Thermal properties from first principles with the use of the Free

EnergySurface concept

Dr inż. Paweł ScharochInstitute of Physics, Wroclaw University of Technology

27th Max Born Symposium, Wroclaw 2010

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

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

The Free Energy Surface Method (FES)Step 1 — constrained relaxation

i

)( iPEStot EE

ii

PESE

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

ijji

PESE

2

generalized elastic constants

PESPESs EE min: stable/metastable phases

0det2

sji

PESE

lack of stability

2. Relaxation of the remaining degrees of freedom

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… -> … -> …

The Free Energy Surface Method (FES)

Step 2 — constrained dynamics

)(RE PES

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: )(ˆ

The dynamical matrix: )(ˆ D

Polarizations and frequencies of normal modes: )(

The Free Energy Surface Method (FES)

Step 3 — constrained thermodynamics Canonical ensemble

Partition function:

Free energy

i

iEZ )](exp[)(

)](ln[)( ZTkF BFES

The Free Energy Surface (FES)

ii

FESF

Features

generalized forces (temperature dependent)

ijji

FESF

2

FESFESs FF min: stable phases

0det2

sji

FESF

lack of stability

generalized elastic constants (temperature dependent)

Example: fcc Al

V

)(VF FES the Free Energy Surface (Helmholtz free energy)

volume

pV

F FES

pressure

BV

FV

FES

2

2

bulk modulus (temperature dependent)

lattice parameters (thermal dilation) FESFESeq FFV min:

(the quasiharmonic approximation)

fcc Al: Potential Energy Surface

LDA

GGA

Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112,  p.513 (2007)

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)

fcc Al: the Free Energy Surface

Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112,  p.513 (2007)

fcc Al: thermal linear expansion curve

Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112,  p.513 (2007)

fcc Al: bulk modulus

Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112,  p.513 (2007)

Al(110) surface – experimental facts

• Temperature-dependent multilayer relaxation

• premelting (anisotropic surface melting)

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)

Mechanisms responsible for the observed effects

1. asymmetry of PES2. thermal expansion of bulk-substrate 3. entropy driven strctural changes

The effect of thermal expansion of bulk-substrate

Choice of constraints

11-atom supercell – examples of constraints α (schematic view)

123

4

A

123 31 2B

The effect of PES asymmetry

dTkE

dTkET

BPES

BPES

)/)(exp(

)/)(exp()(

Thermodynamical average

123 31 2B

(dynamics limited to the configurational space of constraints)

The entropy-driven effect – dynamics

123 31 2B

The entropy-driven effect – Free Energy Surface

123 31 2B

Final result, d12

123 31 2B

ExperimentGobel 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 MDMarzari, D. Vanderbilt, A. De Vita, and M. C. Payne,

Phys.Rev. Lett. 82, 3296 1999.

Bulk-substrate expansion effect dominant

entr.

asym.

bulk

Final result, d23

123 31 2B

Entropy-driven effect dominant

entr.

asym.

bulk

Final result, d34

123 31 2B

All the 3 effects cancel out

entr.asym.

bulk

Electronic density (averaged over the surface cell)

123 31 2B

Anisotropic surface melting

123 31 2B

23d

Polarization of the modes

])110[],011[(],001([ ])110[],011[(],001([]011[

(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), . . .

]001[

])110[],011[],001([

softening:

hardening:

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 ?)

Thank you for your attention

Thank you