# Category Archives: Screening

## LST Relation – The Physical Picture

In 1941, Lydanne, Sachs and Teller wrote a paper entitled “On the Polar Vibrations of Alkali Halides”, where they derived a result now known as the Lydanne-Sachs-Teller (LST) relation. It has wide applicability for polar insulators. I reproduce the relation below:

$\frac{\omega_{LO}^2}{\omega_{TO}^2} = \frac{\epsilon(o)}{\epsilon(\infty)}$

In the equation above, $\omega_{LO}$ and $\omega_{TO}$ refer to the frequencies of the longitudinal and transverse optical phonons respectively. $\epsilon(0)$ and $\epsilon(\infty)$ refer to the static and high frequency (above the phonon frequencies, but below any electronic energy scale) dielectric constants. All these quantities are understood to be the values in the long-wavelength limit (i.e. $q \approx 0$).

The beautiful thing about the LST result is that it is independent of any microscopic description, which is quite unusual in solid-state physics. Therefore, the result can be derived from classical electrodynamics, without resorting to any quantum mechanics. It is an interesting question as to whether or not quantum mechanics plays a role in the long-wavelength optical response in general.

Regardless, it turns out that all quantities in the LST relation are experimentally accessible! I find this relation quite remarkable and deep. Not only that, the agreement with experiment in many polar semiconductors is excellent. Take a look at the table below to get an idea of how well this relation holds for a few materials (reproduced from Mark Fox’s textbook Optical Properties of Solids):

I have found textbook derivations don’t give a good intuition of why this relation holds, so here is my attempt to rectify this situation. First, let me state an important assumption that goes into the LST relation:

The phonons are assumed to be in the harmonic limit (i.e. no phonon anharmonicity) and as a result, the dielectric constant has the following form:

$\epsilon(\omega) = \epsilon(\infty) + \frac{C}{\omega_{TO}^2-\omega^2}$

where $C$ is a constant. This form of the dielectric constant can be arrived at using either classical electrodynamics or quantum mechanics (see e.g. Ashcroft and Mermin, Kittel or Ziman).

Now, with this result under our belts, it turns out that it is quite simple to understand why the LST relation holds. In a simple polar semiconductor, we have two atoms per unit cell that are oppositely charged like so:

Therefore, for the longitudinal optical phonon we have an extra polarization effect due to the long-range nature of the Coulomb interaction. This extra polarization results in an extra restoring force (in addition to the springy restoring force between the ions), yielding a higher longitudinal phonon frequency compared to the transverse optical phonon. I have discussed this a little more extensively in a previous post. This extra restoring force (which is only present for the longitudinal oscillation) is pictured below:

The longitudinal optical phonon is at a higher energy because of the extra Coulombic polarization effect

More precisely, we can write the following when including this extra restoring force:

$\omega_{LO}^2 = \omega_{TO}^2 + \frac{C}{\epsilon(\infty)}$

There is an $\epsilon(\infty)$ in the formula above because this polarization will necessarily be screened by higher energy (electronic) processes. Dividing both sides by $\omega_{TO}^2$, we can write the above equation suggestively as:

$\frac{\omega_{LO}^2}{\omega_{TO}^2} = \frac{\epsilon(\infty)+C/\omega_{TO}^2}{\epsilon(\infty)}$

Looking at the equation for the dielectric constant from earlier, this is precisely the LST relation! In effect, the same extra restoring due to the long-range Coulomb interaction leads to the extra screening in the static limit, yielding, in my mind, a delightful little result.

Using the LST relation, we can deduce a property of ferroelectric materials. Namely, we know that at the transition temperature between the normal state and a ferroelectric ground state, the static dielectric constant, $\epsilon(0)$, diverges. Therefore, we can surmise from the LST relation that a zone center transverse optical phonon must go to zero energy (soften) at the transition temperature (see here for PbTiO3). This is a totally non-trivial consequence of the LST relation, demonstrating again its far-reaching utility.

Did I mention that I think this result is pretty excellent?

I’d like to acknowledge Zhanybek Alpichshev for enlightening some aspects regarding this topic.

## 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.

## Net Attraction à la Bardeen-Pines and Kohn-Luttinger

In the lead up to the full formulation of BCS theory, the derivation of Bardeen-Pines interaction played a prominent role. The Bardeen-Pines interaction demonstrated that a net attractive interaction between electrons in an electron gas/liquid can result in the presence of phonons.

The way that Bardeen and Pines derived this result can be understood by reading this paper. The result is actually quite simple to derive using a random-phase-like approximation or second-order perturbation theory. Regardless, the important result from this paper is that the effective interaction between two electrons is given by:

$V_{eff}(\textbf{q},\omega) = \frac{e^2}{\epsilon_0}\frac{1}{q^2 + k_{TF}^2}(1 + \frac{\omega_{ph}^2}{\omega^2 - \omega_{ph}^2})$

The crucial aspect of this equation is that for frequencies less than the phonon frequency (i.e. for $\omega < \omega_{ph}$), the effective interaction becomes negative (i.e. attractive).

It was also shown by Kohn and Luttinger in 1965 that, in principle, one could also obtain superconductivity in the absence of phonons. The attraction would occur using the phenomenon of Friedel oscillations whereby the effective potential can also become negative. This was quite a remarkable result: it showed that a purely electronic form of superconductivity was indeed theoretically possible.

What makes the effective interaction become attractive in these two models? In the Bardeen-Pines case, the phonons screen the electrons leading to a net attraction, while in the Kohn-Luttinger case, Fermi surface effects can again lead to a net attraction. It is important to note that in both papers, the pre-eminent quantity calculated was the dielectric function.

This is because the effective potential, $V_{eff}(\textbf{q},\omega)$, is equal to the following:

$V_{eff}(\textbf{q},\omega) = \frac{V(\textbf{q},\omega)}{\epsilon(\textbf{q},\omega)}$

In the aforementioned cases, net attraction resulted when $\epsilon(\textbf{q},\omega) < 0$.

This raises an interesting question: is it possible to still form Cooper pairs even when $\epsilon(\textbf{q},\omega) > 0$? It is possible that this question has been asked and answered in the literature previously, unbeknownst to me. I do think it is an important point to try to address especially in the context of high temperature superconductivity.

I welcome comments regarding this question.

Update: In light of my previous post about spin fluctuations, it seems like $\epsilon < 0$ is not a necessary condition to form Cooper pairs. In the s-wave channel, it seems like, barring some pathology, that $\epsilon$ would have to be less than $0$, but in the d-wave case, this need not be so. I just hadn’t put two and two together when initially drafting this post.

## Draw me a picture of a Cooper pair

Note: This is a post by Brian Skinner as part of a blog exchange. He has his own blog, which I heartily recommend, called Gravity and Levity. He is currently a postdoctoral scholar at MIT in theoretical condensed matter physics.

The central, and most surprising, idea in the conventional theory of superconductivity is the notion of Cooper pairing. In a Cooper pair, two electrons with opposite momentum somehow manage to overcome their ostensibly enormous repulsive energy and bind together to make a composite bosonic particle. These composite bosons are then able to carry electric current without dissipation.

But what does a Cooper pair really look like? In this post I’m going to try to draw a picture of one, and in the process I hope to discuss a little bit of physical intuition behind how Cooper pairing is possible.

To begin with, one should acknowledge that the “electrons” that comprise Cooper pairs are not really electrons as God made them. These electrons are the quasiparticles of Fermi liquid theory, which means that they are singly-charged, half-spinned objects that are dressed in excitations of the Fermi sea around them. In particular, each “electron” that propagates through a metal carries with it a screening atmosphere made up of local perturbations in charge density. Something like this:

That distance $r_s$ in this picture is the Thomas-Fermi screening radius, which in metals is on the same order as the Fermi wavelength (generally $\sim 5 - 10$ Angstroms). At distances much longer than $r_s$, the electron-electron interaction is screened out exponentially.

What this screening implies is that as long as the typical distance between electrons inside a Cooper pair is much longer than the Fermi wavelength (which it has to be, since there is really no concept of an electron that is smaller than the Fermi wavelength), the mutual Coulomb repulsion between electrons isn’t a problem. Electrons that are much further apart than $r_s$ simply don’t have any significant Coulomb interaction.

But, of course, this doesn’t explain what actually makes the electron stick together.  In the conventional theory, the “glue” between electrons is provided by the electron-phonon interaction. We typically say that electrons within a Cooper pair “exchange phonons”, and that this exchange mediates an attractive interaction. If you push a physicist to tell you what this exchange looks like in real space, you might get something like what is written in the Wikipedia article:

An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density. The two electrons then become correlated.

This kind of explanation might be accompanied by a picture like this one or even an animation like this one, which attempt to schematically depict how one electron distorts the lattice and creates a positively-charged well that another electron can fall into.

But I never liked these kind of pictures. Their big flaw, to my mind, is that in metals the electrons move much too fast for it to make sense. In particular, the Fermi velocity in metals is usually on the order of $10^6$ m/s, while the phonon velocity is a paltry $(\text{few}) \times 10^3$ m/s. So the idea that one electron can create a little potential well for another to fall into simply doesn’t make sense dynamically. By the time the potential well was created by the slow rearrangement of ions, the first electron would be long gone, and it’s hard to see any meaningful way in which the two electrons would be “paired”.

The other problem with the picture above is that it doesn’t explain why only electrons with opposite momentum can form Cooper pairs. If Cooper pairing came simply from one electron leaving behind a lattice distortion for another to couple to, then why should the pairing only work for opposite-momentum electrons?

So let me advance a different picture of a Cooper pair.

It starts by reminding you that the wavefunction for a (say) right-going electron state looks like this:

$\psi_R \sim e^{i (k x - \omega t)}$.

The probability density for the electron’s position in this state, $|\psi_R(x)|^2$, is uniform in space.

On the other hand, the wavefunction for a left-going electron state looks like

$\psi_L \sim e^{i (-k x - \omega t)}$.

It also has a uniform probability distribution. But if you use the two states (one with momentum $+k$ and the other with momentum $-k$) to make a superposition, you can get a state $\psi_C = (\psi_R + \psi_L)/\sqrt{2}$ whose probability distribution looks like a standing wave: $|\psi_C|^2 \sim \cos^2(k x)$.

In other words, by combining electron states with $+k$ and $-k$, you can arrive at an electron state where the electron probability distribution (and therefore the electron charge density) has a static spatial pattern.

Once there is a static pattern, the positively charged ions in the crystal lattice can distort their spacing to bring themselves closer to the regions of large electron charge density. Like this:

In this way the system lowers its total Coulomb energy.  In essence, the binding of opposite-momentum electrons is a clever way of effectively bringing the fast-moving electrons to a stop, so that the slow-moving ionic lattice can accommodate itself to it.

Of course, the final piece of the picture is that the Cooper pair should have a finite size in space – the standing wave can’t actually extend on forever. This finite size is generally what we call the coherence length $\xi$. Forcing the two electrons within the Cooper pair to be confined within the coherence length costs some quantum confinement energy (i.e., an increase in the electron momentum due to the uncertainty principle), and this energy cost goes like $\sim \hbar v/\xi$, where $v$ is the Fermi momentum. So generally speaking the length $\xi$ should be large enough that $\hbar v / \xi \lesssim \Delta$ where $\Delta$ is the binding energy gained from Cooper pairing.  Usually these two energy scales are on the same order, so that $\xi \sim \hbar v / \Delta$.

Putting it all together, my favorite picture of a Cooper pair looks something like this:

I’m certainly no expert in superconductivity, but this picture makes much more sense to me than the one in Wikipedia.

Your criticisms or refinements of it are certainly welcome.

Author’s note: Thanks to Mike Norman, who taught me this picture over lunch one day.

## 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.