# Tag Archives: Charge Density Waves

## 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).

## Unusual Properties in Rare Earth Tritellurides

Recently, many papers have been revisiting the rare earth tritelluride materials. They were first pointed out to host charge density waves in this paper (sorry paywall!). In present times, the Fisher group at Stanford has revitalized the study of these compounds.

There is a particularly interesting paper here on TbTe$_3$ demonstrating the existence of superconductivity in this material under pressure. Just as a little bit of background, TbTe$_3$ forms a charge density wave (unidirectional in a quasi-square lattice!) at ~340K and orders antiferromagnetically at ~6K. Under pressure, the charge density wave begins to get suppressed, the antiferromagnetism is enhanced slightly and then superconductivity is found at ~2.3 GPa.

Startlingly, at 2.3 GPa, there is a co-existence of charge density wave, antiferromagnetic and superconducting orders. The superconducting T$_c$ peaks at ~4K at 12.4 GPa. One cannot fight the urge to draw comparisons with high-temperature superconductors…

## Was the Higgs Boson Discovered in 1980?

In a tour-de-force experiment in 1980, Klein and Sooryakumar discovered a collective mode in the superconducting phase of NbSe$_2$ in a Raman experiment. This mode interacts with the collective mode of the charge density wave and was interpreted a few year later by Littlewood and Varma to be a Higgs amplitude mode of the superconducting state.

Could it really be said that the Higgs mode was discovered in Urbana in 1980?

Of course, I am being facetious, but it is a rather cute historical curiosity. That being said, I have to also admit that I am not 100% convinced that this interpretation is correct, though it does carry some weight.

This classic experiment has also been driving experiments recently towards observing the Higgs mode in other superconducting systems as well. For example, click here and here.

The other part of this experiment that makes it particularly relevant to current studies is the observation of an interaction between superconductivity and charge density waves as was mentioned in a previous post. With the application of a magnetic field, they were able to suppress the superconductivity and enhance the charge density wave collective excitation as pictured below.

## How do we define states of matter?

Historically, many people seemed to lean towards defining a phase of matter by its (broken) symmetries. For instance, a ferromagnet has broken rotational symmetry and time-reversal symmetry, a solid has broken translational and rotational symmetry, etc. In light of the discoveries of the Quantum Hall Effect and topological insulators, it seems like this symmetry classification does not encompass all states of matter.

The symmetry classification is largely a theoretical construct, however. I would think that one defines a state of matter by particular experimental properties that it exhibits. For example, one could define a superconductor by requiring it to exhibit the following:

1. Zero Resistivity
2. Meissner Effect
3. Zero Peltier Coefficient

Put another way, to verify that one has discovered a superconductor, these three criteria must be satisfied.

Let us take another example: a simple metal. The criterion that must be satisfied for this case is the existence of a Fermi surface. This can be measured by quantum oscillation measurements, angle-resolved photoemission, or a few other probes.

Yet another example: a 2D topological insulator. What one must observe is:

1. The Fermi energy intersects an odd number of topologically protected edge states in half the edge Brillouin zone (which was shown by transport in this classic paper)
2. The existence of a spin-polarization associated with the edge states

While these three examples were chosen because they were simple, I have remaining doubts. Are these observations necessary and sufficient to define these states of matter? Are there cases where one can better define a state of matter theoretically?

For instance, a theorist may define a 2D topological insulator by the existence of a non-trivial topological number, which seems like a perfectly valid criterion to me. This topological number cannot be experimentally observed in a very direct way (to my knowledge) and has to be inferred from the edge states, band structure, etc.

The reason I started thinking about this is because I did not find the definition of a charge density wave in this widely-cited paper by Johannes and Mazin appropriate. It states:

[A charge density wave is a] Peierls-like instabilit[y] that occur[s] due to a divergency in the real part of the electronic susceptibility, so that the electronic subsystem would be unstable per se, even if the ions were clamped at their high symmetry positions.

This definition bothers me in particular because it defines a charge density wave by its cause (i.e. Peierls-like instability due to a divergence in the real part of the electronic susceptibility).  The main qualm I have is that one should not define a state of matter by its origin or cause. This is like trying define a superconductor by the mechanism that causes its existence (i.e. phonon-mediated electron-electron interaction for superconductors, which would exclude unconventional superconductors from its definition). This is obviously problematic. Therefore, shouldn’t one define a charge density wave by its experimentally measured properties?

So I come back to the original question: how does one define a state of matter?

## Density Wave with d-Form Factor an Answer to the Mysterious Pseudogap?

This week, Seamus Davis, a leader in the field of scanning tunneling microscopy (STM) was in town and gave a series of talks on the many materials examined in his laboratory by his various students and postdocs. The most compelling of the talks was presented on the pseudogap phase of the high-temperature cuprate superconductors.

According to Davis, the mysterious nature of the pseudogap phase is due to the presence of a density wave with d-form factor, pictured below. In the topographic STM images of BSCCO and NaCCOC, one can see a vivid visual representation of this density wave (link to the paper PDF!).

Because STM is a surface sensitive probe, the high-Tc community always had niggling doubts about whether this density wave was manifest in the bulk. With increasing evidence that there is a density wave in the bulk from scattering probes such as resonant soft X-ray scattering and from hard x-ray diffraction, the interpretation put forth by Davis seems more plausible than it did a decade ago.

He also claimed in this talk that the presence of this density wave could explain the existence of Fermi arcs seen using angle-resolved photoemission, another surface sensitive technique. However, the observation of quantum oscillations in the underdoped cuprates demonstrating pockets at the Fermi surface complicates matters somewhat. For these two observations to be consistent, the only possible explanation is a phase transition in the underdoped cuprates as a function of magnetic field in the region of ~20 Tesla.

Nonetheless, Davis’ group has done a wonderful set of experiments on these compounds and his talks this week were captivating and enjoyable. This seems like a valuable contribution into the ongoing discussion of the mystery of the pseudogap.

## The Value of a Null Result

In our field, it is unpopular to publish a result where one finds an absence of a particular phenomenon. However, these results can be extremely valuable, as one can see what other authors have tried.

In the study of charge density wave (CDW) systems, which has been undergoing a renaissance of late, there is one particular null result I find quite fascinating. This result (sorry, paywall) was published by F. DiSalvo and R. Fleming in Solid State Communications, demonstrating the inability, even at high electric fields, for a charge density wave to depin and slide in two prototypical quasi-2D transition metal dichalcogenides, 1T-Tantalum Disulphide and 2H-Tantalum Diselenide.

In fact, I am unaware of any report of a sliding CDW in quasi-2D transition metal dichalcogenides. This has pretty vast implications for these materials, as it is difficult to probe the electronic subsystem alone due to the inability to divorce it from the ionic subsystem.

Any comments pointing me in the direction of observations of sliding CDWs in transition metal dichalcogenides are encouraged.