High Temperature Superconductivity

Here at the physics department of the National University of Singapore, Tony Leggett is about to speak on “Cuprate superconductivity: the current state of play”. I’ll take notes and throw them on this blog in a rough form. As always, my goal is to start some interesting conversations. So, go ahead and ask questions, or fill in some more details. Not everything I write here is something I understand!

Certain copper oxide compounds can be superconductive at relatively high temperatures — for example, above the boiling point of liquid nitrogen, 77 kelvin. These compounds consist of checkerboard layers with four oxygen atoms at the corners of each square and one copper in the middle. It’s believed that the electrons move around in these layers in an essentially two-dimensional way. Two-dimensional physics allows for all sorts of exotic possibilities! But nobody is sure how these superconductors work. The topic has been around for about 25 years, but according to Leggett, there’s no one theory that commands the assent of more than 20% of the theorists.

Here’s the outline of Leggett’s talk:

1. What is superconductivity?

2. Brief overview of cuprate structure and properties.

3. What do we know for sure about high-temperature superconductivity (HTS) in the cuprates? That is, what do we know without relying on any microscopic model, since these models are all controversial?

4. Some existing models.

5. Are we asking the right questions?

1. What is superconductivity?

For starters, he asked: what is superconductivity? It involves at least two phenomena that don’t necessarily need to go together, but seem to always go together in practice, and are typically considered together. One: perfect diamagnetism — in the “Meissner effect“, the medium completely excludes magnetic fields. This is an equilibrium effect. Two: persistent currents — this is an incredibly stable metastable effect.

Note the difference: if we start with a ball of stuff in magnetic field and slowly lower its temperature, once it becomes superconductive it will exclude the magnetic field. There are never any currents present, since we’re in thermodynamic equilibrium at any stage.

On the other hand, a ring of stuff with a current flowing around it is not in thermal equibrium. It’s just a metastable state.

The London-Landau-Ginzburg theory of superconductivity is a ‘phenomenological’ theory: it doesn’t try to describe the underlying microscopic cause, just what seems to happen. Among other things, it says that a superconductor is characterized by a ‘macroscopic wave function’ , a complex function with phase . The current is given by

where is a charge (in fact the charge of an electron pair, as was later realized).

This theory explains the Meissner effect and also persistent currents, and it’s probably good for cuprate superconductors.

2. The structure and behavior of cuprate superconductors

The structure of a typical cuprate: there are planes made of CuO₂ and other atoms (typically alkaline earth), and then, between these, a material that serves as a ‘charge reservoir’.

He showed us the phase diagram for a typical cuprate as a function of temperature and the ‘doping’: that is, the number of extra ‘holes’ – missing electrons – per CuO₂. No cuprate has yet been shown to have this phase diagram in its entirety! But different ones have been seen to have different parts, so we may guess the story is like this:

There’s an antiferromagnetic insulator phase at low doping. At higher doping there’s a strange ‘pseudogap’ phase. Nobody knows if this ‘pseudogap’ phase extends to zero temperature. At still higher dopings we see a superconductive phase at low temperature and a ‘strange metal’ phase above some temperature. This temperature reaches a max at a doping of about 0.16 — a more or less universal figure — but the value of this maximum temperature depends a lot on the material. At higher dopings the superconductive phase goes away.

There are over 200 superconducting cuprates, but there are some cuprates that can never be made superconducting — those with multilayers spaced by strontium or barium.

Both ‘normal’ and superconducting states are highly anisotropic. But the ‘normal’ states are actually very anomalous — hence the term ‘strange metal’. The temperature-dependence of various properties are very unusual. By comparison the behaviour of the superconducting phase is less strange!

Most (but not all) properties are approximately consistent with the hypothesis that at a given doping, the properties are universal.

The superconducting phase is highly sensitive to doping and pressure.

3. What do we know for sure about superconductivity in the cuprates?

There’s strong evidence that cuprate superconductivity is due to the formation of Cooper pairs, just as for ordinary superconductors.

The ‘universality’ of high-temperature superconductivity in cuprate with very different chemical compositions suggests that the main actors are the electrons in the CuO₂ planes. Most researchers believe this.

There’s a lot of NMR experiments suggesting that the spins of the electrons in the Cooper pairs are in the ‘singlet’ state:

up ⊗ down – down ⊗ up

Absence of substantial far-infrared absorption above the gap edge suggests that pairs are formed from time-reversed states (despite the work of Tahir–Kheli).

The ‘radius’ of the Cooper pairs is very small: only 3-10 angstroms, instead of thousands as in an ordinary superconductor!

In ordinary superconductor the wave function of a Cooper pair is in an s state (spherically symmetric state). In a cuprate superconductor it seems to have the symmetry of : that is, a d state that’s odd under 90 degree rotation in the plane of the cuprate (the plane), but even under reflection in either the or axis.

There’s good evidence that the pairs in different multilayers are effectively independent (despite the Anderson Interlayer Tunnelling Theory).

There isn’t a substantial dependence on the isotopes used to make the stuff, so it’s believed that phonons don’t play a major role.

At least 95% of the literature makes all of the above assumptions and a lot more. Most models are specific Hamiltonians that obey all these assumptions.

4. Models of high-temperature superconductivity in cuprates

How will we know when we have a ‘satisfactory’ theory? We should either be able to:

A) give a blueprint for building a room-temperature superconductor using cuprates, or

B) assert with confidence that we will never be able to do this, or at least

C) say exactly why we cannot do either A) or B).

No model can yet do this!

Here are some classes of models, from conservative to exotic:

1. Phonon-induced attraction – the good old BCS mechanism, which explains ordinary superconductors. These models have lots of problems when applied to cuprates, e.g. the fact that we don’t see an isotope effect.

2. Attraction induced by the exchange of some other boson: spin fluctuations, excitons, fluctuations of ‘stripes’ or still more exotic objects.

3. Theories starting from the single-band Hubbard model. These include theories based on the postulate of ‘exotic ordering’ in the ground state, e.g. charge-spin separation.

5. What are the right questions to ask?

The energy is the sum of 3 terms: the kinetic energy, the potential energy of the interaction between the conduction electrons and the static lattice, and the potential energy of the interaction of the conduction electrons among each other (both intra-plane and inter-plane). One of these must go down when Cooper pairs must form! The third term is the obvious suspect.

Then Leggett wrote a lot of equations which I cannot copy fast enough… and concluded that there are two basic possibilities, “Eliashberg” and “overscreening”. The first is that electrons with opposite momentum and spin attract each other in the normal phase. The second is that there’s no attraction required in the normal phase, but the interaction is modified by pairing: pairing can cause “screening” of the Coulomb repulsion. Which one is it?

Another good question: Why does the critical temperature depend on the number of layers in a multilayer? There are various possible explanations. The “boring” explanation is that superconductivity is a single-plane phenomenon, but multi-layering affects properties of individual planes. The “interesting” explanations say that inter-plane effects are essential: for example, as in the Anderson inter-layer tunnelling model, or due to a Kosterlitz-Thouless effect, or due to inter-plane Coulomb interactions.

Leggett clearly likes the last possibility, with the energy savings taking place due to increased screening, and with the energy saving taking place predominantly at long wavelengths and mid-infrared frequencies. This gives a natural explanation of why all known high-temperature superconductors are strongly two-dimensional, and it explains many more of their properties, too. Moreover it’s unambiguously falsifiable in electron energy-loss spectroscopy experiments. He has proposed an experimental test, which will be carried out soon.

He bets that with at least a 50% chance some of the younger members of the audience will live to see room-temperature superconductors.