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 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, 2H-NbSe, and 2H-TaSe, 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.

An Integral from the SSH Model

A while ago, I was solving the Su-Schrieffer-Heeger (SSH) model for polyacetylene and came across an integral which I immediately thought was pretty cool. Here is the integral along with the answer:

Just looking at the integral, it is difficult to see why no matter what the value of , the integral will always give +1 or -1, which only depends on the sign of . This means that if or if , you would get the same result, in this case +1, as the answer to the integral! I’ll leave it to you to figure out why this is the case. (Hint: you can use contour integration, but you don’t have to.)

It turns out that the result actually has some interesting topological implications for the SSH model, as there are fractional statistics associated with the domain wall solitons. I guess it’s not so surprising that an integral that possesses topological properties would show up in a physical system with topological characteristics! But I thought the integral was pretty amusing anyhow, so I thought I’d share it.

Aside: For those who are interested in how I arrived at this integral in the SSH model, here are some of my notes. (Sorry if there are any errors and please let me know!) Also, the idea of solitons in the SSH model actually bears a strong qualitative resemblance to the excellent zipper analogy that Brian Skinner used on his blog.

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 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.

A Rather Illuminating Experiment Using Ultrafast Lasers

Historically, in condensed matter physics, there have been generally two experimental strategies: (i) scattering/spectroscopy experiments such as angle-resolved photoemission or X-ray scattering, and (ii) experiments involving macroscopic variables such as specific heat, resistivity, or magnetization. In the past few decades, a qualitatively new frontier opened up. This consisted of experiments that involved kicking a system out of equilibrium (usually with a pulsed femtosecond laser) and monitoring its relaxation back to equilibrium.

There are by now tons of papers using this technique, and I’ve also blogged previously about a particularly elegant use of these methods in a slightly different context than the one I’ll be discussing here. The paper I’m concerned with today uses ultrafast lasers in another rather illuminating context. It involved apparently melting the electronic order in 1T-TiSe2, while the lattice distortion remained in tact.

The importance of this experiment requires a little background. There has been debate for a couple decades now in the literature as to whether excitonic correlations are driving the charge density wave transition in 1T-TiSe2. This experiment claims that one can non-thermally melt (with the ultrafast laser) the excitonic order while the lattice remains distorted. This is done by monitoring the optical response of the sample at time intervals after the intense pulsed laser hits the sample: zone-folded phonons are monitored as evidence of the lattice distortion while the plasmon peak energy is monitored as evidence of excitonic order. The conclusion that the authors come to is that it cannot be purely an excitonic mechanism that is responsible for the charge density wave in this material as the plasmon peak energy is drastically affected by the laser pulse, while the zone-folded phonons do not react.

There is one caveat in this otherwise quite solid piece of work, however. The authors have equated the shift in the plasmon peak frequency (immediately following the arrival of the ultrafast laser pulse on the sample) with the melting of excitonic order. While this interpretation is plausible, it is not necessarily correct considering that the laser is photo-exciting a large number of charge carriers.

Regardless of this last point, the paper is definitely worth the read and highlights the kinds of experiments that can be conducted with these techniques. To my mind, this is one of the more illuminating experiments conducted on 1T-TiSe2 as many other experiments have been quite inconclusive about the mechanism behind the CDW in this material. Despite the aforementioned caveat, this experiment quite definitively demonstrates that one cannot ignore the role that electron-phonon coupling plays in the formation of the CDW in 1T-TiSe2.

What Happens in 2D Stays in 2D.

There was a recent paper published in Nature Nanotechnology demonstrating that single-layer NbSe exhibits a charge density wave transition at 145K and superconductivity at 2K. Bulk NbSe has a CDW transition at ~34K and a superconducting transition at ~7.5K. The authors speculate (plausibly) that the enhanced CDW transition temperature occurs because of an increase in electron-phonon coupling due to the reduction in screening. An important detail is that the authors used a sapphire substrate for the experiments.

This paper is among a general trend of papers that examine the physics of solids in the 2D limit in single-layer form or at the interface between two solids. This frontier was opened up by the discovery of graphene and also by the discovery of superconductivity and ferromagnetism in the 2D electron gas at the LAO/STO interface. The nature of these transitions at the LAO/STO interface is a prominent area of research in condensed matter physics. Part of the reason for this interest stems from researchers having been ingrained with the Mermin-Wagner theorem. I have written before about the limitations of such theorems.

Nevertheless, it has now been found that the transition temperatures of materials can be significantly enhanced in single layer form. Besides the NbSe case, it was found that the CDW transition temperature in single-layer TiSe was also enhanced by about 40K in monolayer form. Probably most spectacularly, it was reported that single-layer FeSe on an STO substrate exhibited superconductivity at temperatures higher than 100K  (bulk FeSe only exhibits superconductivity at 8K). It should be mentioned that in bulk form the aforementioned materials are all quasi-2D and layered.

The phase transitions in these compounds obviously raise some fundamental questions about the nature of solids in 2D. One would expect, naively, for the transition temperature to be suppressed in reduced dimensions due to enhanced fluctuations. Obviously, this is not experimentally observed, and there must therefore be a boost from another parameter, such as the electron-phonon coupling in the NbSe case, that must be taken into account.

I find this trend towards studying 2D compounds a particularly interesting avenue in the current condensed matter physics climate for a few reasons: (1) whether or not these phase transitions make sense within the Kosterlitz-Thouless paradigm (which works well to explain transitions in 2D superfluid and superconducting films) still needs to be investigated, (2) the need for adequate probes to study interfacial and monolayer compounds will necessarily lead to new experimental techniques and (3) qualitatively different phenomena can occur in the 2D limit that do not necessarily occur in their 3D counterparts (the quantum hall effect being a prime example).

Sometimes trends in condensed matter physics can lead to intellectual atrophy — I think that this one may lead to some fundamental and major discoveries in the years to come on the theoretical, experimental and perhaps even on the technological fronts.

Update: The day after I wrote this post, I also came upon an article demonstrating evidence for a ferroelectric phase transition in thin Strontium Titanate (STO), a material known to exhibit no ferroelectric phase transition in bulk form at all.