An overview of vibrations in solids · B. Monserrat { QMC Apuan Alps IX { July 2014 15/47. The...

An overview of vibrations in solids

Bartomeu Monserrat

QMC in the Apuan Alps IX27 July 2014

Why vibrations in a QMC conference?

I If the required accuracy makes QMC necessary, thenvibrations could also be important.

I QMC band gaps are static, the effects of electron-phononcoupling may be important.

B. Monserrat – QMC Apuan Alps IX – July 2014 1 / 47

Outline

The vibrational energy in solidsTheoretical backgroundApplications

Vibrational coupling in solidsTheoretical backgroundApplications

Conclusions


Outline



Conclusions


Why is the vibrational problem difficult?

Hvib = −1

2

∑Rp,α

1

mα∇2pα + V (rα)

I 3N -dimensional function

I Each data point requires an electronic energy calculation


Harmonic approximation

I Vibrational Hamiltonian in {rα} (or {uα}):

Hharvib = −1

2

∑Rp,α

1

mα∇2pα +

1

2

∑Rp,α;Rp′ ,β

upαΦpα;p′βup′β

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

Hharvib =

∑k,s

(−1

2

∂2

∂q2ks+

1

2ω2ksq

2ks

)


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′) + · · ·

I Static lattice DFT total energy.

I DFT total energy along frozen independent mode.

I DFT total energy along frozen coupled modes.

Features:

I Can be improved systematically.

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

I Estimate of error in anharmonic energy.


Independent term (I)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

-30 -20 -10 0 10 20 30Mode amplitude (a.u.)

0.0

0.5

1.0

1.5

2.0

2.5

BO

ene

rgy

surf

ace

(eV

)

|Φanh|2

|Φhar|2

Cmca-12 vibronic mode


Independent term (II)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

-40 -30 -20 -10 0 10 20 30 40Mode amplitude (a.u.)

0

2

4

6

8

10

BO

ene

rgy

surf

ace

(10-2

eV

)

|Φanh|2

|Φhar|2

I41/amd optical mode


Coupled term (I)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

-40

-30

-20

-10

0

10

20

30

40

-40 -30 -20 -10 0 10 20 30 40

q 2 (

a.u.

)

q1 (a.u.)-40 -30 -20 -10 0 10 20 30 40

q1 (a.u.)-40 -30 -20 -10 0 10 20 30 40

q1 (a.u.)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

BO

ene

rgy

surf

ace

(eV

)


Coupled term (II)

V (q) = V (0) +∑k,s

Vks(qks) +1

2

∑k,s

∑k′,s′

′Vks;k′s′(qks, qk′s′) + · · ·

-40

-30

-20

-10

0

10

20

30

40

-40 -30 -20 -10 0 10 20 30 40

q 2 (

a.u.

)

q1 (a.u.)-40 -30 -20 -10 0 10 20 30 40

q1 (a.u.)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

BO

ene

rgy

surf

ace

(eV

)


Vibrational self-consistent field equations

I Vibrational Schrodinger equation:∑k,s

−1

2

∂2

∂q2ks+ V (q)

Φ(q) = EΦ(q)

I Ansatz: Φ(q) =∏

k,s φks(qks)

I 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′)

⟩


Second order perturbation theory

I Second order perturbation theory (similar to MP2).

I Measures the accuracy of the mean-field approach.

I So far small MP2 corrections.

I Can use other methods: whole electronic structure hierarchy.


Anharmonic free energy

I Anharmonic vibrational excited states:

|ΦS(q)〉 =∏k,s

|φSksks (qks)〉

where S is a vector with elements Sks.

I Anharmonic free energy:

Fanh = − 1

βln∑S

e−βES

Single calculation for insulators.


Outline



Conclusions


Solid hydrogen

I Most abundant element in the Universe.

I Hydrogen at high pressure in planetaryinteriors and stars.

I Possibility of exotic phases:high-temperature superconductivity,zero-temperature quantum fluid, . . .


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)


The determination of phase IV

PHYSICAL REVIEW B 86, 059902(E) (2012)

Erratum: Density functional theory study of phase IV of solid hydrogen[Phys. Rev. B 85, 214114 (2012)]

Chris J. Pickard, Miguel Martinez-Canales, and Richard J. Needs(Received 10 August 2012; published 20 August 2012)

DOI: 10.1103/PhysRevB.86.059902 PACS number(s): 62.50.−p, 64.70.K−, 61.50.Ks, 71.15.Mb, 99.10.Cd

We have found an error in some of our results for the Cmca-4 phase, which has resulted in an underestimation of its stability.This error affects Fig. 4 of our paper, and the corrected version is given below. The phonon dispersion relation for Cmca-4 shownin Fig. 3(f) of our paper was calculated correctly. The energies and vibrational data for the other phases were also calculatedcorrectly.

The Pc phase does not have a region of thermodynamic stability in the revised Fig. 4, but our calculations show that at220 GPa and 300 K it is only 0.000 44 eV per proton less stable than Cmca-4, which is the most stable phase under theseconditions. The stability of the Pc phase relative to the other phases increases with temperature, which is consistent with thefact that phase IV is observed at elevated temperatures.1,2 The phase diagram in Fig. 5 of our paper reports results for the C2/c,Cmca-12, and Pc phases that are consistent with those of Liu et al.3

Our main result, that the experimental Raman data for phase IV1,2 are consistent with the formation of a “mixed structure”similar to our Pc structure, still holds. Our conclusion on the thermodynamic stability of the Pc structure is modified so that thePc structure is now predicted to be very close to thermodynamic stability under the conditions explored in recent experiments.1,2

We are grateful to Yanming Ma for bringing the discrepancy between our results and those of Ref. 3 to our attention, and foruseful discussions.

150 200 250 300-2

0

2

4

6

8

10

∆G (

meV

/pro

ton)

DFT

150 200 250 300

Pressure (GPa)

-2

0

2

4

6

8

10ZPE

150 200 250 300-2

0

2

4

6

8

10300 K

(a) (b) (c)C2/cCmca-12PcCmca-4

FIG. 4. (Color online) Enthalpies/free energies of the Cmca-12, Pc, and Cmca-4 structures relative to C2/c for (a) the static latticestructures, (b) with proton ZP motion, and (c) with full vibrational motion at 300 K.

1M. I. Eremets and I. A. Troyan, Nature Mater. 10, 927 (2011).2R. T. Howie, C. L. Guillaume, T. Scheler, A. F. Goncharov, andE. Gregoryanz, Phys. Rev. Lett. 108, 125501 (2012).

3H. Liu, L. Zhu, W. Cui, and Y. Ma, J. Chem. Phys. 137, 074501(2012)

059902-11098-0121/2012/86(5)/059902(1) ©2012 American Physical Society

7→ Pickard, Martinez-Canales, and Needs, Phys. Rev. B 85, 214114 (2012)7→ Pickard, Martinez-Canales, and Needs, Phys. Rev. B 86, 059902 (2012)


Anharmonic vibrations in solid molecular hydrogen

100 150 200 250 300 350 400Pressure (GPa)

0

4

8

12

ZP a

nhar

mon

ic c

orre

ctio

n (m

eV/a

tom

)C2/c-24Cmca-12Cmca-4P21/c-24Pc-48


The phase diagram of solid molecular hydrogen

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

Lloyd-Williams, Monserrat, Lopez Rıos, Drummond, and Needs

See talk on Tuesday at 12:00am:

DFT and QMC calculations of solid molecular hydrogen


Why are snowflakes hexagonal?

0◦C

−5◦C

−10◦C

−15◦C

−20◦C

−25◦C

−30◦CPlates and columns

T < −22◦C

Plates

−22◦C < T < −10◦C

Columns

−10◦C < T < −4◦C

Plates

−4◦C < T < 0◦C

Pictures by Kenneth G. Libbrecht (SnowCrystals.com)


The relative stability of hexagonal and cubic ice

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

Engel, Monserrat, and Needs

See talk on Wednesday at 12:00am:

Anharmonic nuclear motion and the relative stability of hexagonal and cubic ice


Outline



Conclusions


Vibrational coupling to generic physical observables

〈A〉 = 〈Φ(q)|A(q)|Φ(q)〉

I Vibrational wave function: harmonic or anharmonicI Observable coupling:

I Expansion:

A(q) =∑n,k

a(1)nkqnk +

∑n,k

∑n′,k′

a(2)nk;n′k′qnkqn′k′ + · · ·

A(q) =∑n,k

ankq2nk

I Monte Carlo sampling

I Examples: electronic band gaps, chemical shielding tensor



〈A〉 = 〈Φ(q)|A(q)|Φ(q)〉

I Vibrational wave function: harmonic or anharmonic

I Observable coupling:I Expansion:

A(q) =∑n,k

a(1)nkqnk +

∑n,k

∑n′,k′


A(q) =∑n,k

ankq2nk





〈A〉 = 〈Φ(q)|A(q)|Φ(q)〉

I Vibrational wave function: harmonic or anharmonicI Observable coupling:

I Expansion:

A(q) =∑n,k

a(1)nkqnk +

∑n,k

∑n′,k′


A(q) =∑n,k

ankq2nk




Vibrational phase space sampling

〈A〉 = 〈Φ(q)|A(q)|Φ(q)〉

MD/PIMD Random Quadratic


Outline



Conclusions


Electron-phonon coupling in condensed matter

I Coupling to form Cooper pairs in standard superconductivity

− +

−


Electron-phonon coupling in condensed matter

I Coupling to form Cooper pairs in standard superconductivity

I Temperature dependence of band gaps in semiconductors

on April 6, 2014rspa.royalsocietypublishing.orgDownloaded from

7→ Clark, Dean, and Harris, Proc. R. Soc. Lond. A 277, 312 (1964)


Electron-phonon coupling in diamond and silicon

on April 6, 2014rspa.royalsocietypublishing.orgDownloaded from

gap of Ge (see [41] and references therein). They amount to(at Tx6 K): isotope shift Z0.36 meV/amu, renormaliza-tionZK53 meV. The Passler fit to uind(T) yieldsK52 meVfor the renormalization [40], also in good agreement withthe value obtained from the isotope effect. For the direct gapof germanium u0, a value of 0.49 meV/amu is reported in[41]. The corresponding gap renormalization (obtained byusing the MK1/2 law) is K71 meV, whereas a value of K60 meV is obtained by ‘eyeballing’ the linear asymptote ofu0(T) in Fig. 6.44 of [14]. An average renormalization valueof K62 meV has been calculated by LCAO techniques in[17]. This gap exhibits a spin-orbit splitting D0 and thecorresponding isotope effect has been measured for bothcomponents, E0 and E0CD0 [41]. Surprisingly, the effect onthe E0CD0 gap of germanium has been found to be 30%larger than that on E0, a fact which cannot be explained onthe basis of our present understanding of the spin-orbitinteraction. Actually, the authors of [34] have recentlyfound [43] that the spin-orbit split component of the uind ofSi shows the same isotope effects as the main component, afact which suggests that the isotope effect measurements ofE0CD0 presented in [41] should be repeated.

5. Diamond

5.1. Dependence of the edge luminescence frequency ontemperature and isotopic mass

The luminescence at the indirect exciton of diamond

(which is silicon-like) is shown in Fig. 6 [46]. The curve

through the experimental points corresponds to a singleEinstein oscillator. The fitted frequency, 1080 cmK1Z1580 K, is close to the Debye temperature (w1900 K).This figure shows again the fact mentioned in connection

with Fig. 5: in order to obtain the asymptotic behavior of agap for T/N we must either have data for TOTD (not the

case in Fig. 6) or fit the available data to a reliable algebraic

expression, as has been done for Fig. 6. The thick straightline depicts the corresponding asymptote which enables us

to estimate a gap renormalization of 370 meV, much largerthan the corresponding values for Ge and Si !x70 meV". Inorder to corroborate this a priori unexpected result, data onthe corresponding isotope shift come in handy. There are

two stable carbon isotopes, 12C and 13C. In [47] we find aderivative of the gap with respect to M equal to 14G0.8 meV/amu which, using the MK1/2 rule, leads to the

renormalization K2!14!13ZK364 meV, in excellentagreement with the value estimated above. A recent

semiempirical LCAO calculation results in a renormaliza-tion of 600 meV [48], even larger than the experimental

values. The value of the exciton energy shown in Fig. 6,5.79 eV, can be compared with ab initio calculations of the

Fig. 5. Temperature dependence of the indirect gap of silicon. Thepoints are experimental [36], the solid curve represents a single

Einstein oscillator fit to the experimental points. The dashed line

represents the asymptotic behavior at high temperature: its intercept

with the vertical axis allows us to estimate the bare gap and thus thezero-point renormalization due to electron–phonon interaction.

Fig. 6. Energy of the indirect exciton of diamond versus

temperature. The points are experimental. The solid curve

represents an Einstein oscillator fit whereas the dashed linerepresents the asymptotic behavior at high temperature as extracted

from the Einstein oscillator fit. The intercept of the dashed line with

the vertical axis determines the unrenormalized (bare) gap. See text.

M. Cardona / Solid State Communications 133 (2005) 3–18 9

ZP band gap corrections:

I Silicon: −53 meV

I Diamond: about −370 meV

7→ Clark, Dean, and Harris, Proc. R. Soc. Lond. A 277, 312 (1964)7→ Cardona, Solid State Comm. 133, 3 (2005)


Diamond band structure

X Γ L X W L

0

-10

-20

10

20

30

ε nk (

eV)

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


Diamond phonon dispersion

X Γ L X W L0.00

0.04

0.08

0.12

0.16

ωnq

(eV

)



Diamond electron-phonon coupling

DO

S (

arb.

uni

ts)

0 0.04 0.08 0.12 0.16ω (eV)

-20

-15

-10

-5

0

∆ε(ω

) (m

eV)

-38 meV -296 meV



Silicon band structure

X Γ L X W L

0

-10

10

ε nk (

eV)



Silicon phonon dispersion

X Γ L X W L00.00

0.02

0.04

0.06

ωnk

(eV

)



Silicon electron-phonon coupling

DO

S (

arb.

uni

ts)

0 0.01 0.02 0.03 0.04 0.05 0.06ω (eV)

-6

-4

-2

0

∆ε(ω

) (m

eV)

-11 meV -9 meV -40 meV




Diamond

−800

−700

−600

−500

−400

−300

−200

−100

∆E (

meV

)−1

2a∗ 1

2a∗

12b

∗

−12b

∗

Γ

C

Silicon

−140

−120

−100

−80

−60

−40

−20

0

∆E (

meV

)

−12a

∗ 12a

∗

12b

∗

−12b

∗

Γ

Si




Diamond

−800

−700

−600

−500

−400

−300

−200

−100

∆E (

meV

)−1

2a∗ 1

2a∗

12b

∗

−12b

∗

Γ

C




Diamond

−800

−700

−600

−500

−400

−300

−200

−100

∆E (

meV

)−1

2a∗ 1

2a∗

12b

∗

−12b

∗

Γ

C

C

C

C

C

CC

C

C

C

C



Nuclear magnetic resonance (NMR)


Nuclear magnetic resonance (NMR)

Magnetic field

B = Bext −Bind = (1 − σ)Bext

Bind = σBext

Bind(r) =1

c

∫d3r′j(r′) × r− r′

|r− r′|3


L-alanine molecular crystal

L-alanine

N

C

C

HH

H

C

O

O

H

H

H H

1

2

3

4

5

6

7

1 2

3

1

1

2

I Orthorhombic space group P212121 (4 molecules).

I L-alanine (C3H7NO2) has 52 atoms in the primitive cell.


Isotropic shift

1 2 3 4 5 6 7Atom number

-0.8

-0.6

-0.4

-0.2

0.0

∆σis

o (pp

m)

H

1 2 3Atom number

-10

-8

-6

-4

-2

0

∆σis

o (pp

m)

1 1 2

C N O


Shielding anisotropy


-2.0

-1.6

-1.2

-0.8

-0.4

0.0

∆σS

A (

ppm

)

H

1 2 3Atom number

-5

0

5

10

15

20

25

∆σS

A (

ppm

)

1 1 2

C N O


Temperature dependence

Isotropic shift


-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0

∆σis

o (pp

m)

T=0KT=200KT=500K

H

1 2 3Atom number

-20

-16

-12

-8

-4

0

∆σis

o (pp

m)

1 1 2

C N O



-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

∆σS

A (

ppm

)

H

1 2 3Atom number

-5

0

5

10

15

20

25

∆σS

A (

ppm

)

1 1 2

C N O


Anharmonic vibrations

-0.1 -0.05 0 0.05 0.1Hydrogen displacement (A)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Ene

rgy

(eV

)

AnharmonicHarmonic

C

H1

1


Anharmonic coupling to the shielding tensor

Isotropic shift


-0.8

-0.6

-0.4

-0.2

0.0

0.2

∆σis

o (pp

m)

HarmonicAnharmonic

H

1 2 3Atom number

-12

-10

-8

-6

-4

-2

0

∆σis

o (pp

m)

1 1 2

C N O



-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

∆σS

A (

ppm

)

H

1 2 3Atom number

-505

1015202530

∆σS

A (

ppm

)

1 1 2

C N O


Summary

I Anharmonic energy:I Vibrational self-consistent field method for solids.I Phase diagram of high-pressure solid molecular hydrogen.I Relative stability of hexagonal and cubic ice.

I Vibrational coupling:I Quandratic expansion and Monte Carlo.I Electron-phonon coupling in semiconductors.I Vibrational effects on the chemical shielding tensor.


Outlook

I Anharmonic vibrations can be important:I Small enthalpy differences between competing phases.I Harmonic instabilities.I High temperatures.I . . .

I Vibrational coupling can be important:I Zero-point correction and finite temperature.I Any quantity available at the static level can be coupled to

vibrations.I Electronic band gaps, NMR, . . .


Acknowledgements

Richard Needs Neil Drummond Chris Pickard

JonathanLloyd-Williams

Edgar Engel PabloLopez Rıos


Funding


References

I Anharmonic vibrations formalismB. Monserrat, N.D. Drummond, R.J. NeedsPhysical Review B 87, 144302 (2013)

I Metallization of heliumB. Monserrat, N.D. Drummond, C.J. Pickard, R.J. NeedsPhysical Review Letters 112, 055504 (2014)

I Dissociation of hydrogenS. Azadi, B. Monserrat, W.M.C. Foulkes, R.J. NeedsPhysical Review Letters 112, 165501 (2014)

I Electron-phonon couplingB. Monserrat, R.J. NeedsPhysical Review B 89, 214304 (2014)


An overview of vibrations in solids · B. Monserrat { QMC Apuan Alps IX { July 2014 15/47. The...

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Transcript of An overview of vibrations in solids · B. Monserrat { QMC Apuan Alps IX { July 2014 15/47. The...