Self-interstitial Formation Energies Research Articles

Evolution of tensile strength and ductility of the non-equimolar TiVNbMoCr alloys with maximum entropy under the VEC constraint

In this work, a series of non-equimolar TiVNbMoCr alloys were designed according to the maximum entropy principle under the valence electron concentration (VEC) constraint. The effect of the physical parameters on tensile strength and elongation was experimentally investigated and comprehensively discussed. It was found that the lattice constant a and the average grain size d vary with relative changes in the content of each component, and exhibit negative correlations to VEC. Whereas, the self-interstitial formation energy Ei, the atomic radius mismatch δ, the shear modulus mismatch η and the shear modulus G exhibit positive correlations to VEC. These parameters have a significant impact on tensile properties of the alloys. Except for Burgers vector magnitude, the critical physical parameters in the Senkov and Labusch-Maresca-Curtin's strengthening models are positively correlated to the yield strength. The ductility of the solid-solution TiVNbMoCr alloys is related to VEC, a, d and the grain boundary characteristics in strain accommodation, which respectively belong to intrinsic, crystallography structural and microstructural types of parameters.

An atomic scale study on self-interstitial formation and diffusion behaviors in TiVTa and TiVTaNb concentrated solid-solution alloys

The formation and diffusion behaviors of self-interstitial defects in TiVTa and TiVTaNb concentrated solid-solution alloys (CSAs) are studied systematically by first-principles calculations and molecular dynamics simulations. It is found that [111] VV and TiV dumbbells are the most stable configurations in TiVTa-based CSAs, the formation energies of self-interstitials in TiVTaNb are higher than that in TiVTa CSAs. Local chemical environment can affect the formation of interstitials, where interstitials prefer V-rich environment. Larger local lattice distortion in TiVTa can lead to an irregular energy landscape and make it easier to form interstitials than in TiVTaNb. Compared with pure V, the diffusivity of interstitial is much lower than in TiVTa-based CSAs, which is related to the specific stability of interstitial configurations and lattice distortion. The preferential elemental diffusion is also observed which can be demonstrated by the interstitial formation preferences. Significant lattice distortion caused by atomic size mismatch in TiVTa-based CSAs can hinder the formation of interstitials, resulting in chemically biased diffusion and preferential diffusion pathways of interstitials.

Modified extended Finnis Sinclair potential for cubic crystal metal

Unlike the modified embedded atom method (MEAM), which requires consideration of the bond angles, the embedded atom method (EAM) and Finnis Sinclair (FS) potential have achieved a wide range of applications in molecular dynamics simulations of amorphous alloys and disordered systems. In this paper, the potential functions of 10 cubic crystalline metals are reconstructed using the extended Finnis Sinclair (EFS) potential to improve the FS potential in terms of defects, surface energy and thermal properties. First of all, for the BCC crystal structure and FCC crystal structure, the fitted physical quantity expressions suitable for EAM potential and FS potential are deduced, including pressure, bulk modulus, shear elastic modulus elastic constant, vacancy formation energy and surface energy. Then, based on these physical quantities and the Rose equation of state, the least-squares method is used to fit the new potential parameters. Finally, using the current potential, the cohesive energy, elastic constants (C11, C12 and C44), vacancy formation energy, self-interstitial formation energy, surface energy, coefficient of thermal expansion, specific heat capacity and the melting point of 10 kinds of cubic crystals are calculated by molecular dynamics simulation. The reliability of the current potential is verified by comparing with other potential functions, experimental values and first-principles calculations. The results show that the current potential can well reproduce the characteristics of physical quantities such as lattice constant, cohesive energy and elastic constants (C11, C12 and C44). Compared with the previous EFS potential, the current potential is closer to the Rose equation of state. The calculated surface energy is more accurate, and the performance of defects and thermal properties shows the reliability of the current potential function, and these improvements are due to the improvement of the fitting method.

Persistence of negative vacancy and self-interstitial formation energies in atomistic models of amorphous silicon

We use classical potential atomistic modeling to demonstrate that negative vacancy and self-interstitial formation energies are inherently present in amorphous silicon ($a$-Si). Our conclusions are based on comprehensive calculations of vacancy and self-interstitial formation energies in nine different models, ranging in size from 1000 to 2744 atoms and formed by quenching liquid Si at rates between ${10}^{10}$ and ${10}^{12}\phantom{\rule{0.16em}{0ex}}\mathrm{K}/\mathrm{s}$. We account for a range of possible reference energies corresponding to different choices of reservoirs for atoms inserted into or removed from our models. Moreover, we show that successive addition of either vacancies or self-interstitials does not deplete the number of negative formation energy sites. Instead, it further lowers defect formation energies. These findings lead us to conclude that $a$-Si is not a structure at or near a local enthalpy minimum nor can it be brought to a local enthalpy minimum through the introduction of constitutional defects.

Improving radiation-tolerance of bcc multi-principal element alloys by tailoring compositional heterogeneities

Molecular dynamic simulations were performed to investigate the displacement cascade process in refractory bcc complex concentrated alloys, including equi-atomic binary, ternary, and quaternary systems made of the elements Mo, Nb, Ta and W. Our simulation results show that more principal elements do not necessarily mean better radiation resistance. Instead, bcc binary MoNb and NbW CCAs, which have low binding energy of interstitial clusters, can also yield good resistance to the generation of radiation-induced defect clusters. At same time, MoNb also have low self-interstitial formation energy range, so there are more Frenkel Pairs than other bcc binary like MoTa and MoW although number of interstitials in clusters of MoNb is least. More importantly, we find the binding energy of interstitial clusters is highly tunable by changing elements combination and tailoring compositional heterogeneities (such as short-range ordering). Such strategies may pave the way for new design concepts of radiation-tolerant alloys.

An atomistic study of defect energetics and diffusion with respect to composition and temperature in γU and γU-Mo alloys

• Vacancy and self-interstitial formation energies were calculated in γ U and γ U-Mo. • Self-diffusion and interdiffusion coefficients in γ U- x Mo ( x = 7 , 10, 12 wt.%) were calculated. • Results can be used as input parameters in mesoscale and engineering scale fuel modeling. Uranium-molybdenum (U-Mo) alloys are promising candidates for high-performance research and test reactors, as well as fast reactors. The metastable γ phase, which shows acceptable irradiation performance, is retained by alloying U with Mo with specific quenching conditions. Point defects contribute to the atomic diffusion process, defect clustering, creep, irradiation hardening, and swelling of nuclear fuels, all of which play a role in fuel performance. In this work, properties of point defects in γ U and γ U- x Mo ( x = 7, 10, 12 wt. % ) were investigated. Vacancy and self-interstitial formation energies in γ U and γ U- x Mo were calculated with molecular dynamics (MD) simulations using an embedded atom method interatomic potential for the U-Mo system. Formation energies of point defects were calculated in the temperature range between 400 K and 1200 K. The vacancy formation energy was higher than the self-interstitial formation energy in both γ U and γ U- x Mo in the evaluated temperature range, which supports the previous results obtained via first-principles calculations and MD simulations. In γ U- x Mo, the vacancy formation energy decreased with increasing Mo content, whereas the self-interstitial formation energy increased with increasing Mo content in the temperature range of 400 K to 1200 K. The self-diffusion and interdiffusion coefficients were also determined in γ U- x Mo as a function of temperature. Diffusion of U and Mo atoms in γ U- x Mo were negligible below 800 K. The self-diffusion and interdiffusion coefficients decreased with increasing Mo concentration, which qualitatively agreed with the previous experimental observations. Point defect formation energies, self-diffusion coefficients, and interdiffusion coefficients in γ U- x Mo calculated in the present work can be used as input parameters in mesoscale and engineering scale fuel performance modeling.

Beyond two-center tight binding: Models for Mg and Zr

We describe a systematic approach to building ab initio tight-binding models and apply this to hexagonal metals Mg and Zr. Our models contain three approximations to plane-wave density functional theory (DFT): (i) we use a small basis set, (ii) we approximate self-consistency, and (iii) we approximate many-center exchange and correlation effects. We test a range of approximations for many-center exchange-correlation and self-consistency to gauge the accuracy of each in isolation. This systematic approach also allows us to combine multiple approximations in the optimal manner for our final tight-binding models. Furthermore, the breakdown of errors into those from individual approximations is expected to be a useful guide for which approximations to include in other tight-binding models. We attempt to correct any remaining errors in our models by fitting a pair potential. Our final tight-binding model for Mg shows excellent agreement with plane-wave results for a wide range of properties (e.g., errors below 10% for self-interstitial formation energies and below 3% for equilibrium volumes) and is expected to be highly transferable due to the minimal amount of fitting. Calculations with our Zr model also show good agreement with plane-wave results (e.g., errors below 2% for equilibrium volumes) except for properties where self-consistency is important, such as self-interstitial formation energies. However, for these properties we are able to generate a tight-binding model which shows excellent agreement with non-self-consistent DFT with a small basis set (i.e., many-center effects are captured accurately by our approximations). As we understand the source of remaining errors in our Zr model we are able to outline the methods required to build upon it to describe the remaining properties with greater accuracy.

Energetics of intrinsic point defects in aluminium via orbital-free density functional theory

The formation and migration energies for various point defects, including vacancies and self-interstitials, in aluminium are systematically reinvestigated using the supercell approximation in the framework of orbital-free density functional theory. In particular, the finite-size effects and the accuracy of various kinetic energy density functionals are examined. It is demonstrated that as the supercell size increases, the finite-size errors asymptotically decrease as . Notably, the formation energies of self-interstitials converge much more slowly than that of vacancies. With carefully chosen kinetic energy density functionals, the calculated results agree quite well with the available experimental data and those obtained through Kohn–Sham density functional theory, which has an exact kinetic term.

Effect of Strain Field on Threshold Displacement Energy of Tungsten Studied by Molecular Dynamics Simulation**Supported by the National Natural Science Foundation of China under Grant Nos 11375242, 91026002, 91426301 and 11405231, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No XDA03010301. WS and RJK acknowledge the support of the Office of Fusion Energy Sciences, U.S.

The influence of strain field on defect formation energy and threshold displacement energy (Ed) in body-centered cubic tungsten (W) is studied with molecular dynamics simulation. Two different W potentials (Fikar and Juslin) are compared and the results indicate that the connection distance and selected function linking the short-range and long-range portions of the potentials affect the threshold displacement energy and its direction-specific values. The minimum Ed direction calculated with the Fikar potential is 〈100〉 and with the Juslin potential is 〈111〉. Nevertheless, the most stable self-interstitial configuration is found to be a 〈111〉-crowdion for both the potentials. This stable configuration does not change with the applied strain. Varying the strain from compression to tension increases the vacancy formation energy while decreases the self-interstitial formation energy. The formation energy of a self-interstitial changes more significantly than a vacancy such that Ed decreases with the applied hydrostatic strain from compression to tension.

First-principles study of vacancy, interstitial, noble gas atom interstitial and vacancy clusters in bcc-W

Based on first-principles calculations, the vacancy and self-interstitial formation energy in bcc-W are 3.19eV and 9.97eV. Binding energy between the dumbbell interstitials can be up to 2.29eV. Binding energy for the first and second nearest neighbor vacancy pair are −0.12eV and −0.41eV. The migration barrier of vacancy, He, Ne and Ar interstitial are 1.70eV, 0.07eV, 0.15eV and 0.25eV. The migration barrier of self-interstitial along 〈111〉 is 0.05eV. The so-called rotation barrier of self-interstitial is 0.35eV. The formation energy of He, Ne, Ar substitutional and He, Ne, Ar tetrahedral interstitial are 4.85eV, 6.42eV, 9.54eV and 6.23eV, 10.40eV, 15.10eV, respectively. Binding energy for di-gas atom (He, Ne and Ar) interstitial are 0.95eV, 2.28eV and 1.70eV. The binding energy of noble gas atom interstitial and vacancy cluster are obtained and can be used as an input to build a molecular dynamics (MD) W-Ne potential. Then molecular dynamics (MD) simulations can be used to investigate the mechanism and temperature dependence of the surface modification of plasma-facing tungsten in the application of future fusion reactors in the following investigations.

First-principles study of point defects in an fcc Fe-10Ni-20Cr model alloy

The influence of the local environment on vacancy and self-interstitial formation energies has been investigated in a face-centered-cubic (fcc) Fe-10Ni-20Cr model alloy by analyzing an extensive set of first-principle calculations based on density functional theory. Chemical disorder has been considered by designing special quasirandom structures and four different collinear magnetic structures have been investigated in order to determine a relevant reference state to perform point defect calculations at 0 K. Two different convergence methods have also been used to characterize the importance of the method on the results. Although our fcc Fe-10Ni-20Cr would be better represented in terms of applications by the paramagnetic state, we found that the antiferromagnetic single-layer magnetic structure was the most stable at 0 K and we chose it as a reference state to determine the point defect properties. Point defects have been introduced in this reference state, i.e., vacancies and Fe-Fe, Fe-Ni, Fe-Cr, Cr-Cr, Ni-Ni, and Ni-Cr dumbbell interstitials oriented either parallel or perpendicular to the single layer antiferromagnetic planes. Each point defect studied was introduced at different lattice sites to consider a sufficient variety of local environments and analyze its influence on the formation energy values. We have estimated the point defect formation energies with linear regressions using variables which describe the local environment surrounding the point defects. The number and the position of Ni and Cr first nearest neighbors to the point defects were found to drive the evolution of the formation energies. In particular, Ni is found to decrease and Cr to increase the vacancy formation energy of the model alloy, while the opposite trends are found for the dumbbell interstitials. This study suggested that, to a first approximation, the first nearest atoms to point defects can provide reliable estimates of point defect formation energies.

Diffusion quantum Monte Carlo study of the equation of state and point defects in aluminum

The many-body diffusion quantum Monte Carlo (DMC) method with twist-averaged boundary conditions is used to calculate the ground-state equation of state and the energetics of point defects in fcc aluminum using supercells up to 1331 atoms. The DMC equilibrium lattice constant differs from experiment by 0.008 A, or 0.2%, while the cohesive energy using DMC with backflow wave functions with improved nodal surfaces differs by 27 meV. DMC-calculated defect formation and migration energies agree with available experimental data, except for the nearest-neighbor divacancy, which is found to be energetically unstable, in agreement with previous density functional theory (DFT) calculations. DMC and DFT calculations of vacancy defects are in reasonably close agreement. Self-interstitial formation energies have larger differences between DMC and DFT, of up to 0.33eV, at the tetrahedral site. We also computed formation energies of helium interstitial defects where energies differed by up to 0.34eV, also at the tetrahedral site. The close agreement with available experiments demonstrates that DMC can be used as a predictive method to obtain benchmark energetics of defects in metals.

Back Cover: Accuracy of quantum Monte Carlo methods for point defects in solids (Phys. Status Solidi B 2/2011)

Quantum Monte Carlo approaches such as the diffusion Monte Carlo (DMC) method are among the most accurate many-body methods for extended systems. Their scaling makes them well suited for defect calculations in solids. Parker, Wilkins and Hennig (pp. 267–274) review the various approximations needed for DMC calculations of solids and the results of previous DMC calculations for point defects. They present estimates of how approximations affect the accuracy of calculations for self-interstitial formation energies in silicon. As shown on the cover image, including the corrections for the approximations yields DMC values for the X, T and H interstitial defects in silicon on par with the GW and hybrid-functional DFT (HSE) calculations. This Feature Article is part af the article series “Advanced Calculations for Defects in Solids – Electronic Structure Methods” (guest edited by A. Alkauskas, P. Deák, J. Neugebauer, A. Pasquarello, and C. G. Van de Walle) which is being published in the issues of physica status solidi (b) during 2011.

Energetic driving force for preferential binding of self-interstitial atoms to Fe grain boundaries over vacancies

Molecular dynamics simulations of 50 Fe grain boundaries were used to understand their interaction with vacancies and self-interstitial atoms at all atomic positions within 20 Angstroms of the boundary, which is important for designing radiation-resistant polycrystalline materials. Site-to-site variation within the boundary of both vacancy and self-interstitial formation energies is substantial, with the majority of sites having lower formation energies than in the bulk. Comparing the vacancy and self-interstitial atom binding energies for each site shows that there is an energetic driving force for interstitials to preferentially bind to grain boundary sites over vacancies. Furthermore, these results provide a valuable dataset for quantifying uncertainty bounds for various grain boundary types at the nanoscale, which can be propagated to higher scale simulations of microstructure evolution.

Ab initio study of self-interstitials in hcp-Zr

The formation energies of self-interstitials in hcp-zirconium have been obtained from first principles electronic structure calculations in the framework of the density functional theory in the local density approximation using a planewave pseudopotential approach with supercells containing 37 and 97 atoms. For the largest cell, five configurations are found to be nearly degenerate in energy. The most stable ones are the octahedral (O), split 〈0 0 0 1〉 dumbbell (S), and basal octahedral (BO) configurations, with formation energies E f i =2.8 eV, and the basal split (BS) with E f i =2.9 eV. The crowdion (C) configuration is slightly less stable with E f i =3.1 eV, while the basal crowdion (BC) is unstable and decays to BO. The effect of the exchange correlation functional is investigated by repeating the same calculations for 37 atom supercells within the generalized gradient approximation (GGA): the formation energies increase by almost 0.2 eV for the basal configurations and by more than 0.3 eV for the non-basal ones. From the analysis of the present results, the most stable configuration in hcp-Zr at low temperature is predicted to be BO, but BS and to a lesser extend O and S cannot be excluded.

Strain field due to self-interstitial impurity in Ni

The embedded-atom method have been applied to study the strain field produced by the self-interstitial impurity at the octahedral site in Ni. The calculation have been carried out consistently on the basis of discrete lattice theory, using Kanzaki method. The atomic force constants are evaluated using Wills and Harrison interatomic potential. The dynamical matrix and external force are evaluated considering the interaction up to first nearest neighbors. The atomic displacements are tabulated up to 20NN’s. These displacements are of oscillatory nature and of decreasing magnitude with NN’s distance. The physical properties such as self-interstitial formation energy and volume change calculated using atomic displacements are in accordance with the earlier studies.

Simulation of Phosphorus Diffusion Profiles with Different Phosphorus Surface Concentration at the Same Diffusion Temperature in Silicon

Four diffusion models based on the pair diffusion models of vacancy and interstitial mechanisms are adopted for the simulation of six experimental profiles of P surface concentration, CP+ s, from 3×1020 to 2.5×1018 cm-3 at 900°C. Values of fitting parameters are determined to simulate the profiles of CP+ s=3×1020 and 2.5×1018 cm-3 well. Changing only CP+ s, other profiles are simulated. At high CP+ s, a large amount of excess I is generated. For the simulation of CP+ s=3×1020 cm-3, it is necessary to control excess I. For the simulation of CP+ s<3×1020 cm-3, the effect of a main parameter upon the control of excess I should decrease with decreasing CP+ s. Based on this, the decrease in quasi self-interstitial formation energy or the ratio of the diffusion coefficient of a negatively charged P–V pair to that of a neutral one is suitable for the main parameter. To decide the suitable main parameter, it is necessary to obtain a more accurate experimental profile, paying attention to whether or not the profile has a plateau.

Effective Diffusion Coefficient and Controlling Process of P Diffusion in Si Based on the Pair Diffusion Models of Vacancy and Interstitial Mechanisms

Based on the pair diffusion models of vacancy and interstitial (V and I) mechanisms, the V and I components of effective P diffusion coefficient, DP+,Veff and DP+,Ieff, and the controlling process of P diffusion in Si are obtained. Assuming that the I mechanism is dominant, not only the I- concentration, CI- , but also its gradient, d CI- /d λ, is effective on DP+,Ieff at high CP+ . DP+,Ieff is large at d CI- /d λ<0 and small at d CI- /d λ>0. P+ and I- are generated by the dissociation of P–I pair. When excess I- thus generated is removed, d CI- /d λ<0 is obtained. d CI- /d λ<0 is also obtained by the decrease in quasi self-interstitial formation energy. Several diffusion models simulate the P diffusion profile well under an inert atmosphere. Applying the controlling process to them, the reason why they simulate the P profile well is investigated. Because all of them simulate the P profile well, it is difficult to conclude which model is correct. It is suggested that it is possible to conclude which model is correct from the P profile under oxidation at CP+ s >1×1020 cm-3 (s: surface).

Ab initio pseudopotential calculations of point defects and boron impurity in silicon

Ab initio pseudopotential calculations were performed to obtain the formation energies of point defects in crystalline Si. A supercell of 64 Si atoms was used for the calculations. For the self-interstitial defect, a weak bond is formed between two defect Si atoms. However, no evidence of extra bond formation is found in the vacancy study. The formation energies of both defects are comparable, i.e. 3.7 eV for the vacancy and 3.25 eV for the self-interstitials, respectively. Our self-interstitial formation energy agrees well with other theoretical results, while the vacancy formation energy is much smaller. We also present the results of boron substitutional impurities in Si where the charge density plot indicates an ionic character of the BSi bonding.

Ab initio pseudopotential calculations of point defects and boron impurity in silicon

Ab initio pseudopotential calculations were performed to obtain the formation energies of point defects in crystalline Si. A supercell of 64 Si atoms was used for the calculations. For the self-interstitial defect, a weak bond is formed between two defect Si atoms. However, no evidence of extra bond formation is found in the vacancy study. The formation energies of both defects are comparable, i.e. 3.7 eV for the vacancy and 3.25 eV for the self-interstitials, respectively. Our self-interstitial formation energy agrees well with other theoretical results, while the vacancy formation energy is much smaller. We also present the results of boron substitutional impurities in Si where the charge density plot indicates an ionic character of the BSi bonding.