Fault Energy Calculations Research Articles

Thermodynamic modeling of Bi2Te3 in the defect energy formalism

Bi2Te3 is a promising thermoelectric material that is often touted as one of the best-performing low-temperature thermoelectric materials. As a result, it has been widely used commercially, both for clean energy generation and in cooling devices. Like many other thermoelectric materials, defects play a key role in the performance of Bi2Te3. As a result, numerous studies have attempted to experimentally and computationally map out the dominant defects in the phase, these include efforts to determine the dominant defect, estimate defect energies, and predict their concentration. The computer coupling of phase diagrams and thermochemistry (CALPHAD) is one of many tools under the auspices of the materials genome initiative (MGI) that enables the rapid design of new functional materials with improved properties. The Defect energy formalism (DEF) with a charged sublattice, an offshoot of the Compound energy formalism (CEF), provides a way to directly include first-principle defect energy calculations into CALPHAD descriptions of solid phases. The introduction of the charge sublattice enables the estimation of the free carrier concentrations in the phase. Here we apply the DEF to the Bi2Te3 system, emphasizing the robustness of the DEF in describing meaningful endmembers and the elimination of fitting parameters. Unlike previous assessments using the Wagner–Schottky defect model, we include the description of the charged defects in our assessment. The DEF with a charged sublattice provides a good prediction of the non-stoichiometry of the phase when compared with experimental data and also predicts a thermodynamic defect concentration at low temperature that is physically reasonable.

Binomial Spin Glass.

To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the binomial spin glass, a class of models where the couplings are sums of m identically distributed Bernoulli random variables. In the continuum limit m→∞, the class reduces to one with Gaussian couplings, while m=1 corresponds to the ±J spin glass. We demonstrate that for short-range Ising models on d-dimensional hypercubic lattices the ground-state entropy density for N spins is bounded from above by (sqrt[d/2m]+1/N)ln2, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental noncommutativity of the thermodynamic and continuous coupling limits that leads to the presence or absence of degeneracies depending on the precise way the limits are taken. Exact calculations of defect energies reveal a crossover length scale L^{*}(m)∼L^{κ} below which the binomial spin glass is indistinguishable from the Gaussian system. Since κ=-1/(2θ), where θ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite-temperature spin-glass phase.

Determining dilute-limit solvus boundaries in multi-component systems using defect energetics: Na in PbTe and PbS

Defect calculations have untapped potential to quantitatively determine thermodynamics of semiconductors. We present a methodology to determine solve boundaries in multicomponent systems using defect energy calculations.

Crystal structure and non-stoichiometry of cerium brannerite: Ce0.975Ti2O5.95

The crystal structure of cerium brannerite, Ce0.975Ti2O5.95, was determined from Rietveld analysis of high resolution neutron powder diffraction data (space group C2/m, a=9.8320(1)Å, b=3.75287(6)Å, c=6.8852(1)Å, β=119.230(1)°). The presence of both cerium and oxygen vacancies was confirmed from Rietveld analysis and defect energy calculations, combined with careful investigation of the phase diagram. Formation of oxygen vacancies at the O1 site, charge compensated by Ce vacancies, relieves considerable Coulombic repulsion and hence structural strain associated with short O1–O1 contacts forming the shared edge of neighbouring TiO6 polyhedra. Consideration of the bond valence sum of Ce4+ in Ce0.975Ti2O5.95 afforded a revised value of the r0=2.098±0.039Å for the reference bond length, based on analysis of 21 crystal structures mined from the ICSD.

Atomistic simulation of B-type–fluorite structural relationship and B-type large defect clusters in gadolinia-doped ceria

Because initial reports suggested that the nano-domains found in gadolinia-doped ceria, a material used amongst other things as an electrolyte in solid oxide fuel cells, could have a B-type rare earth oxide structure, atomistic models were developed to incorporate the B-type structure into the host fluorite structure. A one-to-one correspondence was discovered between the cation positions in monoclinic B-type structure and those in the cubic fluorite structure, which allows in principle the incorporated B-type structural element to retain its monoclinic symmetry. Defect energy calculations showed, however, that the incorporation of the gadolinia only becomes favorable when the initial B-type structure was partially converted into the C-type structure, thus suggesting that nano-domains are more likely to adopt a C-type arrangement. The calculations also indicated that oxygen diffusion plays a central role in the B- to C-type phase transition in gadolinia.

Density functional theory calculations of defect energies using supercells

Reliable calculations of defect properties may be obtained with density functional theory (DFT) using the supercell approximation. We systematically review the known sources of error and suggest how to perform calculations of defect properties in order to minimize errors. We argue that any analytical error-correction scheme relying on electrostatic considerations alone is not appropriate to derive reliable defect formation energies, certainly not for relaxed geometries. Instead we propose finite size scaling of the calculated defect formation energies, and compare the application of this with both fully converged and ‘Gamma’ (Γ) point only k-point integration. We provide a recipe for practical DFT calculations which will help to obtain reliable defect formation energies and demonstrate it using examples from III–V semiconductors.

Defect Chemistry, Surface Structures, and Lithium Insertion in Anatase TiO2

Atomistic simulation techniques are used to investigate the defect properties of anatase TiO(2) and Li(x)TiO(2) both in the bulk and at the surfaces. Interatomic potential parameters are derived that reproduce the lattice constants of anatase, and the energies of bulk defects and surface structures are calculated. Reduction of anatase involving interstitial Ti is found to be the most favorable defect reaction in the bulk, with a lower energy than either Frenkel or Schottky reactions. The binding energies of selected defect clusters are also presented: for the Ti(3+)-Li(+) defect cluster, the binding energy is found to be approximately 0.5 eV, suggesting that intercalated Li ions stabilize conduction band electrons. The Li ion migration path is found to run between octahedral sites, with an activation energy of 0.45-0.65 eV for mole fractions of lithium in Li(x)TiO(2) of x < or = 0.1. The calculated surface energies are used to predict the crystal morphology, which is found to be a truncated bipyramid in which only the (101) and (001) surfaces are expressed, in accord with the available microscopy data. Calculations of defect energies at the (101) surface suggest that single Ti(3+) defects and neutral Ti(3+)-Li(+) pairs tend to segregate to the surface.

Lanthanide transport in stabilized zirconias: interrelation between ionic radius and diffusion coefficient.

The diffusion of all stable lanthanides was measured both in calcia stabilized zirconia (CSZ) and in yttria stabilized zirconia (YSZ) in the temperature range between 1,286 and 1,600 degrees C. The lanthanide diffusion coefficients obtained increase with increasing ionic radius. The experimental activation enthalpy of diffusion is near 6 eV for CSZ and between 4 and 5 eV for YSZ and is not strongly affected by the type of lanthanide. The results were correlated with defect energy calculations of the lanthanide diffusion enthalpy using the Mott-Littleton approach. An association enthalpy of cation vacancies with oxygen vacancies of about 1 eV (96 kJ/mol) was deduced in the case of CSZ, while there is no association in the case of YSZ. Furthermore, the change in diffusion coefficients can be correlated to the interaction parameter for the interaction between the lanthanide oxide with zirconia: The higher the interaction parameter, the higher the lanthanide diffusion coefficient.

Looking into details of atom kinetics in intermetallics

Information on the atom jump processes during the establishment and destruction of long-range order can be gained from measurement of residual electrical resistivity (REST). By its nature this very sensitive method yields an integral signal, so theoretical efforts are mandatory to properly understand how the single atom jumps work together in generating it. As a valuable approach we report on Monte Carlo simulations of atom kinetics which process the individual jumps into macroscopic configurational changes. In order to create a more reliable quantitative basis for individual jump frequencies we propose ab initio calculations of defect energies and migration profiles, a method which has become increasingly accessible in recent years. When experimental investigation is enlightened by combining these two theoretical methods, we expect deep insight into the hierarchy of atomic movement during changes of the ordered state. First steps towards such an ideal program are discussed. An outlook is given how the theoretical methods described offer structural and kinetic information on various scales and in different systems.

Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions

We present results from Monte Carlo simulations of the one-dimensional Ising spin glass with power-law interactions at low temperature, using the parallel tempering Monte Carlo method. For a set of parameters where the long-range part of the interaction is relevant, we find evidence for large-scale dropletlike excitations with an energy that is independent of system size, consistent with replica symmetry breaking. We also perform zero-temperature defect energy calculations for a range of parameters and find a stiffness exponent for domain walls in reasonable but by no means perfect agreement with analytic predictions.

Structure, point defects and ion migration in LaInO 3

This article deals with the energetics of point defects, oxygen migration and proton incorporation in LaInO 3. We have used atomistic simulation methods to determine the most possible structure for this oxide and compare this result with Rietveld analysis of a sample. Defect energy calculations show that substitution of La by Sr is conducive to the production of oxygen vacancies. We have calculated the energy of oxygen migration between different oxygen sites that takes place through various paths. With the methods utilised, we also found that the incorporation of water into a LaInO 3 cell is favourable being this a necessary condition for the occurrence of proton conduction in this system. These results are compared to both experimental and previous calculations and their validity is discussed.

Diluted Josephson-junction arrays in a magnetic field: Phase coherence and vortex glass thresholds

The effects of random dilution of junctions on a two-dimensional Josephson-junction array in a magnetic field are considered. For rational values of the average flux quantum per plaquette $f$, the superconducting transition temperature vanishes, for increasing dilution, at a critical value $x_S(f)$, while the vortex ordering remains stable up to $x_{VL}>x_S$, much below the value $x_p$ corresponding to the geometric percolation threshold. For $ x_{VL}<x<x_p$, the array behaves as a zero-temperature vortex-glass. Numerical results for $f=1/2$ from defect energy calculations are presented which are consistent with this scenario.

Pseudopotential theory for self-diffusion and impurity diffusion: Defect-energy calculations for fcc metals.

Generalized formulations for the calculation of energies of multivacancy formation and migration in pure and impure lattices and a multi-interstitial formation in otherwise pure lattices have been obtained with use of a pseudopotential scheme. The energies caused by relaxations of the surrounding neighbors are incorporated in these processes. Formulas relating self-diffusion and impurity diffusion via a vacancy mechanism have been discussed. Calculations of the vacancy-formation energy, the migration energy, and hence the activation energy for self-diffusion and impurity diffusion and their changes have been done with use of standard model potentials. My calculations show that the effect of relaxations of the surrounding neighbors of the migrating atom at its activated position is significant while it has negligible effect in vacancy-formation process. The calculations also show that a Heine-Abarenkov model explains successfully all the defect parameters but the Aschroft model does not. A more sophisticated multiparameter model or the Heine-Abarenkov two-parameter model with approximate explanation of all the atomic properties is perhaps a better choice. It is also noted that careful numerical computation is essential for achieving satisfactory agreement with experiment.

Atomic mechanisms of mass transport in ceramic nuclear fuel materials

A critical review is given on self-diffusion, impurity diffusion, chemical diffusion, thermal diffusion and radiation-enhanced diffusion in the ceramic nuclear fuels, i.e. oxides, carbides, nitrides and carbonitrides of uranium and plutonium. Atomic transport in ThO2 is also discussed. Besides experimental results on diffusion, calculations of defect energies, diffusion mechanisms, and the relevant data on kinetic processes related to diffusion, ranging from creep, grain growth and sintering to leaching in water are also treated. Furthermore, recent results on damage-induced defects and their thermal behaviour using ion implantation of UO2 and UN with fission products and Rutherford backscattering/channelling techniques are reported and used to discuss the fission product behaviour in these fuels. Finally, suggestions are made for future work.

Brillouin-scattering and computer-simulation studies of fast-ion conductors: a review

Experimental Brillouin-scattering studies of fast-ion conductors at high temperatures and complementary computer simulations are reviewed. Several different compounds are considered: fluorites, cubic zirconias and rare-earth trifluorides. The behaviour of the acoustic-mode frequencies and elastic constants below and above the transition temperature to the state of high-ionic conductivity is examined. Static-lattice and defect-energy calculations which provide theoretical insights into the diffuse-phase transition, and provide explanations of the anharmonic and defect contributions to the observed reductions in elastic constants, are discussed.

Calculated and experimental defect parameters for silver halides

Our present detailed knowledge of the defect structures of silver chloride and silver bromide results from extensive experimental measurements the interpretation of which has, in recent years, been assisted by calculated values for some defect parameters. Experimental data show that cation Frenkel defects predominate in the materials and that the silver ions move by two kinds of interstitial mechanism as well as through vacancy motion. Both conductance and diffusion measurements indicate that the concentration of Frenkel defects rises more sharply in the 150 K temperature interval below the melting points of these crystals than would be predicted by a temperature-independent defect-formation energy, even when long-range interactions between defects are taken into account. Atomistic simulations techniques, based on the Mott–Littleton approximation, have provided values for defect formation energies and, in the quasi-harmonic approximation, the temperature-dependence of the effect formation energy has been quantified in good agreement with experiment. They have also confirmed the importance of the deformation of the silver ion during its migration, and have provided insights into the motion and clustering of dopant ions. Currently three-body forces are being introduced into the modelling of the silver halides in new interionic potentials and progress in the calculation of defect energies using these potentials is reported.

Defects in alkali-metal halide crystals

This paper discusses classical and recent calculations of defect energies for alkali-metal halide crystals. First, the methods of deriving crystal potentials are summarized and then the application of these in the computer simulation of the lattice, using the Mott–Littleton strategy, are described. The main emphasis is on an assessment of methods and potentials in the light of comparison with experimental data. There is a need for further entropy calculations and also for improvements to the quasiharmonic approximation. Some recent calculations for a lattice in mechanical equilibrium at all temperatures are described. Selected results are given for calculations of defect formation and migration energies, for the relaxation of substitutional impurities from lattice sites, for the formation of impurity–vacancy complexes and for the clustering of impurities to form trimers and Suzuki phases.

Quantum-mechanical calculations of defect energies

This paper describes the circumstances in which it it necessary to go beyond a semiclassical model in computing the energies of defects in polar solids. The emphasis will be on the ways in which a small cluster, treated quantum-mechanically, can be provided with an environment which represents both the long- and short-range potential of the surrounding crystal. Attention will be given to the additional level of self-consistency which is required between the cluster and its environment. The paper also reviews the application of quantum-chemical methods in generating parameters for the potentials used in semiclassical models.

An investigation of the quasiharmonic approximation in the calculation of defect energies for KC1

Abstract Crystal potentials are usually obtained through a combination of theoretical calculation and the fitting of crystal properties. Because of thermal expansion of the crystal, the equilibrium condition is not satisfied at all temperatures unless the crystal potential is made temperature-dependent. A quasiharmonic potential that satisfies the equilibrium condition at all temperatures is derived from KC1 and used to calculate temperature-dependent defect energies (at constant volume). Other thermodynamic defect parameters are then derived by combining these results with the experimental Gibbs energies. The temperature-dependence of these parameters is displayed.

Calculation of defect energies and volumes in some oxides

Abstract Recently derived potential models for the alkaline-earth oxides and four transition-metal oxides are used in calculations of the formation energies and volumes of Schottky and Frenkel defects. Activation energies for vacancy and interstitial migration, including both direct and interstitialcy mechanisms for the latter, are also calculated. Comparisons are made with previously reported parallel studies and with experimental values where these are available.