Lattice Dynamics, Thermal Properties, and Density ...

Lattice Dynamics, Thermal Properties, and Density Functional Perturbation Theory

Ronald CohenGeophysical Laboratory

Carnegie Institution of [email protected]

2007 Summer School on Computational Materials ScienceQuantum Monte Carlo: From Minerals and Materials to MoleculesJuly 9 –19, 2007 • University of Illinois at Urbana–Champaignhttp://www.mcc.uiuc.edu/summerschool/2007/qmc/

Lattice Dynamics, Thermal Properties, and Density Functional Perturbation Theory

• Why you need estimates of thermal corrections• Thermodynamics of collection of oscillators• Mathematical description of crystals (review)• Atomic displacement waves (phonons)• Dynamical matrix• Secular equation (equations of motion)• Densities of states• Kieffer models• Examples• Ionic systems—non-analyticity• Born effective charges• Linear response• Density functional perturbation theory• Process for DFPT computations

Kieffer models1. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals: 1. Mineral heat capcities and their relationships to simple lattice vibrational models. Reviews of Geophysics and Space Physics, 1979. 17: p. 1-19.2. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals: 2. Vibrational characteristics of silicates. Reviews of Geophysics and Space Physics, 1979. 17: p. 20-34.3. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals:3. Lattice dynamics and an approximation for minerals with application to simple substances and framework silicates. Reviews of Geophysics and Space Physics, 1979. 17: p. 35-59.4. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals:4. Application to chain and sheet silicates and orthosilicates. Reviews of Geophysics and Space Physics, 1980. 18: p. 862-886.5. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals:5. Applications to phase equilibria, itostopic fractionation, and high-pressure thermodynamic properties. Reviews of Geophysics and Space Physics, 1982. 20: p. 827-849.

Entro

py

Temperature%

err

or

Matas, J., et al., Thermodynamic properties of carbonates at high pressures from vibrational modelling. Eur J Mineral, 2000. 12(4): p. 703-720.

2

Calculated phonon frequencies of bcc, fcc and hcp Fe all show excellent agreements with experiment

0

5

1 0

1 5

2 0

2 5

3 0

3 5L

0 .00 .0 0 .50 .5 1 .0

ω (m

eV)

[1 1 1 ]

Λ

0 .2 5

X

T A

L A

Γ

T A 2

T A 1

L A

T A

Σ

[1 1 0 ]

Γ

L A

0 .5 [0 0 1 ]

Δ

0

50

100

150

200

250

300

0.00.0 0.50.5 1.0

ω (c

m-1

)

[110]

Ν

0.25

TA

LA

Γ

LM T O -G G A Experim ent

P P-G G A P P -LD A

TA2

TA1

LA

TA

Η

[100]

Γ

LA

0.5 [111]

Ρ

. fcc Fe

bcc Fe

0 5 1 0 1 5 2 0 2 5

V = 4 0 a . u .

V = 6 0 a . u . E x p : 5 0 G P a

E (T H z )

V = 7 0 a . u .hcp Fe

sound velocities along the sound velocities along the HugoniotHugoniot

Experiments: Nguyen and Holmes, Nature 2004 Theory: Sha and Cohen

50 100 150 200 250 300 3507.0

7.5

8.0

8.5

9.0

9.5

10.0

Vp

Sou

nd V

eloc

ity (k

m s

-1)

Pressure (GPa)

VB

Temperature in EarthTemperature in Earth’’s inner Cores inner Core

0 2000 4000 60000

250

500

750

1000

1250

1500

Steinle-Neumann et al. Nature 2001

Elas

tic M

odul

i (G

Pa)

Temperature (K)

Ks

μSha and Cohen

Cohen: November, 2006Cohen: November, 2006MRSMRS 1212

Thermoelasticity of bcc FeThermoelasticity of bcc Fe65 au

70 au

80 au75 au

85 au

0 GPa10 GPa20 GPa30 GPa40 GPa

Change in elastic constants withtemperature at constant V (beyond normal quasiharmonic approximation)

Exp: Dever (1972) J. Appl. Phys. 43: 3293; Isaak and Masuda (1995) JGR 100(B9): 17689.

LiNbO3 Thermal Expansivity from Linear Response (LDA) vs. Experiment

Born Effective Charges

P3(1), z(1)

displacements=0.3%

P3(2), z(2)

)1()2()1()2( 333*

zzPP

ezP

eZ

−−Ω

=∂∂Ω

=

Pb

Ti O1

O3

(1/2 1/2 0.530) (1/2 0 0.610)

(1/2 1/2 0.105)(0 0 0)

-2.61⊥ 0 0

0 -5.18⎪⎪ 0

0 0 -2.16

-2.15 0 0

0 -2.15 0

0 0 -4.38

Born effective charge tensors for tetragonal PbTiO3

3.74 0 0

0 3.74 0

0 0 3.52

Effective charges are tensors

6.2 0 0

0 6.2 0

0 0 5.2

PRB, 58, 6224, 1998

Effective chargesEffective charges can be greatly enhanced compared to nominal charges.