# Category Archives: Superconductivity

## Just a little thought on Aharonov-Bohm Destruction of Superconductivity

Many experimenters in the past have exploited the similarities between superconductors and superfluids to come up with new ideas. One of the most important of these analogies is the Hess-Fairbank effect in a superfluid (pdf!) and the corresponding Meissner effect in a superconductor. The Hess-Fairbank effect is often taken as the pre-eminent experimental signature of a superfluid.

For those who are unfamiliar with the Hess-Fairbank effect, let me describe it briefly. It was found that if one rotates a cylindrical container of helium slowly (i.e. with angular velocity less than half the critical velocity, $\omega < \omega_c/2 \equiv \hbar/2mR^2$), and one cools below the transition temperature into the superfluid phase towards $T \rightarrow 0$, that the liquid in the container remains at rest despite the rotating walls!

We can write the Hamiltonian for the liquid in the rotating container in the rotating frame (not the lab frame!):

$H'(\textbf{r}',\textbf{p}') = \frac{(\textbf{p}'-m \vec{\omega} \times\textbf{r}')^2}{2m} + V'(\textbf{r}')$

Besides an unimportant centrifugal term which I’ve buried in the definition of $V'(\textbf{r}')$, there is a strong mathematical analogy between the Hamiltonian above and that which describes a superconductor in the presence of a vector potential (in the lab frame):

$H(\textbf{r},\textbf{p}) = \frac{(\textbf{p}-e\textbf{A}(\textbf{r}))^2}{2m} + V(\textbf{r})$

One can see that $e \textbf{A}(\textbf{r})$ corresponds to $m \vec{\omega} \times\textbf{r}$. Furthermore, if we assume a constant magnetic field (as in the Meissner effect), we can write for $\textbf{A} = \frac{1}{2} \textbf{B} \times \textbf{r}$ and the correspondence becomes $\vec{\omega} \leftrightarrow e\textbf{B}/2m$.

Consider now what happens when one rotates the container containing the superfluid quickly (faster than the critical angular velocity). I have blogged previously about this scenario and the elegant experiment showing the proliferation of quantized vortices. In fact, if one rotates the cylinder quickly enough, the superfluidity is destroyed entirely.

Now, let us consider switching to a multiply connected geometry. It is important to note that the kinetic energy terms in the Hamiltonians remain the same if we were to use an annular container for the superfluid and a ring in the superconducting case with an Aharonov-Bohm flux through the center. For the annular container, the superfluidity would still be destroyed if the container were rotated quickly enough. This would lead one to conclude that for a superconducting ring, a strong enough Aharonov-Bohm flux would also destroy the superconductivity. There would not be an actual magnetic field applied to the superconductor (i.e. not the Meissner effect in the usual sense), but the superconductivity would be destroyed through a pseudo-Meissner effect nonetheless.

I have to admit that I have not come across an experimental paper demonstrating this effect, so please share if you know of one, as I’m sure this idea has been around for a while! It is just interesting to think that one could destroy superconductivity without actually ‘touching’ it with any kind of measurable field.

## 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.

## Matthias’ List — Check it Twice

Bernd Matthias was a prominent chemist/physicist and who played a major role in the history of superconductivity. He discovered nearly 1,000 superconducting compounds in his career and most notably discovered the NbTi and Nb$_3$Sn superconductors, which found commercial use in MRI instruments.

Using his vast experience in the synthesis of (classic BCS) superconducting materials, he made an empirical list of the rules he followed in searching for new superconductors.

1. High symmetry is good; Cubic symmetry is best
2. High density of electronic states is good
3. Stay away from oxygen
4. Stay away from magnetism
5. Stay away from insulators
6. Stay away from theorists

It is worth reflecting upon this list and thinking about how Matthias came to these conclusions (even the last item). First of all, many of the items in the list are not in any way embodied by BCS theory (1, 3, and 5 in particular). Secondly, the cuprates seems to disobey all of these items (except the last one)! It is interesting to note that Matthias was a vocal opponent of BCS theory for its failure to capture many aspects of superconductors he considered essential and for its inability to predict transition temperatures and new superconducting materials.

Matthias’ list has inspired many in the field of unconventional superconductivity to make lists similar to his. For those of you who read this blog regularly, you are probably aware that I am fond of lists (see here and here for instance), because they help synthesize key experimental observations. You are also likely to find out some of your own biases when making a list. Below is a Matthias-style list of unconventional superconductivity put forth by Igor Mazin, which contains items that I find myself generally agreeing with:

1. Layered structures are good
2. Carrier density should not be too high
3. Transition metals of the fourth period are good
4. Magnetism is essential
5. Proper Fermi surface geometry is essential
• Must match the spin excitation structure
6. Enlist theorists, at least to compute Fermi surface structures

As we near the winter holiday period (in the northern hemisphere!), please feel free to share your own list, add an item to the one above, or even share some misgivings about Mazin’s list.

## Neutron Scattering and Cuprates

Because many of us were born after the cuprates were discovered, and because of the sheer number of papers that have been written on them, it can seem like a daunting task for the young researcher to get a good grip on high-temperature superconductivity. Fortunately, there are some great review papers out there that help synthesize a lot of data and provide references to much of the original work. These review articles tend to concentrate either on one experimental technique or one part of the cuprate phase diagram (e.g. the pseudogap).

In a similar spirit, this blog post will concentrate on some of the significant findings in the cuprates discovered using neutron scattering. It should be kept in mind that most neutron scattering experiments have been done on YBCO and LSCO because large single crystals can be made of these compounds. The list below is a little biased, but I hope some will find it useful.

1. Spin Gap: A low energy gap in the magnetic inelastic neutron scattering  cross section has been observed  below Tc in both optimally doped LSCO and YBCO. For YBCO, the magnitude of the spin gap is larger (consistent with its higher Tc) and the spin gap exists for underdoped and overdoped samples as well. In YBCO the relation $2\Delta_s=3.8k_BT_c$ is approximately observed, where $\Delta_s$ is the magnitude of the spin gap.
2. Dispersive Incommensurate Fluctuations and the Hourglass Spectrum: There exist incommensurate magnetic fluctuations near the antiferromagnetic Bragg point in both underdoped and optimally doped LSCO and YBCO. These peaks show up in the magnetic inelastic neutron scattering cross section and seem to gain in intensity below the superconducting transition. On a $\delta$ vs. energy plot (where $\delta$ is the incommensurability away from the anti-ferromagnetic Bragg point) the spectrum seem to have an hourglass-like shape. It should be noted that (to my knowledge) the hourglass spectrum has not been observed in electron-doped cuprates with studies having been conducted on (NCCO and PLCCO).
3. Resonance Mode: The magnetic resonance mode is a peak at ~40-50meV, which is located at the “pinching point” in the hourglass spectrum. This peak has been observed in many underdoped and optimally doped cuprates, including BSCCO and TBCCO. It is probably the weakest in  LSCO, where it is broad and does not carry much spectral weight. It also only shows up dramatically around Tc. Interestingly, a similar mode has been seen in electron-doped cuprates and some of the Fe-based superconductors.

While there have been some other discoveries in the cuprates with neutron scattering, most of the studies tend to concentrate on one or all of these three general experimental observations.

Here are some links to some papers which discuss these observations in a little more detail:

## Hunches and Feelings

In speaking with other physicists, I have noticed that some of the most productive avenues to go down is to share feelings and hunches about a particular phenomenon. These intuitions are not usually written down anywhere because sometimes they are pure speculation. Therefore, it is important to talk, share and be open to suggestions from others.

Once in a while, there are opinion pieces in journals that are quite interesting to read where physicists do share these very speculations. Here is one example of this from Nature Physics where some prominent theorists working on high-temperature superconductivity have shared some ideas. I have to say that reading this piece gave me a couple ideas for experiments — and these are the kinds of articles that are the most likely to do so. This article is 10 years old, however, and some ideas may be a little dated.

Nevertheless, it is unfortunate that articles like these are not very common, even though they can be of immense value. I think this is a format that journals should pursue more widely, and I urge them to seek out pieces like these in the future.

Comments and opinions regarding this issue are encouraged.

## Macroscopic Wavefunctions, Off-Diagonal Long Range Order and U(1) Symmetry Breaking

Steven Weinberg wrote a piece a while ago entitled Superconductivity for Particular Theorists (pdf!). Although I have to admit that I have not followed the entire mathematical treatment in this document, I much appreciate the conceptual approach he takes in asking the following question:

How can one possibly use such approximations (BCS theory and Ginzburg-Landau theory) to derive predictions about superconducting phenomena that are essentially of unlimited accuracy?

He answers the question by stating that the general features of superconductivity can be explained using the fact that there is a spontaneous breakdown of electromagnetic gauge invariance. The general features he demonstrates are due to broken gauge invariance are the following:

1. The Meissner Effect
2. Flux Quantization
3. Infinite Conductivity
4. The AC Josephson Effect
5. Vortex Lines

Although not related to this post per se, he also makes the following (somewhat controversial) comment that I have to admit I am quoting a little out of context:

“…superconductivity is not macroscopic quantum mechanics; it is the classical field theory of  a Nambu-Goldstone field”

Now, while it may be true that one can derive the phenomena in the list above using the formalism outlined by Weinberg, I do think that there are other ways to obtain similar results that may be just as general. One way to do this is to assume the existence of a macroscopic wave function. This method is outlined in this (illuminatingly simple) set of lecture notes by M. Beasley (pdf!).

Another general formalism is outlined by C.N. Yang in this RMP, where he defines the concept of off-diagonal long range order for a tw0-particle density matrix. ODLRO can be defined for a single-particle density matrix in the following way:

$\lim_{|r-r'| \to \infty} \rho(r,r') \neq 0$

This can be easily extended to the case of a two-particle density matrix appropriate for Cooper pairing (see Yang).

Lastly, there is a formalism similar to that of Yang’s as outlined by Leggett in his book Quantum Liquids, which was first developed by Penrose and Onsager. They conclude that many properties of Bose-Einstein Condensation can be obtained from again examining the diagonalized density matrix:

$\rho(\textbf{r},\textbf{r}';t) = \sum_i n_i(t)\chi_i^*(\textbf{r},t)\chi_i(\textbf{r}',t)$

Leggett then goes onto say

“If there is exactly one eigenvalue of order N, with the rest all of order unity, then we say the system exhibits simple BEC.”

Again, this can easily be extended to the case of a two-particle density matrix when considering Cooper pairing.

The 5-point list of properties of superconductors itemized above can then be subsequently derived using any of these general frameworks:

1. Broken Electromagnetic Gauge Invariance
2. Macroscopic Wavefunction
3. Off-Diagonal Long Range Order in the Two-Particle Density Matrix
4. Macroscopically Large Eigenvalue of Two-Particle Density Matrix

These are all model-independent formulations able to describe general properties associated with superconductivity. Items 3 and 4, and to some extent 2, overlap in their concepts. However, 1 seems quite different to me. It seems to me that 2, 3 & 4 more easily relate the concepts of Bose-Einstein condensation to BCS -type condensation, and I appreciate this element of generality. However, I am not sure at this point which is a more general formulation and which is the most useful. I do have a preference, however, for items 2 and 4 because they are the easiest for me to grasp intuitively.

Please feel free to comment, as this post was intended to raise a question rather than to answer it (which I cannot do at present!). I will continue to think about this question and will hopefully make a more thoughtful post with a concrete answer in the future.

## A Critical Ingredient for Cooper Pairing in Superconductors within the BCS Picture

I’m sure that readers that are experts in superconductivity are aware of this fact already, but there is a point that I feel is not stressed enough by textbooks on superconductivity. This is the issue of reduced dimensionality in BCS theory. In a previous post, I’ve shown the usefulness of the thinking about the Cooper problem instead of the full-blown BCS solution, so I’ll take this approach here as well. In the Cooper problem, one assumes a 3-dimensional spherical Fermi surface like so:

3D Fermi Surface

What subtly happens when one solves the Cooper problem, however, is the reduction from three dimensions to two dimensions. Because only the electrons near the Fermi surface condense, one is really working in a shell around the Fermi surface like so, where the black portion does not participate in the formation of Cooper pairs:

Effective 2D Topology Associated with the Cooper Problem

Therefore, when solving the Cooper problem, one goes from working in a 3D solid sphere (the entire Fermi sea), to working on the surface of the sphere, effectively a 2D manifold. Because one is now confined to just the surface, it enables one of the most crucial steps in the Cooper problem: assuming that the density of states ($N(E)$) at the Fermi energy is a constant so that one can pull it out of the integral (see, for example, equation 9 in this set of lecture notes by P. Hirschfeld).

The more important role of dimensionality, though, is in the bound state solution. If one solves the Schrodinger equation for the delta-function potential (i.e. $V(x,y)= -\alpha\delta(x)\delta(y)$) in 2D one sees a quite stunning (but expected) resemblance to the Cooper problem. It tuns out that the solution to obtain a bound state takes the following form:

$E \sim \exp{(-\textrm{const.}/\alpha)}$.

Note that this is exactly the same function that appears in the solution to the Cooper problem, and this is of course not a coincidence. This function is not expandable in terms of a Taylor series, as is so often stressed when solving the Cooper problem and is therefore not amenable to perturbation methods. Note, also, that there is a bound state solution to this problem whenever $\alpha$ is finite, again similar to the case of the Cooper problem. That there exists a bound state solution for any $\alpha >0$ no matter how small, is only true in dimensions two or less. This is why reduced dimensionality is so critical to the Cooper problem.

Furthermore, it is well-known to solid-state physicists that for a Fermi gas/liquid, in 3D $N(E) \sim \sqrt{E}$, in 2D $N(E) \sim$const., while in 1D $N(E) \sim 1/\sqrt{E}$. Hence, if one is confined to two-dimensions in the Cooper problem, one is able to treat the density of states as a constant, and pull this term out of the integral (see equation 9 here again) even if the states are not confined to the Fermi energy.

This of course raises the question of what happens in an actual 2D or quasi-2D solid. Naively, it seems like in 2D, a solid should be more susceptible to the formation of Cooper pairs including all the electrons in the Fermi sea, as opposed to the ones constrained to be close to the Fermi surface.

If any readers have any more insight to share with respect to the role of dimensionality in superconductivity, please feel free to comment below.