# Tag Archives: Plasmons

## Acoustic Plasmon

In regards to the posts I’ve made about plasmons in the past (see here and here for instance), it seems like the plasmon in a metal will always exist at a finite energy at $q=0$ due to the long-ranged nature of the Coulomb interaction. Back in 1956, D. Pines published a paper, where in collaboration with P. Nozieres, he proposed a method by which an acoustic plasmon could indeed exist.

The idea is actually quite simple from a conceptual standpoint, so a cartoony description should suffice in describing how this is possible. The first important ingredient in realizing an acoustic plasmon is two types of charge carriers. Pines, in his original paper, chose $s$-electrons and $d$-electrons from two separate bands to illustrate his point. However, electrons from one band and holes from another could also suffice. The second important ingredient in realizing the acoustic plasmon is that the masses of the two types of carriers must be very different (which is why Pines chose light $s$-electrons and heavy $d$-electrons).

Screening of heavy charge carrier by light charge carrier

So why are these two features necessary? Well, simply put, the light charge carriers can screen the heavy charge carriers, effectively reducing the range of the Coulomb interaction (see image above). Such a phenomenon is very familiar to all of us who study solids. If, for instance, the interaction between the ions on the lattice sites in a simple 3D monatomic solid were not screened by the electrons, the longitudinal acoustic phonon would necessarily be gapped because of the Coulomb interaction (forgetting, for the moment, about what the lack of charge neutrality would do to the solid!). In some sense, therefore, the longitudinal acoustic phonon is indeed such an acoustic plasmon. The ion acoustic wave in a classical plasma is similarly a manifestation of an acoustic plasmon.

This isn’t necessarily the kind of acoustic plasmon that has been so elusive to solid-state physicists, though. The original proposal and the subsequent search was conducted on systems where light electrons (or holes) would screen heavy electrons (or holes). Indeed, it was suspected that Landau damping into the particle-hole continuum was preventing the acoustic plasmon from being an observable excitation in a solid. However, there have been a few papers suggesting that the acoustic plasmon has indeed been observed at solid surfaces. Here is one paper from 2007 claiming that an acoustic plasmon exists on the surface of beryllium and here is another showing a similar phenomenon on the surface of gold.

To my knowledge, it is still an open question as to whether such a plasmon can exist in the bulk of a 3D solid. This has not stopped researchers from suggesting that electron-acoustic plasmon coupling could lead to the formation of Cooper pairs and superconductvity in the cuprates. Varma has suggested that a good place to look would be in mixed-valence compounds, where $f$-electron masses can get very heavy.

On the experimental side, the search continues…

A helpful picture: If one imagines light electrons and heavy holes in a compensated semimetal for instance, the in-phase motion of the electrons and holes would result in an acoustic plasmon while the out-of-phase motion would result in the gapped plasmon.

## Interactions, Collective Excitations and a Few Examples

Most researchers in our field (and many outside our field that study, e.g. ant colonies, traffic, fish schools, etc.) are acutely aware of the relationship between the microscopic interactions between constituent particles and the incipient collective modes. These can be as mundane as phonons in a solid that arise because of interactions between atoms in the lattice or magnons in an anti-ferromagnet that arise due to spin-spin interactions.

From a theoretical point of view, collective modes can be derived by examining the interparticle interactions. An example is the random phase approximation for an electron gas, which yields the plasmon dispersion (here are some of my own notes on this for those who are interested). In experiment, one usually takes the opposite view where inter-particle interations can be inferred from the collective modes. For instance, the force constants in a solid can often be deduced by studying the phonon spectrum, and the exchange interaction can be backed out by examining the magnon dispersions.

In more exotic states of matter, these collective excitations can get a little bizarre. In a two-band superconductor, for instance, it was shown by Leggett that the two superfluids can oscillate out-of-phase resulting in a novel collective mode, first observed in MgB2 (pdf!) by Blumberg and co-workers. Furthermore, in 2H-NbSe2, there have been claims of an observed Higgs-like excitation which is made visible to Raman spectroscopy through its interaction with the charge density wave amplitude mode (see here and here for instance).

As I mentioned in the post about neutron scattering in the cuprates, a spin resonance mode is often observed below the superconducting transition temperature in unconventional superconductors. This mode has been observed in the cuprate, iron-based and heavy fermion superconducting families (see e.g. here for CeCoIn5), and is not (at least to me!) well-understood. In another rather stunning example, no less than four sub-gap collective modes, which are likely of electronic origin, show up below ~40K in SmB6 (see image below), which is in a class of materials known as Kondo insulators.

Lastly, in a material class that we are actually thought to understand quite well, Peierls-type quasi-1D charge density wave materials, there is a collective mode that shows up in the far-infrared region that (to my knowledge) has so far eluded theoretical understanding. In this paper on blue bronze, they assume that the mode, which shows up at ~8 cm$^{-1}$ in the energy loss function, is a pinned phase mode, but this assignment is likely incorrect in light of the fact that later microwave measurements demonstrated that the phase mode actually exists at a much lower energy scale (see Fig. 9). This example serves to show that even in material classes we think we understand quite well, there are often lurking unanswered questions.

In materials that we don’t understand very well such as the Kondo insulators and the unconventional superconductors mentioned above, it is therefore imperative to map out the collective modes, as they can yield critical insights into the interactions between constituent particles or couplings between different order parameters. To truly understand what is going on these materials, every peak needs to be identified (especially the ones that show up below Tc!), quantified and understood satisfactorily.

As Lestor Freamon says in The Wire:

All the pieces matter.

## Plasma Frequency, Screening Response Time and the Independent Electron Approximation

The plasma frequency in the study of solids arises in many different contexts. One of the most illuminating ways to look at the plasma frequency is as a measure of the screening response time in solids. I’ve discussed this previously in reference to the screening of longitudinal phonons in semiconductors, but I think it is worth repeating and expanding upon.

What I mean by “screening response time” is that in any solid, when one applies a perturbing electric field, the electrons take a certain amount of time to screen this field. This time can usually be estimated by using the relation:

$t_p = \frac{2\pi}{\omega_p}$

Now, suppose I introduce a time-varying electric field perturbation into the solid that has angular frequency $\omega$. The question then arises, will the electrons in the solid be able to respond fast enough to be able to screen this field? Well, for frequencies $\omega < \omega_p,$ the corresponding perturbation variation time is $t = 2\pi/\omega > t_p$. This means the the perturbation variation time is longer than the time it takes for the electrons in the solid to screen the perturbation. So the electrons have no problem screening this field. However, if $\omega > \omega_p$ and $t < t_p$, the electronic plasma in the solid will not have enough time to screen out the time-varying electric field.

This screening time interpretation of the plasma frequency is what leads to what is called the plasma edge in the reflectivity spectra in solids. Seen below is the reflectivity spectrum for aluminum (taken from Mark Fox’s book Optical Properties of Solids):

One can see that below the plasma edge at ~15eV, the reflectivity is almost perfect, resulting in the shiny and reflective quality of aluminum metal in the visible range. However, above $\hbar\omega$=15eV, the reflectivity suddenly drops and light is able to pass through the solid virtually unimpeded as the electrons can no longer respond to the quickly varying electric field.

Now that one can see the effect of the screening time on an external electric field such as light, the question naturally arises as to how the electrons screen the electric field generated from other electrons in the solid. It turns out that much of what I have discussed above also works for the electrons in the solid self-consistently. Therefore, it turns out that the electrons near the Fermi energy also have their electric fields, by and large, screened out in a similar manner. The distance over which the electric field falls by $1/e$ is usually called the Thomas-Fermi screening length, which for most metals is about half a Bohr radius. That the Thomas-Fermi approximation works well is because one effectively assumes that $\omega_p \rightarrow \infty$, which is not a bad approximation for the low-energy effects in solids considering that the plasma frequency is often 10s of eV.

Ultimately, the fact that the low-energy electrons near the Fermi energy are well-screened by other electrons self-consistently permits one to use the independent electron approximation — the foundation upon which band theory is built. Therefore, in many instances that the independent electron approximation is used to describe physical phenomena in solids, it should be kept in mind the hidden role the plasmon actually plays in allowing these ideas to work.

Naively, from my discussion above, it would seem like the independent electron approximation would then break down in a band insulator. However, this is not necessarily so. There are two things to note in this regard: (i) there exists an “interband plasmon” at high energies that plays essentially the same role that a free-carrier plasmon does in a metal for energies $E_g << E < \hbar\omega_p$ and (ii) whether the kinetic or Coulomb energy dominates will determine the low energy phenomenology. An image below is taken from this paper on lithium fluoride, which is a band insulator with a band gap of about 5eV and exhibits a plasmon at ~22eV:

The interband plasmon ultimately contributes to the background dielectric function, $\epsilon$, which reduces the Coulomb energy between the electrons in the form:

$V_{eff} = \frac{e^2}{\epsilon_0 \epsilon r}$

For example, this is the Coulomb interaction felt between an electron and hole when an exciton is formed (with opposite sign), as can be seen for LiF in the above image.

Now, the kinetic energy can be approximated by the band width, $W$, which effectively gives the amount of “wavefunction overlap” between the neighboring orbitals. Now, if $W >> V_{eff}$, then the independent electron approximation remains a good approximation. In this limit, one can get a band insulator, that is adequately described using the independent electron approximation. In the opposite limit, however, often one gets what is called a Mott insulator. Because d- and f-electrons tend to be closely bound to the atomic site, there is usually less wavefunction overlap between the electrons, leading to a small band width. This is why Mott insulators tend to occur in materials that have d- and f-electrons near the Fermi energy

Most studies on strongly correlated electron systems tend to concentrate on low-energy phenomenology.  While this is no doubt important, in light of this post, I think it may be worth looking up from time to time as well.

## Plasmons of a Luttinger Liquid

There’s a quite remarkable experiment in a paper by the Wang group at Berkeley that recently came out in Nature Photonics demonstrating the existence of peculiar plasmons in carbon nanotubes. These are significant because this may constitute the first observations of plasmons in a Luttinger liquid. To observe these plasmons, the group used scattering-scanning near-field optical microscopy (s-SNOM).

The plasmons are novel in that they appear to have a quantized propagation velocity that depends only on the number of conducting channels. Also, the ratios for the propagation velocities appear to be in the form $1:\sqrt{2}:\sqrt{3}:\sqrt{4}$ for one, two, three and four nanotubes respectively. (Each nanotube has one conducting channel).

The plasmons are extremely well-confined spatially ($\lambda_p/\lambda \sim 1/100$, where $\lambda_p$ and $\lambda$ are the plasmon wavelength and the wavelength of free-space light respectively) and also have a quality factor of $\sim$20. This means that there may be important applications in store for these kinds of plasmons as well, though I find the result from a more fundamental perspective quite intriguing.

## 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.

## Plasmons, the Coulomb Interaction and a Gap

In a famous 1962 paper entitled Plasmons, Gauge Invariance and Mass (pdf!), P.W. Anderson described the relationship between the gap in the plasmon spectrum and the idea of spontaneous symmetry breaking. It is an interesting historical note that Higgs cites Anderson’s paper in his landmark paper concerning the Higgs mechanism.

While there are many different formulations, including Anderson’s, of why the plasmon is gapped at zero momentum in a 3D solid, they all rely on one crucial element: the long-range nature of the Coulomb interaction (i.e. the electrons are charged particles). Of these formulations, I prefer one “cartoon-y” explanation which captures the essential physics well.

Before continuing, let me stress that it is quite unusual for a fluid medium (such as the electrons in a metal) to possess no zero frequency excitations at long wavelengths. For instance,  the dispersion relation for surface gravity waves on water (pdf!) is:

$\omega^2(k)=gk \tanh kh$.

Now, in 3D and in the long-wavelength limit, the plasmon sets up opposite charges on the surfaces of the solid as pictured below:

The long-wavelength plasmon therefore sets up the same electric field as in a capacitor. The electric field for a capacitor is $\textbf{E} = \frac{\sigma\hat{x}}{\epsilon_0}$. This expression is surprisingly independent of the distance separating the surfaces of the solid. Therefore, it takes a finite amount of energy to set up this electric field, even in the limit of infinite distance. This finite energy results in the gapping of the plasmon.

This cartoon can be extended further to 2D and 1D solids. In the 2D case, the electric field for the 1D “lines of charge” bounding the solid falls off like $\textbf{E}\sim\frac{1}{\textbf{r}}$. Therefore, in the infinite distance limit, it takes no energy to create this electric field and the plasmon is not gapped at $\textbf{q}=0$. Similarly, for the 1D case, the electric field from the points bounding the solid falls of as $\frac{1}{\textbf{r}^2}$, and the plasmon is again gapless.

This reasoning can be applied further to the phenomenon known as LO-TO splitting in a polar solid. Here, the longitudinal optical phonon (LO) and the transverse optical phonon (TO) branches are non-degenerate down to the very lowest (but non-zero!) momenta. Group theory predicts these modes to be degenerate at $\textbf{q}=0$ for the zincblende crystal structure of typical semiconducting compounds. Below is the phonon dispersion for GaAs demonstrating this phenomenon:

Again, the splitting occurs due to the long-ranged nature of the Coulomb interaction. In this case, however, it is the polar ionic degree of freedom that sets up the electric field as opposed to the electronic degrees of freedom. Using the same reasoning as above, one would predict that the LO-TO splitting would disappear in the 2D limit, and a quick check in the literature suggests this to be the case as reported in this paper about mono-layer Boron Nitride.

I very much appreciate toy models such as this that give one enough physical intuition to be able to predict the outcome of an experiment. It has its (very obvious!) limitations, but is valuable nonetheless.