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Material Property Predictions with the Vienna Ab Initio Simulation Package (VASP)

Material Property Predictions with the Vienna Ab Initio Simulation Package (VASP). L.G. Hector, Jr. GM Technical Fellow Center for Computational Sciences The University of Kentucky Lexington, KY November 10, 2010. Material Property Predictions with Density Functional Theory: Outline.

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Material Property Predictions with the Vienna Ab Initio Simulation Package (VASP)

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  1. Material Property Predictions with the Vienna Ab Initio Simulation Package (VASP) L.G. Hector, Jr. GM Technical Fellow Center for Computational Sciences The University of Kentucky Lexington, KY November 10, 2010

  2. Material Property Predictions with Density Functional Theory: Outline • Density Functional Theory (DFT) and the VASP Code • Vision for Multiscale Materials Design • Elasticity Tensor Components, Cij • Elastic Instability, Lattice Instability, Soft Modes • Thermodynamics: Structure Prediction/ Discrimination • Thermophysical Properties: Cp(T) and aij(T) • Diffusion • Concluding Remarks

  3. Main Computational Engine for Quantum Mechanical Simulations of Solids: VASP (Vienna Ab Initio Simulation Package) – A Plane Wave Density Functional Code

  4. Brief History of Density Functional Theory (DFT) • First suggested by Fermi in 1929 that the total Energy of an electronic system can be determined by the electron density. • Two seminal papers in 1960’s gave the exact relationship between electron density and energy. • Hohenberg-Kohn (1964); Kohn-Sham (1965) • The ground state energy of a system is a unique functional of the electron density. • The ground state energy can be obtained variationally, i.e. the density that minimizes the ground state energy is the exact ground state density. • Walter Kohn’s initial interest in DFT grew out of his interests in metal alloys while at CMU (2001 conversation with LGH) • Kohn was awarded the 1998 Nobel Prize in chemistry for DFT • DFT software, such as VASP, became commercially available (~1996). • Number of publications with “density functional theory” in the title or abstract from ISI Web of Science has increased dramatically since then.

  5. Approximations in DFT: Exchange-Correlation • Kohn-Sham equation is exact but the exchange-correlation functional, mxc, and energy Exc are unknown. • Local Density Approximation (LDA): assumes electron gas is homogenous locally, so that Exc depends ONLY on the local electron density around each volume element dr in the system. • Generalized gradient approximation (GGA): includes gradient corrections to the electron density and does a better job for most things, especially less-dense systems like molecules bonding to oxide surfaces, ionic and covalent crystals, etc. Electron Correlation??? Exchange-Correlation-hole

  6. From Atoms to Autos…OurVision for Multiscale Materials Design – VASP and Material Properties Play Central Role! Competency Continuum Engineering Materials Science Chemistry Physics Microstructural Atomistic Electronic log(Length scale) Philosophy nm mm mm m

  7. We Have Recently Reached Some Milestones Toward Multiscale Materials Design, Especially Design for Alloy Strength • W.A. Curtin, D.L. Olmsted, L.G. Hector, Jr., “A Predictive Mechanism for Dynamic Strain Ageing in Aluminum-Magnesium Alloys,” Nature Materials5 (2006) 875. • C. Woodward, D.R. Trinkle, L.G. Hector, Jr., D.L. Olmsted, “Prediction of Dislocation Cores in Aluminum from Density Functional Theory,” Physical Review Letters100 (2008) 045507. • J. A. Yasi, L.G. Hector, Jr., D.R. Trinkle, “First-principles Data for Solid-Solution Strengthening of Magnesium: From Geometry and Chemistry to Properties,” Acta Materialia 58 (2010) 5704. • G. Leyson, W.A. Curtin, L.G. Hector, Jr., C. Woodward, “Quantitative Prediction of Solute Strengthening in Aluminum Alloys,” Nature Materials published on-line, August, 2010. • S. Ganeshan, L.G. Hector, Jr., Z.-K. Liu, “First-principles Calculations of Impurity Diffusion Coefficients in Dilute Mg alloys with the 8 frequency model, “ Acta • Materialia, in review.

  8. We Need Material Properties in All Cases: So What are Examples of Quantities that can be Predicted with DFT Inputs? • Components of the elasticity tensor (Cij) • Cv(T)and Cp(T): heat capacities • Thermodynamic functions: enthalpy of formation, free energy, zero point energy, vibrational entropy. • Binding energies • Critical Point in Phase Transitions (e.g. Cerium) • Yield strength (as function of solute chemistry) • Dielectric properties (Born charge tensor components, dielectric tensor components) • Thermal expansion tensor (aij) • Magnetism • Transition state barriers • Diffusion coefficients (self- and impurity) • Work of Separation/Adhesion • Negative Possion’s ratio (auxetic materials) T-V Phase Diagram for Ce* *Y Wang, et al. Phys. Rev. B 78 (2008) 104113/1-9.

  9. Components of the Elasticity Tensor, Cij: Why Important? • Fundamental intrinsic material properties • Important components of Hooke’s law/elasticity • If you want to predict dislocation core structures starting with anisotropic elasticity theory, you will need the Cij • Useful for predicting mechanical instability • If one knows the Cij, then one knows the sounds speeds • Indicative of bonding in solids (cf. diamond with Lithium) • With the Cij, one can calculate polycrystalline: Young's Modulus, E (a proportionality constant between uniaxial stress and strain; Poisson's Ratio, n (the ratio of transverse to axial strain); bulk modulus, B (measure of the incompressibility); shear Modulus, G (measure of a resistance to shear) • Thermodynamic stability can influenced by elastic constants* * M. Baldi et al. Phys. Rev. Lett. 102 (2009) 226102

  10. Energy-Based Formulation Expression for the energy U in a harmonic approximation is where i and j run over the 6 components of the strain. However, we don’t actually know the strain! We know what we applied, but the initial structure has some strain typically. Let the initial strain be S and the applied strain be e, then and Cij determined individually or as sums from second derivatives of U w.r.t. displacements!

  11. Stress-Based Formulation* With an arbitrary (unknown) initial strain E and the corresponding stressS, we applied strains e. Linear least-squares readily solves this equation for the unknowns S(E) C(E)! We know S(E+e)from VASP. For P1 (triclinic) symmetry, we have 27 variables, i.e. 6 components of the initial stress, S(E), and 21 independent C(E). • Cubic (3), Hexagonal (5), Trigonal/Hexagonal (6), Orthorhombic (9). *Y. Le Page, P. Saxe, Phys. Rev. B65 (2002) 104104 *Y. Le Page, P. Saxe, Phys. Rev. B63 (2001) 74103 *L.G. Hector, Jr., et al., Phys. Rev. B 76(2007) 014121

  12. Comparison of the Methods

  13. Calculated vs. Experiment: what scatter do we see for selected elements (based upon GGA PW91)? If we were perfect with our predictions, then all of our numbers would lie on top of the black line.

  14. Elasticity Tensor Components, Cij: Observations • Structure used is very important! • If you are off in your prediction of the lattice constants, then your Cij will be inaccurate as well. • K-mesh is very important! • Much finer than for other properties • Tetrahedron method most reliable • Applied strain levels are not very significant • Least squares errors can be useful…why? • Exchange-Correlation functional is important! -LDA overbinds, so the Cij can be too high; GGA underbinds • Thermal expansion can be significant • Errors for the C11 and C33 are greatest for the elements

  15. Elastic Instability, Lattice Instability, Soft Modes For a crystalline system to be elastically stable, the change in its internal energy, U, due to a homogeneous elastic distortion must be positive definite: DU > 0.* This leads to the following condition on the eigenvalue matrix (Born Stability Criterion) : where H is the elastic component of the Hessian matrix: A negative eigenvalue means that DU < 0 and the structure is elasticially unstable. E = E(Cij) *L.G.Hector, Jr., J.F. Herbst, J. Phys.: Condens. Matter 20 (2008) 064229.

  16. Elastic Instability, Lattice Instability, Soft Modes LiGa: Only AGa Binary for Which there is a Known Structure Determination; Also in the Li-Ga Phase Diagram* LiGa has the Zintl fcc NaTl-type structure (Fd-3m, cubic) *J.F. Herbst et al, Phys. Rev. B 82 (2010) 024110.

  17. Elastic Instability, Lattice Instability, Soft Modes CsGa: Does it Exist in the Fd-3m Structure? * Elastic (or cell) instability *J.F. Herbst et al, Phys. Rev. B 82 (2010) 024110.

  18. Elastic Instability, Lattice Instability, Soft Modes CsGa, Fd-3m: Not in Cs-Ga Phase diagram, thermodynamically unstable w.r.t. Cs8Ga11 (Per VASP)

  19. Elastic Instability, Lattice Instability, Soft Modes: LaCo5* P6/mmm (191) LaCo5 (ferromagnetic), 2x2x2 Supercell, 48 atoms, 18 branches, elastically stable Co: 2c, 3g Cccm (orthorhombic, 66) *J.F. Herbst, L.G. Hector, Jr., J. Alloys Comp. 446-447 (2007) 188-194.

  20. Elastic Instability, Lattice Instability, Soft Modes: LaCo5* La –Co Phase diagram: No Known Structure for LaCo5 at Low T

  21. Density Functional Theory: StructureDetermination and Discrimination • Structure of Li2NH (lithium imide): Has recently attracted substantial interest in DFT community for hydrogen storage • H-sites undetermined since 1950’s when initial work began • Model the hydrogen storage reaction LiNH2 + LiH  Li2NH + H2 • two stable compounds react to form a third + H2 • reversible (280°C, 1 bar H2) • 6.5 mass% theoretical H2 capacity Ima2 (46), Li2NH, 32 atom a = 7.12 Å, b = 10.07 Å, c = 7.09 Å *J. F. Herbst, L.G. Hector, Jr., Phys. Rev. B 72 (2005) 125120. *M. Balogh et al., J. Alloys Compd. 420 (2006) . * VASP Code is used for all DFT calculations. See: http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html

  22. Density Functional Theory: StructureDetermination and Discrimination Ima2 Crystal Structure – from neutron powder diffraction at GM and NIST* (now confirmed at 50 K) orthorhombic Ima2 Li-disordered cubic Fd-3m (partially occupied Li 32e sites) VASP-relaxed Ima2 Ima2 structure also, agrees with exp. vibrational data! *M. Balogh, et al. J. Alloys Compd. 420 (2006) 326.

  23. Density Functional Theory: StructureDetermination and Discrimination* DFT-computed energetics of various Li2NH crystal structures in the GGA PW91. All energies in kJ/mol-f.u. (f.u.  formula unit). DF298 = F298 - F298(Ima2) *L.G. Hector, Jr, J.F. Herbst, J. Physics: Condens. Matter 20(6) (2008) 064229,1-11. aJ. F. Herbst, L.G. Hector, Jr., Phys. Rev. B 72 (2005) 125120. b B. Magyari-Kope, et al., Phys. Rev. B 73 (2006) 220101 . c T. Mueller and G. Ceder, Phys. Rev. B. 74 (2006) 134104 .

  24. Mg Ni H Density Functional Theory: StructureDetermination and Discrimination for Mg2NiH4* LTI LTII Zolliker, Yvon, Jorgensen, & Rotella Inorg. Chem. 25 (1986) 3590 C2/c (No. 15) Z = 8 Noreus & Werner J. Less-Comm. Met. 97 (1984) 215 C2/m (No. 12) Z = 4 *J.F. Herbst, L.G. Hector, Jr., Phys. Rev. B 79 (2009) 155113.

  25. Density Functional Theory: StructureDetermination and Discrimination for Mg2NiH4* Phonon Spectrum for LTII [GGA (PW91)] 48 1x2x2 (224 atom) supercells no soft modes; total phonon DoS shows no imaginary excursions Eel(LTII) = – 23.88 eV

  26. Density Functional Theory: StructureDetermination and Discrimination for Mg2NiH4* Phonon Spectrum for LTI [GGA (PW91)] 42 2x2x2 (224 atom) supercells 9 soft modes at high symmetry points: capture & optimize each – all metals except lower Z-pt monoclinic C2/m structure Eel(lower Z-pt C2/m structure) = – 23.62 eV/Mg2NiH4 Eel(LTI) = – 23.13 eV Eel(LTII) = – 23.88 eV lower Z-pt C2/m structure Z = 8

  27. Density Functional Theory: StructureDetermination and Discrimination for Mg2NiH4* Follow Soft Mode: P-1 Structure Low Energy Structure from Path Starting at Z-point c a b triclinic P-1 (No. 2) Z =4 NiH4 tetrahedra irregular, flattened 84 2x2x2 (224 atom) supercells Eel(LTII) = – 23.88 eV Eel(P-1) = – 23.76 eV

  28. Density Functional Theory: StructureDetermination and Discrimination for Mg2NiH4* • DFT structure determination: useful (e.g. for thermochemical calculations) in the instance that a structure is not available in the experimental literature). • DFT clearly discriminates between the Mg2NiH4 LTI and LTII structures: LTII is the preferred structure at both electronic and vibrational levels of analysis • Excellent approximate structure can be derived from pursuit of soft modes in Mg-substituted CaMgNiH4-type structure • PW91 GGA provides most accurate results

  29. Thermophysical Properties: Cp(T) (J/K mol) and aij(T) (1/K) • Required properties for a wide range of applications, e.g. fuel cell and battery • systems, constitutive models of structural alloys, thermometers, building, ship, aircraft construction, microelectronics fabrication, … • Einstein (1907)* got the ball rolling on a QM description of heat capacity: all atoms • vibrate as independent (non-interacting) quantum harmonic oscillators with frequency wE • Debye (1912)** theory of heat capacity: heat (atomic vibrations) due to phonons (or collective vibrations or sound waves) in a box (breaks down at high T due to anharmonicity): • Modern theory based upon lattice dynamics gives the isochoric heat capacity, • Cv(T), but experimentalists always measure Cp(T) • Use quasi-harmonic approach with VASP/DFT/phonon to compute Cp(T) and aij(T) • Neglect electron-phonon many body enhancement factor (1.4 to 2.5) • We have critically evaluated LDA, PBE, PBEsol functionals for selected metals, insulators, • and semiconductors: most continuum mech. theories ignore T-dependence • *A. Einstein, “Planck’s theory of radiation and the theory of the specific heat,” Ann. d. Physik 22, 180 (1907). • **P. Debye, “On the theory of specific heats ,” 344 (1912) 789-839.

  30. Thermophysical Properties: Cp(T) and aij(T) The quasi-harmonic approximation: anharmonic effects are included via the volume dependence of phonon frequencies ! Requires lots of VASP calculations!

  31. Thermophysical Properties: Cp(T) and aij(T) Tungsten, Metallic, Cubic, Tm ~ 3860 K Results from PBEsol [2] [3] [1] present present [1] Cp(T) Thermal expansion [1] A. Debernardi et al., Phys. Rev. B 63 (2001) 064305. [2] A.P. Miiller, A. Cezairliyan, Int. J. Thermophysics 11 (1990) 619-628. [3] G.K. White, M.L. Mingers, Int. J. Thermophysics 15 (1994) 1333-1343

  32. Thermophysical Properties: Cp(T) and aij(T) LiC6: Li-Ion Battery Anode Material, Metallic, Hexagonal, Tm ~ 700 K Results from PBEsol (a,c, are lattice parameters, V is volume) Cp ac data pt. Cv aV Cp-Cv aa Cv (elec) Contributions to the Cp(T) (no experimental data available) Thermal expansion components

  33. Thermophysical Properties: Cp(T) and aij(T) LiC6: Li-Ion Battery Anode Material, Metallic, Hexagonal, Results from PBEsol PBE PBEsol LDA Temperature Dependence of the “c” Lattice Parameter: PBE, PBEsol, LDA

  34. Diffusion: Self-Diffusion HCP Metals • Question: Can we predict Mg and Zn self-diffusion coefficients for vacancy-mediated diffusion? • Answer: Yes, but we need a code to predict transition states, and code to compute vibrational properties with VASP as the engine! • Question: Why do we care about diffusion? • Answer: Macro-scale properties such as creep, strength, and corrosion are controlled by diffusion of solutes or impurities in a solvent or host material.

  35. Self Diffusion in HCP Metals: Mg (GGA and LDA Bound Experimental Data) Within a Basal Plane Between Adjacent Basal Planes Ganeshan et al., Computational Materials Sci. in press (2010)

  36. Self Diffusion in HCP Metals: Zn Within a Basal Plane Between Adjacent Basal Planes Ganeshan et al., Computational Materials Sci. , in press (2010)

  37. Self Diffusion in HCP Transition Metals: Energy Barriers for Ti Between Adjacent Basal Planes Within a Basal Plane: double saddle point 3 4 2

  38. Impurity Diffusion in Mg using Ghate’s* 8-Frequency Model**: Ca, Sn, Zn, Al Between Adjacent Basal Planes *Ghate, Phys Rev 1964;133:A1167. **Ganeshan et al., Acta Materialia, in review (2010)

  39. Summary Remarks DFT/VASP: Well-established computational material science tools Requires a set of scripts that take information from the DFT engine (VASP) and compute material properties Scripts that I use: lattice dynamics; transition state search routines; mechanical properties (Cij); misc. Mathematica Comparison of theoretical predictions with available experimental data is crucial (if nothing more than for guiding future improvements to the theory, or , for revealing potential experimental errors) Choice of exchange-correlation functional is a “Procrustean dilemma”*: one functional does not fit all applications ! My philosophy: we are doing very well if we can “bound” available experimental data with two functionals (e.g. LDA and GGA) *”He who stretches” from Greek Mythology. **Picture of Villa Herwig from: Schrödinger: Life and Thought. W. Moore, Cambridge Univ. Press (1992).

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