Transcript Ferroelectricity Primer

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A PRIMER ON FERROELECTRICITY
AND PIEZOELECTRIC CERAMICS
by Bernard Jaffe
PowerPoint Presentation and editing by
Jon Blackmon
A Simple Picture of
Piezoelectricity
Piezoelectricity is “pressure electricity”.
• Discovered by Pierre and Jacques Curie in the 1880’s.
• Piezoelectricity is a property of certain crystals:
– Quartz (Silicon Dioxide)
– Rochelle salt (Potassium Sodium Tartrate)
– Tourmaline (Aluminum Boron Silicate)
– Barium Titanate
– PZT Ceramic, and many others.
Hierarchy of Phenomena
Piezoelectric
Electricity from pressure
Pyroelectricity
Electricity from heat
Ferroelectricity
Can reverse polarity
• Rochelle salt is all of the above.
Visualize Piezoelectricity
Consider a crystal, each unit of which
has a dipole.
A dipole results from a difference
between the average location of the
+ and - charges in a unit cell.
(The strength of + and - charges are
equal.)
Visualize a crystal composed of
identical unit cells
Each with a dipole.
Squeeze or stretch
crystal parallel to the
dipole.
Charges appear on the
ends of the crystal.
If the same end faces are
electroded
• and charged from a voltage source
– the charges will cause the dipole to stretch or shorten
• depending on polarity
• because of electrostatic attraction or repulsion.
Not necessary to have a dipole in
each unit cell.
If the crystal’s unit cells
each have no center of
symmetry, they will be
piezoelectric.
If they have a center of
symmetry they will be
inert when squeezed or
stretched.
A center of symmetry in a unit cell is an
imaginary center point.
Each atom of such a cell has an exact twin
opposite it on a line through the center point.
There is one unimportant exception
A certain cubic crystal class
has no center of symmetry
is not piezoelectric.
There are 32 crystal classes
• each representing a type of unit cell.
– 20 have no center of symmetry and are thus
piezoelectric.
• Among the piezoelectrics that have no
center of symmetry,
– there are 10 crystal classes that have a dipole in
their unit cell
• These are called pyroelectric.
Pyroelectricity
Just as squeezing or stretching
the crystal, thermal expansion
typically expands or contracts
the dipole.
This causes a charge to appear
on crystal faces near the ends
of the dipoles.
Crystals that are piezoelectric
(no center of symmetry)
but not pyroelectric (no dipole)
• do not show a charge on heating or cooling.
Piezoelectric Fundamental Vibration Modes
Summarizing
• Crystals can vibrate in different modes;
thickness, longitudinal, planar, and shear.
• all pyroelectric crystals are piezoelectric,
• but not all piezoelectric crystals are
pyroelectric.
Ferroelectricity
In all pyroelectric crystals
• the dipoles are influenced by electrostatic
forces when a field is applied to opposite
faces of the crystal.
• In some the dipole can actually be
reversed.
• If a field opposite in sense to the dipole is
applied at higher and higher voltage, the
dipoles of some crystals can reverse their
direction.
• To do this, the atoms, or rather ions that
form the dipole suddenly shift position a
little.
• This phenomenon of a dipole reversing in
an opposing field is called Ferroelectricity.
– Rochelle Salt was the first ferroelectric crystal
known.
Sawyer and Tower study
Displayed charging current of a crystal slab
on the vertical plates of a cathode ray tube,
• and voltage applied to the crystal slab on
the horizontal plates, as a
• Lissajous figure, – a closed figure that is
traced once during each electrical cycle.
Most ordinary crystals would
give a straight line figure
because they obey the familiar
linear relationship,
Q = CV
The Rochelle salt gave a very different figure
a hysteresis loop.
Trace this loop through a cycle
At the extreme right hand corner, high voltage causes
saturation, and we have a linear region.
The low slope represents low incremental capacitance.
As the field is reduced, the charge remains very high.
The field continues through zero
and becomes negative.
Suddenly the charge drops abruptly
and becomes very large the other way.
This is because all dipoles reverse direction.
• Further increase in negative voltage merely
causes saturation again.
• Then the voltage reverses, passes through
zero again, and finally, at the coercive field,
• the dipoles reverse again to their original
direction, and then saturate.
Behavior of this type is found
only in ferroelectric crystals.
The charge at zero field is called
remnant charge.
• To this day, display of a saturatingtype hysteresis loop with finite
remnant charge is prime
experimental evidence of
ferroelectricity in a new material.
Curie Temperature
Most ferroelectric crystals lose their
dipole arrangement and become nonpolar (paraelectric) if they are heated.
The temperature at which they lose
their polar nature and acquire instead
a center of symmetry and linear
capacitance is called the Curie
temperature.
Dielectric constant becomes high
For most ferroelectrics, the dielectric constant
becomes very high at this temperature, as
much as 10,000 to 20,000.
• The dielectric constant is frequently rather
high, too, typically several hundred, at
temperatures in the ferroelectric range.
• There are exceptions to all of these
generalizations.
Electrical Domains
• When a crystal cools from its
paraelectric range through the Curie
temperature, it generally shows
domains.
• The unit cells each have a dipole, but
the direction of this dipole changes
from one region in the crystal to
another.
• It is easier for the crystal to cool with
many compensating domains rather
than one, to minimize free energy.
Multi-domain Crystals
• Each domain contains many millions of unit cells.
• There are only a few discrete orientations that the
dipoles that form domains may assume.
• Domain walls are the boundaries through which
the dipole direction changes.
• Slight pressure or slight applied field can usually
move the domain walls.
Make a spectacular display
With most ferroelectric crystals,
these domains, viewed in
polarized light, form a
spectacular display,
particularly with varying
applied stress or voltage.
Single Domain Crystals
If applied voltage is strong enough, all
dipoles of the crystal will become parallel,
domain walls will disappear, and we will
have a single domain crystal.
• These show strong piezoelectric effects, in
contrast with multidomain crystals where
piezoelectric effects of differently oriented
domains tend to cancel one another.
Ceramic Composition
A ceramic is composed of a multitude of
crystals in random orientation.
• Its properties are the sum of the properties
of all these crystallites.
• If the ceramic is made of ferroelectric
crystals, it will display a dielectric
hysteresis loop just like the one described
for Rochelle salt.
Poled Ceramic is ferroelectric
Such a ceramic can be poled by a strong d.c. field.
• It will then have behavior roughly like that of a
single domain ferroelectric crystal, and will be
piezoelectric.
• Not every domain aligns its dipoles parallel to the
field, but enough of them do so to give an overall
effect.
• For some ceramics, the piezoelectricity is quite
strong, but it does not equal the effect in a singledomain crystal of the same composition.
Stiff Ceramics
An outstanding feature of piezoelectric
ceramics is their great stiffness.
Deflections in response to a driving signal
are very small, but they are very strong
and not easily blocked.
Shake a Stone Wall
A ceramic transducer could shake a stone wall very
violently, but it would move air or water
inefficiently.
To make it able to drive air or water, some sort of
mechanical transformer is necessary, just as a cone
or horn is necessary to couple the voice coil of a
loudspeaker to the air.
Piezoelectric ceramics make
good sensors
• For sensing forces, piezoelectric ceramics
are very sensitive and deliver easily useable
signals in response to a small deflection.
Electromechanical Coupling Coefficient
k
The best measure of strength of the piezoelectric effect
is the electromechanical coupling coefficient, k.
• It measures the ability of the crystal or ceramic to
change energy from one form to another; that is,
to act as a transducer of energy.
• This is not an efficiency; efficiency is concerned with
output and input of power, regardless of form, and
depends only on the losses.
Let us imagine an electroded slab
of crystal or ceramic.
If we squeeze it, it compresses like a spring.
• Energy used to compress it is given back
when the force is released.
• If, however, the slab is piezoelectric, some
of the energy expended in compressing it
will be transduced to electric charge.
Compressed spring and charged
capacitor
The slab will be equivalent to both a compressed
spring and a charged capacitor.
• On release, the crystal returns to its original size
and the charge reduces to zero.
• The following relationship holds:
k2 = mechanical energy converted to electric charge
input mechanic energy
Same with electrical energy
• Conversely, if we take the same slab, attach a battery
and charge it, part of the input energy will be
transduced to mechanical energy and will deform the
slab, making it smaller or larger, dependent on the
polarity of the battery.
• The same relationship holds, and the coupling
coefficient is numerically identical:
k2 = electrical energy converted to mechanical energy
input electrical energy
Dielectric Constant K
• The dielectric constant measures the amount
of charge that an electroded slab of material
can store, relative to the charge that would
be stored by identical electrodes separated
by air or vacuum at the same voltage.
• If we measure the dielectric constant of a
piezoelectric material, first without applied stress,
and then clamped mechanically so firmly that it
cannot deform, the same numerical value of
coupling coefficient can be found:
Kfree(1 - k2) = Kclamped
where capital K is the dielectric constant.
• This method is actually used by first
measuring the dielectric constant at low
frequency, where the slab is free to vibrate,
and then measuring at high frequency,
above all the mechanical resonances, where
the slab crystals effectively clamped by its
own inertia.
Young’s Modulus
• A similar relationship holds for the
elasticity, measured by Young’s modulus
(the stress divided by the strain).
• The modulus is different if the electrodes
are unconnected than it is if they are shorted
together by a wire.
• When shorted, the slab is softer; easier to
deflect.
• This difference can actually be felt in certain
slender shapes when deformed by hand,
particularly in Bimorphs (two piezoelectric slabs
back to back) when flexed.
• The difference in elastic modulus yields the same
numerical value of the coupling coefficient:
Yopen circuit (1 - k2 ) = Yshort circuited
k is the square root of
2
k
• It would make good sense to use k2 as our figure
of merit, but for historical reasons involving
analogies with electrical induction, we actually use
the square root, namely k.
• Inasmuch as k2 is always less than unity, k is
handier to work with for weakly piezoelectric
material, as it is a larger quantity.
• Both k and k2 are dimensionless numerics
Typical values of k
•
•
•
•
0.1 or 10% for quartz and tourmaline
0.5 for ceramic BaTiO3
0.7 for Clevite PZT ceramic
0.9 for Rochelle salt if it is kept at its most
favorable temperature.
Natural Mechanical Resonances
• Most measurements of coupling coefficient are
made by applying small signals to geometrically
shaped crystal and ceramic samples.
• These samples have natural mechanical
resonances.
• Because they are piezoelectric, it is possible to
drive them electrically at these frequencies.
Planar Coupling Coefficient kp
• At resonance, characteristic changes in
electrical impedance can be noted.
• By determining these characteristic
frequencies, the coupling coefficient and
elastic moduli can be evaluated.
• For ceramic samples, thin discs vibrating
radially are frequently used to find the
planar coupling coefficient, kp.
Typical values for kp
• 0.35 for BaTiO3
• 0.50 to 0.55 for Clevite PZT ceramics.
• 0.7 for specially prepared PZT
compositions, a remarkable value.
Piezoelectric Constants
• Are generally written as tensor components with
subscripts, such as d33 or d15.
• Consider a crystal with orthogonal
(right angle) axes X,Y and Z,
represented as 1, 2 and 3.
• First subscript is the electrical direction.
• The second the mechanical direction.
• The numbers 4, 5 and 6 refer to shear around X,Y
and Z, respectively.
Tensor Components
Thus, d31 measures the
deflection along X in
response to a voltage
applied in the Z direction.
The quantity d15 measures the
shear deflection around the
Y axis caused by a voltage
along the X axis.
• Similarly d33 measures the electrical effect
in the Z direction of a stress in the same
direction, or the charge in this direction that
results from a parallel force.
d constant
• The d constant measures the amount of
charge caused by a given force, or the
deflection caused by a given voltage.
• Customary units are micromicrocoulombs
per Newton.
• A Newton is equal to the pull of gravity on
a 102 gram mass, about 3.6 ounces.
• The d constant is numerically the same in
the direct as in the converse effect.
• Quartz has d11 equal to 2 x 10-12 coulombs
per Newton or 2 x 10-12 meters per volt.
Typical values of d
• 2 for Tourmaline
• 150 for Rochelle salt (Rochelle salt has shear d
constants over 500mmC/N)
• 180 for ceramic barium titanate
• 300 for ordinary PZT ceramic.
• Specially prepared PZT compositions have shown
values of 600.
• In practical terms, this is 6 Angstroms per volt, or
a deflection of 1 millimeter for 1.7 million volts.
g Constant
The g constant is another frequently used
piezoelectric measure.
• This denotes the field produced by a given
stress.
g=
volts/meter
newtons/square meter
• Usually simplified to 10-3 meter volts/newton.
Typical values of g33
• 12 x l0-3 meter volts/newton for barium
titanate
• 12 to 35 for different kinds of PZT
• 90 for Rochelle salt
• 50 for the g11 of quartz.
How much is that?
• To realize the magnitude involved, consider that a
newton per square meter is very small, about
1/7000 of a pound per square inch.
• If we work out the conversion, barium titanate,
with g33 of 12 x l0-3 meter volt/newton would
exhibit a field of 2.1 volts per inch in response to a
stress of 1 lb/sq. inch
g and d constants
• The g and d constants of a material are interrelated by their dielectric constant. Thus
g = d/Keo or gKeo=d
• where K is the dielectric constant and eo is the
permittivity of space, 9 x 10-12 Farads per meter.
• For a given level of d constant, g is high if the
dielectric constant is low and vice-versa.
• Thus, quartz has a small d constant, but its g
constant is respectable because the dielectric
constant is so low, about 4.
Two main uses for piezoelectrics
In “motor” applications, large deformations are
desired at minimum voltage, such as in ultrasonic
sound drivers. For these uses, high d constant is
desirable.
In “generator” applications, a strong electrical signal
is desired in response to weak forces that are to be
sensed, as in a microphone or phonograph
cartridge. Here, high g constants are wanted.
Cables act as parallel capacitors
In generator applications, connections to the
element, such as cables, act as parallel
capacitors and lower output voltage.
Therefore, a high dielectric constant is also
desirable to minimize the effect of these
capacitive loads.
g and d should both be high
Since the coupling coefficient (squared) is the
product of g and d, multiplied by the elastic
modulus,
k2 = gdY
• the value of high coupling coefficient for
both types of application becomes evident.
Apologia
The author wishes to apologize in advance for the
oversimplifications and the non-rigorous
pictorializations used.
No claim of originality is made.
For further information, interested readers are
referred to two standard books,
• “Piezoelectricity” by W. G. Cady, and
• “Piezoelectric Crystals and Their Application to
Ultrasonics” by W. P. Mason.