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Thermal properties from first principles with the use of the Free Energy Surface concept

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. Plan. Temperature dependent structural properties from first principles

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Thermal properties from first principles with the use of the Free Energy Surface concept

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

  2. Plan • Temperature dependent structural properties from first principles • The Free Energy Surface Method • Example: fcc Al • Example: Al(110) surface • 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: 2. Relaxation of the remaining degrees of freedom The Potential Energy Surface (PES) The Kohn-Sham total energy Useful features: generalized forces generalized elastic constants stable/metastable phases lack of stability

  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 positionof 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) generalized elastic constants (temperature dependent) stable phases lack of stability

  8. Example: fcc Al volume the Free Energy Surface (Helmholtz free energy) 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 • Direct method (dashed) • DFPT (solid) • Experiment (circles) Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112,  p.513 (2007)

  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 • asymmetry of PES • thermal expansion of bulk-substrate • entropy driven strctural changes The effect of thermal expansion of bulk-substrate

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

  18. B 2 2 1 3 3 1 The effect of PES asymmetry Thermodynamical average (dynamics limited to the configurational space of constraints)

  19. B 2 2 1 3 3 1 The entropy-driven effect – dynamics

  20. B 2 2 1 3 3 1 The entropy-driven effect – Free Energy Surface

  21. B 2 2 1 3 3 1 Final result, d12 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. entr. asym. bulk Bulk-substrate expansion effect dominant

  22. B 2 2 1 3 3 1 Final result, d23 entr. Entropy-driven effect dominant asym. bulk

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

  24. B 2 2 1 3 3 1 Electronic density (averaged over the surface cell)

  25. B 2 2 1 3 3 1 Anisotropic surface melting

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

  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 Thank you for your attention

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