1. Thermal properties from first principles with the use of the Free Energy Surface concept Dr inż. Paweł Scharoch Institute of Physics, Wroclaw University of Technology 27th Max Born Symposium, Wroclaw 2010

2. Plan 1.Temperature dependent structural properties from first principles 2.The Free Energy Surface Method 3.Example: fcc Al 4.Example: Al(110) surface 5.Summary

3. Temperature dependent structural properties from first principles – big challenge Canonical ensemble Partition function Scanning the phase space: deterministic (Molecular Dynamics) or stochastic (Monte Carlo) methods If from first principles: very large computer resourses needed

4. The Free Energy Surface Method (FES) Step 1 — constrained relaxation 1. Imposing on a system the constraints described by the parameters: The Kohn-Sham total energy The Potential Energy Surface (PES) Useful features: generalized forces generalized elastic constants stable/metastable phases lack of stability 2. Relaxation of the remaining degrees of freedom

5. Examples of constraints -> generalized forces -> generalized elastic constants volume -> pressure -> bulk modulus strain tensor -> stress tensor -> elastic tensor surface area (interface) -> surface tension -> surface elastic constant planar position of an adsorbate atom -> force on the atom parallel to the surface -> force constant structural transformation path -> forces along the path -> force constants other constraints… -> … -> …

6. The Free Energy Surface Method (FES) Step 2 — constrained dynamics The ions can move in the configurational space limited by constraints -> dynamics/thermodynamics analysis This can be done within the harmonic approximation The force constants matrix: The dynamical matrix: Polarizations and frequencies of normal modes:

7. The Free Energy Surface Method (FES) Step 3 — constrained thermodynamics Canonical ensemble Partition function: Free energy The Free Energy Surface (FES) Features generalized forces (temperature dependent) stable phases lack of stability generalized elastic constants (temperature dependent)

8. Example: fcc Al the Free Energy Surface (Helmholtz free energy) volume pressure bulk modulus (temperature dependent) lattice parameters (thermal dilation) (the quasiharmonic approximation)

9. fcc Al: Potential Energy Surface LDA GGA Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112, p.513 (2007)

10. fcc Al: phonon dispersion curves Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112, p.513 (2007) Direct method (dashed) DFPT (solid) Experiment (circles)

11. fcc Al: the Free Energy Surface Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112, p.513 (2007)

12. fcc Al: thermal linear expansion curve Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112, p.513 (2007)

13. fcc Al: bulk modulus Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112, p.513 (2007)

14. Al(110) surface – experimental facts Temperature-dependent multilayer relaxation premelting (anisotropic surface melting)

15. Ab initio modelling of Al(110) surface Repeated slab geometry Approximations/computational parameters LDA norm-conserving pseudopotential number of monolayers  11 1 atom per layer vacuum  11 Å cut-off energy  20 Hartree Monkhorst-Pack mesh  (8,12,1) fermi smearing  0.006 Hartree dynamics in the point Γ of BZ polynomial interpolations: (PES- 3rd order, phonons-2nd order) Scharoch Phys.Rev. B80, 125429 (2009)

16. Mechanisms responsible for the observed effects 1.asymmetry of PES 2.thermal expansion of bulk-substrate 3.entropy driven strctural changes The effect of thermal expansion of bulk-substrate

17. Choice of constraints 11-atom supercell – examples of constraints α (schematic view) 1 2 3 4 A 1 2 331 2 B

18. The effect of PES asymmetry Thermodynamical average 1 2 331 2 B (dynamics limited to the configurational space of constraints)

19. The entropy-driven effect – dynamics 1 2 331 2 B

20. The entropy-driven effect – Free Energy Surface 1 2 331 2 B

21. Final result, d12 1 2 331 2 B Experiment Gobel and P. von Blanckenhagen, Phys. Rev. B 47, 2378 (1993) Mikkelsen, J. Jiruse, and D. L. Adams, Phys. Rev. B 60, 7796 (1999) Ab initio MD Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, Phys.Rev. Lett. 82, 3296 1999. Bulk-substrate expansion effect dominant entr. asym. bulk

22. Final result, d23 1 2 331 2 B Entropy-driven effect dominant entr. asym. bulk

23. Final result, d34 1 2 331 2 B All the 3 effects cancel out entr. asym. bulk

24. Electronic density (averaged over the surface cell) 1 2 331 2 B

25. Anisotropic surface melting 1 2 331 2 B

26. Polarization of the modes (0,−0.28,0),(0, 0.31,X),(0, 0.25,X),(0,−0.42,0),(0,−0.06,0),(0, 0.41,0)... (0,0,−0.7),(0,0,X),(0,0,X),(0,0,0.003),(0,0,−0.001),(0,0,0),... softening: hardening:

27. Summary The advantages of the Free Energy Surface method Temperature-dependent structural properties at realistic computational recourses (stable/metastable phases, phase transitions) Different scales (macro, mezo, micro) Different classes of systems (cristal, surface, phase borders) The harmonic approximation often sufficient (even melting !) Relative contribution of different effects visible Can be used at model potentials Can be extended to other perturbations (electric field ?)

28. Thank you for your attention Thank you