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.