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Ab initio Thermodynamics and Structure-Property Relationships

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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

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

First-principles Thermodynamic Calculations

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

Lattice model &

Monte Carlo Simulations

Configurational disorder

Lattice vibrations

Configurational disorder

Electronic excitations

Quantum Mechanical Calculations

The Cluster Expansion Formalism

interactions

correlations

clusters

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

Cross-validation

Example of polynomial fit:

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

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.

Lattice model &

Monte Carlo Simulations

Lattice model &

Monte Carlo Simulations

Lattice vibrations

Configurational disorder

Electronic excitations

Quantum Mechanical Calculations

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

Diffuse Antiphase Boundary

Creation of a plane with easy dislocation motion:

Work softening

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)

Likely source of the discrepancy:

Vibrational entropy.

Temperature scale problemhcp 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…

Lattice model &

Monte Carlo Simulations

Lattice vibrations

Lattice vibrations

Configurational disorder

Electronic excitations

Quantum Mechanical Calculations

Coarse-Graining of the Free Energy

Graphically:

Formally:

where

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

First-principles lattice dynamics

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 onCalculated 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

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

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)

“The” equilibrium reconstruction ofthe 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

- 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

Sb site

Partially occupied

Zn sites

Partially occupied

interstitial Zn sites

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

Zn-Sb (partial) phase diagram

“Zn4Sb3”

ZnSb

Zn

“Zn4Sb3”

- Proves entropy stabilization
- Explains difficulty in n-doping the material.
- Retrograde solubility explains formation of nanovoids upon cooling.

Materials optimization forepitaxial 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

- 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

- 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)

A graphical representation of tensors

f(u)

u

Example:

Strain tensor

–1

+1

Unique representation for any symmetric tensor.

Tetragonal body-centered lattice with symmetric 2nd rank tensor

Unique axis:

Can fcc superstructures be ferroelectric?

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 (GaxIn1-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

Traditional

epitaxial structure

Composition modulations

along epitaxial layer

(Could be induced via strain-configuration coupling.)

Structure-Property Relationships

Useful input for design and optimization of optoelectronic devices

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

Conclusion & outlook

- 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.

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