# Tag Archives: 2D Physics

## Strontium Titanate – A Historical Tour

Like most ugly haircuts, materials tend to go in and out of style over time. Strontium titanate (SrTiO3), commonly referred to as STO, has, since its discovery, been somewhat timeless. And this is not just because it is often used as a substitute for diamonds. What I mean is that studying STO rarely seems to go out of style and the material always appears to have some surprises in store.

STO was first synthesized in the 1950s, before it was discovered naturally in Siberia. It didn’t take long for research on this material to take off. One of the first surprising results that STO had in store was that it became superconducting when reduced (electron-doped). This is not remarkable in and of itself, but this study and other follow-up ones showed that superconductivity can occur with a carrier density of only ~$5\times 10^{17} cm^{-3}$.

This is surprising in light of BCS theory, where the Fermi energy is assumed to be much greater than the Debye frequency — which is clearly not the case here. There have been claims in the literature suggesting that the superconductivity may be plasmon-induced, since the plasma frequency is in the phonon energy regime. L. Gorkov recently put a paper up on the arXiv discussing the mechanism problem in STO.

Soon after the initial work on superconductivity in doped STO, Shirane, Yamada and others began studying pure STO in light of the predicted “soft mode” theory of structural phase transitions put forth by W. Cochran and others. Because of an anti-ferroelectric structural phase transition at ~110K (depicted below), they we able to observe a corresponding soft phonon associated with this transition at the Brillouin zone boundary (shown below, taken from this paper). These results had vast implications for how we understand structural phase transitions today, when it is almost always assumed that a phonon softens at the transition temperature through a continuous structural phase transition.

Many materials similar to STO, such as BaTiO3 and PbTiO3, which also have a perovskite crystal structure motif, undergo a phase transition to a ferroelectric state at low (or not so low) temperatures. The transition to the ferroelectric state is accompanied by a diverging dielectric constant (and dielectric susceptibility) much in the way that the magnetic susceptibility diverges in the transition from a paramagnetic to a ferromagnetic state. In 1978, Muller (of Bednorz and Muller fame) and Burkard reported that at low temperature, the dielectric constant begins its ascent towards divergence, but then saturates at around 4K (the data is shown in the top panel below). Ferroelectricity is associated with a zone-center softening of a transverse phonon, and in the case of STO, this process begins, but doesn’t quite get there, as shown schematically in the image below (and you can see this in the data by Shirane and Yamada above as well).

Taken from Wikipedia

The saturation of the large dielectric constant and the not-quite-softening of the zone center phonon has led authors to refer to STO as a quantum paraelectric (i.e. because of the zero-point motion of the transverse optical zone-center phonon, the material doesn’t gain enough energy to undergo the ferroelectric transition). As recently as 2004, however, it was reported that one can induce ferroelectricity in STO films at room temperature by straining the film.

In recent times, STO has found itself as a common substrate material due to processes that can make it atomically flat. While this may not sound so exciting, this has had vast implications for the physics of thin films and interfaces. Firstly, this property has enabled researchers to grow high-quality thin films of cuprate superconductors using molecular beam epitaxy, which was a big challenge in the 1990’s. And even more recently, this has led to the discovery of a two-dimensional electron gas, superconductivity and ferromagnetism at the LAO/STO interface, a startling finding due to the fact that both materials are electrically insulating. Also alarmingly, when FeSe (a superconductor at around 7K) is grown as a monolayer film on STO, its transition temperature is boosted to around 100K (though the precise transition temperature in subsequent experiments is disputed but still high!). This has led to the idea that the FeSe somehow “borrows the pairing glue” from the underlying substrate.

STO is a gem of a material in many ways. I doubt that we are done with its surprises.

## Precision in Many-Body Systems

Measurements of the quantum Hall effect give a precise conductance in units of $e^2/h$. Measurements of the frequency of the AC current in a Josephson junction give us a frequency of $2e/h$ times the applied voltage. Hydrodynamic circulation in liquid 4He is quantized in units of $h/m_{4He}$. These measurements (and similar ones like flux quantization) are remarkable. They yield fundamental constants to a great degree of accuracy in a condensed matter setting– a setting which Murray Gell-Mann once referred to as “squalid state” systems. How is this possible?

At first sight, it is stunning that physics of the solid or liquid state could yield a measurement so precise. When we consider the defects, impurities, surfaces and other imperfections in a macroscopic system, these results become even more astounding.

So where does this precision come from? It turns out that in all cases, one is measuring a quantity that is dependent on the single-valued nature of the (appropriately defined) complex scalar  wavefunction. The aforementioned quantities are measured in integer units, $n$, usually referred to as the winding number. Because the winding number is a topological quantity, in the sense that it arises in a multiply-connected space, these measurements do not particularly care about the small differences that occur in its surroundings.

For instance, the leads used to measure the quantum Hall effect can be placed virtually anywhere on the sample, as long as the wires don’t cross each other. The samples can be any (two-dimensional) geometry, i.e. a square, a circle or some complicated corrugated object. In the Josephson case, the weak links can be constrictions, an insulating oxide layer, a metal, etc. Imprecision of experimental setup is not detrimental, as long as the experimental geometry remains the same.

Another ingredient that is required for this precision is a large number of particles. This can seem counter-intuitive, since one expects quantization on a microscopic rather than at a macroscopic level, but the large number of particles makes these effects possible. For instance, both the Josephson effect and the hydrodynamic circulation in 4He depend on the existence of a macroscopic complex scalar wavefunction or order parameter. In fact, if the superconductor becomes too small, effects like the Josephson effect, flux quantization and persistent currents all start to get washed out. There is a gigantic energy barrier preventing the decay from the $n=1$ current-carrying state to the $n=0$ current non-carrying state due to the large number of particles involved (i.e. the higher winding number state is meta-stable). As one decreases the number of particles, the energy barrier is lowered and the system can start to tunnel from the higher winding number state to the lower winding number state.

In the quantum Hall effect, the samples need to be macroscopically large to prevent the boundaries from interacting with each other. Once the states on the edges are able to do that, they may hybridize and the conductance quantization gets washed out. This has been visualized in the context of 3D topological insulators using angle-resolved photoemission spectroscopy, in this well-known paper. Again, a large sample is needed to observe the effect.

It is interesting to think about where else such a robust quantization may arise in condensed matter physics. I suspect that there exist similar kinds of effects in different settings that have yet to be uncovered.

Aside: If you are skeptical about the multiply-connected nature of the quantum Hall effect, you can read about Laughlin’s gauge argument in his Nobel lecture here. His argument critically depends on a multiply-connected geometry.

## 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.

## What Happens in 2D Stays in 2D.

There was a recent paper published in Nature Nanotechnology demonstrating that single-layer NbSe$_2$ exhibits a charge density wave transition at 145K and superconductivity at 2K. Bulk NbSe$_2$ 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$_2$ case, it was found that the CDW transition temperature in single-layer TiSe$_2$ 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$_2$ 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.