1. SLACS-INFM/CNR Sardinian Laboratory for Computational Materials Science www.slacs.it SLACS Atomically informed modeling of the microstructure evolution of nanocrystalline materials A. Mattoni alessandro.mattoni@dsf.unica.it

2. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS SLACS CNR-INFM CRS LN LR Regional Laboratories Atomistic investigation: large scale molecular dynamics simulations Large scale electronic structure calculations Continuum modeling: models for growth, interface mobilities http://www.slacs.it Division: Material Physics (Microstructure evolution of nanostructured materials) (6 members, www.dsf.unica.it/colombowww.dsf.unica.it/colombo)

3. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS OUTLINE The microstructure of interest for nanocrystalline materials Boundaries between order/disordered phase The theoretical framework Molecular dynamics atomistic simulations Modeling the growth of nanocrstals embedded into an amorphous matrix

4. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Molecular dynamics The material of interest is described as an assembly of molecular constituents

5. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Molecular dynamics An interatomic depending on atomic positions The interatomic forces are calculated accordingly Newton’s equations of motion are integrated numerically (“Verlet velocity”) Choose dt “judiciously” (~1fs) and iterate in time (“ad nauseam”)

6. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Interatomic potentials ”6-12” Lennard-Jones potential: repulsive core 1/r 12 ; VdW attraction 1/r 6 r>r eq ”6-12” Lennard-Jones potential: prototypical interatomic force model for close-packed metals Professor Sir John Lennard-Jones (FRS), one of the founding fathers of molecular orbital theory

7. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Interatomic potentials Stillinger-Weber potential for anysotropic covalent bonding (1985) F. Stillinger Department of Chemistry Princeton University Princeton, NJ 08540 T.A. Weber (EDIP) Environment dependent interatomic potential (1998)

8. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS MD comes of age… K. Kadau et al. Int. Journal of Modern Physics C 17 1755 (2006) B. J. Alder and T. E. Wainwright, J. Chem. Phys.27,1208(1957) Stillinger-Weber Lennard-Jones Tersoff EDIP 320 BILLION ATOM SIMULATION ON BlueGene/L Los Alamos National Laboratory

9. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS MD comes of age… more or less Compromise between accuracy and computational workload The bottleneck of standard molecular dynamics: time and length scales In order to properly reproduce fracture related properties of covalent materials of group IV materials (Si, Ge, C) it is necessary to take into account interactions as long as the second nearest neighbors distance A. Mattoni, M. Ippolito and L. Colombo, B 76, 224103 (2007) Reliability of the model potentials

10. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Computational Effort CMPTool: a set of highly efficient parallel numerical libraries for computational materials science developed in collaboration with Caspur, Rome Group of materials science (M. Rosati, S. Meloni, L. Ferraro, M. Ippolito) Typical simulation parameters number of atoms > 10 5 Runs as long as 6 10 6 iterations (6 ns) 1ns annealing of 100000 atoms requires of the order of 1000 CPU hours on state-of-the-art AMD - Opteron Dual core Linux cluster A. Mattoni et al. Comp. Mat. Sci. 30 143 (2004) S. Meloni et al. Comp. Phys. Comm. 169 462 (2005)

11. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Nanocrystalline materials Crystalline materials 0-D Points: I,V, clusters, dots Lines: Dislocations 1-D Interfaces: Grain boundaries 2-D 3-D Amorphous materials In the amorphous phase (isotropic) the concept of dislocation is lost The microstructure evolution is controlled by: Recrystallization, normal grain growth Plastically deformed materials Ion implantation

12. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Mixed phase nanocrystalline systems Nanocrystalline materials (nc-Si) may be prepared through the crystallization of amorphous (disordered) nc grains are embedded into a second phase matrix Experimentally it is found that the smallest grain size is obtained when the amorphous samples are annealed at a crystallization temperature that is close to half the bulk melting temperature Q. Jiang, J. Phys.: Condens. Matter 13 (2001) 5503– 5506 nc Embedding amorphous matrix

13. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Nc-Si for photovoltaics Nano-crystalline silicon (nc-Si) consists in a distribution of grains embedded into an amorphous matrix Observation of domains separated by amorphous boundaries and (in some cases texturing) Bright field TEM micrograph S. Pizzini et al.Mat. Sci. Eng. B 134 p. 118 (2006)

14. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Modeling the a-/nc- evolution During annealing of amorphous bulk it is difficult to deconvolve nucleation from growth (impurities, control the temperatures, grains impingement) C. Spinella et al. J. Appl. Phys. 84 5383 (1998) Atomistic simulation as a tool to perform numerical experiment under perfectly controlled conditions of temperature and purity What is the equation of motion of an isolated a-c boundary (planar or curved)? Silicon as a prototype of a covalently bonded material Mattoni and Colombo, Phys. Rev. Lett. 99, 205501 (2007)

15. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Why does a grain grow? a-Si/c-Si is a metastable system M. G. Grimaldi et al. Phys. Rev. B 44 1546 (1991) ~ 0.1 eV/atom 1 kJ/mole=1.03 10 -2 eV/atom

16. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Driving force p a-c Driving force : specific free-energy difference

17. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Equation of motion Transition state theoryInterface limited growth Equation of motion of the a-c displacement a-Si c-Si

18. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Transition State Theory a-Si c-Si

19. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Curved a-c boundary The capillarity is expected to be sizeable up to R~R * and there give rise to an Accelerated -> uniform growth In silicon R * < 1 nm

20. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Planar a-c boundary Uniform motion: the a-c velocity is constant A. Mattoni et al. EPL 62 862 (2003) Exponential dependence on T with E b =2.6eV EXP G. L. Olson Mater Sci. Rep. 3, (1988) AS N. Bernstein et al. PRB 61 6696 (2000 )

21. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Curved a-c boundary c-Si/a-Si: Isolated Crystalline fiber embedded into the amorphous phase nc-Si/a-Si: Crystalline fiber embedded into an amorphous phase [1 0 0] case

22. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Characterization of the a-nc system

23. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Structure Factor T/T m 1.00.51.50.0 amorphous

24. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Analysis

25. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Crystallinity Crystallinity of a mixed a-Si/nc-Si: relative number of crystalline atoms

26. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Fiber recrystallization

27. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Power law model Power law model the model describes both decreasing and increasing nonuniform growth

28. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Fiber recrystallization

29. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Fiber recrystallization There is a dependence of the growth exponents on temperature and there is a clear transition close to the amorphous melting

30. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Fiber recrystallization

31. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Fiber recrystallization

32. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Characterization of defects

33. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS A simple explanation

34. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Conclusions Molecular dynamics simulation are emerging as a powerfool tool to help the characterization of the microstructure evolution of nanostructured materialsMolecular dynamics simulation are emerging as a powerfool tool to help the characterization of the microstructure evolution of nanostructured materials An atomically informed continuum model is found to describe recrystallization in both the cases of isolated grain and distribution of grainsAn atomically informed continuum model is found to describe recrystallization in both the cases of isolated grain and distribution of grains Contact: alessandro.mattoni@dsf.unica.it EU-STREP “NANOPHOTO” CASPUR-ROME and CINECA-BOLOGNA computational support A. Mattoni and L. Colombo, Phys. Rev. Lett. 99, 205501 (2007) M. Fanfoni and M. Tomellini, Phys. Rev. B 54, 9828 (1996) www.dsf.unica.it/colombowww.dsf.unica.it/colombo) C. Spinella et al. J. Appl. Phys. 84 5383 (1998)

35. SLACS-INFM/CNR Rome, December 13, 2007 MATHEMATICAL MODELS FOR DISLOCATIONS Recrystallization