material property predictions with the vienna ab initio simulation package (vasp) l.g. hector, jr....

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, KYNovember 10, 2010

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, SoftModes

• Thermodynamics: Structure Prediction/Discrimination

• Thermophysical Properties: Cp(T) and ij(T)

• Diffusion• Concluding Remarks

Main Computational Engine for Quantum Mechanical Simulations of Solids: VASP (Vienna Ab Initio Simulation

Package) – A Plane Wave Density Functional Code

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.

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1990 1992 1994 1996 1998 2000 2002 2004

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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

0 2 4 6 8

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Ni1La

H2Ni3

Ni2

Ni3

La

H1

H1

H2

H

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LaNi5H7 (P63mc)

[1230]

Philosophy nm m mm m

log(Length scale)

Engineering

MaterialsScience

Chemistry

Physics

Competency

Microstructural

LaNi5H7hexagonal

Electronic

Atomistic

From Atoms to Autos…Our Vision for Multiscale Materials Design – VASP and Material Properties Play Central Role!

Continuum

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 Materials 5 (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 Letters 100 (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.

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 (ij)

• Magnetism• Transition state barriers• Diffusion coefficients (self- and impurity)• Work of Separation/Adhesion• Negative Possion’s ratio (auxetic materials)

*Y Wang, et al. Phys. Rev. B 78 (2008) 104113/1-9.

T-V Phase Diagram for Ce*

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, (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

Energy-Based Formulation

Expression for the energy U in a harmonic approximation is

i jjiijCUU 02)(2

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

)()(2)(2 0 SeCSe TUeU

eSε

and

Cij determined individually or as sums from second derivatives of U w.r.t. displacements!

Stress-Based Formulation*With an arbitrary (unknown) initial strain E and the

corresponding stress , we applied strains e.

C(E)eΣ(E)

sΣ(E)e)Σ(E

Linear least-squares readily solves this equation for the unknowns (E) C(E)! We know from VASP. For P1 (triclinic) symmetry, we have 27 variables, i.e. 6 components of the initial stress, (E), and 21 independent C(E).

- Cubic (3), Hexagonal (5), Trigonal/Hexagonal (6), Orthorhombic (9).

*Y. Le Page, P. Saxe, Phys. Rev. B 65 (2002) 104104*Y. Le Page, P. Saxe, Phys. Rev. B 63 (2001) 74103 *L.G. Hector, Jr., et al., Phys. Rev. B 76 (2007) 014121

Comparison of the Methods

C11 C12 C13 C33 C44 a (Å) c (Å) # CalculationsStress-based 49 31 21 58 14 3.1906 5.1800 11Experiment 59 26 21 61 17 3.2089 5.2101Energy-based 70 31 24 74 22 3.1354 5.0909 28

Elastic Constants (GPa)Mg (hcp, P63/mmc)

C11 C12 C13 C33 C44 a (Å) c (Å) # CalculationsStress-based 290 23 12 364 162 2.2639 3.5663 11Experiment 299 27 11 342 166 2.2826 3.5836Energy-based 336 81 21 443 181 2.1851 3.4710 19

Be (hcp, P63/mmc)Elastic Constants (GPa)

Calculated vs. Experiment: what scatter do we see for selected elements (based upon GGA PW91)?

LiBeNaMgAlSiKCaScTiVCrCoNiCuZnGeRbSrYZrNbMoRuPdAgCdCsPrNdGdDyHoErHfTaWReIrPtAuTlPbTh

If we were perfect with our predictions, then all of our numbers would lie on top of the black line.

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

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: U > 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 U < 0 and the structure is elasticially unstable. E = E(Cij)

*L.G.Hector, Jr., J.F. Herbst, J. Phys.: Condens. Matter 20 (2008) 064229.

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.

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.

Elastic Instability, Lattice Instability, Soft Modes CsGa, Fd-3m: Not in Cs-Ga Phase diagram, thermodynamically

unstable w.r.t. Cs8Ga11 (Per VASP)

Elastic Instability, Lattice Instability, Soft Modes: LaCo5*P6/mmm (191) LaCo5 (ferromagnetic), 2x2x2 Supercell, 48 atoms, 18 branches, elastically stable

Cccm (orthorhombic, 66)

Co: 2c, 3g

*J.F. Herbst, L.G. Hector, Jr., J. Alloys Comp. 446-447 (2007) 188-194.

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

• 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

b

N Li H

c a

b

N Li H

c a

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

Density Functional Theory: Structure Determination and Discrimination

Ima2 Crystal Structure – from neutron powder diffraction at GM and NIST* (now confirmed at 50 K)Li2NH_imide_MPB_disordered_cubic_F4132

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Li-disordered cubic Fd-3m (partially occupied Li 32e sites)

Li2NH_imide_orthorhombic_Ima2_from_disordered_cubic_POSCAR_AE

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Li2NH_imide_Ima2_VASP_relax

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VASP-relaxed Ima2

*M. Balogh, et al. J. Alloys Compd. 420 (2006) 326.

Ima2 structure also, agrees with exp. vibrational data!

Density Functional Theory: Structure Determination and Discrimination

Li2NHStructure

Space Group

Eel Eel ZPE E0 E0 F298 F298 F298

1 (Structure V in the

next slide)

Ima2a -1698.84 0.00 46.38 -1652.46 0.00 -7.03 -1659.49 0.00

2 Pnma* -1699.93 -1.09 46.62 -1653.31 -0.85 -6.57 -1659.88 -0.39

3 C2/m* -1701.22 -2.38 46.32 -1654.90 -2.44 -5.56 -1660.46 -0.97

4 Pnmab -1701.82 -2.98 47.06 -1654.76 -2.30 -6.16 -1660.92 -1.43

5 Pbcac -1703.37 -4.53 46.51 -1656.86 -4.40 -6.47 -1663.33 -3.84

DFT-computed energetics of various Li2NH crystal structures in the GGA PW91. All energies in kJ/mol-f.u. (f.u. formula unit). F298 = F298 - F298(Ima2)

*L.G. Hector, Jr, J.F. Herbst, J. Physics: Condens. Matter 20(6) (2008) 064229,1-11.a J. 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 .

Density Functional Theory: Structure Determination and Discrimination*

Noreus & WernerJ. Less-Comm. Met. 97 (1984) 215

C2/m (No. 12) Z = 4

Zolliker, Yvon, Jorgensen, & Rotella Inorg. Chem. 25 (1986) 3590

C2/c (No. 15) Z = 8

LTI LTII

Mg

Ni

H

*J.F. Herbst, L.G. Hector, Jr., Phys. Rev. B 79 (2009) 155113.

Density Functional Theory: Structure Determination 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


Phonon Spectrum for LTI [GGA (PW91)]

42 2x2x2 (224 atom) supercells9 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 structureZ = 8


Follow Soft Mode: P-1 Structure Low Energy Structure from Path Starting at Z-point

c

b

a

84 2x2x2 (224 atom) supercells

triclinic P-1 (No. 2) Z =4

NiH4 tetrahedra irregular, flattened

Eel(LTII) = – 23.88 eVEel(P-1) = – 23.76 eV


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


Thermophysical Properties: Cp(T) (J/K mol) and ij(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 E

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 ij(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.

Thermophysical Properties: Cp(T) and ij(T)

The quasi-harmonic approximation: anharmoniceffects are included via the volume dependence ofphonon frequencies !

Requires lots of VASP calculations!

elec elec phonon ZPE mag

p V V o

elec vibV V V

elec FV B

B

vibV

VP

F(V,T) U V F V,T E V,T E V F

C T C T V,T TB V,T V T

where

C T C T C T

and

f ,TC T k T n d

k T

and

C T is taken from phonons

V(T)T

V (T) T

2

22

0

1

L VP

For cubic systems

V(T)T T

V (T) T

Note that for hexagonal, tetragonal, and orthorhombic systems,

there are three components of the

thermal expansion coefficient tensor \

that must be compu

0

1 1

3 3

ted!


Tungsten, Metallic, Cubic, Tm ~ 3860 KResults from PBEsol

Cp(T) Thermal expansion

[1]

[2]

presentpresent

[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

[1]

[3]


LiC6: Li-Ion Battery Anode Material, Metallic, Hexagonal, Tm ~ 700 KResults from PBEsol (a,c, are lattice parameters, V is volume)

Contributions to the Cp(T)(no experimental data available)

Cp

Cv

Cv (elec)

Cp-Cv

data pt.c

a

V

Thermal expansion components


LiC6: Li-Ion Battery Anode Material, Metallic, Hexagonal, Results from PBEsol

Temperature Dependence of the “c” Lattice Parameter: PBE, PBEsol, LDA

PBE

PBEsol

LDA

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.

Self Diffusion in HCP Metals: Mg (GGA and LDABound Experimental Data)

Ganeshan et al., Computational Materials Sci. in press (2010)

Between Adjacent Basal PlanesWithin a Basal Plane

Self Diffusion in HCP Metals: Zn

Ganeshan et al., Computational Materials Sci. , in press (2010)

Within a Basal Plane Between Adjacent Basal Planes

Within a Basal Plane: double saddle pointBetween Adjacent Basal Planes

Self Diffusion in HCP Transition Metals: Energy Barriers for Ti

2 3 4

*Ghate, Phys Rev 1964;133:A1167.**Ganeshan et al., Acta Materialia, in review (2010)

Impurity Diffusion in Mg using Ghate’s* 8-Frequency Model**: Ca, Sn, Zn, Al

Between Adjacent Basal Planes

Summary RemarksDFT/VASP: Well-established computational material science toolsRequires a set of scripts that take information from the DFT engine (VASP) and compute material propertiesScripts 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).

material property predictions with the vienna ab initio simulation package (vasp) l.g. hector, jr....

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