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