# Category Archives: Charge Density Waves

## 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.

## 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:

$\int_{0}^{2\pi} \frac{\delta(1+\mathrm{tan}^2(x))}{1+\delta^2\mathrm{tan}^2(x)}\frac{dx}{2\pi} = \mathrm{sgn}(\delta)$

Just looking at the integral, it is difficult to see why no matter what the value of $\delta$, the integral will always give +1 or -1, which only depends on the sign of $\delta$. This means that if $\delta=1,000,000$ or if $\delta=0.00001$, 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$^{-1}$ 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.

## 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.

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.

## 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.

## 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.

## Transport Signatures in Charge Density Wave Systems

This post is inspired in part by Inna’s observation that a Josephson junction can act as a DC-AC converter. It turns out that CDWs can also act in a similar manner.

Sometimes I feel like quasi-1D charge density waves (CDWs) are like the lonely neglected child compared to superconductors, the popular, all-star athlete older sibling. Of course this is so because superconductors carry dissipationless current and exhibit perfect diamagnetism. However, quasi-1D CDWs can themselves exhibit pretty stunning transport signatures associated with the CDW condensate. Note that these spectacular properties are associated with incommensurate CDWs, as they break the translational symmetry of the crystal.

To make a comparison with superconductivity (even though no likes to be compared to their older sibling), here is a cartoon of the frequency-dependent conductivity (taken from G. Gruner’s Review of Modern Physics entitled Dynamics of Charge Density Waves):

Frequency-dependent conductivity for (a) a superconductor and (b) an incommensurate CDW

In the superconducting case, there is a delta function at zero frequency, indicative of dissipationless transport. For the CDW, there is also a collective charge transport mode, but in this case it is at finite energy (as it is pinned by impurities), and it is dissipative (indicated by the finite width).

This collective charge transport mode can be “depinned” and results in a nonlinear conductivity known as  a sliding CDW. This is evidenced below in the I-V characteristics. Below a threshold electric field/voltage, usual Ohmic characteristics are observed, associated with the “normal” non-condensed electrons. However, above the threshold electric field/voltage the collective mode is depinned and contributes to the I-V characteristics.

Non-linear IV characteristics indicative of collective charge transport in the CDW phase

Even more amazingly, once this CDW has been depinned, applying a DC field results in an AC response. Below is an image from a famous paper by Fleming and Grimes showing the Fourier transformed AC response with several harmonics. As the voltage is turned up, the fundamental frequency increases markedly (the voltage is highest in (a) and is decreased slowly until (e) where the CDW is no longer sliding).

AC response to a DC applied voltage in order of decreasing DC voltage in NbSe3. (a) V=5.81mV, (b) V=5.05mV, (c) 4.07mV, (d) V=3.40mV (e) V=0

The observed oscillation frequency is due to the collective mode getting depinned from its impurity site and then getting  weakly pinned successively by impurities, though this picture is debated. N.P. Ong, who did some great early work on CDW transport, has noted that the CDW “sings”. A nice cartoon of this idea is presented in the ball-and-egg-crate model shown below. One can imagine the successive “hits in the road” at periodic time intervals resulting in the AC response seen above.

Ball and egg crate model of CDW transport

Hopefully this post will help people appreciate more the shy younger sibling that is the charge density wave.

All images taken from G. Gruner RMP 60, 1129 (1988).