First-Principles Calculations Of Vacancy Formation Energy
First-Principles Calculations ofVacancy Formation EnergyTao Wang, Long-Qing Chen andZi-Kui LiuApril 01, 20041
Phases Research LabThe Pennsylvania State UniversityLed by Professor Zi-Kui LiuComputational Thermodynamics,Phase Transformations, and Materials Design ofAluminum Alloys :Magnesium Alloys :Nickel Alloys :Platinum Alloys :Metal Oxide :William Golumbfskie, Chao Jiang, Dongwon Shin, Shengjun ZhangDr. Edwin Garcia, Dr. Ravi ChinnapanYu Zhong, Dr. Raymundo ArroyaveTao Wang, Dr. Jinzhi Zhu, Dr. Yi Wang, Dr. Shihuai ZhouSara PrinsMei Yang, Dr. Zhi-Jie Liu,Through First-Principles Calculations,CALPHAD Modeling, and Experiments
Why Vacancy? Vacancy is the simplest and commonestlattice defect;Vacancy migration is the dominantmechanism behind atomic transport;Vacancy plays very important role forsurface morphology *; * K. F. McCarty, et al., Nature, 20013
Experimental Measurement Experimental methods regarding themeasurement of vacancies are difficult;Many technical questions anduncertainties exist for thoseexperiments;Different experiments result in a largerange of values.4
Theoretical Calculation Calculate the total energy of a crystalwith N atoms, E0;Calculate the total energy of a crystalwith N-1 atoms and one defect, E1;Defect formation energy:N 1 E f E1 E0N5
First-Principles Calculations FP calculations always underestimatevacancy formation energies.LDA is better than GGA for the vacancyformation energy calculation?Unrelaxed calculation is better thanrelaxed calculation for the vacancyformation energy calculation?6
FP Calculation on Surface EnergyObservation: FP calculations alwaysunderestimate surface energies.Reason: Both the LDA and GGA haveintrinsic errors at surface because themany-body interaction is approximated.7
Correction for CalculatedSurface Energy The intrinsic errors were estimated by jellium model.The correction is a function of electron density.LDA has smaller intrinsic error than GGA.T. R. Mattsson and A. E. Mattson, Physical Review B, 20028
Charge Density near VacancyK. M. Carling, et al., Physical Review B, 20039
Correction on VacancyFormation Energy Calculation As the first approximation, treat the vacancyas an interior surface: E c 4πR 2 σwhere R is the radius of the “hole”. The corrected vacancy formation energy: E corrected E f EcfT. R. Mattsson and A. E. Mattsson, Physical Review B, 200210
Determination of R The radius, R, can be estimated by fitting theelectron density profile of jellium model to the that ofDFT’s.In fcc Al, R is about 1.2 A.Because the above calculation is very complex, thedifferent vacancy formation volumes in differentelements are usually ignored.K. Carling, et al., Physical Review Letters, 200011
Our Treatment on R1) Vacancy formation volumeN 1V0 V V1 N2) Case I: consider a sphere of that volume,upper boundRu 33 V4π12
Our Treatment on R (cont’d)3) Case II: consider a sphere embedded intothat volume, low boundRl 0.5 * 3 V4) Using the V and R of Al and assuming Ras a weighted average of Rl and Ru, weobtain a relation between R and Rl and Ru.13
Relation between Rand Rl and Ru V depends on size of supercell.For 32 and 54 atom supercells:R 0.82 Rl 0.18Ru For 108 and 128 atom supercells:R 0.71Rl 0.29 Ru Assume the same relations for different elements.14
R of Some Elements V ( A 3 )Rl ( A)Ru ( A)R ( A)Al (32)12.231.151.431.20Al (108)11.371.121.401.20Ni (32)7.120.961.191.00Ni (108)6.790.941.171.01Mo (54)9.781.071.331.12Mo (128)8.991.041.291.11Ta (54)10.021.081.341.13Ta (128)9.811.071.331.1415
Vacancy Formation Energyof Some Fcc PhasesElementCellSize, NE0Al (fcc)32-118.0108-398.5Exp.Ni (fcc)E1-113.8-394.3 E f E c E correctedf0.5680.1510.7190.5110.6620.64, 0.71, 0.66, 0.62, 0.69, 0.67, 0.66, 0.66, .491.88Exp.1.72, 1.73, 1.73, 1.55, 1.54Energies in eV.16
Vacancy Formation Energyof Some Bcc PhasesElementCellSize, bcc) E f E c E correctedf2.580.282.862.690.272.962.24, 3.0, 3.0, 2.9, 0.203.11Exp.2.54, 2.8Energies in eV.17
Summary Some corrections should be applied to FPcalculation on vacancy formation energy;Mattsson’s correction can improve thecalculated vacancy formation energy in Al, Ni,Mo, but not for Ta.18
Correction on Vacancy Formation Energy Calculation As the first approximation, treat the vacancy as an interior surface: where R is the radius of the "hole". The corrected vacancy formation energy: E 4πR2 σ c f c corrected Ef E E T. R. Mattsson and A. E. Mattsson, Physical Review B, 2002
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Table 9. Net Rental Vacancy Rate by Borough, Asking Rent, and Type of Housing 27 Figure 7. Net Rental Vacancy Rate by Asking Rent 28 Figure 8. Net Rental Vacancy Rate in Condos and Coops by Borough 29 Figure 9. Household Incomes Needed to Afford Vacant Units Available for Rent 31 Figure 10. Household Income of Renters 31 Table 10.
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 4 Self-Diffusion & Vacancy Diffusion Diffusion of Vacancy vs. Substitutional Atoms Continue from p. 7 2 Therefore, Diffusion coefficient of vacancy vs. substitutional atom For self-diffusion 2 The relationship between jump frequency is Since the jump distance is the same
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