Modern Theory of Polarization

It is quite curious that the simple concept of polarization in a solid was not understood until the early to mid-90s. The solution to the problem actually came from the computational physics community because of their inability to calculate accurately the polarization in solids. Prior to the solution, there were papers seriously discussing whether polarization was a bulk phenomenon or whether it was a property of a crystal surface. The solution to the problem is commonly associated with these papers by Resta (pdf!) and King-Smith and Vanderbilt (pdf!) .

There are a few startling realizations that arose from the modern theory of polarization:

1. Polarization is not a well-defined quantity in that it is multi-valued.
2. Only polarization difference has any physical meaning.
3. Experiments only measure differences in polarization.
4. Polarization is deeply rooted in the concept of the Berry phase.
5. (not startling) Polarization is a bulk phenomenon.

There is a fantastic pedagogical introduction to the subject by Spaldin entitled A beginner’s guide to the modern theory of polarization.

As just a little preview of the Spaldin paper, let me outline the issues as she does. The problem with defining polarization as the dipole moment per unit cell can easily be understood using the picture of the one-dimensional chain below:

Using the box on the left, one would calculate the dipole moment per unit length as:

whereas the box on the right gives:

.

That these don’t match has to do precisely with the fact that polarization in a solid is multi-valued, as alluded to earlier. Now consider the following one-dimensional lattice with a distortion:

The polarization now for the distorted lattice (second row in the image above) for the left and right boxes are respectively:

Left Box:

Right Box:

We can therefore see that for both boxes, the change in polarization is , which is a single-valued and experimentally well-defined quantity.

While this illustration was classical, when one includes the wavefunction of the electrons, one is forced to consider the Berry phase of the Bloch electrons. While I have known about this result for some time now, I still find it quite surprising that the simple concept of polarization in a solid has any relationship to the Berry phase at all. I strongly recommend Spaldin’s eminently readable article as an excellent introduction to the subject.