# Excitons in Cuprous Oxide

There is an excellent pair of papers here and here demonstrating the existence of a Rydberg series of excitons in Cu$_2$O, a material that has long been the poster child for observing excitons. The data in both papers are pretty stunning — in the first one, they see a Rydberg series up to n=25! In the second, they see higher angular momentum ($f$-like and $h$-like) excitons apart from the usual $p$-like excitons.

Excitons can occur in semiconductors and insulators and are the result of the Coulomb interaction between electrons and holes. For those familiar with superconductivity, an exciton, in many ways, bears a qualitative resemblance to Cooper pairs, except here the quasi-particles are of opposite charge. Hence, the binding between the electron and hole arises from the usual attractive Coulomb interaction. However, it helps to think of the exciton (similar to Cooper pairs) as possessing a center of mass wavefunction and a relative wavefunction.

The relative wavefunction is then qualitatively and quantitatively similar to the quantum mechanical solution of the hydrogen atom. The main differences are that (1) the reduced mass of the electron-hole pair is usually much less than the reduced mass of the electron-proton system and also that (2) there exists a background dielectric constant, $\epsilon$, that further reduces the energy of the Rydberg series in an insulator. In the simple hydrogen-like picture of the exciton, the energy levels (with center of mass momentum taken to be zero) then have the following energies:

$E_n =- \frac{\mu}{m\epsilon^2}\frac{R}{n^2}$

where $\mu$ is the reduced mass of the electron-hole system, $m$ is the proton mass, and $R$ is the hydrogenic Rydberg.

It turns out that this perfectly spherically symmetric model does not encapsulate completely the exciton in a solid, however, because the potential in a crystal is not perfectly rotation-symmetric. Because of the lattice, the “angular momentum” of the exciton cannot be labeled using an $s, p, d, f$, etc. kind of scheme and we must resort to some group theory.

For the case of Cu$_2$O discussed in the papers above, the symmetry of the crystal is cubic — therefore the symmetry is still relatively high. However, because continuous rotational symmetry is broken, the light used to detect the excitons in this experiment are no longer forbidden from exciting $f$-like and $h$-like excitons and it is remarkable that they are observable!

Even though the observations in these papers can be accounted for in a quite simple theoretical framework, the experimental results are nonetheless quite elegant and remarkable.

As an interesting aside, the reason that this experiment is possible is because of the extreme purity of the Cu$_2$O crystals used for these studies. It turns out the cleanest of these samples are naturally-occurring, rather than man-made.