# Category Archives: Superconductivity

## 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.

## Transition Metal Dichalcogenide CDWs

There is an excellent review paper by K. Rossnagel on the origin of charge density waves (CDWs) in the transition metal dichalcogenide compounds (such as 2H-NbSe$_2$, 1T-TiSe$_2$, 2H-TaSe$_2$, etc.) . A lot of the work on these materials was undertaken in the 70s and 80s, but there has been a recent revival of interest because of the nature of superconductivity in a few of these compounds.

By “nature”, I mean that  the phase diagrams in these materials bear a striking resemblance to the phase diagram in the cuprates, except that the anti-ferromagnetism is replaced by a CDW phase. Shown below is the phase diagram for 1T-TiSe$_2$ under pressure and with copper intercalation (taken from this paper).

Strangely, with copper intercalation, the Hall resistance is negative, while  it is positive under pressure. This is interesting because like the cuprates, superconductivity can be brought about with either electrons or holes as majority carriers. A similar phase diagram is also observed for another TMD 1T-TaS$_2$ (see here for instance). 1T-TaS$_2$ has also been shown to exhibit Mott physics at low temperature in the parent compound.

It is suspected that the origin of the CDWs in 1T-TiSe$_2$ and 1T-TaS$_2$ are at least in part electronically driven (see Rossnagel’s review article and references therein). This makes the observation of the superconductivity in these compounds all the more interesting — as the superconductivity may also be primarily electronically driven. I have also blogged previously about another set of CDW materials (the rare earth tritellurides) that exhibit cuprate-like phase diagrams including an antiferromagnetic phase, and also about the interplay between CDWs and superconductivity in NbSe$2$.

It seems to me that there are some really quite fundamental open questions in the study of these compounds, which is in part why I keep re-visiting this topic myself.

## The Prescience of Ginzburg

In 1977, before the discovery of high-temperature superconductivity, V. Ginzburg wrote:

“On the basis of general theoretical considerations, we believe at present
that the most reasonable estimate is Tc$\lesssim$300 K, this estimate being, of course, for materials and systems under more or less normal
conditions (equilibrium or quasi-equilibrium metallic systems in the absence
of pressure or under relatively low pressures, etc.). In this case, if we exclude
from consideration metallic hydrogen and, perhaps, organic metals, as well
as semimetals in states near the region of electronic phase transitions, then it is suggested that we should use the exciton [electronic] mechanism of attraction between the conduction electrons.

In this scheme, the most promising – from the point of view of the possibility of raising T$_c$-materials are, apparently, layered compounds and dielectric-metal-dielectric sandwiches. However, the state of the theory, let alone of experiment, is still far from being such as to allow us to regard as closed other possible directions, in particular, the use of filamentary compounds. Furthermore, for the present state of the problem of high-temperature superconductivity, the soundest and most fruitful approach will be one that is not preconceived, in which attempts are made to move forward in the most diverse directions.”

I took the quote out of this paper here, though many of the ideas are echoed from and better expressed in one of his previous papers, linked here. It is amusing that for at least 15 years prior to the discovery of the cuprates, Ginzburg stressed looking for high-temperature superconductors (with T$_c$s above the boiling point of liquid nitrogen) in layered, quasi-2D materials that could host superconductivity with an electronically driven Cooper pairing.

The papers linked above are very readable and he reached these conclusions on startlingly general grounds — by discussing the inverse dielectric function.

## What DO We Know About High Tc?

Many papers on cuprate superconductors start out by saying that we don’t know much about them. About ten years ago, A.J. Leggett wrote down a laundry list of things we do know. Looking at this list, it seems like things are not as bleak as the introductions to papers on cuprates make them out to be! Here is his list of things we knew back in 2006:

1. Superconductivity in the cuprates is a result of Cooper pairing.
2. The main driver of superconductivity in the cuprates is the copper-oxide plane.
3. To a good approximation, Cooper pairs form independently on each layer (even in a multilayered compound) in the cuprates.
4. The net saving of energy in the transition to the superconducting state is not from ionic kinetic energy as in the classic BCS superconductors. (This surprising result in BCS superconductors was shown to be true in this paper.)
5. The spin state is a singlet.
6. The order parameter is of $d_{x^2-y^2}$ symmetry.
7. The size of Cooper pairs is somewhere between 10-30$\text{\AA}$.
8. The pairs are formed from time-reversed partners as in BCS theory.

There is one more aspect of this paper that I think is significant. Leggett stresses the importance of asking questions that are model-independent, such as, (1) What is the pairing symmetry? (2) Does the macroscopic Ginzburg-Landau theory work? and (3) Where is the energy saved in the transition to the superconducting state? Posing questions like these are the “tortoise” method versus developing microscopic theories, which are the “hare’s” method. With the existence of so many microscopic models, it seems to me that taking the “tortoise” path may yield fruit in the long term.

I have been told by a number of my peers that I apparently like lists, which is probably why this article sticks out in my mind. Between the time this article was written and the present, are there any additional items to add to the list? I’m not an expert in high-Tc, so I’m wondering if there is more we now know. Many experimental signatures spring to mind, but none that I would necessarily say that we know “for sure”.

## Let there be (THz) light

The applications of scientific discoveries is sometimes not what you would expect, and high temperature superconductivity is no different.  When high-temperature superconductivity was discovered in copper-oxides (cuprates) in 1986, the envisioned applications were power lines, electromagnets, and maglev trains, all cooled by cheap-as-milk liquid nitrogen.  While applications involving high-temperature superconductors’ dissipationless and diamagnetic properties are slowly coming online, there are other potential technologies which most people are less aware of.  The one I want to discuss here is using the layered structure of cuprate high temperature superconductors to produce coherent THz emitters.  Creating light sources and detectors for the THz portion of the electromagnetic spectrum—the notorious THz gap—has been a pressing challenge for decades.

The Josephson effect

The Josephson effect underlies many important applications of superconductors, such as sensitive magnetometers, qubits for quantum computing, and the SI definition of the volt.  The starting point for the Josephson effect is a superconductors’ complex order parameter, $\Psi=\Psi_0 e^{\imath\varphi}$.  The amplitude, $\Psi_0$, is related to some measure of the robustness of the superconducting state–either the superfluid density or the superconducting gap.  The phase, $\varphi$, reflects that a superconductor is a phase-coherent state–a condensate.  At $T_c$, a superconductor chooses an arbitrary phase, and a current in a superconductor (a supercurrent) corresponds to a gradient in this phase.  A Josephson junction, sketched below, consists of two superconductors separated by a non-superconducting barrier.  Because each superconductor chooses an arbitrary phase and the superconducting wavefunctions can penetrate into the barrier, a phase gradient develops in the barrier region, and a supercurrent can flow.  This supercurrent is given by $I_s=I_c\sin(\delta\varphi)$, where  $I_c$ is the critical current which causes the Josephson junction to become resistive (different from the critical current which makes the superconductor resistive) and  $\delta\varphi$ is the phase difference between the two superconductors.  This is the DC Josephson effect.  In the resistive regime ($I>I_c$), one encounters the AC Josephson effect, in which the Josephson junction supports an oscillating current with AC Josephson frequency $\omega=\frac{2\pi V}{\Phi_0}$, where V is the voltage across the junction and $\Phi_0$ is the magnetic flux quantum.  The current in this regime is given by: $I(t)=I_c\sin(\delta\varphi + \frac{2\pi V}{\Phi_0}t)$

Thus, a Josephson junction can convert a DC voltage to an AC current (and vis versa).

Schematic of a Josephson junction, consisting of two superconductors with a barrier in between. The barrier may be an insulator, a metal, or a constricted piece of superconductor. Each superconducting slab has a complex wavefunction with an arbitrarily chosen phase, $\varphi_{1,2}$. Supercurrent through a Josephson junction depends on the phase difference, $\delta\varphi=\varphi_1-\varphi_2$

Schematic of IV curve of Josephson junction (solid line), from Ref [1]. For sufficiently small bias currents, a supercurrent flows through the junction and no voltage is sustained–the regime of the DC Josephson effect. At currents exceeding $I_c$, the junction becomes resistive and is able to sustain a voltage across it, even though each superconducting slab remains superconducting. This is where the AC josephson effect is realized. The dashed line is an ohmic resistance, which a Josephson junction approaches in the limit of high bias voltage.

More is better

While a Josephson junction in its resistive regime is a perfect DC to AC converter with frequency proportional to voltage, the amount of power it can output is limited by the fact that device performance (and eventually superconductivity) degrade if you crank the voltage up too high.  However, it turns out that if you have multiple Josephson junctions in series, the available power scales with the number of junctions, and if all of these junctions oscillate in phase, they can form a coherent radiation source.  This is where high temperature superconductors come in.

Calling all cuprates

The crystal structure of cuprate high temperature superconductors consists of $CuO_2$ sheet where superconductivity originates, separated by insulating layers.  While the $CuO_2$ sheets are coupled with each other, the coupling can be weak in some cuprates, such that the material behaves like an array of intrinsic Josephson junctions in series.  Thus, a structure which has to be specially manufactured for other superconductors, the cuprates give for free.

Crystal structure of the high temperature superconductor, $Bi_2Sr_2CaCu_2O_{8+\delta}$ (BSCCO) which is most commonly used to make THz emitters. The layered structure of cuprates—superconducting $CuO_2$ layers separated by insulating intervening layers– permits the material itself to be a series of Josephson junctions. Adapted from Ref. [2].

The first step to making a cuprate superconductor into an emitter of coherent THz radiation is to pattern a single crystal into a smaller structure called a mesa. The mesa behaves as a resonant cavity such that a half-integer number of wavelengths ($\lambda/2$) of radiation fit into the width, w, of the device.  The lowest-order resonance condition is met when the AC Josephson frequency is equal to the frequency of a cavity mode, $\omega_c=\frac{\pi c_0}{n w}$, where $c_0/n$ is the mode propagation velocity in the medium and n is the far-infrared refractive index.  For a given mesa width, the resonance condition is met for a specific value of applied voltage for each Josephson junction, $V_{jj}=\frac{c_0 \Phi_0}{2 w n}$.  For a stack of Josephson junctions in series, the applied voltage scales with the number of junctions (N): $V=NV_{jj}$

A schematic of such a device is shown below.  The mesa, produced by ion milling, is 1-2 microns high (corresponding to ~1000 intrinsic Josephson junctions), 40-100 microns wide (setting the resonance emission frequencies), and several hundred microns in length.  A voltage is applied along the height of the stack and THz radiation is emitted out the side of the stack.  Devices have been fabricated with emissions at frequencies between 250 GHz and 1THz.  Linewidths of ~10MHz have been achieved as have radiation powers of 80 microwatts, though it is predicted that the latter figure can be pushed to 1mW [2,3].  The emission frequency can be tuned either by fabricating a new device with a different width, or by fabricating a device shaped like a trapezoid or a stepped pyramid and varying the bias voltage [4].  The latter corresponds to different numbers of Josephson junctions in the stack oscillating coherently.

THz emitter made out of high-temperature superconducting cuprates. A ‘mesa’ is ion-milled from a single crystal of BSCCO with a restricted width dimension, w. THz radiation is emitted out of the side, with frequency depending on the width of the mesa and the applied voltage. From Refs [2-3].

Emission spectra of three devices with different widths,w, made out of high temperature superconductors operated at T~25K. Inset shows linear relationship between frequency and 1/w. From Ref [3]

Implications

Successful fabrication of coherent THz emittors out of high temperature superconductors is a relatively new achievement and there is additional progress to be made, particularly towards increasing the emitted power.  This technology is promising for filling in a portion of the THz gap outside the capabilities of quantum cascade lasers, whose lowest emission frequency is presently 1.6THz.  In the future, one can imagine a light source consisting of an array of BSCCO mesas of different dimensions producing a narrow-bandwidth lightsource which is tuneable between 250GHz and 1.5THz for security and research applications.

References

[1] J. Annett. Superconductivity, Superfluids, and Condensates, Oxford University Press (2003)

[2] U. Welp et alNature Photonics 7 702 (2013)

[3] L. Ozyuzer et al, Science 318 1291 (2007)

[4] T. M. Benseman et al, Phys. Rev. B 84 064523 (2011)

## An Important Note on Cuprates

Two high-Tc researchers are sentenced to death but each of them is allowed to express one last wish in front of the king. Researcher number one says: “His Majesty, I will readily disappear from mother earth, but before that, please, let me explain you my point of view about the cuprates.” This causes researcher number two to jump up and beg: “Oh no, please kill me first!”

Source: Roman Schuster’s Thesis (pdf!)