# Tag Archives: Avoided Crossing

## Screening, Plasmons and LO-TO Splitting: One Last Time

I hope you are not by now fed up with my posts on this topic, but there is a great paper by Mooradian and Wright, which I’ve actually linked before in a different context, that is worth tackling. In this paper, they discuss the concept of plasmon-longitudinal optical (LO) phonon coupling.

To my mind, there is a significant aspect of the data which they do not explicitly address in their paper. Of course, I’m referring to degeneracy, LO-TO splitting, and screening. An image of their data is shown below (click to enlarge):

A quick run-down of the experiment: they are using Raman scattering on several different samples of GaAs with different doping levels. The carrier density can be read off in the image above.

It can be seen that for lower doped samples of GaAs, that there is strong LO-TO splitting. This is because of the long-ranged nature of the Coulomb interaction, as detailed here. As the carrier concentration is increased beyond the plasmon-phonon mixing region, the LO-TO splitting starts to disappear. This observation is noteworthy because there exists a “critical carrier density”, beyond which the LO and TO phonons are degenerate.

One can think of this in the following way: the plasmon energy is a measure of how quickly the free carriers can respond to an electric field. Therefore, for the highly doped GaAs samples, where the plasmon is at a significantly higher energy than the phonons, the free carriers can quickly screen the Coulomb field set up by the polar lattice. The electric field that is set up by the phonons can hence be approximated by a screened electric field (of the Thomas-Fermi kind) in this limit, and the Coulomb interaction is hence no longer long-ranged.

While the points I have made above will be quite obvious to many of you, I still find the data and its implications from a historical perspective quite profound.

Aside: I was heartened by Sarang’s post on the concept of emergence and upward heritability. One tends to think harder about one’s stance when there is an opposing view. He made some extremely important points regarding this topic, though I have to admit that I still lean towards Wilczek-ian concepts at present.

## A Little More on LO-TO Splitting

In my previous post, I addressed the concept of LO-TO splitting and how it results from the long-ranged nature of the Coulomb interaction. I made it a point to emphasize that while the longitudinal and transverse optical phonons are non-degenerate near $\textbf{q}=0$, they are degenerate right at $\textbf{q}=0.$ This scenario occurs because of the retarded nature of the Coulomb interaction (i.e. the finite speed of light).

What exactly goes on? Well, it so happens that in a very narrow momentum window near $\textbf{q}=0$, the transverse optical phonon is strongly coupled to light and forms a polariton. This is a manifestation of the avoided crossing or level repulsion principle that I have blogged about previously. Since light is a transverse wave, it interacts with the transverse optical phonon (but not the longitudinal one).

In a tour-de-force experiment at Bell Labs by Henry and Hopfield (the same Hopfield of Hopfield neural networks), Raman scattering was conducted at grazing incident angles to measure the dispersion of the lower polariton branch as shown below:

The dispersing solid lines represent the transverse optical (TO) phonon interacting with light. The straight solid line is the unaffected longitudinal optical (LO) phonon branch. The dotted line labelled with the angles are the incident beam angles in the Raman experiment. The remaining dotted lines represent the non-interacting TO phonon and the non-interacting light dispersion.

Usual Raman measurements are taken in a backscattering as opposed to a grazing incidence geometry, hence the momentum transfers are ordinarily too high to observe the low-$\textbf{q}$ dispersion. Because of this, the authors mentioned that some Raman exposures in this experiment took up to seven hours!

The takeaway from the plot above is that the transverse optical phonon at $\textbf{q}=0$ is degenerate with the longitudinal one and “turns into a photon” at higher momenta, while the photon branch at $\textbf{q}=0$ “turns into the transverse optical phonon” at higher momenta.

Unfortunately, the paper does not contain their raw data, only the dispersion. Publishing standards seem to have been different back then. Nonetheless, this is a very clever and illuminating experiment.

## Avoided Crossings, Level Repulsion or Anti-Crossing

The question of avoided crossing arises frequently in the study of condensed matter systems and always seems surprising to younger graduate students. In condensed matter physics, avoided crossings are ubiquitous, arising in phonon spectra, electronic band spectra and often when different types of quasi-particles interact.

Let me first say that avoided crossings are the norm and degenerate energies only occur because of some strict constraint (usually due to symmetry). As an example, let’s take the simple s-d Hamiltonian of a solid:

$-t\sum_i c^+_i c_{i+1} + h.c. + E_0 \sum_i d^+_i d_i + \sum_i \Delta c^+_id_i + h.c.$

where the $c^+_i$ creates an electron in an s-orbital, $d^+_i$ creates an electron in a d-orbital, $t$ is the hopping parameter, $E_0$ is the energy of an electron in a d-orbital and $\Delta$ is the hopping amplitude from an s- to a d-orbital. One can obtain the energy dispersion of this Hamiltonian by diagonalizing the following matrix:

$h(k) = \left( \begin{array}{cc} 2t\cos(ka) & \Delta \\ \Delta^* & E_0 \end{array} \right)$

Here are the resulting bands for a couple different values of $\Delta$ (click to enlarge):

While in the literature this is usually called a gap, this is nothing more than an avoided crossing and will occur for any finite $\Delta$. Again, this avoided crossing is not limited to electron energy bands, but can occur in many situations. Here is an old experiment by Mooradian and Wright from the 60’s which shows coupling between a plasmon and an optical phonon (click to enlarge):

One can very clearly see the plasmon (the broad peak) “becomes a logitudinal optical phonon” (the narrow peak) and the optical phonon “becomes the plasmon”. Notably, one can see that there is no coupling to the other (transverse) optical phonon because of symmetry reasons (plasmons cannot couple to transverse phonons), and it therefore stays put.