Ferroelectricity and ferromagnetism are two different things, but they are analogous. Materials are ** ferroelectric **if they have a spontaneous

**polarization that can be changed or reversed by the application of an external**

*electric**electric*field.

**, in contrast, refers to materials which exhibit a permanent**

*Ferromagnetism**magnetic*moment.

The materials are very different. In fact, most ferroelectric materials do *not* contain any iron and, hence, the *ferro *in the term is somewhat misleading. Ferroelectric materials are a special class of *crystals*, like barium or lead titanate (BaTiO_{3 }or PbTiO_{3}). Lead zirconate titanate (LZT) is another example. These materials are also *piezoelectric*: when applying some *mechanical* stress, they will generate some *voltage*. In fact, the process goes both ways: when *applying* some voltage to them, it will also create mechanical deformation, as illustrated below (credit for this illustration goes to Wikipedia).

Ferroelectricity has to do with *electric *dipoles, while ferromagnetism has to do with *magnetic *dipoles. We’ve only discussed *electric dipoles *so far (see the section on dielectrics in my post on capacitors) and so we’re only in a position to discuss ferroelectricity right now, which is what I’ll do here. However, before doing so, let me briefly quote from the Wikipedia article on ferromagnetism, because that’s really concise and to the point on this:

“One of the fundamental properties of an electron (besides that it carries charge) is that it has a magnetic dipole moment, i.e. it behaves like a tiny magnet. This dipole moment comes from the more fundamental property of the electron that it has quantum mechanical spin. Due to its quantum nature, the spin of the electron can be in one of only two states; with the magnetic field either pointing “up” or “down” (for any choice of up and down). The spin of the electrons in atoms is the main source of ferromagnetism, although there is also a contribution from the *orbital *angular momentum of the electron about the nucleus.”

In short, ferro*magnetism *was discovered and known much before ferroelectricity was discovered and studied, but it’s actually more complicated, because it’s a quantum-mechanical thing really, unlike ferroelectricity, which we’ll discuss now. Before we start, let me note that, in many ways, this post is a continuation of the presentation on dielectrics, which I referred to above already, so you may want to check that discussion in that post I referred to if you have trouble following the arguments below.

**Molecular dipoles**

Let me first remind you of the basics. The (electric) ** dipole moment** is the product of the distance between two equal but opposite charges q

_{+}and q

_{−}. Usually, it’s written as a vector so as to also keep track of its

*direction*and use it in

*vector equations*, so we write

**p**= q

**d**, with

**d**the

*vector*going from the negative to the positive charge, as shown below.

Now, molecules like water molecules have a permanent dipole moment, as illustrated below. It’s because the center of ‘gravity’ of the positive and negative charges do *not *coincide, so that’s what makes the H_{2}O molecule polar, as opposed to the O_{2} molecule, which is non-polar.

Now, if we place *polar *molecules in some electric field, we’d expect them to line up, to some extent at least, as shown below (the second illustration has *more *dipoles pointing vaguely north).

However, at ordinary temperatures and electric fields, the collisions of the molecules in their thermal motion keeps them from lining up *too* much. In fact, we can apply the principles of statistics mechanics to calculate *how much* exactly. You can check out the details in Feynman’s *Lecture *on it, but the result is that the *net dipole moment per unit volume *(so that’s the polarization)* *is equal to:

So the polarization is proportional to the number of molecules per unit volume (N), the square of their dipole moment (p_{0}) and, as we’d might expect, the electric field *E*, and *inversely *proportional to the temperature (T). In fact, the formula above is a sort of first-order approximation, in line with what we wrote on the *electric susceptibility* χ (*chi*) in our post on capacitors, where we also assumed the relation between P and E was linear, so we wrote: P = ε_{0}·χ·E. Now, engineers and physicists often use different symbols and definitions and so you may of may not have heard about another concept saying essentially the same thing: the *dielectric constant*, which is denoted by *κ* (*kappa*) and is, quite simply, equal to *κ =* 1 + χ. Combining the expression for P above, and the P = ε_{0}·χ·E = ε_{0}·(*κ*−1)·E expression, we get:

This doesn’t say anything new: it just states the dependence of χ on the temperature. Now, you can imagine this linear relationship has been verified experimentally. As it turns out, it’s sort of valid, but it is *not *as straightforward as you might imagine. There’s a nice post on this on the University of Cambridge’s Materials Science site. But this blog is about physics, not about materials science, so let’s move on. The only thing I should add to this section is a remark on the energy of dipoles.

You know charges in a field have energy, *potential *energy. You can look up the detail behind the formulas in one of my other posts on electromagnetism, so I’ll just remind you of them: the energy of a charge is, quite simply, the product of the charge (q) and the electric *potential *(Φ) at the location of the charge. Why? Well… The potential is the amount of work we’d do when bringing the *unit *charge there from some other (reference) point where Φ = 0. In short, the energy of the positive charge is q·Φ(1) and the energy of the negative charge is −q·Φ(2), with 1 and 2 denoting their respective location, as illustrated below.

So we have U = q·Φ(1) − q·Φ(2) = q·[Φ(1)−Φ(2)]. Now, we’re talking tiny little dipoles here, so we can approximate ΔΦ = Φ(1)−Φ(2) by ΔΦ = **d**•**∇**Φ = Δx·(∂Φ/∂x) + Δy·(∂Φ/∂y). Hence, also noting that **E** = −**∇**Φ and q**d** = **p**_{0}, we get:

U = q·Φ(1) − q·Φ(2) = q**d**•**∇**Φ = −**p**_{0}•**E **= −p_{0}·E·cosθ, with θ the angle between **p**_{0 }and **E**

So **the energy is lower when the dipoles are lined up with the field**, which is what we would expect, of course. However, it’s an interesting thing so I just wanted to show you that. 🙂

**Electrets, piezoelectricity and ferroelectricity**

The analysis above was very general, so we actually haven’t started our discussion on ferroelectricity yet! All of the above is just a necessary introduction to the topic. So let’s move on. Ferroelectrics are solids, so let’s look at solids. Let me just copy Feynman’s introduction here, as it’s perfectly phrased:

“The first interesting fact about solids is that there can be a permanent polarization built in—which exists even without applying an electric field. An example occurs with a material like wax, which contains long molecules having a permanent dipole moment. If you melt some wax and put a strong electric field on it when it is a liquid, so that the dipole moments get partly lined up, they will stay that way when the liquid freezes. The solid material will have a permanent polarization which remains when the field is removed. Such a solid is called an ** electret**. An electret has permanent polarization charges on its surface. It is the electrical analog of a magnet. It is not as useful, though, because free charges from the air are attracted to its surfaces, eventually cancelling the polarization charges. The electret is “discharged” and there are no visible external fields.”

Another example (i.e. other than wax) of an electret is the crystal lattice below. As you can see, all the dipoles are pointing in the same direction even with no applied electric field. Many crystals have such polarization but, again, we do not normally notice it because the external fields are discharged, just as for the electrets.

Now, this gives rise to the phenomena of *pyroelectricity *and *piezoelectricity*. Indeed, as Feynman explains: “If these internal dipole moments of a crystal are changed, external fields appear because there is not time for stray charges to gather and cancel the polarization charges. If the dielectric is in a condenser, free charges will be induced on the electrodes. The moments can also change when a dielectric is heated, because of thermal expansion. The effect is called ** pyroelectricity**. Similarly, if we change the stresses in a crystal—for instance, if we bend it—again the moment may change a little bit, and a small electrical effect, called

**, can be detected.”**

*piezoelectricity*But, still, piezoelectricity is not the same as *ferroelectricity*. In fact, there’s a hierarchy here:

- Out of all crystals, some will be piezoelectric.
- Among all piezoelectric crystals, some will also be pyroelectric.
- Among the pyroelectric crystals, we can find some
*ferroelectric*crystals.

The defining characteristic of* ferroelectricity* is that the built-in permanent moment

*can be reversed*by the application of an external electric field. Feynman defines them as “nearly cubic crystals, whose moments can be turned in different directions, so we can detect a large change in the moment when an applied electric field is changed: all the moments flip over and we get a large effect.”

Because this is a blog, *not *a physics handbook, I’ll refer you to Feynman and/or the Wikipedia article on ferroelectricity for an explanation of the *mechanism*. Indeed, the objective of this post is to explain *what *it is, and so I don’t want to go off into the weeds. The two diagrams below, which I took from the mentioned Wikipedia article, illustrate the difference between your average dielectric material as opposed to a ferroelectric material. The first diagram shows you the linear relationship between P and E we discussed above: if we reverse the field, so E becomes negative, then the polarization will be reversed as well, but gradually, as shown below.

In contrast, the illustration below shows a ** hysteresis** effect, which can be used as a memory function, and ferroelectric materials are indeed used for ferroelectric RAM (FeRAM) memory chips for computers! I’ll let you

OK. That’s it for today. More tomorrow. 🙂

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