Radiant Technologies, Inc.
2835D Pan American Freeway NE
Albuquerque, NM 87107
Tel:
505-842-8007
Fax:
505-842-0366
www.ferrodevices.com
e-mail: radiant@ferrodevices.com
Chapter 3: Theory of Paraelectric Capacitance
Joe T. Evans, Jr.
Introduction:
Linear capacitors are not very interesting: they just make straight lines on a graph of
charge and voltage. Fortunately, we can put highly complex materials between the plates
of a capacitor and generate highly complex behaviors from those capacitors. These nonlinear capacitors are critical to the operation of our society where they are used for
sensors, variable tuning devices, filters, and transducers. The discussion below will
contrast linear capacitors with a type of commonly used non-linear device.
A Review of Linear Capacitance:
All capacitors can be charged by the application of a voltage.
Q
= CV
eq(1.3)
We can rewrite the equation as a ratio where the capacitance C is the “slope” of the
function:
Q/V = C
eq(3.1)
For the purpose of this discussion the distinction between linear and non-linear devices is
simple: a linear device has a constant capacitance at all voltages at a constant temperature
and pressure. A non-linear capacitor does not.
Hysteresis of a 1nF Ceramic Disk Capacitor
0.005
0.004
C = ∆Q/∆V
0.003
Charge (µC)
0.002
∆Q1
0.001
∆V1
0.000
-0.001
∆Q3
-0.002
∆Q2
-0.003
∆V2
∆V3
Linearity means:
C1 = C2 = C3
-0.004
-0.005
-5
-4
-3
-2
-1
0
Voltage
1
2
3
4
5
Figure 3.1
The slope of a linear capacitor is constant at all voltages.
It is important to notice that the conditions here are for constant temperature and
pressure. The reason for this is that there is no truly linear dielectric except a vacuum.
All materials change their relative dielectric constant at higher electric fields. They also
change their capacitance at higher temperatures and higher pressures. The easiest way to
visualize this concept is to remember the equation for capacitance:
∴Capacitance
= C = εo • A / t
eq(1.2)
The thickness of the capacitor is part of the equation. If we compress a capacitor with
even a millionth of a newton of force (about the force of the spoken voice), “t” changes
and the capacitance increases a very small amount. If we increase the temperature, the
capacitor increases its size due to the coefficient of thermal expansion, again changing “t”
in the denominator of equation (1.2).
The situation is even further complicated by friction. For example, a coat hanger heats
up when you bend it back and forth. A capacitor has a little “friction” when you charge it
up and discharge it. Its temperature will rise, changing its capacitance a little bit. You
can even make the case that the temperature of the capacitor will increase even as you are
increasing the voltage across the capacitor, causing its charge vs voltage curve to arc
downwards slightly at each higher voltage. Of course, the capacitor will lose excess heat
to its package and the air around it so that if you charge it slowly, the average
2
temperature will remain roughly constant and the change in capacitance effect of
temperature from “friction” will be less.
We have now reached an amazing level of theoretical complexity while analyzing the
charge vs voltage curve of a linear capacitor. We just proved that any capacitor with an ε
greater than εo is not truly linear. In reality, it is nearly impossible to measure the
theoretical non-linearity because there are real limitations to measurement accuracy due
to noise, time, and sensitivity. As far as we are concerned in this discussion, a traditional
linear capacitor is linear because our best measurements indicate that it is so. Our
concern is with capacitors having dielectric materials that have large non-linearities by
their nature. These materials show large changes to their dielectric constants that are
easy to see in real devices.
Non-Linear Capacitors:
Let’s jump right into it. Figure 3.2 is a plot of a non-linear capacitor called a paraelectric
capacitor.
Radiant 9/65/35 PLZT
[ 1700A ]
60
50
∆Q2
∆V2
40
30
Polarization
20
10
∆Q1
0
∆V1
-10
-20
-30
-40
Non-linear Capacitance: C1 ≠ C2
-50
-60
-30
-25
-20
-15
-10
-5
0
Volts
5
10
15
20
25
30
Figure 3.2
A Paraelectric Capacitor
How did this happen? Why does the capacitance change with voltage? Remember that
in the lesson on Chapter 1 - Linear Capacitance, I described how the relative dielectric
3
constant arising from the electric field inside the capacitor causes the electrons and atoms
of the dielectric material between the plates to separate a little bit. Well, the material
used for the dielectric material in the capacitor of Figure 3.2 has a lot of atoms in its
crystal lattice stuffed into a very small space, about 4 angstroms on a side. An angstrom
is equal to 1/10th of a nanometer. An atom usually occupies a sphere about 1 angstrom in
diameter and this material has an entire titanium or lanthanum atom, ½ of six oxygen
atoms each, and 1/8th of eight lead atoms occupying that 4Å x 4Å x 4Å cube. There is
not a lot of room so that when the electric field is applied, the atoms in their entirety
move to compensate. This gives the material a high relative dielectric constant, ranging
from a value of 100 on the low side for some materials to 30,000 on the high side.
Note that there are no units given for the relative dielectric constant. εr has no
units because it multiplies εo which does have units.
When the atoms move in an electric field, some move towards the positive side of the
field and some move away. The material stretches. But, it can only stretch so far. Its
initial movement is large, giving us C1 in Figure 3.2. But, it runs out of room to move at
higher voltages, giving us C2 in Figure 3.2.
Capacitor Plate
E
Dielectric material
The dielectric material
expands in one direction and
shrinks in the other.
Figure 3.3
Non-linear Capacitor
Another issue in Figure 3.2 is that the loop is open, not closed. The cause for this effect
is very complex. Suffice it to say that this is one of those “friction” effects mentioned in
the earlier section of this chapter about linear capacitors and we will see it again during
our experiments.
4
The material shown in Figure 3.2 is called a “paraelectric” material and is identified by
the classic “S” shape of its charge vs voltage curve. This particular material goes by the
acronym PLZT, which stands for Lead (Pb) Lanthanate (La) Zirconate (Zr) Titanate (Ti).
Its chemical formula is (PbLa)(ZrTi)O3. PLZT and related materials like Bismuth
Titanate (BiTiO3) are ceramics descended from the Native American pottery of the
American Southwest where Radiant makes its home. The various metal oxides are mixed
together and heated to transform the mixture into a ceramic. The Native Americans used
the ceramics to hold food and water. We use these ceramics in electronics and sensors.
Small amounts of dopants are added to the mixture to adjust the properties of the final
material. In the case of the PLZT in Figure 3.2, the lanthanum is a dopant to PZT that
changes the capacitor from a “ferroelectric” material with memory, which we will discuss
a little later, to a “paraelectric” material with no memory.
The lanthanum makes the PLZT transparent. PLZT is very fast electrically. Its change in
capacitance means that its dielectric constant changes with the applied voltage. As we
learned in Chapter 2 - The Importance of Dielectric Constant, the speed of light in PLZT
changes with the applied voltage so we can use PLZT to modulate light. Another
paraelectric material, Lithium Niobate (LiNbO3), is used to make the electro-optic
modulators that feed information into the optical fibers that form the backbone of the
Internet from which you may have downloaded this document!
Let’s introduce another method of plotting the measured data, a method that provides
information about the inner workings of the capacitor. In Figures 3.1 and 3.2, the
measured charge (or polarization) was plotted against the voltage. Both figures also
show specific points where the slope of the measured data is calculated. As equation
(3.1) shows, the slope of the measured data is the capacitance at that point of the
measurement. On a paraelectric capacitor, the slope is not constant, meaning that its
capacitance changes as a predictable function of the voltage profile. Using calculus, we
can calculate the slope of the measured data at each point of the measurement and then
make a plot of the slope vs the voltage, as in Figures 3.5 and 3.6. Figure 3.5 shows the
hysteresis measurement of a commercial high voltage capacitor capable of withstanding
6,000 volts, a device used in control circuitry for manufacturing. High voltage capacitors
are usually made with a form of PZT. A picture of this particular capacitor is in Figure
3.4.
5
Figure 3.4
6,000V 3.3nF Commercial High Voltage Capacitor
High Voltage Paraelectric Capacitor
[ 3.3nF 6000V Rating ]
10.0
7.5
5.0
Charge (µC)
2.5
0.0
-2.5
-5.0
-7.5
-10.0
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
Voltage
Figure 3.5
5,000V Hysteresis of the Commercial High Voltage Capacitor in Figure 3.4
6
Note that the maximum voltage of our test in Figure 3.4 is 5,000V. Now, in Figure 3.6,
we plot the capacitance vs voltage function of the same sample. Figure 3.6 shows the
slope of the hysteresis at each voltage in Figure 3.5.
High Voltage Paraelectric Capacitor
[ 3.3nF 6000V Rating ]
0.008
0.007
Capacitance (µF)
0.006
0.005
Linear Capacitance
0.004
0.003
0.002
0.001
0.000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
Voltage
Figure 3.6
The Measured Capacitance vs Voltage Function of the Capacitor in Figure 3.5
Figure 3.6 is a magnificent demonstration of the paraelectric function. The dashed line
across the plot represents what a linear capacitor would look like on the same plot. It is a
horizontal line because its capacitance never changes. This paraelectric capacitor
changes its capacitance by at least a factor of four over the ±5,000 volt test. It is rated as
a 3.3n F capacitor. Note that it actually starts at 4nF (0.004µF) but then drops to 1nF by
5,000 volts.
Why use a paraelectric material in place of a linear capacitor when the linear capacitor
will be constant? The reason is that dielectric materials that act in a linear fashion usually
have low dielectric constants. Paraelectric materials can have very large relative
dielectric constants, sometimes approaching 30,000. To get the desired capacitance
7
value, a linear capacitor would have to be hundreds or even thousands of times larger in
area than the capacitor made with paraelectric material. That my not be practical.
Most ferroelectric materials derive their properties from the same crystal lattice
characteristics as paraelectric capacitors. In the next chapter, we will see how adding
“memory” to a paraelectric capacitor gives us the classic ferroelectric hysteresis curve.
8