# Monthly Archives: September 2015

## 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.

## Guest Post

The reason that I haven’t made a post recently was that Brian Skinner and I decided to do blog exchange. His blog is called Gravity and Levity is one of the few blogs out there where every article is worth reading. He blends great physical insight with a genuine personal touch. Anyway, the post I made for his site can be read here. Brian will also make a post here in the coming days — keep a look out for it.

## So Much for New Principles at Higher Scales…

Source: https://xkcd.com/435/

## General Aspects of Topology in Quantum Mechanics

Condensed matter physics has, in the past ten years or so, made a left turn towards studying topological properties of materials. Following the discovery of the Quantum Hall Effect (QHE) in 1980, it took about 25 years to experimentally discover that similar phenomenology could occur in bulk samples in the absence of a magnetic field in topological insulators. In the current issue of Nature Physics, there are three papers demonstrating the existence of a Weyl semimetal in TaAs and NbAs. These states of matter bear a striking similarity to quantum mechanical effects such as the Aharonov-Bohm effect and the Dirac monopole problem.

So what do all of these things have in common? Well, I vaguely addressed this issue in a previous post concerning Berry phases, but I want to elaborate a little more here. First it should be understood that all of these problems take place on some sort of manifold. For instance, the Aharonov-Bohm effect takes place in a plane, the Dirac monopole problem on a 3D sphere and the problems in solid-state physics largely on a torus due to periodic boundary conditions.

Now, what makes all of these problems exhibit a robust topological quantization of some sort is that the Berry connection in these problems cannot adequately be described by a single function over the entire manifold. If one were to attempt to write down a function for the Berry connection, there would necessarily exist a singularity somewhere on the manifold. But because the Berry connection is not an observable, one can just write down two (or more) different functions on different parts (or “neighborhoods”) of the manifold. The price one has to pay is that one has to “patch” the functions together at the boundary of the neighborhoods. Therefore, the existence of the topological quantization in most of the problems described above arise because of a singularity in the Berry connection somewhere on the manifold that cannot be gotten rid of with a gauge transformation.

For instance, for the Aharonov-Bohm effect, the outside of the solenoid and the inside of the solenoid must be described by different functions, or else the “outside function” would be singular at the center of the solenoid.  Qualitatively, one can think of the manifold as a plane with a hole punched in the middle of it. In the case of the Dirac monopole, the magnetic monopole itself is the position of the singularity and there is a hole punched in 3-dimensional space.

There is an excellent discussion on both these problems in Sakurai’s quantum mechanics textbook. I particularly like the approach he takes to the Dirac monopole problem, which he adapted from Wu and Yang’s elegant solution. The explanation of the QHE using similar ideas was developed in this great (but unfortunately quite mathematical) paper by Kohmoto (pdf!). I realize that this post only sketches the main point (with perhaps too much haste), but I hope that it will be illuminating to some.

Update: I have written a guest post for Brian Skinner’s blog Gravity and Levity where I discuss the topics here in a little more detail. You can read the post here if you’re interested.