Thor WikfeldtScience Institute, Univ. of IcelandNordita, Stockholm

NQEs in Organic H-bonded Ferroelectrics

kOO

kHH

kOO

kHH

V(s)

s

http://meetings.aps.org/Meeting/MAR12/Event/164320http://meetings.aps.org/Meeting/MAR12/Event/164320http://meetings.aps.org/Meeting/MAR12/Event/164320http://meetings.aps.org/Meeting/MAR12/Event/164320

Ferroelectrics

S. Horiuchi and Y. Tokura Nature Mat. 7, 357 (2008)

Inorganic

OrganicS. Horiuchi et al. Nature 463, 789 (2010)

• Polarity reversal (FeRAM), optoelectronics, pyroelectricity, piezoelectricity• Organic ferroelectrics: low mass, cheap, flexible, non-toxic

H-bonded ferroelectrics

S. Horiuchi and Y. Tokura Nature Mat. 7, 357 (2008)

Inorganic

OrganicS. Horiuchi et al. Nature 463, 789 (2010)

• Unresolved fundamental questions‣ Nature of ferroelectric-paraelectric transition at TC:

displacive vs order-disorder, cooperative effects, domain dynamics

‣ Isotope effects, TC shifts by ~100 K upon deuteration: quantum tunneling or geometrical isotope effect?

• Polarity reversal (FeRAM), optoelectronics, pyroelectricity, piezoelectricity• Organic ferroelectrics: low mass, cheap, flexible, non-toxic

H-bonded ferroelectrics

• Squaric acid (H2SQ) is an ideal simple model system• Croconic acid (H2CR) potentially useful high-T ferroelectric

cb

a

S. Horiuchi et al. Nature 463, 789 (2010)

• Simulate H2SQ and H2CR by density functional theory and PIMD!K.T. Wikfeldt & A. Michealides, JCP 140, 041103 2014

DFT: Importance of dispersion interactions

Dion et al. PRL 92 246401 (2004)Klimes et al. J. Phys. Cond. Matt. 22, 022201 (2010)

Squaric acid Croconic acid

• PBE overestimates interlayer separation by 10-15%• vdW-functionals more accurate, perform similarly

Proton transfer barriers - squaric

Functional PBE vdW-DF optPBEvdW vdW-DF2 MP2 RPA

Barrier (meV) 18 31 66 87 110 148

• Compare DFT barriers with higher-level theory: - random phase approximation (RPA) - Møller-Plesset 2nd order perturbation theory (MP2)

0 10 20 30 40 50 60 70 80 90

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E [m

eV/p

roto

n]

d [Å]

vdW-DF2

Proton transfer barriers - squaric

Functional PBE vdW-DF optPBEvdW vdW-DF2 MP2 RPA

Barrier (meV) 18 31 66 87 110 148

• Compare DFT barriers with higher-level theory: - random phase approximation (RPA) - Møller-Plesset 2nd order perturbation theory (MP2)

0 10 20 30 40 50 60 70 80 90

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E [m

eV/p

roto

n]

d [Å]

vdW-DF2

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3 3.5

E [m

eV/p

roto

n]

d [Å]

vdW-DF2PBE

optPBE-vdW

• Both H-bonding chains must reverse to reach second ground state!

Proton transfer barriers - croconic

Ab initio path-integral molecular dynamics

• Feynman’s path integrals powerful approach to simulate nuclear quantum effects

Quantum effects on H-bond geometry

0 100 200 300 400 500 600 700T [K]

vdW-DF2

PBE 2.46 2.48 2.50 2.52 2.54 2.56 2.58 2.60 2.62 2.64 2.66

d OO [

Å]

expt(H2SQ)

expt(D2SQ)

H2SQD2SQMD

dOH

dOO

Semmingsen et al. J. Chem. Phys., 66, 4405, (1977) McMahon et al. Z. Kristallogr. 195, 231 (1991)

Squaric acid

• What is the mechanism behind geometrical isotope effects?

Proton/deuteron ordering

• Larger delocalization of H than D, stronger attraction, shorter H-bonds

2δ

100K 200K 300K 400K 500K

-0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4δ [Å] δ [Å]

0

1

2

3

4

5

P(δ)

0 2 4 6 8

10

H2SQ D2SQ

MD

P(δ)

(a) (b)

0 0.4 0.2

Squaric acid

X. Z. Li, B. Walker & A. Michaelides, PNAS 108, 6369 (2011)

Proton/deuteron ordering

• Ordering behavior affected, ~200 K diff between H2SQ and D2SQ • Experiments: TC(H2SQ) = 373 K, TC(D2SQ) = 520 K

2δ

100K 200K 300K 400K 500K

-0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4δ [Å] δ [Å]

0

1

2

3

4

5

P(δ)

0 2 4 6 8

10

H2SQ D2SQ

MD

P(δ)

(a) (b)

0 0.4 0.2

Squaric acid

Proton/deuteron ordering

• Does quantum tunneling play a role?

2δ

100K 200K 300K 400K 500K

-0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4δ [Å] δ [Å]

0

1

2

3

4

5

P(δ)

0 2 4 6 8

10

H2SQ D2SQ

MD

P(δ)

(a) (b)

0 0.4 0.2

Squaric acid

Quantum tunneling

RG

2δ

-0.4-0.2 0.0 0.2

0.06 0.08 0.1

0.12 0.14

1000 2000 3000

R g [

Å]δ c

[Å]

-0.3 -0.2 -0.1 0 0.05 0.06 0.07 0.08 0.09

0.1 0.11 0.12 0.13 0.14 0.15

-0.3 -0.2 -0.1 0

100K200K500K

δc [Å] δc [Å]

[Å]

H2SQ D2SQ

PIMD step

(a) (b)

RgRg

4000

0.4

H2SQ200 K

H1H2H3

δc0

Rg

RgH1 H3H2

• vdW functionals crucial for interlayer interactions and proton transfer barriers in H-bonded ferroelectrics. vdW-DF2 appears best

• Two uncoupled H-bonding chains in squaric acid, 1-D problem • Coupled H-bonding chains in croconic acid • Squaric acid PIMD reproduces geometrical isotope effects and Tc shift• Quantum tunneling contributes directly to geometrical effect by

tunneling of H/D into central barrier

• Combined thermal excitations and tunneling in collective H atom jumps in paraelectric phase

• Future work: PIMD momentum distributions vs experiment, simple Hamiltonian model for larger and longer simulations of H-bond chains

Summary

This work published in: K.T. Wikfeldt & A. Michealides, JCP 140, 041103 2014