What is Ferroelectric?

Ferroelectrics are materials which possess a “spontaneous”

electric polarization Ps which can be reversed by applying a
suitable electric field E.

This process is known as “switching”, and is followed by “hysteresis”.

Ferroelectrics are electrical analogues of “ferromagnetics” (P-E and M-H

relations).
 Ferroelectric Characteristics
Three important characteristics of ferroelectrics:
• Reversible polarization
• “Anomalous” properties (i.e. ferroelectric disappears
above a temperature Tc known as “Curie Point”
• Non-linearities
 Ferroelectric Characteristics
• “Anomalous” properties (i.e. ferroelectric disappears
above a temperature Tc known as “Curie Point”
• Above Tc, the anomaly is frequently of the “ Curie-
Weiss” form: (Curie-Weiss Relation)

e = C / (T-T0)

C ~ Curie-Weiss constant
T0 is called “Curie-Weiss Temperature”
T0 < Tc in materials with first-order transitions
T0 = Tc in materials with second-order transitions

!! Some materials do not follow Curie-Weiss Relation

!!
 Ferroelectric Characteristics
• Dielectric non-linearities
Measured dielectric permittivity changes with
change of applied (bias field)
 What is Piezoelectricity?
Piezoelectrics are
 materials which acquire electric polarization under
external mechanical stresses (Direct Effect),
OR
 materials that change size or shape when subject to
external electric field E (Converse Effect).
! (Piezo ~ Pressure or Stress) !

Many piezoelectric materials are NOT ferroelectric

All ferroelectrics are piezoelectric
Above T0, some ferroelectrics are STILL piezoelectric
applications
Applications
 Applications in ceramic power transformers

Examples provided courtesy of Prof. Kenji Uchino, Penn State University, USA.
 Applications in ceramic disc transformers

Examples provided courtesy of Prof. Kenji Uchino, Penn State University, USA
 Applications in Ultrasonic Tubular Motors

Examples provided courtesy of Prof. Kenji Uchino, Penn State University, USA
 Structural Symmetry
Crystals in Nature 7 Crystal Systems 14 Bravais Unit Cells
Triclinic, Monoclinic, Orthorhombic,
Tetragonal, Trigonal, Hexagonal, Cubic

Symmetry Elements
1,1,2, 2,3, 3,4, 4,6, 6, m, translation 

230 Space Groups (Microscopic)

IF Translation Symmetry Removed

32 Point Groups (Macroscopic)

 Structural Symmetry
Crystal Point Groups Centro Non-Centrosymmetry
Structure Symmetry
Piezoelectric Pyroelectric
Triclinic 1, 1 1 1 1

Monoclinic 2, m, 2/m 2/m 2, m 2, m

Orthorhombic 222, mm2, mmm mmm 222, mm2 mm2

Tetragonal 4, 4, 4/m, 422, 4mm, 4/m, (4/m)mm 4, 4, 422, 4mm, 4, 4mm

42m, (4/m)mm 42m
Trigonal 3, 3, 32, 3m, 3m 3, 3m 3, 32, 3m 3, 3m

Hexagonal 6, 6, 6/m, 622, 6mm, 6/m, (6/m)mm 6, 6, 622, 6mm, 6, 6mm

6m2, (6/m)mm 6m2
Cubic 23, m3, 432, 43m, m3, m3m 23, 43m None
m3m

Point Groups for Seven Crystal Systems

Note that: underlined numbers represent inversion symmetry
 Structural Symmetry
32 Crystal Classes
(all crystalline materials are electrostrictive)

11 Classes 21 Classes
Centro-Symmetric Non-Centrosymmetric

1 Class 20 Classes
Non-Piezoelectric Piezoelectric

10 Classes
10 Classes
Unique Polar Axis
NO Unique Polar Axis
(Pyroelectric)

1, 2, m, 2mm, 4, 4mm, 3, 3m, 6, 6mm

Pyroelectrics: Spontaneous polarization upon heating or cooling

Ferroelectrics: Reversible or re-orientable spontaneous polarization
Ferroelectrics are a subgroup of the polar materials and
are BOTH pyroelectric and piezoelectric
 Polarization (P)
Polarization (P) = Values of the dipole moment per unit volume
= Values of the charge per unit surface area

P = Nm / V = Nqd/Ad = Nq/A

N = number of dipole moment per unit volume

m = dipole moment = qd
q = charge
d = distance between positive and negative charges

V = volume AND A = surface area

 Spontaneous Polarization (Ps)
Spontaneous polarization (Ps) exists in 10 classes of polar crystals
with a unique polar axis (out of 20 piezoelectric classes)

BaTiO3 Single Crystal

Cubic (T > Tc) Tetragonal (T < Tc)

Ps = 0 Ps  0
 Pyroelectric Effect

BaTiO3 Triglycine Sulfate (TGS)

Pyroelectric Effect = Change of spontaneous polarization
(Ps) with temperature (T) (Discovered in Tourmaline by
Teophrast (314 B.C.) and named by Brewster in 1824);

p = pyroelectric coefficient = Ps/T

Notice that BaTiO3 and TGS (and most crystals) has a negative
pyroelectric coefficient
Spontaneous Polarization (Ps) Re-Orientation
Ceramics  a large number of randomly oriented crystallites 
 polarization re-orientation “Poling Process”

Unpoled Poled
Changes in Ps-directions require small ionic movements
 Larger number of possible directions of polar axes 
 Closer to poling direction  Easily poled
Tetragonal 4mm  6 possible polar axes
Rhombohedral 3m  8 possible polar axes
 better alignment (poled)
 Ferroelectric Domains
Ferroelectric Domains = A region with uniform alignment (same direction)
of spontaneous polarization (Ps)
Domain Walls = The interface between the two domains
 very thin ( < a few lattice cells)

A ferroelectric single crystal, when grown, has multiple ferroelectric domains


Applying appropriate electric field

Possible single domain through domain wall motion

Too large electric field

Reversal of the polarization in the domain “domain switching”

Hysteresis Loop
 Ferroelectric Hysteresis Loop
D

Starting from very small E-field  Linear P-E relationship (OA)

E  leads to domain re-alignment in the positive direction along E
rapid increase in P (OB) until it reaches the saturation value (Psat)
E  results in  P, but NOT all to Zero P as E = 0 (BD) because some domains remain aligned in positive direction
 Remnant OR Remanent Polarization (Pr) 
Certain opposite E is needed to completely depolarize the domain  Coercive Field (Ec)
As E  in negative direction  direction of domains flip
Hysteresis Loop 
Spontaneous Polarization (Ps) is obtained through extrapolation 
Hysteresis Loop is observed by a Sawyer-Tower Circuit 
Ferroelectric Curie Point and Phase Transitions
Curie Point (Tc) = Phase transition temperature between
non-ferroelectric and ferroelectric phases

T < Tc = Ferroelectric Phase

T > Tc = Paraelectric (Non-ferroelectric) Phase

Transition Temperature = Other phase transition temperature

between one ferroelectric phase to another
Ferroelectric Curie Point and Phase Transitions
Near Curie Point (Tc)  Thermodynamic properties
(dielectric, elastic, optical, thermal)
show “ anomalies” and structural changes
Ferroelectric Curie Point and Phase Transitions
In most ferroelectrics, er above Curie Point (Tc) obeys
Curie-Weiss Relation

e = e0 + C/(T-T0)

C = Curie-Weiss constant
T0 = Curie-Weiss Temperature
(different from Curie Point Tc)
T0 < Tc for first-order phase transition
T0 = Tc for second-order phase transition

Tc = actual temperature when crystal structure changes

T0 = formula constant obtained by extrapolation
(Usually e0 term is neglected because e0 << e near T0)
 Ferroelectric Curie Point and Phase Transitions

In relaxor ferroelectrics, such

as Pb(Mg1/3Nb2/3)O3 (PMN),
and Tungsten-Bronze type
compounds,
such as (Sr1-xBax)Nb2O6,
er does NOT obey
Curie-Weiss Relation
(1/e) – (1/em) = C’/(T-Tm)n

C’ = constant
Tm = Temperature with em
em = Maximum dielectric constant
1<n<2
 Equilibrium Properties of Crystals
Heckmann’s Diagram

(1)

(1)

(2) (0)

(2) (0)

Relations Between Thermal, Electrical, and Mechanical Properties of Crystals

(Rank of Tensors in Parenthesis)
 Equilibrium Properties of Crystals
Heckmann’s Diagram
Three Outer Corners: Temperature (T), Electric Field (Ei), and Stress (ij)  “Forces”

Three Inner Corners: Entropy (S), Electric Displacement (Di), and Strain (ij)  “Results”

Lines Joining These Corner Pairs


“Principal Effects”

Relations Between Thermal, Electrical, and Mechanical Properties of Crystals

 Equilibrium Properties of Crystals
I. An increase of temperature produces a change of entropy dS:

dS = (C/T)dT

where C ( a scalar) is the “heat capacity per unit volume”

T is the absolute temperature

II. A small change of electric field dEi produces a change of electric

displacement dDi

dDi = ijdEj

where ij is the “permittivity” tensor

III. A small change of stress dkl produces a change of strain dxij

dxij = sijkl dkl

where sijkl is the “elastic compliance”

 Equilibrium Properties of Crystals
Coupled Effects : Lines joining pairs not on the same corner
Bottom : Thermoelastic Effects
Right : Electrothermal Effects (Pyroelectric Effects)
Left : Electromechanical Effects (Piezoelectric Effects)
Direct and Converse Piezoelectric Effects
(Third-Rank Tensors)
dDi = dijkdjk  “Direct Effect”

dxij = dijkdEk  “Converse Effect”

where dijk is the “piezoelectric coefficient ”

Thermoelastic Effects : dxij = jjdT

Pyroelectric Effects : dDi = pidT