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65 years of the BCS theory

Thanks to an arithmetic mistake, I thought 2022 was the 75th anniversary of the invention (or discovery?) of the BCS theory of superconductivity. It’s really the 65th anniversary, but since I’d worked myself up to write about it, I’m going to. πŸ€·πŸ½β€β™‚οΈ It also helps that the theory is a remarkable fact of nature that make sense of what is weirdly a macroscopic effect of microscopic causes.

There are several ways to classify superconductors – materials that conduct electricity with zero resistance under certain conditions. One of them is as conventional or unconventional. A superconductor is conventional if BCS theory can explain its superconductivity. ‘BCS’ are the initials of the theory’s three originators: John Bardeen, Leon Cooper and John Robert Schrieffer. BCS theory explains (conventional) superconductivity by explaining how the electrons in a material enter a collective superfluidic state.

At room temperature, the valence electrons flow around a material, being occasionally scattered by the grid of atomic nuclei or impurities. We know this scattering as electrical resistance.

An illustration of a lattice of sodium and chlorine atoms in a sodium chloride crystal. Credit: Benjah-bmm27, public domain

The electrons also steer clear of each other because of the repulsion of like charges (Coulomb repulsion).

When the material is cooled below a critical temperature, however, vibrations in the atomic lattice encourage the electrons to become paired. This may defy what we learnt in high school – that like charges repel – but the picture is a little more complicated, and it might make more sense if we adopt the lens of energy instead.

A system will favour a state in which it has lower energy than one in which it has more energy. When two carriers of like charges, like two electrons, approach each other, they repel each other more strongly the closer they get. This repulsion increases the system’s energy (in some form, typically kinetic energy).

In some materials, conditions can arise in which two electrons can pair up – become correlated with each other – across relatively long distances, without coming close to each other, rendering the Coulomb repulsion irrelevant. This correlation happens as a result of the electrons’ effect on their surroundings. As an electron moves through the lattice of positively charged atomic nuclei, it exerts an attractive force on the nuclei, which respond by tending towards the electron. This increases the amount of positive potential near the electron, which attracts another electron nearby to move closer as well. If the two electrons have opposite spins, they become correlated as a Cooper pair, kept that way by the attractive potential imposed by the atomic lattice.

Leon Cooper explained that neither the source of this potential nor its strength matter – as long as it is attractive, and the other conditions hold, the electrons will team up into Cooper pairs. In terms of the system’s energy, the paired state is said to be energetically favourable, meaning that the system as a whole has a lower energy than if the electrons were unpaired below the critical temperature.

Keeping the material cooled to below this critical temperature is important: while the paired state is energetically favourable, the state itself arises only below the critical temperature. Above the critical temperature, the electrons can’t access this state altogether because they have too much kinetic energy. (The temperature of a material is the average kinetic energy of its constituent particles.)

Cooper’s theory of the electron pairs fit into John Bardeen’s theory, which sought to explain changes in the energy states of a material as it goes from being non-superconducting to superconducting. Cooper had also described the formation of electron pairs one at a time, so to speak, and John Robert Schrieffer’s contribution was to work out a mathematical way to explain the formation of millions of Cooper pairs and their behaviour in the material.

The trio consequently published its now-famous paper, ‘Microscopic Theory of Superconductivity’, on April 1, 1957.

(I typo-ed this as 1947 on a calculator, which spit out the number of years since to be 75. πŸ˜‘ One could have also expected me to remember that this is India’s 75th year of independence and that BCS theory was created a decade after 1947, but the independence hasn’t been registering these days.)

Anyway, electrons by themselves belong to a particle class called fermions. The other known class is that of the bosons. The difference between fermions and bosons is that the former obey Pauli’s exclusion principle while the latter do not. The exclusion principle forbids two fermions in the same system – like a metal – from simultaneously occupying the same quantum state. This means the electrons in a metal have a hierarchy of energies in normal conditions.

However, a Cooper pair, while composed of two electrons, is a boson, and doesn’t obey Pauli’s exclusion principle. The Cooper pairs of the material can all occupy the same state – i.e. the state with the lowest energy, more popularly called the ground state. This condensate of Cooper pairs behaves like a superfluid: practically flowing around the material, over, under and through the atomic lattice. Even when a Cooper pair is scattered off by an atomic nucleus or an impurity in the material, the condensate doesn’t break formation because all the other Cooper pairs continue their flow, and eventually also reintegrate the scattered Cooper pair. This flow is what we understand as electrical superconductivity.

“BCS theory was the first microscopic theory of superconductivity,” per Wikipedia. But since its advent, especially since the late 1970s, researchers have identified several superconducting materials, and behaviours, that neither BCS theory nor its extensions have been able to explain.

When a material transitions into its superconducting state, it exhibits four changes. Observing these changes is how researchers confirm that the material is now superconducting. (In no particular order:) First, the material loses all electric resistance. Second, any magnetic field inside the material’s bulk is pushed to the surface. Third, the electronic specific heat increases as the material is cooled before dropping abruptly at the critical temperature. Fourth, just as the energetically favourable state appears, some other possible states disappear.

Physicists experimentally observed the fourth change only in January this year – based on the transition of a material called Bi-2212 (bismuth strontium calcium copper oxide, a.k.a. BSCCO, a.k.a. bisko). Bi-2212 is, however, an unconventional superconductor. BCS theory can’t explain its superconducting transition, which, among other things, happens at a higher temperature than is associated with conventional materials.

In the January 2022 study, physicists also reported that Bi-2212 transitions to its superconducting state in two steps: Cooper pairs form at 120 K – related to the fourth sign of superconductivity – while the first sign appears at around 77 K. To compare, elemental rhenium, a conventional superconductor, becomes superconducting in a single step at 2.4 K.

A cogent explanation of the nature of high-temperature superconductivity in cuprate superconductors like Bi-2212 is one of the most important open problems in condensed-matter physics today. It is why we still await further updates on the IISc team’s room-temperature superconductivity claim.