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)

= C / (T-T0)

C ~ Curie-Weiss constantT0 is called “Curie-Weiss Temperature”

T0 < Tc in materials with first-order transitionsT0 = Tc in materials with second-order transitions

!! Some materials do not follow Curie-Weiss Relation !!

Ferroelectric Characteristics

• Dielectric non-linearitiesMeasured 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

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Structural Symmetry

14 Bravais Unit Cells Crystals in Nature 7 Crystal Systems

Symmetry Elements

Triclinic, Monoclinic, Orthorhombic, Tetragonal, Trigonal, Hexagonal, Cubic

ntranslatiom,,6,6,4,4,3,3,2,2,1,1

230 Space Groups (Microscopic)

32 Point Groups (Macroscopic)

IF Translation Symmetry Removed

Structural SymmetryCrystal

StructurePoint Groups Centro

SymmetryNon-Centrosymmetry

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, 42m, (4/m)mm

4/m, (4/m)mm 4, 4, 422, 4mm, 42m

4, 4mm

Trigonal 3, 3, 32, 3m, 3m 3, 3m 3, 32, 3m 3, 3m

Hexagonal 6, 6, 6/m, 622, 6mm, 6m2, (6/m)mm

6/m, (6/m)mm 6, 6, 622, 6mm, 6m2

6, 6mm

Cubic 23, m3, 432, 43m, m3m

m3, m3m 23, 43m None

Point Groups for Seven Crystal Systems

Note that: underlined numbers represent inversion symmetry

Structural Symmetry

Pyroelectrics: Spontaneous polarization upon heating or coolingFerroelectrics: Reversible or re-orientable spontaneous polarizationFerroelectrics are a subgroup of the polar materials and

are BOTH pyroelectric and piezoelectric

32 Crystal Classes(all crystalline materials are electrostrictive)

11 ClassesCentro-Symmetric

21 ClassesNon-Centrosymmetric

1 ClassNon-Piezoelectric

20 ClassesPiezoelectric

10 ClassesUnique Polar Axis

(Pyroelectric)

10 ClassesNO Unique Polar Axis

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

Polarization (P)

Polarization (P) = Values of the dipole moment per unit

volume

= Values of the charge per unit surface area

P = NV = Nqd/Ad = Nq/A

N = number of dipole moment per unit volume

= 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) Ps = 0

Tetragonal (T < Tc)Ps 0

Pyroelectric Effect

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

BaTiO3 Triglycine Sulfate (TGS)

Spontaneous Polarization (Ps) Re-Orientation

Changes in Ps-directions require small ionic movements Larger number of possible directions of polar axes

Closer to poling direction Easily poled

Unpoled Poled

Ceramics a large number of randomly oriented crystallites polarization re-orientation “Poling Process”

Tetragonal 4mm 6 possible polar axesRhombohedral 3m 8 possible polar axes

better alignment (poled)

E

Ferroelectric DomainsFerroelectric 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

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 PhaseT > 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, r above Curie Point (Tc) obeys Curie-Weiss Relation

= 0 + C/(T-T0)

C = Curie-Weiss constantT0 = Curie-Weiss Temperature (different from Curie Point Tc)

T0 < Tc for first-order phase transitionT0 = Tc for second-order phase transition

Tc = actual temperature when crystal structure changes

T0 = formula constant obtained by extrapolation(Usually 0 term is neglected because 0 << 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,

r does NOT obey Curie-Weiss Relation

(1/) – (1/m) = C’/(T-Tm)n

C’ = constantTm = Temperature with m

m = Maximum dielectric constant1 < n < 2

Equilibrium Properties of Crystals

Relations Between Thermal, Electrical, and Mechanical Properties of Crystals(Rank of Tensors in Parenthesis)

Heckmann’s DiagramHeckmann’s Diagram

(1)

(2) (0)

(1)

(0)(2)

Equilibrium Properties of Crystals

Relations Between Thermal, Electrical, and Mechanical Properties of Crystals

Heckmann’s DiagramHeckmann’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”

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 dkl produces a change of strain dxij

dxij = sijkl dkl

where sijkl is the “elastic compliance”

Equilibrium Properties of Crystals

Coupled Effects : Lines joining pairs not on the same cornerBottom : Thermoelastic Effects

Right : Electrothermal Effects (Pyroelectric Effects)Left : Electromechanical Effects (Piezoelectric Effects)

Direct and Converse Piezoelectric Effects(Third-Rank Tensors)

dDi = dijkdjk “Direct Effect”

dxij = dijkdEk “Converse Effect”

where dijk is the “piezoelectric coefficient ”

Thermoelastic Effects : dxij = jjdT

Pyroelectric Effects : dDi = pidT