first-principles thermodynamic calculations in the...

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

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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?

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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

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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

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• 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

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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