Ab initio Thermodynamics and Structure-Property Relationships

1. 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/

2. 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.

3. Automated Material Discovery

4. 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

5. Thermodynamic data Lattice model & Monte Carlo Simulations Configurational disorder Lattice vibrations Configurational disorder Electronic excitations Quantum Mechanical Calculations

6. The Cluster Expansion Formalism interactions correlations clusters Sanchez, Ducastelle and Gratias (1984), Tepesch, Garbulski and Ceder (1995), A. van de Walle (2009)

7. Cluster expansion fit

8. Cross-validation Example of polynomial fit: A. van de Walle and G. Ceder, J. Phase Equilibria 23, 348 (2002)

9. Automated Cluster Expansion Construction

10. O 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.

11. Convex hull construction E x[Ce] x[O]

12. Ground State Search charge-balanced line CeSm2O5 Sm2O3 CeO2 (Vac) (Vac)

13. Thermodynamic data Lattice model & Monte Carlo Simulations Lattice model & Monte Carlo Simulations Lattice vibrations Configurational disorder Electronic excitations Quantum Mechanical Calculations

14. x 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)

15. Application to Ti-Al Alloys Antiphase Boundary Diffuse Antiphase Boundary Creation of a plane with easy dislocation motion: Work softening

16. 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)

17. Likely source of the discrepancy: 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…

18. Thermodynamic data Lattice model & Monte Carlo Simulations Lattice vibrations Lattice vibrations Configurational disorder Electronic excitations Quantum Mechanical Calculations

19. The Cluster Expansion Formalism

20. Coarse-Graining of the Free Energy Graphically: Formally: where van de Walle & Ceder, Rev. Mod. Phys.74, 11 (2002).

21. 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

22. 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)

23. Beyond the cluster expansion…

24. 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)

25. 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)

26. Predicted structures Side view Top view Simulated STM images Actual STM image

27. “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!

28. 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).

29. Structural complexity Sb site Partially occupied Zn sites Partially occupied interstitial Zn sites Snyder, Christensen, Nishibori, Caillat, and Iversen, Nature Mater.3, 458 (2004).

30. Computational Method Independent cell approximation (works well if cell is big): + + + + + + +

31. Formation Energies & Convex Hull

32. 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.

33. 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)

34. 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).

35. 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)

36. Bases and Tensors

37. “Tensor-product” basis binary case

38. Exploiting symmetry Not equivalent Equivalent

39. Symmetry restrictions Example 1 Example 2 Example 3

40. Tensorial Cluster Expansion Same as in conventional cluster expansion

41. A graphical representation of tensors f(u) u Example: Strain tensor –1 +1 Unique representation for any symmetric tensor.

42. fcc with symmetric 2nd rank tensor

43. Tetragonal body-centered lattice with symmetric 2nd rank tensor Unique axis:

44. 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

45. 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)

46. Superlattices Traditional epitaxial structure Composition modulations along epitaxial layer (Could be induced via strain-configuration coupling.)

47. Structure-Property Relationships Useful input for design and optimization of optoelectronic devices A. van de Walle, Nature materials7, 455 (2008)

48. 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.