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A PRIMER ON FERROELECTRICITY AND PIEZOELECTRIC CERAMICS
by Bernard Jaffe
PowerPoint Presentation and editing by
A Simple Picture of Piezoelectricity
P iezoelectricity 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)
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.
C onsider 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
E ach with a dipole.
S queeze or stretch crystal parallel to the dipole.
C harges 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.
I f the crystal’s unit cells each have no center of symmetry, they will be piezoelectric.
I f 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.
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
Crystals can vibrate in different modes; thickness, longitudinal, planar, and shear.
all pyroelectric crystals are piezoelectric,
but not all piezoelectric crystals are pyroelectric.
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
D isplayed 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
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.
T he charge at zero field is called remnant charge.
To this day, display of a saturating-type hysteresis loop with finite remnant charge is prime experimental evidence of ferroelectricity in a new material.
M ost ferroelectric crystals lose their dipole arrangement and become non-polar (paraelectric) if they are heated.
T he 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
F or 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.
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.
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
W ith most ferroelectric crystals, these domains, viewed in polarized light, form a spectacular display, particularly with varying applied stress or voltage.
Single Domain Crystals
I f 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.
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
S uch 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 single-domain crystal of the same composition.
A n outstanding feature of piezoelectric ceramics is their great stiffness.
D eflections 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.
T o 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:
k 2 = 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:
k 2 = 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:
K free (1 - k 2 ) = K clamped
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.
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:
Y open circuit (1 - k 2 ) = Y short circuited
k is the square root of k 2
It would make good sense to use k 2 as our figure of merit, but for historical reasons involving analogies with electrical induction, we actually use the square root, namely k.
Inasmuch as k 2 is always less than unity, k is handier to work with for weakly piezoelectric material, as it is a larger quantity.
Both k and k 2 are dimensionless numerics
Typical values of k
0.1 or 10% for quartz and tourmaline
0.5 for ceramic BaTiO 3
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 k p
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, k p .
Typical values for k p
0.35 for BaTiO 3
0.50 to 0.55 for Clevite PZT ceramics.
0.7 for specially prepared PZT compositions, a remarkable value.
Are generally written as tensor components with subscripts, such as d 33 or d 15 .
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.
T hus, d 31 measures the deflection along X in response to a voltage applied in the Z direction.
T he quantity d 15 measures the shear deflection around the Y axis caused by a voltage along the X axis.
Similarly d 33 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.
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 d 11 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 500 C/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.
The g constant is another frequently used piezoelectric measure.
This denotes the field produced by a given stress.
g = volts/meter
Usually simplified to 10 -3 meter volts/newton.
Typical values of g 33
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 g 11 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 g 33 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 inter-related by their dielectric constant. Thus
g = d/K o or gK o =d
where K is the dielectric constant and o 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,
k 2 = gdY
the value of high coupling coefficient for both types of application becomes evident.
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,