vibrational properties of solids: theory and applications

Vibrational properties of solids:theory and applications

Bartomeu Monserrat

Fritz-Haber-Institut27 November 2014

Outline

1 Anharmonic vibrational energy and free energyTheoretical frameworkApplications

2 Vibrational coupling in solidsTheoretical frameworkApplications

3 Conclusions

1 / 27

Anharmonic vibrational energy and free energy Theoretical framework

Outline



3 Conclusions

2 / 27


Why is the vibrational problem difficult?

Hvib = −1

2

∑Rp,α

1

mα∇2pα + V (rpα)

3N -dimensional function

Each data point requires an electronic energy calculation

2 / 27


Harmonic approximation

Vibrational Hamiltonian in {rpα} (or {upα}):

Hharvib = −1

2

∑Rp,α

1

mα∇2pα +

1

2

∑Rp,α;Rp′ ,β

upαΦpα;p′βup′β

Normal mode analysis: {upα} −→ {qks}Vibrational Hamiltonian in {qks}:

Hharvib =

∑k,s

(−1

2

∂2

∂q2ks+

1

2ω2ksq

2ks

)

3 / 27


Principal axes approximation to the BO energy surface

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

Static lattice DFT total energy.

DFT total energy along frozen independent mode.

DFT total energy along frozen coupled modes.

Features:

Can be improved systematically.

Subspace with higher N -body terms (e.g. perovskites).

Estimate of error in anharmonic energy.

4 / 27


Vibrational self-consistent field equations

Vibrational Schrodinger equation:∑k,s

−1

2

∂2

∂q2ks+ V (q)

Φ(q) = EΦ(q)

Ansatz: Φ(q) =∏

k,s φks(qks)

Self-consistent equations:(−1

2

∂2

∂q2ks+ V ks(qks)

)φks(qks) = λksφks(qks)

V ks(qks) =

⟨∏k′,s′

′φk′s′(qk′s′)

∣∣∣∣∣∣V ({qk′′s′′})

∣∣∣∣∣∣∏k′,s′

′φk′s′(qk′s′)

⟩

5 / 27


Second order perturbation theory

Second order perturbation theory (similar to MP2).

Measures the accuracy of the mean-field approach.

So far small MP2 corrections.

Can use other methods: whole electronic structure hierarchy.

6 / 27


Anharmonic free energy

Anharmonic vibrational excited states:

|ΦS(q)〉 =∏k,s

|φSksks (qks)〉

where S is a vector with elements Sks.

Anharmonic free energy:

Fanh = − 1

βln∑S

e−βES

Single calculation for insulators.

7 / 27


One-body term (I)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

-5 -4 -3 -2 -1 0 1 2 3 4 5Mode amplitude (1/ 2ω )

0.0

0.5

1.0

1.5

2.0

2.5

BO

ene

rgy

surf

ace

(eV

)

√

Cmca-12

8 / 27


One-body term (II)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

-4 -3 -2 -1 0 1 2 3 4Mode amplitude (1/ 2ω )

0.00

0.02

0.04

0.06

0.08

0.10

BO

ene

rgy

surf

ace

(eV

)

√

I41/amd

z

y

x

9 / 27


Two-body term (I)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

q 2 (1

/}}}32t

2)

q1 (1/}}}32t1)

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4q1 (1/}}}32t1)

-4 -3 -2 -1 0 1 2 3 4q1 (1/}}}32t1)

-4 -3 -2 -1 0 1 2 3 4 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

BO e

nerg

y su

rface

(eV)

10 / 27


Two-body term (II)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

q 2 (1

/}}}32t

2)

q1 (1/}}}32t1)

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4q1 (1/}}}32t1)

-4 -3 -2 -1 0 1 2 3 4 0.000.020.040.060.080.100.120.140.160.180.20

BO e

nerg

y su

rface

(eV)

11 / 27

Anharmonic vibrational energy and free energy Applications

Outline



3 Conclusions

12 / 27


The phase diagram of high pressure hydrogen

Tem

pera

ture

(K

)

Pressure (GPa)

0

200

400

600

800

1000

1200

0 100 200 300 400 500 600 700 800

I

IIIII

IV

Atomic solid

Molecularfluid

Atomic fluidMolecular

Solid

7→ Goncharov et al., Phys. Rev. Lett. 80, 101 (1998)7→ Datchi et al., Phys. Rev. B 61, 6535 (2000)7→ Gregoryanz et al., Phys. Rev. Lett. 90, 175701 (2003)7→ Deemyad and Silvera, Phys. Rev. Lett. 100, 155701 (2008)7→ Howie et al., Phys. Rev. Lett. 108, 125501 (2012)

12 / 27


Phases III and IV candidates

C2/c-24

(phase III)

Pc-48(phase IV)

13 / 27


Molecular high-pressure hydrogen phase diagram

Experiment

100 150 200 250 300 350 400Pressure (GPa)

0

100

200

300

400

500

Tem

pera

ture

(K

) Phase I

PhaseII

Phase III

Phase IV

DFT + Harmonic

100 150 200 250 300 350 400Pressure (GPa)

0

100

200

300

400

500

Tem

pera

ture

(K

)

Cmca-4

C2/c-24

Phase I

models phase III

models ?!

7→ N.D. Drummond, B. Monserrat, J.H. Lloyd-Williams, P. Lopez Rıos, C.J. Pickard, R.J. Needs

14 / 27


Molecular high-pressure hydrogen phase diagram

Experiment

100 150 200 250 300 350 400Pressure (GPa)

0

100

200

300

400

500

Tem

pera

ture

(K

) Phase I

PhaseII

Phase III

Phase IV

DMC + Anharmonic

100 150 200 250 300 350 400Pressure (GPa)

0

100

200

300

400

500

Tem

pera

ture

(K

)

C2/c-24

Pc-48

P21/c-24

Phase I

models phase II

models phase IV

models phase III

7→ N.D. Drummond, B. Monserrat, J.H. Lloyd-Williams, P. Lopez Rıos, C.J. Pickard, R.J. Needs

14 / 27


Why are snowflakes hexagonal?

Ener

gy [m

eV/H

2O]

# of proton-ordering

Ic IhI41m

d

Pc Cmc2

1

Ghar

Ganh

Ghar

Ganh

0

5

10

15

20

0 5 10 15 20 25 30

6minIhAIcGanh

Hexagonal and cubic ice almost degenerate in energyQMC + harmonic vibrations lead to degenerate energiesExperimental estimates: 0.5 – 1.5 meV/H2O energy difference

7→ E.A. Engel, B. Monserrat, R.J. Needs15 / 27

Vibrational coupling in solids Theoretical framework

Outline



3 Conclusions

16 / 27


Vibrational coupling

L0

O(L0)

L1

O(L1)

L2

O(L2)

O depends on L

L follows distribution P (L)

Expectation value:

〈O〉 =

∫dLP (L)O(L)

16 / 27


Vibrational expectation value

〈O〉 = 〈Φ(q)|O(q)|Φ(q)〉

Observable coupling:

Quadratic expansion:

O(q) = O(0) +3N∑i=1

c(1)i qi +

3N∑i=1

3N∑j=1

c(2)ij qiqj + · · ·

〈O〉 = O(0) +3N∑i=1

c(2)ii 〈φi|q

2i |φi〉+O(q4)

Single calculation for all temperatures.

Monte Carlo sampling:

〈O〉 =1

Ns

Ns∑i=1

O(qi) distributed according to |Φ|2

One calculation at each temperature.

17 / 27



〈O〉 = 〈Φ(q)|O(q)|Φ(q)〉Observable coupling:


O(q) = O(0) +

3N∑i=1

c(1)i qi +

3N∑i=1

3N∑j=1

c(2)ij qiqj + · · ·

〈O〉 = O(0) +

3N∑i=1

c(2)ii 〈φi|q

2i |φi〉+O(q4)

Single calculation for all temperatures.

Monte Carlo sampling:

〈O〉 =1

Ns

Ns∑i=1


One calculation at each temperature.

17 / 27



〈O〉 = 〈Φ(q)|O(q)|Φ(q)〉Observable coupling:


O(q) = O(0) +

3N∑i=1

c(1)i qi +

3N∑i=1

3N∑j=1

c(2)ij qiqj + · · ·

〈O〉 = O(0) +

3N∑i=1

c(2)ii 〈φi|q

2i |φi〉+O(q4)

Single calculation for all temperatures.Monte Carlo sampling:

〈O〉 =1

Ns

Ns∑i=1


One calculation at each temperature.17 / 27


Vibrational phase space sampling

〈O〉 = 〈Φ(q)|O(q)|Φ(q)〉

MD/PIMD Random Quadratic

18 / 27

Vibrational coupling in solids Applications

Outline



3 Conclusions

19 / 27


Diamond thermal band gap

0 200 400 600 800 1000Temperature (K)

5.2

5.4

5.6

5.8

6.0E

g (eV

)

Static latticeIncluding el-ph coupling0.462 eV

19 / 27


Diamond thermal band gap

0 200 400 600 800 1000Temperature (K)

5.1

5.2

5.3

5.4

Eg (

eV)

Theory Experiment

7→ Exp. data from Clark, Dean and Harris Proc. R. Soc. London, Ser. A 277, 312 (1964)

19 / 27


Electron-phonon coupling in diamond and silicon

Coupling strength much larger in diamond than in silicon.

Difference arises from localised pockets of the BZ.

C

C

C

C

CC

C

C

C

C

−800

−700

−600

−500

−400

−300

−200

−100

∆E (

meV

)

−12a

∗ 12a

∗

12b

∗

−12b

∗

Γ

C

7→ B. Monserrat, R.J. Needs, Phys. Rev. B 89, 214304 (2014)

20 / 27


Metallisation of helium and the age of the UniverseWhite dwarf cooling:

0 2 4 6 8 10 12 14 16 18 20Time (10

9 years)

0

2

4

6

8

10

Tef

f (10

3 K)

Age

of t

he U

nive

rse

Black dwarfe−

e− e−

γ γ γ

Degenerateelectron gas

Insulating Helium

Metallic Helium

Higher metallisation pressurewhen atomic vibrations areincluded.

Pre

ssur

e (T

Pa)

Temperature (K)

22

24

26

28

30

32

34

36

38

40

0 2000 4000 6000 8000 10000

Insulating solid

Metallic solid

Plasma

No el-ph coupling

7→ B. Monserrat, N.D. Drummond, C.J. Pickard, R.J. Needs, Phys. Rev. Lett. 112, 055504 (2014)21 / 27


Nuclear magnetic resonance: chemical shielding tensor

Bext Bext

B = Bext −Bind = (1 − σ)Bext

Bind = σBext

Bind(r) =1

c

∫d3r′j(r′) × r− r′

|r− r′|3

22 / 27


Finite temperature nuclear magnetic resonance

Strong interplay theory/experiment

Computationally inexpensive temperature effects

1 2 3 4 5-7 8 9 10-12Atom number

-0.6-0.4-0.20.00.20.40.60.81.0

δ theo

ry −

δ exp (

ppm

) StaticT=293K

Hydrogen

1 2 3 4 5 6 7Atom number

-6

-4

-2

0

2

4

6

δ theo

ry −

δex

p (pp

m) Carbon

RMSD=0.49 ppm

RMSD=0.21 ppm

RMSD=3.5 ppm

RMSD=2.5 ppm

MgO L-alanine bDA0

2

4

6

8

10

Com

puta

tiona

l fac

tor

(103

TN

MR)

QuadraticMC (0.10)MC (0.05)

7→ B. Monserrat, R.J. Needs, C.J. Pickard, J. Chem. Phys. 141, 134113 (2014)

23 / 27

Conclusions

Conclusions

Summary:

Anharmonic vibrational energy of solids.Examples: high-pressure hydrogen, hexagonal ice.Vibrational coupling to physical properties.Examples: electron-phonon coupling, NMR.

Conclusions:

Anharmonicity is important when different phases have small enthalpydifferences.Vibrational expectation values have not been extensively explored andcan be very important.

24 / 27

Conclusions

Vibrations in solids: what next?

Methods:

Use of forces.Better choice of supercell (J.H. Lloyd-Williams).Exchange symmetry in wave function: nuclear spin.

Applications:

NMR: J-coupling, electric field gradient.Thermal expansion/contraction.El-ph: can we go beyond Born-Oppenheimer?

25 / 27

Conclusions

Acknowledgements

Richard Needs Chris Pickard Neil Drummond

JonathanLloyd-Williams

Edgar Engel PabloLopez Rıos

26 / 27

Conclusions

Funding

27 / 27

vibrational properties of solids: theory and applications

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