Polarization in Piezoceramic Materials
White Paper from Piezo Technologies
One can think of a
piezoceramic as a rock which generates electrical current if squeezed
and which moves if driven by an electrical current. For these wonderful
properties to occur electrodes must be connected to the piezoceramic and
the piezoceramic must be polarized. Electrodes are metal connections
typically covering opposite surfaces of the device. Polarization is a
vector quantity (i.e. having magnitude and direction) imparted to the
piezoceramic during the manufacturing process.
A molecule is polarized
if the average position of all of its positive ions is not the same as
the average position of all of its negative ions. For example, a water
molecule (see Figure 1) has a bond angle of 104.5° between the oxygen
and the two hydrogen atoms. As a result, the average position of the two
positive ions is not centered on the oxygen ion as it would be if the
bond angle were 180° and so the water molecule is strongly polarized.
At a molecular level,
the piezoelectric material is polarized when it is below its
Curie. Above the Curie temperature, the molecular structure has a
symmetric structure with the average central position of positive and
negative ions overlapping. As the structure cools through the Curie
temperature, the crystalline structure distorts and separates the
average positions of the positive and negative ions (see Figure 2).
Figure 1: Polarized Water Molecule. A:
The hydrogen atoms form bonds separated by 104.5°. B: The average
position of the two positive ions is on a line about 0.587 angstroms
away from the oxygen ion.
The energy of this
system can be described with a double well potential (see Figure 3-1).
If an ion has energy greater than the small peak separating the two
sides of the well (corresponding to a temperature above the Curie
temperature), then the ion bounces between the steep walls at the outer
limits of the well and the time averaged position is in the
center. However if the energy of the ion is below the level of the small
peak, then the ion will be trapped in one of the two wells of the
structure. The ion will be found on one side or the other, but it is
very unlikely to be found at the center.
Figure 2: Unit Cell Polarization. Polarization of a piezoelectric can be explained with the mechanical model of four balls (positive ions) located on the corners of a square frames (unit cell), which are connected by springs to a single central ball (negative ion). The bond length pushes the at-rest position of the central ion out of the plane of the four corner ions. Increasing temperature corresponds to increasing the energy of the system.
A: Above the Curie temperature, the energy of the system allows the central ion to bounce from above to below the plane and the time average position is in the middle.
B: If the energy of the system is too low, the central ion will be confined to stay either above or below the plane. As a result, the time average position of the ion is not in the plane and the system has a net polarization (depicted with an arrow), which points from the central negative ion to the plane of the four positive ions.
The energy of adjacent
molecules is reduced when polarization in the two molecules are
aligned. For this reason, entire domains or areas with a common
direction of polarization, spontaneously arise. The size of these
domains is limited both by imperfections in the material and by the
field energy. Piezoceramics are not formed by a single crystal, but by
grains of crystal separated by glassy interfaces and containing various
flaws. The energy associated with the electric fields increases with the
size of a domain, so eventually it is energetically advantageous for
the field directions to flip (just like two bar magnets placed side by
side will align head to tail) even in a perfect crystal.
Figure 3-1: Double well model of piezoelectric polarization.
A: The energy of the central negative ion corresponds to a well with two minima. The two minima are separated by a peak in energy E = kTC, where k is the Boltzmann constant and TC is the Curie temperature.
B: When the energy of the system is greater than kTC the central ion readily moves across the central peak and the time averaged probability (red dashed line) has a single peak in the middle.
C: If the temperature is below the Curie temperature, then the central ion will be trapped in one of the two basins. The time average curve has two peaks, corresponding to being trapped in one or the other basin.
The direction of
polarization of a domain can be switched by a sufficiently strong
external electric field. In the spring model shown in Figure 2, this is
analogous to pushing down on the central ion which was resting above the
plane until it pops over to the other side and finds its other rest
position below the plane. In the double well model, it is analogous to
distorting the shape of the two wells (see Figure 3-2).
At best, the total
polarization of a piezoceramic is less than about 60% of the
polarization which could be found in a single crystal of the same
formulation. The local domains of the ceramic are as likely to be
aligned side to side as to be aligned with the axis of polarization (see
Figure 4A). The force exerted by the external electrical field on the
polarization is proportional to the cosine of the angle between the two
vectors. This means that the force drops to zero when the domain is
aligned perpendicularly to the axis, and thus, these domains are
relatively unaffected by the polarization process. The result, as shown
in Figure 4B, is that the polarization of only a small fraction of the
domains is realigned during the polarization process.
Figure 3-2: Distorted Double Well Model of Piezoelectric Polarization. Applying an external field distorts the energy wells, and shifts the average probability of finding the central ion to the lower well.
Polarization of the
ceramic is not permanent. The polarized ceramic has higher energy and
lower entropy than it did prior to polarization. Random variations due
to heat, stress, noise and quantum tunneling will, on average, decrease
the polarization of the ceramic. Depoling of ceramic is usually a
logarithmic process. For example, a part may lose 2% of its net
polarization in the first hour after being polarized, followed by
another 2% lose in the next ten hours. Then, it will lose another 2% in
the next four days, another 2% in the next year, and then another 2% in
the next decade. The rate of depolarization of the piezoceramic
increases rapidly as the temperature approaches the Curie temperature.
Figure 4: Polarization.
Prior to polarization, the material is filled with many domains with
strong net polarization. However the total polarization of the sample is
zero because the domains point in different directions.
B: A sufficiently large external field can flip the polarization direction of some domains (red arrows). The result is a non-zero polarization, even if the external field did not change the local polarization of all of the domains.