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
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
1. Imposing on a system the constraints described by the parameters:
i 
2. Relaxation of the remaining degrees of freedom
The Kohn-Sham total energy
E tot  E PES (i )
The Potential Energy
Surface (PES)
Useful features:
E PES
 i
 i
generalized forces
 2 E PES
 ij
 i  j
generalized elastic constants

s
PES
: E PES  Emin
  2 E PES
det 
  i  j


0

 s  

stable/metastable phases
lack of stability
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
EPES
  (R)
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:
ˆ ( )

Dˆ ( )
Polarizations and frequencies of normal
modes:
 ( )
The Free Energy Surface Method (FES)
Step 3 — constrained thermodynamics
Canonical ensemble
Partition function:
Z ( )   exp[Ei ( )]
i
Free energy
F
FES
( )  kBT ln[Z ( )]
The Free Energy Surface (FES)
Features
F FES
 i
 i
generalized forces (temperature dependent)
 2 F FES
 ij
 i  j
generalized elastic constants (temperature dependent)

s
FES
: F FES  Fmin
  2 F FES
det 
  i  j


0

 s  

stable phases
lack of stability
Example: fcc Al
 V
F FES (V )
F FES

p
V
volume
the Free Energy Surface (Helmholtz free energy)
pressure
 2 F FES
V
 B bulk modulus (temperature dependent)
2
V
V

FES
FES
:
F

F
 lattice parameters (thermal dilation)
eq
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
•
•
•
Direct method (dashed)
DFPT (solid)
Experiment (circles)
Scharoch P, Peisert J, Tatarczyk K; Acta Phys Pol A, 112, p.513 (2007)
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
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)
Repeated slab geometry
Mechanisms responsible for the observed effects
1.
2.
3.
asymmetry of PES
thermal expansion of bulk-substrate
entropy driven strctural changes
The effect of thermal expansion
of bulk-substrate
Choice of constraints
11-atom supercell – examples of constraints α (schematic view)
4
3
2
1
A
B
1
2
3
3
2
1
The effect of PES asymmetry
B
1
2
3
Thermodynamical average
 (T ) 
PES

exp(

E
( ) / k BT )d

PES
exp(

E
( ) / k BT )d

(dynamics limited to the configurational
space of constraints)
3
2
1
The entropy-driven effect – dynamics
B
1
2
3
3
2
1
The entropy-driven effect – Free Energy Surface
B
1
2
3
3
2
1
Final result, d12
B
1
2
3
3
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)
2
1
entr.
asym.

Ab initio MD
Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne,
Phys.Rev. Lett. 82, 3296 1999.
Bulk-substrate expansion effect dominant
bulk
Final result, d23
B
1
2
3
3
2
1
entr.
Entropy-driven effect
dominant
asym.
bulk
Final result, d34
B
1
2
All the 3 effects
cancel out
3
2
3
entr.
1
asym.
bulk
Electronic density (averaged over the surface cell)
B
1
2
3
3
2
1
Anisotropic surface melting
B
1
2
3
3
2
1
  d23
Polarization of the modes
[001]
[1 1 0]
([001],[(1 1 0],[110])
([001],[(1 1 0],[110])
([001],[1 1 0],[110])
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), . . .
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
Thank you for your attention