What is the thermal Hall effect?

One thing that physics and mechanical engineering students learn early on is that there are often analogies between charge flow and heat flow, and this is reflected in the mathematical models we use to describe charge and heat transport.  We use Ohm's law, \(\mathbf{j}=\tilde{\sigma}\cdot \mathbf{E}\), which defines an electrical conductivity tensor \(\tilde{\sigma}\) that relates charge current density \(\mathbf{j}\) to electric fields \(\mathbf{E}=-\nabla \phi\), where \(\phi(\mathbf{r})\) is the electric potential.  Similarly, we can use Fourier's law for thermal conduction, \(\mathbf{j}_{Q} = - \tilde{\kappa}\cdot \nabla T\), where \(\mathbf{j}_{Q}\) is a heat current density, \(T(\mathbf{r})\) is the temperature distribution, and \(\tilde{\kappa}\) is the thermal conductivity.  

We know from experience that the electrical conductivity really has to be a tensor, meaning that the current and the electric field don't have to point along each other.  The most famous example of this, the Hall effect, goes back a long way, discovered by Edwin Hall in 1879.  The phenomenon is easy to describe.  Put a conductor in a magnetic field (directed along \(z\)), and drive a (charge) current \(I_{x}\) along it (along \(x\)), as shown, typically by applying a voltage along the \(x\) direction, \(V_{xx}\).  Hall found that there is then a transverse voltage that develops, \(V_{xy}\) that is proportional to the current.  The physical picture for this is something that we teach to first-year undergrads:  The charge carriers in the conductor obey the Lorentz force law and curve in the presence of a magnetic field.  There can't be a net current in the \(y\) direction because of the edges of the sample, so a transverse (\(y\)-directed) electric field has to build up.  

There can also be a thermal Hall effect, when driving heat conduction in one direction (say \(x\)) leads to an additional temperature gradient in a transverse (\(y\)) direction.  The least interesting version of this (the Maggi–Righi–Leduc effect) is in fact a consequence of the regular Hall effect:  the same charge carriers in a conductor can carry thermal energy as well as charge, so thermal energy just gets dragged sideways.   

Surprisingly, insulators can also show a thermal Hall effect.  That's rather unintuitive, since whatever is carrying thermal energy in the insulator is not some charged object obeying the Lorentz force law.  Interestingly, there are several distinct mechanisms that can lead to thermal Hall response.  With phonons carrying the thermal energy, you can have magnetic field affecting the scattering of phonons, and you can also have intrinsic curving of phonon propagation due to Berry phase effects.  In magnetic insulators, thermal energy can also be carried by magnons, and there again you can have Berry phase effects giving you a magnon Hall effect.  There can also be a thermal Hall signal from topological magnon modes that run around the edges of the material.  In special magnetic insulators (Kitaev systems), there are thought to be special Majorana edge modes that can give quantized thermal Hall response, though non-quantized response argues that topological magnon modes are relevant in those systems.  The bottom line:  thermal Hall effects are real and it can be very challenging to distinguish between candidate mechanisms. 

(Note: Blogger now compresses the figures, so click on the image to see a higher res version.)

Some interesting recent papers - lots to ponder

As we bid apparent farewell to LK99, it's important to note that several other pretty exciting things have been happening in the condensed matter/nano world.  Here are a few papers that look intriguing (caveat emptor:  I have not had a chance to read these in any real depth, so my insights are limited.)

• Somehow I had never heard of Pines' Demon until this very recent paper came out, and the story is told briefly here.  The wikipedia link is actually very good, so I don't know that I can improve upon the description.  You can have coupled collective modes for electrons in two different bands in a material, where the electrons in one band are sloshing anti-phase with the electrons in the other band.  The resulting mode can be "massless" (in the sense that its energy is linearly proportional to its momentum, like a photon's), and because it doesn't involve net real-space charge displacement, to first approximation it doesn't couple to light.  The UIUC group used a really neat, very sensitive angle-resolved electron scattering method to spot this for the first time, in high quality films of Sr2RuO4.  (An arxiv version of the paper is here.) 
• Here is a theory paper in Science (arxiv version) that presents a general model of so-called strange metals (ancient post on this blog).  Strange metals appear in a large number of physical systems and are examples where the standard picture of metals, Fermi liquid theory, seems to fail.  I will hopefully write a bit more about this soon.  One of the key signatures of strange metals is a low temperature electrical resistivity that varies like \(\rho(T) = \rho_{0} + AT\), as opposed to the usual Fermi liquid result \(\rho(T) = \rho_{0} + AT^{2}\).  Explaining this and the role of interactions and disorder is a real challenge.  Here is a nice write-up by the Simons Foundation on this.
• Scanning tunneling microscopy is a great spectroscopic tool, and here is an example where it's been possible to map out information about the many-body electronic states in magic-angle twisted bilayer graphene (arxiv version).  Very pretty images, though I need to think carefully about how to understand what is seen here.
• One more very intriguing result is this paper, which reports the observation of the fractional quantum anomalous Hall effect (arxiv version).  As I'd mentioned here, the anomalous Hall effect (AHE, a spontaneous voltage appearing transverse to a charge current) in magnetic materials was discovered in 1881 and not understood until recently.  Because of cool topological physics, some materials show a quantized AHE.  In 2D electron systems, the fractional quantum Hall effect is deeply connected to many-body interaction effects.  Seeing fractional quantum Hall states spontaneously appear in the AHE is quite exciting, suggesting that rich many-body correlations can happen in these topological magnetic systems as well.  Note: I really need to read more about this - I don't know anything in depth here.
• On the more applied side, this article is an extremely comprehensive review of the state of the art for transistors, the critical building block of basically every modern computing technology.  Sorry - I don't have a link to a free version (unless this one is open access and I missed it).  Anyway, for anyone who wants to understand modern transistor technology, where it is going, and why, I strongly encourage you to read this.  If I was teaching my grad nano class, I'd definitely use this as a reference.
• Again on the applied side, here is a neat review of energy harvesting materials.  There is a lot of interest in finding ways to make use of energy that would otherwise go to waste (e.g. putting piezo generators in your clothing or footwear that could trickle charge your electronics while you walk around).  
• In the direction of levity, in all too short supply these days, xkcd was really on-point this week.  For condensed matter folks, beware the quasiparticle beam weapon.  For those who do anything with electronics, don't forget this handy reference guide. 

Neutrality and experimental detective work

One of the remarkable aspects of condensed matter physics is the idea of emergent quasiparticles, where through the interactions of many underlying degrees of freedom, new excitations emerge that are long-lived and often can propagate around in ways very different than their underlying constituents.  Of course, it’s particularly interesting when the properties of the quasiparticles have quantum numbers or obey statistics that are transformed from their noninteracting counterparts.  For example, in the resonating valence bond model, starting from electrons with charge \(-e\) and spin 1/2, the low energy excitations are neutral spin-1/2 spinons and charge \(e\) holons.  It’s not always obvious in these situations whether the emergent quasiparticles act like fermions (obeying the Pauli principle and stacking up in energy) or bosons (all falling into the lowest energy state as temperature is reduced).  See here for an example.

Suppose there is an electrically insulating system that you think might host neutral fermionic excitations.  How would you be able to check?  One approach would be to look at the low temperature specific heat, which relates how much the temperature of an isolated object changes when a certain amount of disorganized thermal energy is added.  The result for (fermionic) electrons in a metal is well known:  because of the Pauli principle, the specific heat scales linearly with temperature, \(C \sim T\).  (In contrast, for the vibrational part of the specific heat due to bosonic phonons, \(C \sim T^3\) in 3D.).  So, if you have a crystalline(*) insulator that has a low temperature specific heat that is linear in temperature (or, equivalently, when you plot \(C/T\) vs. \(T\) and there is a non-zero intercept at \(T=0\)), then this is good evidence for neutral fermions of some kind.  Such a system should also have a linear-in-\(T\) thermal conductivity, too, and an example of this is reported here. This connects back to a post that I made a month ago.  Neutral fermions (presumably carrying spin) can lead to quantum oscillations in the specific heat (and other measured quantities). 

This kind of detective work, considering which techniques to use and how to analyze the data, is the puzzle-solving heart of experimental condensed matter physics.  There is a palette of measurable quantities - how can you use those to test for complex underlying physics?

(*) It’s worth remembering that amorphous insulators generally have a specific heat that varies like \(T^{1.1}\) or so, because of the unreasonably ubiquitous tunneling two-level systems.  The neutral fermions I’m writing about in this post are itinerant entities in nominally perfect crystals, rather than the localized TLS in disordered solids.  

What is a metal-insulator transition?

The recent excitement about the alleged high temperature superconductor "LK99" has introduced some in the public to the idea of a metal-insulator or insulator-metal transition (MIT/IMT).  For example, one strong candidate explanation for the sharp drop in resistance as a function of temperature is a drastic change in the electronic (and structural) properties of Cu2S at around 328 K, as reported here.  

In condensed matter physics, a metal is usually defined as a material with an electrical conductivity that increases with decreasing temperature.  More technically, in a (macroscopic) metal it is possible to create an electronic excitation (moving some electron from one momentum to another, for example) at arbitrarily small energy cost.  A metal is said to have "gapless excitations" of the electrons.  Even more technically, a metal has a nonzero electronic density of states at the electronic chemical potential.   

In contrast, an insulator has an electronic conductivity that is low and decreases with decreasing temperature.  In an insulator, it costs a non-zero amount of energy to create an electronic excitation, and the larger that energy cost, the more insulating the material.  An insulator is said to have an "energy gap".  If that energy gap is small compared to the thermal energy available (\( \sim k_{\mathrm{B}}T\)), there will be some conduction because of thermally excited electrons (and holes).  One way to classify insulators is by the reason for the energy gap, though knowing the mechanism for certain is often challenging.  A material is a "band insulator" if that gap comes about just because of how the atoms are stacked in space and how each atom shares its electrons.  This is the case for diamond, for example, or for common semiconductors like Si or GaAs (called semiconductors because their energy gaps are not too large).  A material can be an insulator due primarily to electron-electron interactions (a Mott insulator or the related charge transfer insulator); a material can be an insulator primarily because of interactions between the electrons and the lattice structure (a Peierls insulator); a material can be an insulator because of disorder, which can lead to electrons being in localized states (an Anderson insulator).

In some materials, there can be a change between metallic and insulating states as a function of some physically tunable parameter.  Common equilibrium control knobs are temperature, pressure, magnetic field, and density of charge carriers.  It's also possible to drive some materials between insulating and metallic states by hitting them with light or applying large electric fields.  

Sudden changes in properties can be very dramatic, as the Cu2S case shows.  That material tends to be in one crystal structure at high temperatures, in which it happens to be a band insulator with a large gap.  Then, as the temperature is lowered, the material spontaneously changes into a different crystal structure in which there is much more conduction.  There are other materials well known for similar transitions (often between a high temperature conducting state and a low temperature insulating state), such as VO2 and V2O3, in which the electrical conductivity can abruptly change by 5 orders of magnitude over a small temperature range.  

MIT/IMT materials can be of technological interest, particularly if their transitions are readily triggered.  For example, vanadium oxides are used in thermochromic and electrochromic switchable windows, because the optical properties of the material are drastically different in the conducting vs insulating phases.   The fundamental interest in MIT/IMTs systems is clear as well, especially when electronic interactions are thought to be responsible - for example, the rich array of conducting, superconducting, and insulating states that show up in twisted bilayer graphene as a function of carrier density (a representative paper here).  It's always interesting to consider how comparatively simple ingredients can lead to such rich response, through energetic (and entropic) competition between different states with wildly disparate properties.

Desirable properties for a superconductor

Given the present interest, let's talk about what kind of properties one wants in a superconductor, as some people on social media seem ready to jump straight on the "what does superconductivity mean for bitcoin?" train.

First, the preliminaries.  Superconductivity is a state of matter in which the conduction electrons act collectively in an interesting way.   In the superconductors we know about, electrons pair up and can be described by a single collective quantum state (with a well-defined phase - the quantum state can be written as a complex quantity that has an amplitude and a phase angle, as in \(A \exp{i\phi}\), where \(\phi\) is the phase).  A consequence of this is that there is an "energy gap" - it costs a certain amount of energy to create individual unpaired electrons.  It's this energy gap that allows dc current to flow without electrical resistance in a superconductor. There is a length scale, the coherence length, over which the superconducting state tends to vary, like at the boundary of a material.  There is also a length scale, the penetration depth, over which magnetic field can penetrate into a superconductor.  Magnetic field is expelled from the bulk of a superconductor because the material spontaneously develops surface currents such that the field from those currents cancels out the external field in the bulk of the material.  Depending on the ratio of the coherence length and the penetration depth, a superconductor can be Type I (expels all magnetic field until the field \(H\) exceeds some critical value \(H_{c}\), at which point superconductivity dies) or Type II (allows magnetic field above a critical field \(H_{c1}\) to penetrate in the form of vortices, with a non-superconducting core and surrounded by screening currents, until superconductivity is eventually killed above some upper critical field \(H_{c2}\)).   Motion of vortices actually leads to energy losses, so it is desirable for applications involving AC currents especially to have the vortices be pinned in place somehow in the material, often by disorder.  It is this pinning that leads to superconducting levitation in fixed orientations relative to a magnet, even with the SC hanging below the magnet.   Superconductivity tends to die either by the pairs falling apart (common in Type I superconductors as temperature is increased until thermal energy exceeds the attractive pairing interaction) or by the loss of global phase coherence (a crude analogy:  the dance partners are still paired up, but each pair is dancing to their own tune).  

Superconductors have a critical temperature above which global superconductivity is lost.  They also have critical field scales, as mentioned above.  Clearly, for many applications, it would be greatly desirable for a superconductor to have both a high critical temperature (obviously) and high critical fields.  Similarly, superconductors have a critical current density - some combination of the local field (from the current) exceeding the critical field and current-driven phase slips can lead to loss of superconductivity.  It would be great to have a high critical current density.  The relationship between critical temperature, critical field, and critical current density is not always simple, though they tend to correlate, because if SC is very robust all three quantities will tend to be larger.

It would also be wonderful if a new superconducting family of materials was ductile.  The higher temperature superconductors (cuprates, pnictides, nickelates) are all ceramics, meaning that they are brittle and not readily formed into wires.  It's taken 36 years or so for people to get good at making wires and ribbons that incorporate the cuprate superconductors, typically by encasing them in powder form inside Cu or Ag tubes, then squeezing appropriately and annealing.  

Lastly, and perhaps not well appreciated, from a practical perspective, it'd be nice if superconductors were air stable.  That is, it's annoying to work with materials that react poorly to oxygen, humidity in the air, O2 or water in the presence of UV light from the sun, etc.  Having a material that is chemically very stable with a clearly known and set stoichiometry would be great.  Along with this, it would be nice if the material was easily made, at scale, without having to resort to crazy conditions (super high temperatures or pressures; weird rare or hazardous constituents).

How useful any candidate superconductor will be and on what timescale is set by the combination of these properties.  A room temperature superconductor that turns into goo in the presence of damp air would not be nearly as useful as one that is chemically stable sitting on a bench.  

For all the people who seem to be jumping to the conclusion that room temperature superconductivity will suddenly lead to breakthroughs in quantum information processing, that is far from clear.  Lots of processes that screw up superconducting qubits happen more at higher temperatures, even if superconductivity is robust.  I'm not aware of anyone peddling qubits based on copper oxide superconductors right now, even though the transition temperature is 10 times higher than that of Nb.

In short:  usefulness does not flow instantly from materials discovery, even if the material parameters all seem good.  Patience is hard to come by yet essential in trying to adapt new materials to applications.