# Tag Archives: J. Bardeen

## A Matter of Definitions

When one unearths a new superconductor, there exist three experimental signatures one hopes to observe to verify this discovery. These are:

1. D.C. resistance is zero
2. Meissner Effect (expulsion of magnetic field)
3. Zero Peltier coefficient or thermopower

The last item is a little finical, but bear with me for a second. The Peltier coefficient effectively measures the transport of heat current that accompanies the transport of electric current. So in a superconductor, there is no heat transport (condensate carries zero entropy!), when there is electrical transport. For instance, here is a plot of the thermopower for a few iron pnictides:

Let us ask a similar, seemingly benign, question: what are the experimental signatures one hopes to observe when one discovers a charge density wave (CDW) material?

If we are to use the superconductor as a guide, one would probably say the following:

1. Non-linear conductivity
2. CDW satellite reflections in a diffraction pattern
3. An almost zero Peltier coefficient or thermopower once the CDW has been depinned

I have posted about the non-linear I-V characteristics of CDWs previously. Associated with the formation of a charge density wave is, in all known cases to my knowledge, a periodic lattice distortion. This can be observed using X-rays, neutrons or electrons. Here is an image from 1T-TaS$_2$ taken from here:

Now, startlingly, once the charge density wave is depinned in a large enough electric field, the thermopower decreases dramatically. This is plotted below as a function of electric field along with the differential conductivity:

This indicates that there is very little entropy transport associated with the charge density wave condensate. Personally, I find this result to be quite stunning. I suspect that this was one of the several signatures that led John Bardeen to suggest that the origin of the charge density wave in low-dimensional materials was essentially quantum mechanical in origin.

Having outlined these three criteria, one should ask: do many of the materials we refer to as charge density waves actually exhibit these experimental signatures?

For many of the materials we refer to as charge density waves today, notably the transition metal dichalcogenides, such as 1T-TaS$_2$, 2H-NbSe$_2$, and 2H-TaSe$_2$, items (1) and (3) have not been observed! This is because it has not been possible to definitively depin the charge density wave. This probably has to do with the triple-q structure of the charge density wave in many of these materials, which don’t select a preferential direction.

There exist many clues that the latter materials do indeed exhibit a charge density wave transition similar to others where a depinning has been observed. It is interesting to note, though, that there are some glaring experimental absences in the transition metal dichalcogenides,  which are often considered prototypical examples of a charge density wave transition.

## Envisioning the Future Technological Landscape

I recently read the well-written and prescient piece entitled As We May Think by Vannevar Bush, which was published in The Atlantic magazine in July of 1945. With World War II coming to a close, and with many physicists and engineers involved in the war effort, Bush outlines what he sees as the future work of physical scientists when they return to their “day jobs”. Many of his predictions concentrate on technological advancements. Reading it today, one is struck by how visionary this article has turned out to be (though it may be argued that some of the prophesies were self-fulfilling). It should be pointed out that this article was written before the discovery of the transistor, which Bardeen and Brattain discovered in 1947.

The most stunning of his predictions to my mind were the following:

1. Personal computers
2. Miniature storage capable of holding vast amounts of data (including encyclopedias)
3. Something akin to digital photography, which he calls dry photography
4. The internet and world wide web
5. Speech recognition (though he foresaw people using this more widely than is currently used)
6. Portable or easily accessible encyclopedias with hyperlinked text
7. Keyboard- and mouse-controlled computers

Reading about how he saw the future makes it less surprising that Bush was Claude Shannon‘s thesis advisor. For those of you who don’t know, Shannon’s work gave rise to the field now known as information theory and also to the idea that one could use transistors (or binary logic/Boolean algebra) to implement numerical relationships. His ideas underpin the language of the modern computer.

It is amazing the clarity with which Bush saw the technological future. I heartily recommend the article as some eye-opening bedtime reading, if that makes sense.

## 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.

## Bardeen, CDWs and Macroscopic Quantum Phenomena

There is a well-written 1990 Physics Today article by John Bardeen entitled Superconductivity and Other Macroscopic Quantum Phenomena (pdf!). For those who are unaware, Bardeen was a two-time Nobel Laureate in Physics for inventing the transistor and secondly for the BCS theory of superconductivity.

Later in his career, Bardeen focused on the theory of transport in quasi-1D charge density wave materials. Bardeen was vocal in advocating that the transport in these materials must be understood in a quantum mechanical manner whereas most other physicists working on the problem treated it as a classical one (see True Genius by Daitch and Hoddeson). In the Physics Today article, he describes why he believes that the CDW sliding in these quasi-1D materials must be viewed as a manifestation of a macroscopic quantum phenomenon similar to that in superconductors and superfluids.

While Bardeen seemed to have lost his battle against the mainstream condensed matter physics community on this point upon his death in 1991, some interesting work has taken place since his death that has started to provide evidence for his perspective. In 1997, Monceau and co-workers showed the presence of Aharonov-Bohm-like oscillations in the CDW compound NbSe$_3$, with an oscillation period of, interestingly, $hc/2e$. While his tunneling theory of CDW transport may have been incorrect, his view of CDW transport as a macroscopic quantum phenomenon may yet be vindicated.

A lot of interest in these problems dissipated as scientists shifted to work on the problem of high temperature superconductivity following the discovery of the cuprates in 1986. However, it seems to me that there are still many unresolved issues in these compounds that persist to the present day that were cast aside rather than figured out.

As Paul Valery once said:

A poem is never finished, only abandoned.

The same can aptly be said about scientific problems.