Theory of Relaxor Ferroelectrics - Argonne National Laboratory

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Theory of Relaxor Ferroelectrics
Gian G. Guzmán-Verri 1,2 & Chandra M. Varma 2
1 Materials Science Division, Argonne National Laboratory
2 Department of Physics and Astronomy, UC Riverside
MTI Non-conventional Insulators Workshop
11/14/2012
Work partially funded by the UC Lab Fees Research Program
09-LR-01-118286-HELF

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Why are relaxors interesting? New physics & technologically important
E ⩵0
1 Ε, relaxors
Ε
E E c
T fe T max T CW T B
Broad region of fluctuations (diffuse phase transition).
Several special temperatures.
Neutron diffuse scattering exhibits unusual line shapes.
Widely used for capacitors.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Outline
1 Ferroelectrics vs Relaxor Ferroelectrics
Unit Cell
Dielectric Constant
Soft Modes
Diffuse Scattering
2 Theory of Relaxor Ferroelectrics
Model Hamiltonian
Lorentz and Onsager Approximations
3 Results
Soft Mode
Diffuse Scattering
Dielectric Constant
4 Conclusions

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Outline
1 Ferroelectrics vs Relaxor Ferroelectrics
Unit Cell
Dielectric Constant
Soft Modes
Diffuse Scattering
2 Theory of Relaxor Ferroelectrics
Model Hamiltonian
Lorentz and Onsager Approximations
3 Results
Soft Mode
Diffuse Scattering
Dielectric Constant
4 Conclusions

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Unit Cell
Unit Cell
Complex perovskite: ABO 3 , B-site disordered.
e.g.: PbMg 1/3 Nb 2/3 O 3 . (PMN).
Mg +2 and Nb +5 are non-isovalent.
Ionic radii: 0.72Å (Mg +2 ) and 0.64Å (Nb +5 ).
Ferroelectrics: B-site ordered.
e.g.: PbTiO 3 (PTO), BaTiO 3 (BTO).
Compositional disorder is essential.
Pb 4
Mg 2 Nb 5
O 2

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Dielectric Constant
Dielectric Constant
J. Remeika et al., Mater. Res. Bull. 5, 37 (1970); V. Bovtun et al., Ferroelectrics 298, 23 (2004)
Ε10 3
8.0
6.0
4.0
2.0
PbTiO 3
T 715 K
5.0 10 5 K
0.8
0.6
0.4
0.2
Ε 1 10 3
Ε10 3
15.0
10.0
5.0
PMN
10 2 Hz
10 3 Hz
10 5 Hz
10 9 Hz
T 410 K
1.2 10 5 K
2.5
2.0
1.5
1.0
0.5
Ε 1 10 3
500 550 600 650 700 750 800 850
T K
200 300 400 500 600 700
T K
sharp peak
follows Curie-Weiss law
little frequency dispersion (< 10 12 Hz)
broad peak
deviates Curie-Weiss law (T < T B )
strong frequency dispersion

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Soft Modes
Soft Modes (Ferroelectrics)
P. W. Anderson, Fizika Dielektrikov, Akad. Nauk. (1960); W. Cochran, Phys. Rev. Lett. 3 412, (1959);
G. Shirane et al., Phys. Rev. B. 2, 155 (1970)
60
ħTO 2 meV 2
50
40
30
20
10
PTO, q ⩵0
T c
700 800 900 1000 1100
Inversion symmetry must be broken.
Lattice instability (TO mode softens).
T K
Lyddane-Sachs-Teller: ɛ(0)
ɛ(∞) = ( ΩLO
Ω TO
) 2
∝ 1
T T c
Pb 4
Ti 2
O 2
T −Tc .
T T c

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Soft Modes
Soft Modes (Relaxors)
C. Stock et al., J. Phys. Soc. Jpn. 74, 3002 (2005).
ħTO 2 meV 2
120
100
80
60
40
20
T CW
PMN,q ⩵0
T B
0
0 200 400 600 800 1000
T K
Inversion symmetry is not broken macroscopically.
Minimum at T CW .
Overdamped for 200 K T 620 K.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering
Diffuse Scattering: Ferroelectrics vs Relaxors
Y. Yamada et al., Phys. Rev. Lett. 177, 848 (1969); S. N. Gvasaliya et al., J. Phys.: Condens. Matter 17, 4343 (2005)
Internsitya.u.
Internsitya.u.
Internsitya.u.
BTO,T 393K
10 6 PMN,T ⩵ 450 K
10 5 Bragg Peak
10 5
Diffuse Scatt.
10 4
10 3
10 3
10 2
10 1
10 1
100.50.40.30.20.10.00.10.20.30.40.5
7 0.50.40.30.20.10.00.10.20.30.40.5
BTO,T0,1q,0 ≃393K r.l.u. 10 6 PMN,T ⩵1,1,q 300 K r.l.u.
10 7 10 7 10 5
10 5
10 3
10 4
10 3
10 2
10 1
10 1
100.50.40.30.20.10.00.10.20.30.40.5
7 0.50.40.30.20.10.00.10.20.30.40.5
BTO,T0,1q,0 ≲393K r.l.u. 10 6 PMN,T ⩵1,1,q 150 K
r.l.u.
10 5
10 5
10 4
10 3
10 3
10 2
10 1
0.5 0.25 0.0 0.25
10 1
0.5 0.5 0.25 0.0 0.25 0.5
0,1q,0 r.l.u.
1,1,q r.l.u.
Internsitya.u.
Internsitya.u.
Internsitya.u.
Elastic diffuse scattering in PMN does not behave as a simple Lorentzian.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering
Diffuse Scattering: Ferroelectrics vs Relaxors
Y. Yamada et al., Phys. Rev. Lett. 177, 848 (1969); S. N. Gvasaliya et al., J. Phys.: Condens. Matter 17, 4343 (2005)
Intensity a.u.
T c
BTO
Yamada et al . 101
397 420 450 480
T K
Intensity a.u.
12.0
9.0
6.0
3.0
0.0
0 100 200 300 400 500 600 700
T K
PMN
Hiraka110 ,100
Gvasaliya110
FE fluctuations extend over narrow temperature range.
RFE fluctuations extend over broad temperature range and
saturate.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Outline
1 Ferroelectrics vs Relaxor Ferroelectrics
Unit Cell
Dielectric Constant
Soft Modes
Diffuse Scattering
2 Theory of Relaxor Ferroelectrics
Model Hamiltonian
Lorentz and Onsager Approximations
3 Results
Soft Mode
Diffuse Scattering
Dielectric Constant
4 Conclusions

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Model Hamiltonian
Model Hamiltonian
Anharmonic potential (V (|u i |))+ dipole forces (v ij ) + local quenched random fields (h i )
H = ∑ i
( |Πi | 2
2M
)
+ V (|u i|) − ∑ u i · v ij · u j − ∑
i

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Lorentz and Onsager Approximations
Lorentz and Onsager Fields
Onsager, J. Am. Chem. Soc. 58,1486 (1936); Van Vleck, J. Chem. Phys. 5 320 (1937); Brout and Thomas., Phys. 3, 317 (1967)
Lorentz Field (LF):
E local
i
≃ E Lorentz
i
= ∑ j
Clausius-Mossotti (CM):
ɛ − 1
ɛ + 2 = 4π 3
Water is FE at ∼ 100 ◦ C!
np 2
3k B T
v ij 〈u j 〉 th
Onsager Field (OF):
E local
i
≃ E Onsager
i
= E cavity
i
Onsager formula (Of):
ɛ − 1 =
No FE at finite T .
ɛ 4πnp 2
2ɛ + 1 k B T
+ E reaction
i
Onsager: LF overestimates the local field.
Van Vleck: from high T expansion, CM (1/T 2 ), Of (1/T 4 ).
Brout and Thomas: OF guarantees fluctuation-dissipation theorem.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Lorentz and Onsager Approximations
Model Hamiltonian (without compositional disorder)
M 0
⊥
2 v0
⊥
0.3
0.2
0.1
Onsager
Lorentz
t Ons
∝
Ln1 t Ons
0.0
0.0 0.5 1.0 1.5 2.0
T T c
Lorentz
∝t L
3Γ kBTc v 0
⊥2
0.30
0.25
0.20
0.15
0.10
0.05
Lorentz
Onsager
0.00
0.0 0.2 0.4 0.6 0.8 1.0
v 0 ⊥ Κ
v 0
⊥
Dipole forces + OF → logarithmic corrections.
Fluctuations suppress T c .

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Lorentz and Onsager Approximations
Model Hamiltonian (with compositional disorder)
Compositional disorder, Vilfan and Cowley (1985).
〈u i 〉 th = ∑ 〈 〉
χij
c h j
j
Self-consistent equations
〈 〈
u 2 i
〉 0
th〉
c
) 2
+
(v ⊥ 0 − vq )
,
MΩ 2 q
(Ω = M ⊥ 0
( ) (〈 2 〈 〉 0
M Ω ⊥ 0 = κ + 3γ ui
th〉
2 c
〈
− 〈u i 〉 0 2 〉
th = 1 ∑
c N
q
2MΩ q
coth
〈
〈u i 〉 0 2 〉
th = 1 ∑ ∆ 2
( )
c N
q MΩ 2 2
.
q
〈
+ 〈u i 〉 0 2 〉
th
c
( βΩq
2
)
,
)
− v ⊥ 0 ,

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Outline
1 Ferroelectrics vs Relaxor Ferroelectrics
Unit Cell
Dielectric Constant
Soft Modes
Diffuse Scattering
2 Theory of Relaxor Ferroelectrics
Model Hamiltonian
Lorentz and Onsager Approximations
3 Results
Soft Mode
Diffuse Scattering
Dielectric Constant
4 Conclusions

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering (PMN)
a 2 10 3
3.0
2.0
Hiraka 110 ,100
Gvasaliya 110
Present Model
S 0.05
⊥
1.0
0.0
0.0 100 200 300 400 500 600 700
T K
S q =
2MΩ q
coth
( βΩq
2
)
+ ( ∆2
( ) 2 )
)
MΩ 2 2
, MΩ 2 q = M Ω ⊥ 0 +
(v 0 ⊥ − vq
q

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering (PMN)
S q
⊥ a
2
10 6
a 150K
10 5
10 4
10 3
10 2
10 1
q rlu
S q
⊥ a
2
10 6
b 300 K
10 5
10 4
10 3
10 2
q rlu
10 1
0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 0.3
S q =
2MΩ q
coth
( βΩq
2
)
+ ( ∆2
) 2 )
)
MΩ 2 2
, MΩ 2 q
(Ω = M ⊥ 0 +
(v 0 ⊥ − vq
q

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Dielectric Constant (PMN)
4 Π
10 3
Ε 1
4 Π
3
6.0
5.0
4.0
3.0
2.0
1.0
a PMN
10 2 Hz
10 3 Hz
10 9 Hz
Present Model w
ConstantDamping
0.0
100 200 300 400 500 600 700
T K
4 Π
10 3
Ε 1
4 Π
3
6.0
5.0
4.0
3.0
2.0
1.0
b PMN
10 2 Hz
10 3 Hz
10 9 Hz
Present Model w
Variable Damping
0.0
100 200 300 400 500 600 700
T K
χ −1
0 (ω) = M(Ω⊥ 0 )2 + iωΓ[ω], Γ[ω]: damping function.
(
ɛ(ω) = 1 + 4π nz ∗2) χ 0 (ω).
Discrepancy will be discussed at the end.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Soft Mode
Results: soft mode
M 0
⊥
2 v0
⊥ 10
1
3.0
2.0
1.0
2 v 0 ⊥ a 2 ⩵1.00
2 v 0 ⊥ a 2 ⩵0.50
2 v 0 ⊥ a 2 ⩵0.25
2 v 0 ⊥ a 2 ⩵0.00
0.0 0.5 1.0 1.5 2.0
T T c
Dipole forces and random fields suppress long-range order
for arbitrarily small disorder.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering
Results: elastic diffuse scattering
400
S0a 2
300
200
2 v 0 ⊥ a 2 ⩵0.0
2 v 0 ⊥ a 2 ⩵0.8
2 v 0 ⊥ a 2 ⩵0.9
2 v 0 ⊥ a 2 ⩵1.0
2 v 0 ⊥ a 2 ⩵1.1
2 v 0 ⊥ a 2 ⩵1.5
2 v 0 ⊥ a 2 ⩵2.0
100
S q =
2MΩ q
coth
0
0.0 0.5 1.0 1.5 2.0 2.5
( βΩq
2
T T c
)
+ ( ∆2
( ) 2 )
)
MΩ 2 2
, MΩ 2 q = M Ω ⊥ 0 +
(v 0 ⊥ − vq
q

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering
Results: elastic diffuse scattering
Sqa 2
500
100
50
10
5
T T c ⩵0.0
T T c ⩵1.0
T T c ⩵2.0
2
Sqa 2
v 0 ⊥ a 2 ⩵ 0.8 2 1 0 1 2
500
100
50
10
5
T T c ⩵0.0
T T c ⩵1.0
T T c ⩵2.0
2
v 0 ⊥ a 2 ⩵2.0
1
2 1 0 1 2
q x
,0,0
2 Π a
1
q x
,0,0
2 Π a
S q =
2MΩ q
coth
( βΩq
2
)
+ ( ∆2
( ) 2 )
)
MΩ 2 2
, MΩ 2 q = M Ω ⊥ 0 +
(v 0 ⊥ − vq
q

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering
Results: correlation functions of polarization
Mukamel and Pytte, Phys. Rev. B 25, 4779 (1981); Lines, Phys. Rev. B 9, 950 (1974)
Short-range forces without disorder Short-range forces with disorder
(
) 2
〈 〉 k uq u −q ∝ B T
χ −1
∆
+|qa|
0
2 χ −1 +|qa|
0
2
〉
[ ] 〈u i u j ∝
k B T Exp − √ (r/a)
χ0
/r ∆ 2√ [ ]
χ 0 Exp − √ (r/a)
χ0
Dipole forces without disorder
〈
uq u −q
〉
∝
k B T
〈u i u j
〉
∝
(χ ⊥ 0 )−1 +4π q2 z
|q| 2 +η|(qa)|2 −3η(qz a) 2
{
k B T (χ ⊥ 0 )2 /r 3 , r ‖ ẑ
−k B T (χ ⊥ 0 )1/2 /r 3 , r ⊥ ẑ
Dipole forces with disorder
⎛
⎞2
⎜
∆
⎟
⎝
(χ ⊥ 0 )−1 +4π q2 z
|q| 2 +η|(qa)|2 −3η(qz a) 2 ⎠
{
∆ 2 (χ ⊥ 0 )3 /r 3 , r ‖ ẑ
−∆ 2 (χ ⊥ 0 )3/2 /r 3 , r ⊥ ẑ
slowly varying and highly anisotropic.
positive (ferroelectric) correlations stronger than negative
(antiferroelectric) correlations.
similar behavior for near-neighbor correlations (next slide).

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Diffuse Scattering
Results: near-neighbor correlations (polar nanoregions)
uiu j thc ui
2 thc
uiu j thc ui
2 thc
uiu j thc ui
2 thc
⊥ a
a ⩵1
a
⊥ a
10 0
10 1
10 2
10 3
0
10 0
1
2
3
4
a
5
z ija
T
6
Tc⩵10.0
v 0
v ⊥ 0
v ⊥ 0
v 0
7
2 2 2 2
8
⩵0
⩵10
⩵100
9 10
b T Tc⩵1.1
10 1
10 2
10 3
4 5 6 7 8 9 10
0 1 2 3
z ija
c T Tc⩵0.0
10 1
10 2
10 3
0 1 2 3 4 5 6 7 8 9 10
10 0
zij a
uiu j thc ui
2 thc
uiu j thc ui
2 thc
uiu j thc ui
2 thc
10 0
10 1
10 2
10 3
10 4
0
10 0
1
2
3
4
d
5
z ija
T
6
Tc⩵10.0
7 8
9 10
e T Tc⩵1.1
10 1
10 2
10 3
10 4
0 1 2 3 4 5 6 7 8 9 10
z ija
f T Tc⩵0.0
10 1
10 2
10 3
10 4
0 1 2 3 4 5 6 7 8 9 10
10 0
xij a

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Dielectric Constant
Results: dielectric susceptibility
Χ0Ωv 0
⊥
120
100
80
60
40
20
Ω v 0 ⊥ ⩵ 5 10 3
Χ0Ωv 0
⊥
600
300
Ω v 0 ⊥ ⩵ 0
0
0.0 1.0 2.0
T T c
Ω v 0 ⊥ ⩵1 10 2 Ω v 0 ⊥ ⩵ 5 10 2
0
0.0 0.5 1.0 1.5 2.0
χ −1
0 (ω) = M(Ω⊥ 0 )2 + iωΓ(ω).
(
ɛ(ω) = 1 + 4π nz ∗2) χ 0 (ω).
T T c
Curie-Weiss constant ∝ γ −1 ; γ: anharmonic coefficient.
M(Ω ⊥ 0 [Tmax])2 = ω (Re[Γ(ω)] − Im[Γ(ω)]).
χ max(ω) = (2ωRe[Γ(ω)]) −1 .

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Dielectric Constant
Results: T B and T max
TCWTc
1.0
1.0
0.5
0.5
0.0
0.0
0.5
1.0
0.5
1.0
0.0 0.2 0.4 0.6 0.8 1.0
v 0 ⊥ 2
TBTc
TmaxTc
1.0
0.8
0.6
0.4
2 v ⊥ 0 a 2 ⩵0.10
0.2
2 v ⊥ 0 a 2 ⩵0.15
2 v ⊥ 0 a 2 ⩵0.25
0.0
0.0 1.0 2.0 3.0 4.0 5.0
Ωv 0 ⊥ 10 2
M(Ω ⊥ 0 [T max]) 2 = ω (Re[Γ(ω)] − Im[Γ(ω)]).

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Dielectric Constant
Dielectric Constant (PMN)
4 Π
10 3
Ε 1
4 Π
3
6.0
5.0
4.0
3.0
2.0
1.0
a PMN
10 2 Hz
10 3 Hz
10 9 Hz
Present Model w
ConstantDamping
0.0
100 200 300 400 500 600 700
T K
4 Π
10 3
Ε 1
4 Π
3
6.0
5.0
4.0
3.0
2.0
1.0
b PMN
10 2 Hz
10 3 Hz
10 9 Hz
Present Model w
Variable Damping
0.0
100 200 300 400 500 600 700
T K
More elaborate damping function needed.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Outline
1 Ferroelectrics vs Relaxor Ferroelectrics
Unit Cell
Dielectric Constant
Soft Modes
Diffuse Scattering
2 Theory of Relaxor Ferroelectrics
Model Hamiltonian
Lorentz and Onsager Approximations
3 Results
Soft Mode
Diffuse Scattering
Dielectric Constant
4 Conclusions

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Conclusions
Minimal model for relaxor ferroelectricity: local anharmonic forces,
dipole forces, quenched random fields.
Any solution must consider fluctuations of polarization and disorder at
the replica level.
Compositional disorder and dipolar forces extend the region of
fluctuations down to T = 0 K.

Ferroelectrics vs Relaxor Ferroelectrics Theory of Relaxor Ferroelectrics Results Conclusions
Acknowledgments
Albert Migliori - Los Alamos National Laboratory
Alexandra Navrotsky - UC Davis
Frances Hellman - UC Berkeley