1. 16. Dielectrics and Ferroelectrics
Maxwell Equations
Polarization
Macroscopic Electric Field
Depolarization Field, E1
Local Electric Field at An atom
Lorentz Field, E2
Field of Dipoles Inside Cavity, E3
Dielectric Constant And Polarizability
Electronic Polarizability
Classical Theory
Examples
Structural Phase Transitions
Ferroelectric Crystals
Classification of Ferroelectric Crystals
Displacive Transitions
Soft Optical Phonons
Landau theory of the Phase Transition
Second-Order Transition
First-Order Transition
Antiferroelectricity
Ferroelectric Domains
Piezoelectricity

2. Maxwell Equations

3. Polarization Polarization P  dipole moment per unit volume.
Total dipole moment
For a neutral system, p is independent of the choice of the coordinate origin.
H2 O
Dipole field:

4. Macroscopic Electric Field
E0 = external (applied) field
e(r) = microscopic field
Macroscopic field
VC = volume of crystal cell
E due to a volume of uniform P is equal to that due to a surface charge density
For any point between the plates & far from the edges
E1 = field due to σ = n  P.

5. Depolarization Field, E1
If P is uniform, then
E1 = depolarization field
Inside an ellipsoidal body, P uniform → E1 uniform
Along the principal axes
Nj = depolarization factor
ellipsoid of revolution

6. χ = dielectric susceptibility
Along a principal axis of an ellipsoid: →

7. Local Electric Field at an Atom
In general, E  Eloc.
Consider a simple cubic crystal of spherical shape.
The macroscopic field in a sphere is
If all dipoles are equal to
the dipole field at the center of the sphere is
Cubic symmetry → → →
For an abitrary symmetry
E2 = Lorentz cavity field
(due to charges on surface of cavity)
E3 = field of atoms inside cavity
E1 + E2 = field of body with hole.

8. E1 + E2 + E3= field of all other atoms at one atom.
Sites 10a (~50A) away can be replaced by 2 surface integrals:
1 over the outer ellipsoidal surface,
the other over the cavity defining E2 .

9. Lorentz Field, E2 

10. Field of Dipoles inside Cavity, E3
E3 is only field that depends on crystal structure.
For cubic crystals →
Lorentz relation

11. Dielectric Constant and Polarizability
For an isotropic / cubic medium, ε is a scalar:
For a non-cubic medium, ε & χ are tensors:
Polarizability α of an atom:
α is in general a tensor
Polarization:
For cubic medium, Lorentz relation applies : →
Clausius-Mossotti relation

12. Electronic Polarizability
Dipolar: re-orientation of molecules with permanent dipoles
Ionic: ion-ion displacement
Electronic: e-nucleus displacement
In heterogeneous materials, there is also an interfacial polarization.

13. At high frequencies, electronic contribution dominates.
e.g., optical range:

14. Classical Theory of Electronic Polarizability
Bounded e subject to static Eloc :
Steady state:
Static electronic polarizability:
Bounded e subject to oscillatory Eloc :
Oscillatory solution :
Electronic polarizability:
Quantum theory:
f i j = oscillator strength of dipole transition between states i & j.

15. Structural Phase Transitions
At T = 0, stable structure A has lowest free energy F = U for a given P.
High P favors close-packing structures which tend to be metallic.
E.g., H & Xe becom metallic under high P.
Let B has a softer (lower ω) phonon spectrum than A.
→ SB > SA due to greater phonon occupancy for B.
→  TC s.t. FB = UB –T SB > FA = UA –T SA  T > TC
( phase transition A → B unless TC > Tmelt )
FB (TC ) = FA (TC )
Near TC , transition can be highly stress sensitive.
Ferroelectrics: spontaneous P.
Unusual ε(T).
Piezoelectric effect.
Pyroelectric effect.
Electro-optical effects such as optical frequency doubling.

16. Ferroelectric Crystals
Ferroelectric  TC → Paraelectric
Ferroelectric : P vs E plot shows hysteresis.
Pyroelectric effects (P  T ) are often found in ferroelectrics where P is not affected by E less than the breakdown field.
E.g., LiNbO3 is pyroelectric at 300K.
High TC = 1480K.
Large saturation P = 50 μC/cm2 .
Can be “poled”
(given remanent P by E at T >1400K).
PbTiO3

17. Classification of Ferroelectric Crystals

18. 2 main classes of ferroelectrics:
order-disorder: soft (lowest ωTO ) modes diffusive at transition.
e.g., system with H-bonds: KH2PO4 .
displacive: soft modes can propagate at transition.
e.g., ionic crytsls with perovskite, or ilmenite structure.
Most are in between
Order-disorder
TC nearly doubled on H→D.
Due to quantum effect involving mass-dependent de Broglie wavelength.
n-diffraction → for T < TC , H+ distribution along H-bond asymmetric.
Displacive
T > TC
T < TC : displaced
BaTiO3

19. At 300K, PS = 8104 esu cm–2 .
VC = (4 10–8 )3 = 64 10–24 cm3. →
p  510–18 esu cm
Moving Ba2+ & Ti4+ w.r.t. O2– by δ = 0.1A gives p /cell = 6e δ  310–18 esu cm
In LiNbO3, δ is 0.9A for Li & 0.5A for Nb → larger p.

20. Displacive Transitions
2 viewpoints on displacive transitions:
Polarization catastrophe
( Eloc caused by u is larger than elastic restoring force ).
Condensation of TO phonon
(t-indep displacement of finite amplitude)
Happens when ωTO = 0 for some q  0.
ωLO > ωTO & need not be considered .
In perovskite structures, environment of O2– ions is not cubic → large Eloc.
→ displacive transition to ferro- or antiferro-electrics favorable.
Catastophe theory:
Let Eloc = E + 4 π P / 3 at all atoms.
In a 2nd order phase transition, there is no latent heat.
The order parameter (P) is continuous at TC .
C-M relation:
Catastophe condition:

21. →
for s → 0 →
(paraelectric)

22. Soft Optical Phonons LST relation ωTO → 0  ε(0) → 
no restoring force: crystal unstable
E.g., ferroelectric BaTiO3 at 24C has ωTO = 12 cm–1 .
Near TC , →
if ωLO is indep of T
SbSI
from Raman scatt
SrTiO3
from n scatt

23. Landau Theory of the Phase Transition
Landau free energy density:
Comments:
Assumption that odd power terms vanish is valid if crystal has center of inversion.
Power series expansion often fails near transition (non-analytic terms prevail) .
e.g., Cp of KH2PO4 has a log singularity at TC .
The Helmholtz free energy F(T, E) is defined by
Transition to ferroelectric is facilitated by setting
(This T dependence can be explained by thermal expansion & other anharmonic effects )
g2 ~ 0+ → lattice is soft & close to instability.
g2 < 0 → unpolarized lattice is unstable.

24. Second-Order Transition
For g4 > 0, terms g6 or higher bring no new features & can be neglected.
E = 0 →
→ PS = 0 or
Since γ , g4 > 0, the only real solution when T > T0 , is PS = 0 (paraelectric phase).
This also identifies T 0 with TC .
minimizes F ( T, 0 ) (ferroelectric phase).
For T < T0 ,
LiTaO3

25. First-Order Transition
For g4 < 0, the transition is 1st order and term g6 must be retained.
E = 0 →
→ PS = 0 or
BaTiO3 (calculated)
For E  0 & T > TC , g4 & higher terms can be neglected:
T0 = TC for 2nd order trans.
T0 < TC for 1st order trans.

26. Fundamental types of structural phase transitions from a centrosymmetric prototpe

27. Perovskite
Lead zirconate-lead titanate (PZT) system
Widely used as ceramic piezoelectrics.

28. Ferroelectric Domains
Atomic displacements of oppositely polarized domains.
Domains with 180 walls
BaTiO face  c axis.
Ea // c axis.

29. Piezoelectricity Ferroelectricity → Piezoelectricity (not vice versa)
BaTiO3 :
d = 10−5 cm/statvolt
Unstressed
σ = stress (tensor)
d = piezoelectric constant (tensor)
χ = dielectric susceptibility
e = elastic compliance constant (tensor)
α = 1,∙∙∙, 6
PiezoE not FerroE
e.g., SiO2
A+3 B3−
d  10−7 cm/statvolt
Unstressed: 3-fold symmetry
PVF2 films are flexible & often used as ultrsonic transducers