Download Piezoelectric Materials

Introduction to Ferroelectric
Materials and Devices
426415
1
Objectives
• To have basic knowledge of ferroelectric
material
• To understand piezoelectric effects
• To describe its application
• To understand lead zirconate titanate
(PZT) solid solution system
2
What is this material ?
3
Piezoelectric effects
(3) (1)
(2)
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Piezoelectric effects
Direct
effect
D = dT + TE
Converse
effect
S = sET + dE
D is dielectric displacement = polarization, T is the stress, E is the electric
field, S = the strain, s = the material compliance (inverse of modulus of
elasticity),  = dielectric constant, d = piezoelectric (charge) constant
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Piezoelectric constants
Piezoelectric Charge Constant (d)
The polarization generated per unit of mechanical stress
applied to a piezoelectric material
alternatively
The mechanical strain experienced by a piezoelectric
material per unit of electric field applied
Electro-Mechanical Coupling Factor
(i) For an electrically stressed component
k2 = stored mechanical energy
total stored energy
(ii) For a mechanically stressed component
k2 = stored electrical energy
total stored energy
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Polarization
Dipole moment
5 Basic polarizations
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Polarization
Five basic types of polarisation:
(a) electronic polarisation of an atom,
(b) atomic or ionic polarisation of an ionic
crystal,
(c) dipolar or orientational polarisation of
molecules with asymmetry structure (H2O),
(d) spontaneous polarisation of a perovskite
crystal,
and (e) interface or space charge polarisation
of a dielectric material.
(Left-hand-side pictures illustrate the
materials without an external electrical field
and the right-hand-side pictures with an
external electrical field, E.)
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Crystallographic point group
32 crystallographic point groups. The remark “ i ” represents centrosymmetric crystal
which piezoelectric effect is not exhibited, both remark “ * ” and “ + ” represent
noncentrosymmetric crystal where the remark “ * ” indicates that piezoelectric effect may
be exhibited and the remark “ + ” indicates that pyroelectric and ferroelectric effects may
be exhibited.
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Symmetry elements
There are 3 types of symmetry operations:
1. Rotation
2. Reflection
3. Inversion
An example of 4-fold rotation symmetry
http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
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Symmetry elements
1-Fold Rotation Axis
2-fold Rotation Axis
3-Fold Rotation Axis
6-Fold Rotation Axis
12
http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
Symmetry elements
Mirror Symmetry
Mirror symmetry
No mirror symmetry
13
http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
Symmetry elements
Center of Symmetry
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http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
Symmetry elements
Center of Symmetry
In this operation lines are drawn from
all points on the object through a
point in the center of the object,
called a symmetry center
(symbolized with the letter "i").
If an object has only a center of
symmetry, we say that it has a 1 fold
rotoinversion axis. Such an axis has
the symbol , as shown in the right
hand diagram above
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http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
Symmetry elements
Rotoinversion
Combinations of rotation with a center of symmetry
perform the symmetry operation of rotoinversion.
2-fold Rotoinversion - The operation of 2fold rotoinversion involves first rotating the
object by 180o then inverting it through an
inversion center.
This operation is equivalent to having a
mirror plane perpendicular to the 2-fold
rotoinversion axis. A 2-fold rotoinversion
axis is symbolized as a 2 with a bar over the
top, and would be pronounced as "bar 2".
But, since this the equivalent of a mirror
plane, m, the bar 2 is rarely used.
16
http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
Symmetry elements
Rotoinversion
3-fold Rotoinversion - This involves
rotating the object by 120o (360/3 = 120),
and inverting through a center. A cube is
good example of an object that possesses
3-fold rotoinversion axes. A 3-fold
rotoinversion axis is denoted as
(pronounced "bar 3"). Note that there are
actually four axes in a cube, one running
through each of the corners of the cube. If
one holds one of the axes vertical, then note
that there are 3 faces on top, and 3 identical
faces upside down on the bottom that are
offset from the top faces by 120o.
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http://www.tulane.edu/~sanelson/eens211/introsymmetry.htm
Symmetry elements
Combinations of Symmetry Operations
As should be evident by now, in three dimensional objects, such as crystals,
symmetry elements may be present in several different combinations. In fact, in
crystals there are 32 possible combinations of symmetry elements. These 32
combinations define the 32 Crystal Classes. Every crystal must belong to one
of these 32 crystal classes.
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Crystal system
(a) A simple square lattice. The unit cell is a square with a side a.
(b) Basis has two atoms.
(c) Crystal = Lattice + Basis. The unit cell is a simple square with two atoms.
(d) Placement of basis atoms in the crystal unit cell.
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Crystal system
(a) A simple square lattice. The unit cell is a square with a side a.
(b) Basis has two atoms.
(c) Crystal = Lattice + Basis. The unit cell is a simple square with two atoms.
(d) Placement of basis atoms in the crystal unit cell.
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7 crystal systems and 14 bravais lattices
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The seven crystal systems (unit cell geometries) and fourteen Bravais lattices.
Crystal system and symmetry elements
crystal system
axis lengths
angles between
axes
common symmetry
elements
triclinic
a≠b≠c
 ≠  ≠  ≠ 90°
1-fold rotation w/ or
w/out i
monoclinic
a≠b≠c
 =  = 90°, β >90°
2-fold rotation and/or
1m
orthorhombic
a≠b≠c
 =  =  = 90°
3 2-fold rotation axes
and/or 3 m
hexagonal
a1 = a2 = a3, a ≠ c
60° btw a’s,  = 90o
1 3-fold or 6-fold axis
tetragonal
a=b≠c
 =  = = 90°
1 4-fold rotation or
rotoinversion axis
cubic
a=b≠c
 =  = = 90°
4 3-fold axes
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Crystal system and symmetry elements
23
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Crystal directions
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Crystal planes
Labeling of crystal planes and typical examples in the cubic lattice
Fig 1.41
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Crystal planes
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Zr+4 or Ti+4
Pb+2
O-2
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Polarization
Classification of piezoelectric, pyroelectric and ferroelectric effects based on the symmetry 28
system.
Crystal with a centre of symmetry
A NaCl-type cubic unit cell has a center of symmetry.
(a) In the absence of an applied force, the centers of mass for positive and negative
ions coincide.
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(b) This situation does not change when the crystal is strained by an applied force.
Noncentrosymmetric crystal
A hexagonal unit cell has no center of symmetry. (a) In the absence of an applied force the centers of
mass for positive and negative ions coincide. (b) Under an applied force along y the centers of mass for
positive and negative ions are shifted which results in a net dipole moment P along y. (c) When the force
is along a different direction, along x, there may not be a resulting net dipole moment in that direction
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though there may be a net P along a different direction (y).
Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
Cubic
Rhombohedral
High
Temperature
Zr+4 or Ti+4
Perovskite
Tetragonal
Pb+2
P
O2
Rhombohedral
Low
Temperature
MPB
PZT Solid Solution Phase Diagram
Zr/Ti ratio 52/48 MPB (Morphotropic Phase Boundary)
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Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
PZT (xPbZrO3 – (1-x)PbTiO3)

 Binary Solid Solution 
PbZrO3 (antiferroelectric matrial with orthorhombic structure)
and
PbTiO3 (ferroelectric material with tetragonal perovskite structure)

Perovskite Structure (ABO3) with Ti4+ and Zr4+ ions “randomly” occupying
the B-sites

Important Transducer Material (Replacing BaTiO3)
• Higher electromechanical coupling coefficient (K) than BaTiO3
• Higher Tc results in higher operating and fabricating temperatures
• Easily poled
• Wider range of dielectric constants
• relatively easy to sinter at lower temperature than BaTiO3
• form solid-solution compositions with several additives which results in a
wide range of tailored properties
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Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
Composition dependence of
dielectric constant (K) and
electromechanical planar
coupling coefficient (kp) in PZT
system

This shows enhanced dielectric
and electromechanical properties
at the MPB

Increased interest in PZT
materials with MPBcompositions for applications
33
Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
Advantages of PZT Solid-Solution System
• High Tc across the diagram leads to more stable ferroelectric states
over wide temperature ranges
• There is a two-phase region near the Morphotropic Phase Boundary
(MPB) (52/48 Zr/Ti composition) separating rhombohedral (with 8
domain states) and tetragonal (with 6 domain states) phases
• In the two-phase region, the poling may draw upon 14 orientation
states leading to exceptional polability
• Near vertical MPB results in property enhancement over wider
temperature range for chosen compositions near the MPB
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Compositions and Modifications of PZT System
1. Effects of composition and grain size on properties
MPB compositions
(Zr/Ti = 52/48)

Maximum dielectric and
piezoelectric properties

Selection of Zr/Ti can be used to
tailor specific properties

High kp and r are desired
 Near MPB compositions
OR
High Qm and low r are desired
 Compositions away from
MPB
Grain Size
(composition and processing)

Fine-Grain ~ 1 mm or less
Coase-Grain ~ 6-7 mm

Some oxides are grain growth
inhibitor (i.e. Fe2O3)

Some oxides are grain growth
promoter (i.e. CeO2)

Dielectric and piezoelectric
properties are grain-size
dependent
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Compositions and Modifications of PZT System
2. Influences of low level “off-valent” additives (0-5
mol%) on dielectric and piezoelectric properties
Two main groups of additives:
1. electron acceptors (charge on the replacing cation is smaller)
(A-Site:K+, Rb+ ; B-Site: Co3+, Fe3+, Sc3+, Ga3+, Cr3+, Mn3+, Mn2+, Mg2+, Cu2+)
(Oxygen Vacancies)


Reduce both dielectric and piezoelectric responses
Increase highly asymmetric hysteresis and larger coercivity

Much larger mechanical Q
“Hard PZT”
2. electron donors (charge on the replacing cation is larger)
(A-Site: La3+, Bi3+, Nd3+; B-Site: Nb5+, Ta5+, Sb5+)
(A-Site Vacancies)

Enhance both dielectric and piezoelectric responses at room temp

Under high field, symmetric unbiased square hysteresis loops

low electrical coercivity
“Soft PZT”
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Conclusion
Piezoelectric effects
1. mechanical energy  electrical energy :sensors
2. electrical energy  mechanical energy :actuators
3. noncentrosymmetric crystal, perovskite structure
Lead Zirconate Titanate Pb(Zrx,Ti1-x)O3  PZT
near MPB  high piezoelectric response (high K and d)
Hard PZT  additive = electron acceptors (A-Site:K+,
Rb+ ; B-Site: Co3+, Fe3+, Sc3+, Ga3+, Cr3+,
Mn3+, Mn2+, Mg2+, Cu2+)
low piezoelectric response
Soft PZT  additive = electron donors (A-Site: La3+, Bi3+,
Nd3+; B-Site: Nb5+, Ta5+, Sb5+)
high piezoelectric response
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Modified PZT System
“Hard PZT” Materials
 Curie temperature above 300 C
 NOT easily poled or depoled except at high temperature
 Small piezoelectric d constants
 Good linearity and low hysteresis
 High mechanical Q values
 Withstand high loads and voltages
“Soft PZT” Materials
 Lower Curie temperature
 Readily poled or depoled at room temperature with high
field
 Large piezoelectric d constants
 Poor linearity and highly hysteretic
 Large dielectric constants and dissipation factors
 Limited uses at high field and high frequency
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Domain structure
Zr+4 or Ti+4
Pb+2
O-2
Tc ~ 350 oC
(a) Domain structure of a tetragonal ferroelectric
ceramic (lead zirconate titanate) with 180o and 90o
domain walls is revealed by etching in HF and HCl
solution. The formation of parallel lines in the
grains of the ceramic (a) is due to 90o orientation of
the polar direction. (b) A schematic drawing of 90o
and 180o domains in a ferroelectric ceramic.
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ferroelectric domain switching
The application of an electric field
causes (a) the reorientation of a
spontaneous polarisation, Ps, in a
unit cell to the field direction; (b)
the sum of the microscopic
piezoelectricity of the unit cells
result in the macroscopic
piezoelectricity of piezoceramics.
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Compositions and Modifications of PZT System
2. Modification by element substitution
Element substitution  cations in perovskite lattice (Pb2+, Ti4+, and Zr4+)
are replaced partially by other cations with the same chemical valence
and similar ionic radii and solid solution is formed
 Pb2+ substituted by alkali-earth metals, Mg2+, Ca2+, Sr2+, and Ba2+ 

PZT replaced partially by Ca2+or Sr2+
Tc  BUT kp, 33 , and d31  
Shift of MPB towards the Zr-rich side 
 Density  due to fluxing effect of Ca or Sr ions 
Ti4+ and Zr4+ substituted by Sn4+ and Hf4+ , respectively 

Ti4+ replaced partially by Sn4+
c/a ratio decreases with increasing Sn4+ content 
 Tc  and  stability of kp and 33 
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Electrode
(3)
Electrode
Electrode
Electrode
Polarization
Electrode
Electrode
Electrode
Electric Field
Electrode
Electrode
Reference:
3. Ferroelectric Materials, n.d. DoITPoMs teaching and learning package, viewed 8 December 2008,
<http://www.doitpoms.ac.uk/tlplib/ferroelectrics/printall.php>
Electrode
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42
Electrode
Electrode
Electrode
Electrode
Polarization
Electrode
Electrode
Electrode
Electric Field
Electrode
Electrode
Electrode
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Dielectric Hysteresis loop
A 2 dimensional schematic
sketch of an ideal polarisation
hysteresis loop. The dashed line
presents the initial polarisation
process of the thermally unpoled
ceramic. The domain orientation
state is represented by arrows in
the boxes.
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Butterfly Hysteresis loop
A 2 dimensional schematic sketch
of an ideal butterfly hysteresis loop.
The dashed line presents the initial
polarisation process of the
thermally unpoled ceramic. The
domain orientation state is
represented by arrows in the boxes
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ferroelastic domain switching
(a) 90o switching of a polarisation in a
unit cell induced by a mechanical
loading; the polarisation direction of each
unit cell is represented by a black arrow
beside it. (b) 90o domain switching under
a compressive loading in a polycrystal;
the polarisation direction is represented
by the black arrow in each grain.
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ferroelastic domain switching
represents unsysmetric ferroelastic hysteresis and domain switching states
after applying tensile and compressive loading. The sketch simply
simulates the possible position of c-axis orientation with in a unit cell.
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ferroelastic domain switching
?
?
Compressive stress-strain curve of poled soft and hard PZT
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Piezoelectric Effect
Di =Pi= dij Tj
Pi = induced polarization along the i direction, dij = piezoelectric
coefficients, Tj = mechanical stress along the j direction
Converse Piezoelectric Effect
Sj = dij Ei
Sj = induced stain along the j direction, dij = piezoelectric
coefficients, Ei = electric field along the i direction
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Electro-Mechanical Coupling Factor
Electrical energy converted to mechanical energy
k 
Input of electrical energy
2
k = electromechanical coupling factor
Electro-Mechanical Coupling Factor
Mechanical energy converted to electrical energy
k 
Input of mechanical energy
2
k = electromechanical coupling factor
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Piezoelectric Equations and Constants
Piezoelectric Charge Constant (d)
The polarization generated per unit of mechanical stress
applied to a piezoelectric material
alternatively
The mechanical strain experienced by a piezoelectric
material per unit of electric field applied
Piezoelectric Voltage Constant (g)
The electric field generated by a piezoelectric material per
unit of mechanical stress applied
alternatively
The mechanical strain experienced by a piezoelectric
material per unit of electric displacement applied.
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Piezoelectric Materials
52
Piezoelectric Figures of merit*
*A figure of merit is a quantity used to
characterize the performance of a device
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Piezoelectric Figures of merit
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Piezoelectric Figures of merit
Coupling factor K
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Piezoelectric Figures of merit
56
Piezoelectric Figures of merit
The mechanical quality factor , QM
= (Strain in phase with stress)/(Strain out of phase
with stress)
High QM  low energy lost to mechanical damping.
So piezoelectric material with high QM is desirable
in a piezoelectric driver or resonator
57
Piezoelectric constants
Permittivity 
58
Piezoelectric constants
59
Basic Piezoelectric mode
The piezoelectric constants of a ferroelectric material poled in 3-direction. (a) shows d33
and d31-effect and (b) shows d15-effect.
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Basic Piezoelectric mode
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Piezoelectric transducers are widely used to generate ultrasonic waves in solids and also to
detect such mechanical waves. The transducer on the left is excited from an ac source and
vibrates mechanically. These vibrations are coupled to the solid and generate elastic waves.
When the waves reach the other end they mechanically vibrate the transducer on the right which
converts the vibrations to an electrical signal.
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Piezoelectric Voltage Coefficient
E = gT
E = electric field, g = piezoelectric voltage coefficient
T = applied stress
g = d/(or)
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Piezoelectric Spark Generator
The piezoelectric spark generator, as used in various applications such as lighters and
car ignitions, operates by stressing a piezoelectric crystal to generate a high voltage
which is discharged through a spark gap in air as schematically shown in picture (a).
Consider a piezoelectric sample in the form of a cylinder as in this picture. Suppose that
the piezoelectric coefficient d = 250 x 10-12 mV-1 and r = 1000. The piezoelectric cylinder
has a length of 10 mm and a diameter of 3 mm. The spark gap is in air and has a
breakdown voltage of about 3.5 kV. What is the force required to spark the gap? Is this a
realistic force?
The piezoelectric spark generator.
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Piezoelectric Quartz Oscillators
When a suitably cut quartz crystal with
electrodes is excited by an ac voltage as
(a), it behaves as if it has the equivalent
Circuit in (b).
(c) and (d) The magnitude of the
impedances Z and reactance (both
between A and B) versus frequency,
neglecting losses.
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Mechanical Resonant Frequency
fs 
1
2 LC
fs = mechanical resonant frequency, L = mass of the transducer, C = stiffness
Antiresonant Frequency
1
1
1 1
fa 
; where 

C CO C
2 LC
fa = antiresonant frequency, L = mass of the transducer, C is Co and C in
parallel, where Co is the normal parallel plate capacitance between
electrodes
For oscillators, the circuit is designed so that oscillations can take place
only when the crystal in the circuit is operated at fs
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Design of Buzzer
67
Design of Buzzer
68
A typical 1 MHz quartz crystal has the following properties:
fs = 1 MHz, fa = 1.0025 MHz, Co = 5 pF, R = 20 .
What is C in the equivalent circuit of the crystal? What is the quality
factor Q of the crystal, given that
1
Q
2f s RC
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Piezoelectric measurement
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