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Ab initio Thermodynamics and Structure-Property Relationships. Axel van de Walle Applied Physics and Materials Science Department Engineering and Applied Sciences Division. http://www.its.caltech.edu/~avdw/. From a virtual to a real material. Wonderful compound. Useless material.

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Ab initio thermodynamics and structure property relationships
Ab initio Thermodynamics and Structure-Property Relationships

Axel van de Walle

Applied Physics and Materials Science Department

Engineering and Applied Sciences Division

http://www.its.caltech.edu/~avdw/


From a virtual to a real material
From a virtual to a real material... Relationships

Wonderful compound

Useless material

We need a way to predict, not only structure-property relationships, but also thermodynamic stability.



Ab initio thermodynamics and structure property relationships

First-principles Thermodynamic Calculations Relationships

Thermodynamic data

  • Large number of atoms

  • Many configurations

Lattice model &

Monte Carlo Simulations

Lattice vibrations

Configurational disorder

Electronic excitations

  • Small number of atoms

  • Few configurations

Quantum Mechanical Calculations

http://www.its.caltech.edu/~avdw/atat


Ab initio thermodynamics and structure property relationships

Thermodynamic data Relationships

Lattice model &

Monte Carlo Simulations

Configurational disorder

Lattice vibrations

Configurational disorder

Electronic excitations

Quantum Mechanical Calculations


The cluster expansion formalism
The Cluster Expansion Formalism Relationships

interactions

correlations

clusters

Sanchez, Ducastelle and Gratias (1984), Tepesch, Garbulski and Ceder (1995), A. van de Walle (2009)


Cluster expansion fit
Cluster expansion fit Relationships


Cross validation
Cross-validation Relationships

Example of polynomial fit:

A. van de Walle and G. Ceder, J. Phase Equilibria 23, 348 (2002)



Application example

O Relationships

Application Example

  • (Sm/Ce)O2 Superlattices have been shown to exhibit enhanced Oxygen conductivity*.

CexSm1-xOy

~50 nm

  • Goal: study interface thermodynamics to help understand origin of enhanced conductivity.

*I. Kosackia, T. Christopher, M. Rouleau, P. F. Becher, J. Bentley, D. H. Lowndes, Solid State Ionics 176, 1319 (2005) and I. Kosacki, C.M. Rouleau, P.F. Becher and D.H. Lowndes, in press.


Convex hull construction
Convex hull construction Relationships

E

x[Ce]

x[O]


Ground state search
Ground State Search Relationships

charge-balanced

line

CeSm2O5

Sm2O3

CeO2

(Vac)

(Vac)


Ab initio thermodynamics and structure property relationships

Thermodynamic data Relationships

Lattice model &

Monte Carlo Simulations

Lattice model &

Monte Carlo Simulations

Lattice vibrations

Configurational disorder

Electronic excitations

Quantum Mechanical Calculations


Equilibrium composition profile

x Relationships

Optimal composition

Forbidden composition

O

y

Equilibrium Composition profile

~25 nm

Superlattices

CexSm1-xOy

70000-atom simulation

Interface

Material: CexSm1-xOyVac2-y

van de Walle & Ellis, Phys. Rev. Lett.98, 266101 (2007)


Antiphase boundary

Application to Ti-Al Alloys Relationships

Antiphase Boundary

Diffuse Antiphase Boundary

Creation of a plane with easy dislocation motion:

Work softening


Short range order and diffuse antiphase boundary energy calculations
Short-range order and diffuse antiphase boundary energy calculations

Neeraj (2000)

Energy cost of creating a diffuse anti-phase boundary in a Ti-Al hcp short-range ordered alloy by sliding k dislocations

Calculated diffuse X-ray scattering in Ti-Al hcp solid-solution

van de Walle & Asta. Metal. and Mater. Trans. A, 33A, 735 (2002)


Temperature scale problem

Likely source of the discrepancy: calculations

Vibrational entropy.

Temperature scale problem

hcp Ti

Fultz, Nagel, Antony, et al. (1993-1999)

Ceder, Garbulsky, van de Walle (1994-2002)

de Fontaine, Althoff, Morgan (1997-2000)

Zunger, Ozolins, Wolverton (1998-2001)

DO19

Ti3Al

van de Walle, Asta and Ceder (2002),

Murray (1987) (exp.)

Many other examples…


Ab initio thermodynamics and structure property relationships

Thermodynamic data calculations

Lattice model &

Monte Carlo Simulations

Lattice vibrations

Lattice vibrations

Configurational disorder

Electronic excitations

Quantum Mechanical Calculations



Coarse graining of the free energy
Coarse-Graining of the Free Energy calculations

Graphically:

Formally:

where

van de Walle & Ceder, Rev. Mod. Phys.74, 11 (2002).


First principles lattice dynamics
First-principles lattice dynamics calculations

First-principles data

Least-squares fit to

Spring model

Phonon density of states

Direct force constant method

(Wei and Chou (1992), Garbuski and Ceder (1994),

among many others)

Thermodynamic

Properties


Effect of lattice vibrations on calculated phase diagrams
Effect of lattice vibrations on calculationsCalculated Phase Diagrams

Ti-Al

Ohnuma et al.

(2000)

van de Walle, Ghosh, Asta (2007)

Adjaoud, Steinle-Neumann, Burton and A. van de Walle (2009)



Surface reconstruction problem
Surface reconstruction problem calculations

Example: SrTiO3 (100) c(6x2) surface

(Applications: Catalyst, Gate oxide in integrated circuits,

Substrate for thin-film growth)

?

O

Known “bulk”

crystal structure

Sr

Ti

Solution: Combine experimental and computational methods

Lanier, van de Walle, Erdman, Landree, Warschkow, Kazimirov, Poeppelmeier, Zegenhagen, Asta, Marks, Phys. Rev. B76, 045421 (2007)


Automated screening
Automated Screening calculations

Sr, Ti coordinates from e- and X-ray diffraction

Locate candidate O sites

nb of config

Enumerate every possible O configuration

~240

Discard configurations with “too many bonds”

~17000

Discard configuration with large electrostatic energy

~100

Quantum Mechanical Calculations

~4

Predicted structure(s)


Predicted structures
Predicted structures calculations

Side view

Top view

Simulated

STM images

Actual STM image


The equilibrium reconstruction of the srtio 3 100 c 6x2 surface
“The” equilibrium reconstruction of calculationsthe SrTiO3 (100) c(6x2) surface

  • A dynamic random “solid solution” of many different atomic motifs.

  • Each structure enters the refined model with fractional occupation.

  • Solved an exceptionally complex surface reconstruction problem!


Thermoelectrics phase stability in the zn sb system
Thermoelectrics: calculationsPhase stability in the Zn-Sb system

  • Zn-Sb has been of interest for many years in the search for efficient and low-cost thermoelectric materials:

    • Environmentally benign and relatively abundant elements

    • “Zn4Sb3” phase exhibits a high thermoelectric efficiency.

  • However the “Zn4Sb3” phase has a positive formation energy (20 meV/atom):

    • is it really thermodynamically stable?

    • if so, why?

  • Entropy could play a role!

G. S. Pomrehn, E. S. Toberer, G. J. Snyder & A. van de Walle, PRB (2011, forthcoming).


Structural complexity
Structural complexity calculations

Sb site

Partially occupied

Zn sites

Partially occupied

interstitial Zn sites

Snyder, Christensen, Nishibori, Caillat, and Iversen, Nature Mater.3, 458 (2004).


Computational method
Computational Method calculations

Independent cell approximation (works well if cell is big):

+

+

+

+

+

+

+



Zn sb partial phase diagram
Zn-Sb (partial) phase diagram calculations

“Zn4Sb3”

ZnSb

Zn

“Zn4Sb3”

  • Proves entropy stabilization

  • Explains difficulty in n-doping the material.

  • Retrograde solubility explains formation of nanovoids upon cooling.


Materials optimization for epitaxial optoelectronic device design
Materials optimization for calculationsepitaxial optoelectronic device design

  • Essential features

    • Anisotropy

    • Experimental control over structure

  • Goal:

    • Guide experimental efforts to produce high-performance solar cells and LEDs.

Nasser et al. (1999)


Structure property relationships
Structure-Property Relationships calculations

  • Goal: Relate atomic-level structure to macroscopic properties.

  • For scalar properties of crystalline alloys, tool already exists:

  • Used for representing the structural dependence of

    • bulk modulus

    • equation of state

    • phonon entropy

    • electronic density of state

    • band gap

    • defect level energy

    • Currie temperature

  • The cluster expansion forms a basis for scalar functions of configurations.

The Cluster Expansion.

M. Asta, R. McCormack, and D. de Fontaine (1993)

H. Y. Geng, M. H. F. Sluiter, and N. X. Chen, (2005)

G. D. Garbulsky and G. Ceder, (1994).

H. Y. Geng, M. H. F. Sluiter, and N. X. Chen, (2005).

A. Franceschetti and A. Zunger (1999).

S. V. Dudiy and A. Zunger (2006).

A. Franceschetti et al (2006).


The tensorial cluster expansion
The Tensorial Cluster Expansion calculations

  • Needed to express configurational-dependence of many important properties in epitaxial systems:

    • elastic constants, equilibrium strain/stress

    • dielectric constant

    • carrier effective masses

    • ferroelectric vector

    • piezoelectric tensor

    • strain-gap coupling

    • optoelectric coupling, etc.

A. van de Walle, Nature materials7, 455 (2008)


Bases and tensors
Bases and Tensors calculations


Tensor product basis
“Tensor-product” basis calculations

binary

case


Exploiting symmetry
Exploiting symmetry calculations

Not equivalent

Equivalent


Symmetry restrictions
Symmetry restrictions calculations

Example 1

Example 2

Example 3


Tensorial cluster expansion
Tensorial Cluster Expansion calculations

Same as in conventional

cluster expansion


A graphical representation of tensors
A graphical representation of tensors calculations

f(u)

u

Example:

Strain tensor

–1

+1

Unique representation for any symmetric tensor.


Fcc with symmetric 2 nd rank tensor
fcc with symmetric 2 calculationsnd rank tensor


Tetragonal body centered lattice with symmetric 2 nd rank tensor
Tetragonal body-centered lattice with symmetric 2 calculationsnd rank tensor

Unique axis:


Can fcc superstructures be ferroelectric
Can fcc superstructures be ferroelectric? calculations

empty

point

pairs

Do not couple with vector-valued quantities on the fcc lattice.

  • All pairs are also lattice vectors “v” (in fcc).

  • v and –v are equivalent

  • pair has inversion symmetry

  • no unique axis possible

Triplets do couple with vector-valued quantities:

Yes, ferroelectricity possible


Example configuration strain coupling in ga x in 1 x n
Example configuration-strain coupling in (Ga calculationsxIn1-xN)

ECI

  • Bond between alike atoms causes

  • contraction along bond

  • expansion perpendicular to bond

-0.0075

0.1806

-0.0015

0.0064

0.0048

-0.0057

0.0035

0.0048

(Ga-In sublattice shown)


Superlattices
Superlattices calculations

Traditional

epitaxial structure

Composition modulations

along epitaxial layer

(Could be induced via strain-configuration coupling.)


Structure property relationships1
Structure-Property Relationships calculations

Useful input for design and optimization of optoelectronic devices

A. van de Walle, Nature materials7, 455 (2008)


Conclusion outlook
Conclusion & outlook calculations

  • Ab initio materials design is becoming a reality and requires

    • methods to assess phase stability

    • methods to uncover structure-property relationships

  • There is the need to develop methods that break free of the “known lattice” assumption.