1
First-principles thermodynamic calculations in the
harmonic and quasi-harmonic approximations using
Quantum Espresso
M. Palumbo
1 ICAMS, STKS, Ruhr University Bochum, Bochum, Germany
"From First Principles to Multi-Scale Modeling of Materials"
Rio, 22-27th May 2011
2
Introduction
The harmonic approximation
• Basic principles
• Equations of motions
• Density Functional Perturbation Theory (DFPT) or Linear
Response Method
• Examples and limitations
The Quasi Harmonic Approximation (QHA)
• Basic principles
• Examples
The Quantum-Espresso code
Outline
3
Introduction
We will consider solids...
Most solids are crystalline (but not all! amorphous solids...)
Atoms are periodically distributed in a crystalline solid:
phasephase
phase
• Unit cell (type, lattice parameters)
• Basis (number of atoms, atomic type, positions in the unit cell)
4
Born Von-Karmàn rule:
Bloch theorem: , g = l a1 + m a2 + n a3
unit cell
g
a1
a2
a3
crystalline orbitals
Periodic treatment for crystals
5
Atoms move!
Animation by K. Parlinski (Phonon Software Package)
primitive cubic crystal AB
Introduction
Is it samba? Tango? Or maybe salsa?
6
Cp
C
G
FE
DR3
H
A, B
From G. Grimvall, Thermophysical properties of materials, North-Holland 1986
Dulong-Petit law
Debye or
Einstein
(B) phonons contribution (Debye or Einstein
model, Debye approximation for low T)
(A) electronic heat capacity (linear in T,
enhancement factor for phonon coupling)
(C) main contribution from harmonic
vibrational heat capacity. Asymptotic value
3R at high T.
(D) CP-CV contribution. Usually dominated
by vibrational part, linear in T. Higher order
contributions come from higher order
anharmonicity and electronic high T
excitations.
(E) explicit anharmonic contributions to
CV, linear in T to leading order.
(F) electronic contribution, linear in T. No
enhancement factor (λ=0, no phonons-
electrons coupling). Low T data cannot be
extrapolated at high T
(G) correction factor to Cel because of non
constant N(EF) at high T. For some metals,
can be 50% or more of Cel at Tm
(H) contribution to heat capacity because
of vacancies formation, varies exponentially
with Tm/T. Usually low for metals, but
significant close to Tm
Temperature dependence of heat capacity in real metals
7
Two fundamental assumptions:
1. atoms oscillates around a mean equilibrium position which is a
Bravais lattice site
2. typical excursions of each atom from its equilibrium position is
small compared to the interatomic spacing (how much small?)
The harmonic approximation
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The harmonic approximation
9
Linear system of 3N 2nd order differential equations of motion
The equations of motion
10
Phonons solutions
11
The dynamical matrix elements can be calculated by first
principles calculations with 2 methods:
Supercell method (or Small Displacement Method)
Linear Response Method (Density Functional
Perturbation Theory, DFPT)
The dynamical matrix
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• The dynamic matrix is a 3Nat*3Nat Hermitean matrix
• Symmetrical with respect to q → -q change:
• If all atoms are moved by the same displacement d from equilibrium,
the dynamical matrix is zero:
Properties of the dynamical matrix
13
• Set up a “large” supercell
• Displace atoms by a “small” amount
• Linearize and fit atom forces
• Assemble the dynamical matrix
• Diagonalize the dynamical matrix (solve the equations of motion)
- Works with any DFT total energy code
- Needs to check supercell is large enough,
atomic displacements are small enough,…
- Code available: PHON (Dario Alfe’)
- PHONONS (K. Parliski)
The supercell method
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Density Functional Perthurbation Theory (DFPT)
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Density Functional Perthurbation Theory (DFPT)
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Density Functional Perthurbation Theory (DFPT)
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Density Functional Perthurbation Theory (DFPT)
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The DFPT self-consistent cycle
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The ph.x code
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The ph.x code
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The ph.x code
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The ph.x code
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The ph.x code
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The ph.x code
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• The vibrational free energy and all other thermodynamic properties in the
harmonic approximation can be determined from the phonon frequencies
calculated at the equilibrium volume V0
• These frequencies, phonon dispersion curves in the BZ and phonon DOS
can be calculated by first principles by:
Supercell method (or Small Displacement Method)
Linear Response Method (Density Functional Perturbation)
Thermodynamic functions in the harmonic approximation
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Phonons dispersion and DOS calculated by first principles with Quantum-Espresso and the Linear
Response Method
PBE potential, ultrasoft pseudopotentials, Ecut=40 Ry, k-points mesh 8*8*8, q-points mesh (4*4*4)
Exp. Data from R. J. Birgenau et al., Phys Rev 136 (1964) 1359
First principles phonons for Ni: harmonic
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Phonons dispersion and DOS calculated by first principles with Quantum-Espresso and the Linear
Response Method
PBE potential, ultrasoft pseudopotentials, Ecut=100 Ry, k-points mesh 16*16*16, q-points mesh 8*8*8
Exp. data from Trampenau et al. Phys Rev B 47 (1993) 3132 and Shaw et al., Phys Rev B 4 (1971) 969
First principles phonons for Cr: harmonic
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0 250 500 750 1000 1250 1500 1750 2000
-100
-80
-60
-40
-20
0
20
Chromium
Nickel
Fvib
/ J
mol-1
T / K
0 250 500 750 1000 1250 1500 1750 2000
0
5
10
15
20
25
Chromium
Nickel
CV / J
mol-1
K-1
T / K
0 250 500 750 1000 1250 1500 1750 2000
-80
-60
-40
-20
0 Chromium
Nickel
Evib / J
mol-1
T / K
0 250 500 750 1000 1250 1500 1750 2000
0
20
40
60
80
Chromium
Nickel
S / J
mol-1
K-1
T / K
Thermodynamics of Cr and Ni: harmonic approximation
29
0 200 400 600 800 1000 1200 1400
0
5
10
15
20
25
30
35
40
45
35Kee
36Ewe
25Rod
52Bus
55Kra
64Gup
36Syk 1
36Syk 2
Calc. harmonic
34Ahr
27Umi
29Lap
40Per
36Clu
36Bro
65Paw
36Mos
36Bro2
Cp
(J/m
ol K
)
T (K)
The harmonic crystal is quite different from a real crystal:
o Phonons are independent (no phonon-phonon interactions)
o Infinite phonon lifetime
o Infinite thermal conductivity
o No thermal expansion (V=constant)
0 500 1000 1500 2000
0
10
20
30
40
50
60
[34Jae]
[37And]
[79Kem] 10 K/min
[79Kem] 5 K/min
[79Kem] 20 K/min
[65Koh]
[65Kal]
[58Kra]
[52Est]
[79Wil]
[60Bea]
[50Arm]
[58Mar]
[62Clu]
[27Sim]
[56Ray]
Einst+T+T2+T
3 fitted
QE harmonic
Cp
/ J
mol K
-1
T (K)
Limitations of the harmonic approximation
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Phonons calculations are performed at different volumes, then F is minimized
Bulk modulus Thermal expansion Heat capacity
The quasi-harmonic approximation (QHA)
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Thermophysical functions calculated by first principles with Quantum-Espresso and the Linear
Response Method in the quasi-harmonic approximation
0 250 500 750 1000 1250 1500 1750 2000
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
Chromium
Nickel
B0
/ G
Pa
T / K
Bulk modulus
0 250 500 750 1000 1250 1500 1750 2000
0
10
20
30
40
50
60
70
Chromium
Nickel
/ 1
0-6 K
-1
T (K)
Thermal Expansion
0 250 500 750 1000 1250 1500 1750 2000
2.8
3.0
3.2
3.4
3.6
3.8
Chromium
Nickel
a0 / A
T / K
Lattice parameter
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Chromium
Nickel
CP-C
V/ J m
ol-1
K-1
T / K
QHA contribution to heat capacity
Thermophysical properties for Cr and Ni in the QHA
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0 250 500 750 1000 1250 1500 1750
100
120
140
160
180
200
Calc. QHA
52Nei
55Dek
53Lev
51Boz
60Ale
79Lan
89Caq
B0 / G
Pa
T / K
Thermophysical functions calculated by first principles with Quantum-Espresso and the Linear
Response Method in the quasi-harmonic approximation
0 250 500 750 1000 1250 1500 1750
0
10
20
30
40
50
60
70
64Arb
69Pat
69Ric
71Tan
70Gur
71Sha
Calc. QHA
41Nix
54Alt
65Whi
61Fra
65Fra
/ 1
0-6 K
-1
T (K)
Thermal expansion Bulk modulus
Quasi Harmonic Approximation:
comparison with experimental data for Ni
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Thermodynamic functions calculated by first principles with Quantum-Espresso and the Linear
Response Method in the quasi-harmonic approximation
0 200 400 600 800 1000 1200 1400
0
5
10
15
20
25
30
35
40
45
34Ahr
27Umi
29Lap
40Per
36Clu
36Bro
65Paw
36Mos
36Bro2
35Kee
36Ewe
25Rod
52Bus
55Kra
64Gup
36Syk 1
36Syk 2
Calc. harm
Calc. harm+QHA
Cp
(J/m
ol K
)
T (K)
Nickel
0 500 1000 1500 2000
0
10
20
30
40
50
Calc. harm
Calc. harm+QHA
[34Jae]
[37And]
[79Kem] 10 K/min
[79Kem] 5 K/min
[79Kem] 20 K/min
[65Koh]
[65Kal]
[58Kra]
[52Est]
[79Wil]
[60Bea]
[50Arm]
[58Mar]
[62Clu]
[27Sim]
[56Ray]
Cp
/ J
mo
l K
-1
T / K
Chromium
First principles heat capacity for Cr and Ni
(Quasi Harmonic Approximation)
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0 200 400 600 800 1000 1200 1400
0
5
10
15
20
25
30
35
40
45
34Ahr
27Umi
29Lap
40Per
36Clu
36Bro
36Ewe
65Paw
36Mos
36Bro2
35Kee
25Rod
52Bus
55Kra
64Gup
36Syk 1
36Syk 2
Calc. harmonic
Calc. harm+QHA
Calc. harm+QHA+el
Cp
(J/m
ol K
)
T (K)
0 500 1000 1500 2000
0
10
20
30
40
50
60
Calc. harm
Calc. harm+QHA
Calc. harm+QHA+el
[34Jae]
[37And]
[79Kem] 10 K/min
[79Kem] 5 K/min
[79Kem] 20 K/min
[65Koh]
[65Kal]
[58Kra]
[52Est]
[79Wil]
[60Bea]
[50Arm]
[58Mar]
[62Clu]
[27Sim]
[56Ray]
Cp
/ J
mol K
-1
T (K)
The electronic contribution to the heat capacity can be obtained by integrating the first
principles calculated DOS (at constant equilibrium volume).
Nickel Chromium
Computational details:
• Quantum Espresso code
• PBE potential
• Cut off Energy= 100 Ry (Cr), 40 Ry (Ni)
• K points mesh 24*24*24 (Cr), 32*32*32 (Ni)
Electronic contribution
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http://www.quantum-espresso.org/
The Quantum Espresso open source code
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Many modules and tools are available:
pwscf.x
ph.x
matdyn.x
q2r
plotdos
….
GUI
(Graphical User Interface)
The Quantum Espresso open source code
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References
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Introductory tests:
o N. W. Ashcroft, N. D. Mermin, Solid State Physics, Holt, Reinhart and
Winston, 1976
o C. Kittel, Introduction to Solid State Physics, Wiley; 8 edition (November
11, 2004)
o S. Baroni, S. De Gironcoli, A. Dal Corso, P. Giannozzi, Rev. Mod. Phys. 73
(2001) 515
o S. Baroni, P. Giannozzi, E. Isaev, in Theoretical & Computational Methods
in Mineral Physics: Geophysical Applications (2009)
Other useful references
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o PHON, Dario Alfe’, http://www.homepages.ucl.ac.uk/~ucfbdxa/phon/
o PHONONS (K. Parlinski) http://wolf.ifj.edu.pl/phonon/
o VASP 5.2, http://cms.mpi.univie.ac.at/vasp/
o ABINIT, http://www.abinit.org/
Other software codes for phonons calculations
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For support…
ICAMS, International Centre for Advanced
Materials Simulations, Ruhr University Bochum
For comments, discussions, slides,…
Stefano Baroni, SISSA, Trieste, Italy
Andrea Dal Corso, SISSA, Trieste, Italy
Suzana G. Fries, ICAMS, Bochum, Germany
You for listening…
Many thanks to Andrea Dal Corso for providing
some of these slides
Aknowledments
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Characteristic of phonons
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Characteristic of phonons
