic structure and point-defect thermodynamics of the model Σ5(310)[001] grain boundary in CeO2 were explored with atomistic simulations. An interface with a double-diamond-shaped structural repeat unit was found to have the lowest energy. Segregation energies were calculated for oxygen vacancies, electron polarons, gadolinium and scandium acceptor cations, and tantalum donor cations. These energies deviate strongly from their bulk values over the same length scale, thus indicating a structural grain-boundary width of approximately 1.5 nm. However, an analysis revealed no unambiguous correlation between segregation energies and local structural descriptors, such as interatomic distance or coordination number. From the segregation energies, the grain-boundary space-charge potential in Gouy–Chapman and restricted-equilibrium regimes was calculated as a function of temperature for dilute solutions of (i) oxygen vacancies and acceptor cations and (ii) electron polarons and donor cations. For the latter, the space-charge potential is predicted to change from negative to positive in the restricted-equilibrium regime. For the former, the calculation of the space-charge potential from atomistic segregation energies is shown to require the inclusion of the segregation energies for acceptor cations. Nevertheless, the space-charge potential in the restricted-equilibrium regime can be described well with an empirical model employing a single effective oxygen-vacancy segregation energy.

How to Adequately Describe Full Range Intercalation-A Two-Sided Approach.

One of the key challenges in battery research is to quantitatively describe the intercalation storage capacity as a function of the reversible cell voltage. The reason that such endeavors are not yet very successful, lies in the lack of an adequate charge carrier treatment. Using the most challenging example of nanocrystalline lithium iron phosphate, where the full range from FePO4 to LiFePO4 is accessible without miscibility gap, this study shows how a quantitative description of literature results can be achieved even for such a huge window. For this purpose, point-defect thermodynamics is applied and the problem is tackled from the two end-member sides including saturation effects. A first, rather heuristic treatment interpolates in-between using the safe thermodynamic criterion of local phase stability. Already this straightforward approach works very satisfactorily. In order to also gain mechanistic insight, interactions among and between ions and electrons have to be taken account of. This study shows how to implement them into the analysis.

Configurational entropy significantly influences point defect thermodynamics and diffusion in crystalline silicon

It has long been suggested that the familiar intrinsic point defects (vacancies and self-interstitials) encountered in crystals at low temperatures ($T$) transform into extended domains characterized by a missing or excess atom compared with the same-sized region in the perfect crystal so that such extended defects may be viewed as dropletlike regions of enhanced or diminished density. However, the implications of such a transformation, or whether it even occurs in crystalline Si, remain uncertain. To address this fundamental problem, we consider a comprehensive thermodynamic analysis of the thermodynamics of vacancy and self-interstitial formation over a broad $T$ range based on thermodynamic integration with a focus on entropic contributions. In cooled liquids, it is well known that the form of the intermolecular potential can greatly influence the configurational entropy ${S}_{c}$, and correspondingly, we analyze several empirical Si potentials to determine how the potential influences both the $T$ dependence of ${S}_{c}$ and the enthalpy and entropy of defect formation. We indeed find that the ${S}_{c}$ associated with point defects increases significantly upon heating, consistent with the existence of extended defects. Moreover, each type of defect species gives a significantly different contribution to ${S}_{c}$ at elevated $T$ and to a qualitive difference in the $T$ dependence of the entropy of defect formation in the extended defect regime. We discuss some potential consequences of these thermodynamic changes of defect formation on the $T$ dependence of diffusion in heated crystals.

Thermodynamics of Point Defects in Solids and Relation with the Bulk Properties: Recent Results

For several decades, the crucial question has arisen as to whether there exists any direct interconnection between the thermodynamic parameters of point defects in solids with the bulk properties of the solid under investigation. To answer this important question, an interrelation of the defect Gibbs energy gi in solids with bulk properties has been proposed almost half a century ago. Considering that gi corresponds to an isobaric and isothermal process, this interrelation states that, for different processes (defect formation, self-diffusion activation, and heterodiffusion), gi is proportional to the isothermal bulk modulus B and the mean volume per atom Ω, termed cBΩ model. Here, we review several challenging applications of this interrelation that appeared during the last decade (2011–2021), including high pressure diamond anvil measurements, high Tc superconductors, nuclear fuels, and materials for micro-electronics devices, applications of usefulness in Geophysics and Seismology, a problem of major technological interest, search for compositions of better target properties in Cu-Co-Si alloys via machine learning as well as two independent studies on the physical origin of this interrelation that has been further strengthened during the last few years.

Thermodynamics of carbon point defects in hexagonal boron nitride

We present a first-principles computational study of the thermodynamics of carbon defects in hexagonal boron nitride (hBN). The defects considered are carbon monomers, dimers, trimers, and larger carbon clusters, as well as complexes of carbon with vacancies, antisites, and substitutional oxygen. Our calculations show that monomers ($\text{C}_{\text{B}}$, $\text{C}_{\text{B}}$), dimers, trimers, and $\text{C}_{\text{N}}\text{O}_{\text{N}}$ pairs are the most prevalent species under most growth conditions. Compared to these defects, larger carbon clusters, as well as complexes of carbon with vacancies and antisites, occur at much smaller concentrations. Our results are discussed in view of the relevance of carbon defects in single-photon emission in hBN.

Thermodynamics and kinetics of intrinsic point defects in plutonium dioxides

Thermodynamic and kinetic properties of intrinsic point defects in plutonium dioxides (PuO2), including their formation, migration, and effects on oxygen (O) self-diffusion, are systematically studied using atomic simulations in this work. Coulomb interactions among charged defects and ions are found play a key factor in determining the energetics and structures of defects in PuO2, where point defects are energetically prefer to binding together forming defect pairs and less relaxation volumes are introduced by these bound point defects. Further calculations of defect migration properties reveal different migration mechanisms of O and Pu point defects, where O point defects prefer to migrate along [100] direction with an energy barrier of 0.31 eV by the vacancy mechanism, while Pu point defects prefer to migrate along [100] direction with an energy barrier of 2.57 eV by the interstitial mechanism. This confirms the cation sublattice is much more stable than the anion sublattice in the fluorite structure PuO2. Given the dominance of O defects in PuO2, their effects on O self-diffusion are investigated. Both vacancies and interstitials could significantly enhance O self-diffusivity. Under the Meyer-Neldel rule, activation energy and pre-exponential factors of O diffusion show a dependence of point defect concentrations. Specifically, internal stresses introduced by defects are found responsible for the different dependency of the activation energy on the vacancy and interstitial concentrations respectively. Finally, an empirical equation is derived to connect the point defect concentration, i.e. O/Pu ratio, and activation energy of O self-diffusion in PuO2.

Thermodynamics, kinetics and electronic properties of point defects in β-FeSi2

β-FeSi2, a semiconductor material made of two of the most earth-abundant elements, has important applications in thermoelectrics, photovoltaics and optoelectronics owing to its attractive properties such as suitable band gap and air stability over a wide temperature range. While point defects always play a vital role in semiconductor materials, only sporadic studies have been dedicated to the defects in β-FeSi2. Here, using first-principles calculations we systematically investigate the intrinsic point defects in β-FeSi2. Our results reveal that the formation energies of the intrinsic defects in β-FeSi2 are high enough to prevent them from forming in a significant concentration under thermal equilibrium growth conditions. As a possible kinetic process generating intrinsic defects, we study the α-to-β phase transition of FeSi2. We find that the phase transition is a slow process occurring on the time scale of an hour. Incomplete phase transition may lead to kinetically formed intrinsic defects. We further calculate the activation energies of the intrinsic defects and show that the experimentally observed conductivity of pure β-FeSi2 should be a result of unintentional doping. Possible extrinsic impurities that may lead to n-type and p-type conductivity and their activation energies are calculated, which are in good agreement with available experiments. Our results provide guidance for optimizing the doping strategy of β-FeSi2 for device applications.

(Invited) Measuring and Tailoring Chemo-Mechanical Coupling in Mixed Ionic and Electronic Conducting Oxides

Stoichiometric chemical expansion is the chemical strain accompanying small, non-integer changes in stoichiometry, in the absence of a phase change. For example, both oxygen loss (Δδ, eq. 1) and hydration (eq. 2) can cause chemical strain (εC) in mixed ionic and electronic conducting oxides. OO X → vO •• +1/2 O2 (g) + 2e' (1) vO •• + OO X + H2O → 2(OH)O • (2) The occurrence of chemical expansion in solid state ionic materials and devices can cause large chemical stresses, leading to mechanical failure. To prevent the formation of cracks and delamination, this chemical stress should be minimized. One way to achieve this goal involves minimization of stoichiometry gradients within materials, which is dependent on processing and operating conditions, their relationship to the material’s point defect thermodynamics, and the chemical diffusivity of the material. Another approach to minimize chemical stress, which we take, is to focus on an intrinsic material property, the coefficient of chemical expansion. For oxides that change their oxygen stoichiometry, the CCE can be defined as: CCE = εC / Δδ (3) while for oxides that change their hydration or proton content, the CCE can be defined as: CCE = εC / Δ[(OH)O •]. (4) We seek to identify and understand the factors that affect the CCE, so that its magnitude may be rationally tailored. Measurements have focused on a wide array of bulk perovskite-structured titanate, stannate, zirconate, cerate, and gallate compositions as well as on nanoparticles of fluorite-structured (Ce,Pr)O2- δ. Chemical strains have been measured macroscopically by dilatometry and at the crystal-structure level by in situ X-ray and neutron diffraction as a function of gas atmosphere. For nanoparticle measurements, a specially-designed cell was employed for simultaneous dilatometric and impedance measurements, leading to possibly the first assessment of nanopowder chemical expansion by this approach. Stoichiometry changes in corresponding conditions have been assessed by thermogravimetric analysis. In initial studies, factors including the size of the oxygen vacancy radius, charge localization, and temperature were identified as variables that could be tailored to modify the CCE. In more recent work, we have been focusing on the effect of nanostructuring, the impact of the choice of multivalent cation, and the role of crystal symmetry in perovskites. Regarding the latter variable, the results are consistent with a general pattern where deviation of the tolerance factor away from 1 (the ideal A- to B-site cation ratio giving cubic symmetry) correlates with lower CCE values. For nanopowders, despite well-documented differences in their point defect chemistry vs. bulk materials, no measurable deviation from bulk CCE behavior was observed. Implications for durable processing and use of these mixed conductors will be addressed.

Energetics of native defects, solute partitioning, and interfacial energy of Q precipitate in Al-Cu-Mg-Si alloys

The compound Al3Cu2Mg9Si7, is known as the Q-phase and forms as a thermodynamically stable precipitate during aging in the quaternary Al-Cu-Mg-Si system. We perform atomic-scale density functional theory (DFT) calculations of defect properties, solute partitioning, and interfacial stability of the Al3Cu2Mg9Si7 (Q) precipitate. We find: (i) simple native point defect (i.e., vacancies and anti-sites) thermodynamics can partially explain the experimentally observed off-stoichiometry, such as the observed variation of compositions, Al3+δCu2Mg9-δSi7 (Mg-deficient and Al-rich) in experiment. (ii) Calculated solute-partitioning energies of common solutes allow us to define general rules for site-preference in the Q-phase in terms of electronic structure and atomic radius. To validate our DFT predictions, we perform atom-probe tomography (APT) experiments for six-different elements (Zn, Ni, Mn, Ti, V, and Zr). The results show that the partitioning behavior of solutes Ni, Zn, and Mn are consistent with DFT predictions, but the transition elements (Ti, V, and Zr), which are anomalously slow diffusers in Al, partition to the Q-phase in constrast to DFT partitioning energies. (iii) For the low energy interface (112¯0)Q//(510)Al observed in needle shaped Q-precipitate, we survey various terminations and orientations and derive a low-energy interfacial structure. We find this low-energy interfacial model has Cu atoms nearest to the interface, which is in agreement with previous literature on Cu interfacial segregation at the Q′//α-Al interface. The computed interfacial energy (0.52 J/m2) and the corresponding structure will be useful input to future multi-scale modeling of microstructural evolution.

As-vacancy complex in Zn-diffused GaAs: Positron lifetime spectroscopy study

Positron annihilation lifetime spectroscopy is applied to study the point defects formed upon Zn diffusion in semi-insulating GaAs. The diffusion was performed by annealing the samples at 950°C for different times under different As vapor pressures. Deep and shallow positron traps were observed by lifetime spectroscopy. Zn acceptors are assumed to act as shallow traps. A complex containing the As vacancy and Zn atoms is suggested to be the deep positron traps. The observation of As vacancy complexes is elucidated by studying the influence of As vapor pressure applying the mass action law of thermodynamics of point defects.

Space charge storage in composites: thermodynamics.

Contacts of two phases, which allow for synergistic dissociative storage of a component in two space charge zones ("job-sharing storage"), are considered from the viewpoint of point defect thermodynamics. The respective relations between charge and component activity (chemical potential of the component) are derived, or - for more complex cases - the recipes for their derivation are given. These relations describe - according to different experimental conditions - the connection between mass storage and outer partial pressure or between mass storage and the cell voltage in a coulometric titration cell. They also reflect theoretical charge/discharge curves in battery cells when job-sharing storage predominates, and thus are also significant for supercapacitors. In addition to explicitly worked-out cases, it is pointed out how more general situations, such as simultaneous storage in bulk and in boundaries, specific adsorption or size effects, are to be treated.

The behavior of Fe3+/∑Fe during partial melting of spinel lherzolite

This study presents an internally consistent model for the behavior of Fe3+/∑Fe during partial melting of spinel lherzolite. The Fe3+/∑Fe ratio for olivine is calculated on the basis of point defect thermodynamics, and the oxidation states of iron in the other solid phases are calculated using Fe3+/Fe2+ distribution between olivine and orthopyroxene, clinopyroxene, or spinel. Conservation of mass is used to relate the Fe3+/Fe2+ ratio of partial melt to the concentrations of Fe3+ and Fe2+ in the initial and residual solids as a function of pressure, temperature, and oxygen fugacity. Results from isobaric batch melting calculations demonstrate that the Fe3+/∑Fe ratio of the partial melt decreases with increasing melt fraction. Conversely, the Fe3+/∑Fe ratio of the partial melt increases with increasing melt fraction during decompression batch melting. The relative oxygen fugacity of the upper mantle depends on both the oxidation state of iron and mantle potential temperature. Results from incremental decompression melting calculations in which 1% melt is produced for each 100MPa of decompression and then removed from the residual solid indicate that relative oxygen fugacity calculated from the oxidation state of iron in basaltic glass does not represent a unique value for the oceanic upper mantle but, rather, reflects conditions in the lower portion of the melting regime. A 100°C change in mantle potential temperature produces a change in relative oxygen fugacity of ∼0.8logunits, similar to the global range inferred from mid-ocean ridge basalt glasses. It is necessary, therefore, to compare relative oxygen fugacity calculated from basaltic glass with proxies for potential temperature before drawing conclusions on heterogeneity of the oxidation state of iron in the oceanic upper mantle. Results from model calculations also suggest that the sub-arc mantle is intrinsically more oxidizing than the oceanic mantle because it is cooler. The global correlation between Fe3+/∑Fe and H2O in basaltic glasses may, in part, reflect the solidus-lowering influence of the latter. Secular variations in relative oxygen fugacity of the upper mantle due to cooling of the Earth over geologic time do not appear to be large enough to have influenced the nature of volcanic outgassing significantly enough to have contributed to the Great Oxidation Event.

First-Principles Analysis of Defect Thermodynamics and Ion Transport in Inorganic SEI Compounds: LiF and NaF.

The formation mechanism and composition of the solid electrolyte interphase (SEI) in lithium ion batteries has been widely explored. However, relatively little is known about the function of the SEI as a transport medium. Such critical information is directly relevant to battery rate performance, power loss, and capacity fading. To partially bridge this gap in the case of inorganic SEI compounds, we report herein the results of first-principles calculations on the defect thermodynamics, the dominant diffusion carriers, and the diffusion pathways associated with crystalline LiF and NaF, which are stable components of the SEI in Li-ion and Na-ion batteries, respectively. The thermodynamics of common point defects are computed, and the dominant diffusion carriers are determined over a voltage range of 0-4 V, corresponding to conditions relevant to both anode and cathode SEI's. Our analyses reveal that for both compounds, vacancy defects are energetically more favorable, therefore form more readily than interstitials, due to the close-packed nature of the crystal structures. However, the vacancy concentrations are very small for the diffusion processes facilitated by defects. Ionic conductivities are calculated as a function of voltage, considering the diffusion carrier concentration and the diffusion barriers as determined by nudged elastic band calculations. These conductivities are more than ten orders of magnitude smaller in NaF than in LiF. As compared to the diffusivity of Li in other common inorganic SEI compounds, such as Li2CO3 and Li2O, the cation diffusivity in LiF and NaF is quite low, with at least three orders of magnitude lower ionic conductivities. The results quantify the extent to which fluorides pose rate limitations in Li and Na batteries.

First-principles study of codoping in lanthanum bromide

Co-doping of Ce-doped LaBr$_3$ with Ba, Ca, or Sr improves the energy resolution that can be achieved by radiation detectors based on these materials. Here, we present a mechanism that rationalizes of this enhancement that on the basis of first principles electronic structure calculations and point defect thermodynamics. It is shown that incorporation of Sr creates neutral $V_\text{Br}$-Sr$_\text{La}$ complexes that can temporarily trap electrons. As a result, Auger quenching of free carriers is reduced, allowing for a more linear, albeit slower, scintillation light yield response. Experimental Stokes shifts can be related to different Ce$_\text{La}$-Sr$_\text{La}$-$V_\text{Br}$ triple complex configurations. Co-doping with other alkaline as well as alkaline earth metals is considered as well. Alkaline elements are found to have extremely small solubilities on the order of 0.1 ppm and below at 1000 K. Among the alkaline earth metals the lighter dopant atoms prefer interstitial-like positions and create strong scattering centers, which has a detrimental impact on carrier mobilities. Only the heavier alkaline earth elements combine matching ionic radii with sufficiently high solubilities. This provides a rationale for the experimental finding that improved scintillator performance is exclusively achieved using Sr, Ca, or Ba. The present mechanism demonstrates that co-doping of wide gap materials can provide an efficient means for managing charge carrier populations under out-of-equilibrium conditions. In the present case dopants are introduced that manipulate not only the concentrations but the electronic properties of intrinsic defects without introducing additional gap levels. This leads to the availability of shallow electron traps that can temporarily localize charge carriers, effectively deactivating carrier-carrier recombination channels. The principles of this ... [continued]

Finite-size versus interface-proximity effects in thin-film epitaxialSrTiO3

The equilibrium electrical conductivity of epitaxial ${\text{SrTiO}}_{3}$ (STO) thin films was investigated as a function of temperature, $950\ensuremath{\le}$ $T$/K $\ensuremath{\le}1100$, and oxygen partial pressure, ${10}^{\ensuremath{-}23}\ensuremath{\le}$ $p$${\text{O}}_{2}$/bar $\ensuremath{\le}1$. Compared with single-crystal STO, nanoscale thin-film STO exhibited with decreasing film thickness an increasingly enhanced electronic conductivity under highly reducing conditions, with a corresponding decrease in the activation enthalpy of conduction. This implies substantial modification of STO's point-defect thermodynamics for nanoscale film thicknesses. We argue, however, against such a finite-size effect and for an interface-proximity effect. Indeed, assuming trapping of oxygen vacancies at the STO surface and concomitant depletion of oxygen vacancies---and accumulation of electrons---in an equilibrium surface space-charge layer, we are able to predict quantitatively the conductivity as a function of temperature, oxygen partial pressure, and film thickness. Particularly complex behavior is predicted for ultrathin films that are consumed entirely by space charge.

Perspectives on point defect thermodynamics

We review and discuss methods for including the role of point defects in calculations of the free energy, composition and phase stability of elements and compounds. Our principle aim is to explain and to reconcile, with examples, the perspectives on this problem that are often strikingly different between exponents of CALPHAD, and others working in the overlapping fields of physics, chemistry and materials science. Current methodologies described here include the compound energy formalism of CALPHAD, besides the rather different but related canonical and grand‐canonical formalisms. We show how the calculation of appropriate defect formation energies should be formulated, how they are included in the different formalisms and in turn how these yield equilibrium defect concentrations and their contribution to free energies and chemical potentials. Furthermore, we briefly review the current state‐of‐the‐art and challenges in determining point defect properties from first‐principles calculations as well as from experimental measurements.

On the Defect Structure and Transport Properties of Co3O4 Spinel Oxide

AbstractDefect structure and transport properties of Co3O4 cobalt oxide have been studied as a function of temperature and oxygen pressure, using marker and thermogravimetric techniques. It has been found that the oxidation of CoO to form Co3O4 follows parabolic kinetic, being thus diffusion controlled. Marker experiments have demonstrated that cation sublattice of Co3O4 oxide is predominantly disordered, but the defect structure is rather complex. At very low oxygen pressures, close to the dissociation pressure of the oxide, interstitial cations are the predominant point defects, while at high pressures cation vacancies predominate. This behavior is reflected in complex dependence of the parabolic rate constant of CoO oxidation on oxygen pressure. At low pressures, namely, parabolic rate constant of the reaction increases with oxygen pressure, reaching then virtually constant value in intermediate pressure range and increases again in highest pressure range. Theoretical analysis of kinetic results in terms of point defect thermodynamics confirmed the above conclusion, concerning complex defect structure of Co3±yO4 oxide. Finally, self-diffusion coefficient of cations in the discussed oxide has been calculated as a function of temperature and oxygen pressure from kinetic rate measurements, using Wagner's theory of metals oxidation.

The effect of native point defect thermodynamics on off-stoichiometry in β-Mg17Al12

The mechanical strength of Mg–Al–Zn alloys can be affected by a fine spatial dispersion of β-Mg17Al12 precipitates in the Mg matrix. In an effort to understand the phase stability and the unusual asymmetric off-stoichiometry observed in β-Mg17Al12, we have performed a series of first-principles density functional theory (DFT) calculations of bulk and defect properties of Mg17Al12. Specifically, we consider native point defects (i.e. vacancies and anti-sites) in all four sublattices of Mg17Al12, i.e. 2a, 8c, 24g (Mg) and 24g (Al). The T=0K static energies of defect Mg17Al12 supercells indicate that anti-site defects are energetically favored over vacancies, and the lowest anti-site defect formation energies are in 24g sites for both AlMg and MgAl. These Al-rich and Mg-rich anti-site defect formation energies are similar in magnitude, and thus do not explain the asymmetric off-stoichiometry of Mg17Al12. We also investigate the effect of atomic vibrations via DFT phonon calculations on native point defect free energies of Mg17Al12 and combine these entropic contributions with the point defect formation energies to evaluate the thermodynamics of off-stoichiometry in this phase. We find that the formation of the AlMg anti-site is not strongly stabilized by vibrational entropy. Thus, we conclude that the observed asymmetry in the off-stoichiometry of the β-Mg17Al12 phase in the Mg–Al phase diagram is not explained by simple native point defect thermodynamics, and must involve a more complicated defect formation mechanism, such as multi-defect clustering.