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Visit theMorgan Electro Ceramics Web Sitewww.morgan-electroceramics.com

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.

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 saturating-type 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 non-polar (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 single-domain 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 k2

- 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 inter-related 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.

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