Ab initio Thermodynamics and Structure-Property Relationships

Axel van de WalleApplied Physics and Materials Science Department

Engineering and Applied Sciences Division

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

From a virtual to a real material...

Useless material

Wonderful compound

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

thermodynamic stability.

Automated Material Discovery

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrations

•Large number of atoms•Many configurations

•Small number of atoms•Few configurations

First-principles Thermodynamic Calculations

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

Configurational disorder

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrationsConfigurational disorderConfigurational disorder

The Cluster Expansion Formalism

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

correlationsinteractions

clusters

Cluster expansion fit

Cross-validation

Example of polynomial fit:

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

Automated Cluster Expansion Construction

Application Example• (Sm/Ce)O2 Superlattices have been shown

to exhibit enhanced Oxygen 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.

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

~50 nm

CexSm1-xOy

O

Convex hull construction

x[O]

x[Ce]

E

Ground State Search

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 0.05 0.1 0.15 0.2 0.25 0.3

x[O]

x[Ce]

charge-balancedline

CeO2CeSm2O5Sm2O3

(Vac) (Vac)

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrationsConfigurational disorder

Lattice model &Monte Carlo Simulations

x

y

Equilibrium Composition profile

Superlattices

70000-atom simulation

Material: CexSm1-xOyVac2-y

Optimal composition

~25 nm

CexSm1-xOy

O

Interface

Forbidden composition

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

Application to Ti-Al Alloys

Short-range order and diffuse antiphase boundary energy calculations

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

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

Neeraj (2000)

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

van de Walle, Asta and Ceder (2002),Murray (1987) (exp.) Many other examples…

hcp Ti

DO19

Ti3Al

Likely source of the discrepancy: Vibrational entropy.

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)

Temperature scale problem

Thermodynamic data

Quantum Mechanical Calculations

Lattice model &Monte Carlo Simulations

Electronic excitationsLattice vibrationsConfigurational disorderLattice vibrations

The Cluster Expansion Formalism

E 1, , n

F 1, , n J T

Coarse-Graining of the Free EnergyGraphically:

Formally:

F 1 ln ie E i 1 ln i e E i

1 ln e F

F 1 ln i e E iwhere

kBT 1

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

First-principles lattice dynamics

Least-squares fit toSpring model

First-principles data

ThermodynamicProperties

Phonon density of states

Direct force constant method(Wei and Chou (1992), Garbuski and Ceder (1994), among many others)

Ohnuma et al.(2000)

van de Walle, Ghosh, Asta (2007)

Ti-Al

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

Effect of lattice vibrations onCalculated Phase Diagrams

Beyond the cluster expansion…

Surface reconstruction problem

Known “bulk”crystal structure

?

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

Solution: Combine experimental and computational methods

(Applications: Catalyst, Gate oxide in integrated circuits, Substrate for thin-film growth)

Ti

Sr

O

Lanier, van de Walle, Erdman, Landree, Warschkow, Kazimirov, Poeppelmeier, Zegenhagen, Asta, Marks, Phys. Rev. B 76, 045421 (2007)

Automated Screening

Locate candidate O sites

Enumerate every possible O configuration

Discard configurations with “too many bonds”

Discard configuration with large electrostatic energy

Quantum Mechanical Calculations

~240

~17000

~100

~4

Predicted structure(s)

nb of config

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

Predicted structures

SimulatedSTM images

Actual STM image

Side view

Top view

“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

Partially occupiedinterstitial Zn sites

Partially occupiedZn sites

Sb site

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

Computational Method

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

T, T,

x T,

Equilibrium condition between phase and :

( )

reduces to ( ) with all quantities expressed per cell.

where

Es s1 s2+ s3

s4 s5 s6+

+

++ + +

T, k BTN

ln s exp Fs Ns

k BT

kB : Boltzmann’s constantN : total number of atomsFs : vibrational free energy of configurational state sNs : number of each atomic species in state s.

Grand canonical potential of phase at temperature T and chemical potential :

Equilibrium composition:

Formation Energies & Convex Hull

Zn-Sb (partial) phase diagram

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

“Zn4Sb3”ZnSb Zn

“Zn4Sb3”

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.

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 Cluster Expansion.

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 materials 7, 455 (2008)

Bases and TensorsQ

c

Qijk

c ijk

The form a basis for the space of tensors.

1 0 0

0 0 0

0 0 0

,

0 0 0

0 1 0

0 0 0

,

0 0 0

0 0 0

0 0 1

,

0 1/ 2 0

1/ 2 0 0

0 0 0

,

0 0 1/ 2

0 0 0

1/ 2 0 0

,

0 0 0

0 0 1/ 2

0 1/ 2 0

Example: A basis for symmetric rank 2 tensors in 3D is:

is a “basis tensor” c are coefficients to be determined

where

“Tensor-product” basis

ic ia i j

c jb j

Basis ai for space A Basis b j for space B

Basis a ibj for space A B

i jc ijaibj

Still need to exploit symmetry

form every possible pairwise product of ’s and :

To obtain a basis for tensor-function Q of configuration,

c ic j

Q

J

i

ibinarycase

Exploiting symmetry

Not equivalent

Equivalent

J J if , is equivalent by symmetry to , .

Symmetry restrictions

Example 1

Example 2

Example 3

Tensorial Cluster Expansion

Q

C

J m ,

Outer sum: over all symmetrically distinct i i (“cluster functions”) m : multiplicities : basis tensor C : set of basis tensors compatible with point group of cluster , : average over all , equivalent to , by symmetry J are coefficients to be determined.

Same as in conventionalcluster expansion

A graphical representation of tensors

fu i,j,k,

Qijk uiujuk

Plot fu as a function of unit vector u.

Unique representation for any symmetric tensor.

–1 +1

u f(u)

Example:Strain tensor

fcc with symmetric 2nd rank tensor

Tetragonal body-centered lattice with symmetric 2nd rank tensor

Unique axis:

Can fcc superstructures be ferroelectric?empty

pointpairs

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

Yes, ferroelectricity possible

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:

Example configuration-strain coupling in (GaxIn1-xN)

Bond between alike atoms causes• contraction along bond• expansion perpendicular to bond

(Ga-In sublattice shown)

-0.0075

0.1806

-0.0015

0.0064

0.0048

ECI

-0.0057

0.0048

0.0035

Superlattices

Traditionalepitaxial structure

Composition modulationsalong 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 materials 7, 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.