Cooper pairs – Close Read

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. 🤷🏽‍♂️ 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. 😑 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.

Physicists observe long-expected helium superfluid phase

Physicists have reported that they have finally observed helium 3 existing in a long-predicted type of superfluid, called the ß phase.

This is an important discovery, if it’s borne out, for reasons that partly have to do with its isotope, helium 4. Helium 4 is a fascinating substance because the helium 4 atom is a boson – a type of particle whose quantum properties and behaviour are explained by rules called Bose-Einstein statistics. Helium 3, on the other hand, is a fermion, and fermions are governed by Fermi-Dirac statistics.

Bosons and fermions have one important difference: bosons are allowed to disobey Pauli’s exclusion principle, and by doing so they can assume exotic states of matter rarely found in nature, with many unusual properties.

For example, when helium 4 is cooled below a certain temperature, it becomes a superfluid: a liquid that flows without experiencing any resistance. If you poured a superfluid into a bowl, it will be able to climb the walls of the bowl and spill out without any help. But helium 3 atoms are fermions, so they are bound to obey Pauli’s exclusion principle and can’t become a superfluid.

At least this is what physicists believed for a long time, until the early 1970s, when two independent groups of physicists found – one in theory and the other in experiments – that helium 3 could indeed enter a superfluid phase, but at a temperature 1,000-times lower than the critical temperature of helium 4. The theory group, led by Anthony Leggett at the University of Sussex, had in fact made a significant discovery.

Today, we know that the flow of superfluid helium 4 is analogous to the flow of electrons in a conventional superconductor, which also move around as if they face no resistance from the surrounding atoms. Leggett and co. found that the theory used to explain these superconductors could also be used to explain helium 3 superfluidity. This theory is called Bardeen-Cooper-Schrieffer (BCS) theory, and the materials whose superconductivity it can explain are called BCS superconductors.

Electrons are fermions and cannot ‘super-flow’. But in a BCS superconductor that has been cooled below its critical temperature, some forces in the material cause the electrons to overcome their mutual repulsion (“like charges repel”) and pair up. These electron pairs, while being made of two individual fermions, actually behave like bosons. Similarly, Leggett and co. found that helium 3 atoms could pair up to form a bosonic composite and super-flow.

Over many years, physicists used what they had learnt through these discoveries to expand our understanding of this substance. They found, among other things, that superfluid helium 3 can exist in many phases. The superfluidity would persist in each phase but with different characteristics.

Superfluid helium 3 was first thought to have two phases, called A and B. The temperature-pressure plot below clearly shows the conditions in which each phase emerges.

Credit: E.V. Thuneberg, *Encyclopedia of Condensed Matter Physics*, 2005

When physicists subjected superfluid helium 3 in its A phase to a strong magnetic field, they found another phase that they called A1, whose atom-pairs had different spin characteristics.

In 2015, a group of researchers led by Vladimir Dmitriev, at the P.L. Kapitza Institute for Physical Problems, Moscow, discovered a fourth phase, which they called the polar, or P, phase. Here, they confined helium 3 in a nematic aerogel and exposed the setup to a low magnetic field. Aerogels are ultra-light materials that are extremely porous; nematic means its molecules were arranged in parallel. The aeorogel in the Dmitriev and co. experiment was 98% porous, and whose pores “were much longer than they were wide” (source). That is, the team had found that the shape of the container in which helium 3 was confined also affected the phase of its superfluidity.

In August 2021 (preprint), the same team reported that it had observed a long-expected-to-exist fifth phase called the ß phase.

They reported that they took the setup they used to force superfluid helium 3 into the P phase, but this time exposed it to a high magnetic field. According to their paper, they found that while the superfluid earlier moved into the P phase through a single transition, as the temperature was brought down, this time it did so in two steps. First, it moved into an intermediate phase and then into the P phase. The intermediate is the ß phase.

(If this sounds simple, it wasn’t: the discoveries were each limited by the availability of specially designed instruments capable of picking up on very small-scale changes unavailable to the naked eye. Second, researchers also have had to know in advance what changes they should expect to happen in each phase, and this requires the corresponding theoretical clarity.)

The temperatures at which the phase transition between the two polar phases differ as the magnetic field strength increases. The gap between the two phases is bridged by the ß phase. Source: https://doi.org/10.1103/PhysRevLett.127.265301

I have considerably simplified helium 3’s transition from the ‘normal’ to the superfluid phase in this post. To describe it accurately, physicists use advanced mathematics and associated concepts in high-energy physics. One such concept is symmetry-breaking. When a helium 3 atom pairs up with another to form a bosonic composite, the pair must have a ‘new’ spin and orbital momentum; and their combined wavefunction will also have a ‘new’ phase. All these steps break different symmetries.

There’s a theory called Grand Unification in particle physics, in which physicists expect that at higher and higher energies, the three fundamental forces that affect subatomic particles – the strong-nuclear, the weak-nuclear and the electromagnetic – will combine into a single unified force. Physicists have found in their mathematical calculations that the symmetries that will break in this super-transition resemble those broken by helium 3 during its transition to superfluidity.

Understanding helium 3 can also be rewarding for insights into the insides of neutron stars. Neutron stars are extreme objects – surpassed in their extremeness only by black holes, which exist at the point where known theories of gravitational physics collapse into meaninglessness. A few lakh years after a neutron star is born, it is expected to have cooled sufficiently for its interiors to be composed of superfluids and superconductors.

We may never be able to directly observe these materials in their natural environment. But by studying helium 3’s various phases of superfluidity, we can get a sense of what a neutron star’s innards could be like, and whether their interactions among themselves and the neutrons on the surface could explain these objects’ still-mysterious characteristics.

Featured image: The liquid helium is in the superfluid phase. A thin invisible film creeps up the inside wall of the cup and down on the outside. A drop forms. It will fall off into the liquid helium below. This will repeat until the cup is empty – provided the liquid remains superfluid. Caption and credit: Alfred Leitner, public domain.

The awesome limits of superconductors

On June 24, a press release from CERN said that scientists and engineers working on upgrading the Large Hadron Collider (LHC) had “built and operated … the most powerful electrical transmission line … to date”. The transmission line consisted of four cables – two capable of transporting 20 kA of current and two, 7 kA.

The ‘A’ here stands for ‘ampere’, the SI unit of electric current. Twenty kilo-amperes is an extraordinary amount of current, nearly equal to the amount in a single lightning strike.

In the particulate sense: one ampere is the flow of one coulomb per second. One coulomb is equal to around 6.24 quintillion elementary charges, where each elementary charge is the charge of a single proton or electron (with opposite signs). So a cable capable of carrying a current of 20 kA can essentially transport 124.8 sextillion electrons per second.

According to the CERN press release (emphasis added):

The line is composed of cables made of magnesium diboride (MgB2), which is a superconductor and therefore presents no resistance to the flow of the current and can transmit much higher intensities than traditional non-superconducting cables. On this occasion, the line transmitted an intensity 25 times greater than could have been achieved with copper cables of a similar diameter. Magnesium diboride has the added benefit that it can be used at 25 kelvins (-248 °C), a higher temperature than is needed for conventional superconductors. This superconductor is more stable and requires less cryogenic power. The superconducting cables that make up the innovative line are inserted into a flexible cryostat, in which helium gas circulates.

The part in bold could have been more explicit and noted that superconductors, including magnesium diboride, can’t carry an arbitrarily higher amount of current than non-superconducting conductors. There is actually a limit for the same reason why there is a limit to the current-carrying capacity of a normal conductor.

This explanation wouldn’t change the impressiveness of this feat and could even interfere with readers’ impression of the most important details, so I can see why the person who drafted the statement left it out. Instead, I’ll take this matter up here.

An electric current is generated between two points when electrons move from one point to the other. The direction of current is opposite to the direction of the electrons’ movement. A metal that conducts electricity does so because its constituent atoms have one or more valence electrons that can flow throughout the metal. So if a voltage arises between two ends of the metal, the electrons can respond by flowing around, birthing an electric current.

This flow isn’t perfect, however. Sometimes, a valence electron can bump into atomic nuclei, impurities – atoms of other elements in the metallic lattice – or be thrown off course by vibrations in the lattice of atoms, produced by heat. Such disruptions across the metal collectively give rise to the metal’s resistance. And the more resistance there is, the less current the metal can carry.

These disruptions often heat the metal as well. This happens because electrons don’t just flow between the two points across which a voltage is applied. They’re accelerated. So as they’re speeding along and suddenly bump into an impurity, they’re scattered into random directions. Their kinetic energy then no longer contributes to the electric energy of the metal and instead manifests as thermal energy – or heat.

If the electrons bump into nuclei, they could impart some of their kinetic energy to the nuclei, causing the latter to vibrate more, which in turn means they heat up as well.

Copper and silver have high conductance because they have more valence electrons available to conduct electricity and these electrons are scattered to a lesser extent than in other metals. Therefore, these two also don’t heat up as quickly as other metals might, allowing them to transport a higher current for longer. Copper in particular has a higher mean free path: the average distance an electron travels before being scattered.

In superconductors, the picture is quite different because quantum physics assumes a more prominent role. There are different types of superconductors according to the theories used to understand how they conduct electricity with zero resistance and how they behave in different external conditions. The electrical behaviour of magnesium diboride, the material used to transport the 20 kA current, is described by Bardeen-Cooper-Schrieffer (BCS) theory.

According to this theory, when certain materials are cooled below a certain temperature, the residual vibrations of their atomic lattice encourages their valence electrons to overcome their mutual repulsion and become correlated, especially in terms of their movement. That is, the electrons pair up.

While individual electrons belong to a class of particles called fermions, these electron pairs – a.k.a. Cooper pairs – belong to another class called bosons. One difference between these two classes is that bosons don’t obey Pauli’s exclusion principle: that no two fermions in the same quantum system (like an atom) can have the same set of quantum numbers at the same time.

As a result, all the electron pairs in the material are now free to occupy the same quantum state – which they will when the material is supercooled. When they do, the pairs collectively make up an exotic state of matter called a Bose-Einstein condensate: the electron pairs now flow through the material as if they were one cohesive liquid.

In this state, even if one pair gets scattered by an impurity, the current doesn’t experience resistance because the condensate’s overall flow isn’t affected. In fact, given that breaking up one pair will cause all other pairs to break up as well, the energy required to break up one pair is roughly equal to the energy required to break up all pairs. This feature affords the condensate a measure of robustness.

But while current can keep flowing through a BCS superconductor with zero resistance, the superconducting state itself doesn’t have infinite persistence. It can break if it stops being cooled below a specific temperature, called the critical temperature; if the material is too impure, contributing to a sufficient number of collisions to ‘kick’ all electrons pairs out of their condensate reverie; or if the current density crosses a particular threshold.

At the LHC, the magnesium diboride cables will be wrapped around electromagnets. When a large current flows through the cables, the electromagnets will produce a magnetic field. The LHC uses a circular arrangement of such magnetic fields to bend the beam of protons it will accelerate into a circular path. The more powerful the magnetic field, the more accurate the bending. The current operational field strength is 8.36 tesla, about 128,000-times more powerful than Earth’s magnetic field. The cables will be insulated but they will still be exposed to a large magnetic field.

Type I superconductors completely expel an external magnetic field when they transition to their superconducting state. That is, the magnetic field can’t penetrate the material’s surface and enter the bulk. Type II superconductors are slightly more complicated. Below one critical temperature and one critical magnetic field strength, they behave like type I superconductors. Below the same temperature but a slightly stronger magnetic field, they are superconducting and allow the fields to penetrate their bulk to a certain extent. This is called the mixed state.

A hand-drawn phase diagram showing the conditions in which a mixed-state type II superconductor exists. Credit: Frederic Bouquet/Wikimedia Commons, CC BY-SA 3.0

Say a uniform magnetic field is applied over a mixed-state superconductor. The field will plunge into the material’s bulk in the form of vortices. All these vortices will have the same magnetic flux – the number of magnetic field lines per unit area – and will repel each other, settling down in a triangular pattern equidistant from each other.

An annotated image of vortices in a type II superconductor. The scale is specified at the bottom right. Source: A set of slides entitled ‘Superconductors and Vortices at Radio Frequency Magnetic Fields’ by Ernst Helmut Brandt, Max Planck Institute for Metals Research, October 2010.

When an electric current passes through this material, the vortices are slightly displaced, and also begin to experience a force proportional to how closely they’re packed together and their pattern of displacement. As a result, to quote from this technical (yet lucid) paper by Praveen Chaddah:

This force on each vortex … will cause the vortices to move. The vortex motion produces an electric field¹ parallel to [the direction of the existing current], thus causing a resistance, and this is called the flux-flow resistance. The resistance is much smaller than the normal state resistance, but the material no longer [has] infinite conductivity.

1. According to Maxwell’s equations of electromagnetism, a changing magnetic field produces an electric field.

Since the vortices’ displacement depends on the current density: the greater the number of electrons being transported, the more flux-flow resistance there is. So the magnesium diboride cables can’t simply carry more and more current. At some point, setting aside other sources of resistance, the flux-flow resistance itself will damage the cable.

There are ways to minimise this resistance. For example, the material can be doped with impurities that will ‘pin’ the vortices to fixed locations and prevent them from moving around. However, optimising these solutions for a given magnetic field and other conditions involves complex calculations that we don’t need to get into.

The point is that superconductors have their limits too. And knowing these limits could improve our appreciation for the feats of physics and engineering that underlie achievements like cables being able to transport 124.8 sextillion electrons per second with zero resistance. In fact, according to the CERN press release,

The [line] that is currently being tested is the forerunner of the final version that will be installed in the accelerator. It is composed of 19 cables that supply the various magnet circuits and could transmit intensities of up to 120 kA!

While writing this post, I was frequently tempted to quote from Lisa Randall‘s excellent book-length introduction to the LHC, Knocking on Heaven’s Door (2011). Here’s a short excerpt:

One of the most impressive objects I saw when I visited CERN was a prototype of LHC’s gigantic cylindrical dipole magnets. Event with 1,232 such magnets, each of them is an impressive 15 metres long and weighs 30 tonnes. … Each of these magnets cost EUR 700,000, making the ned cost of the LHC magnets alone more than a billion dollars.
The narrow pipes that hold the proton beams extend inside the dipoles, which are strung together end to end so that they wind through the extent of the LHC tunnel’s interior. They produce a magnetic field that can be as strong as 8.3 tesla, about a thousand times the field of the average refrigerator magnet. As the energy of the proton beams increases from 450 GeV to 7 TeV, the magnetic field increases from 0.54 to 8.3 teslas, in order to keep guiding the increasingly energetic protons around.
The field these magnets produce is so enormous that it would displace the magnets themselves if no restraints were in place. This force is alleviated through the geometry of the coils, but the magnets are ultimately kept in place through specially constructed collars made of four-centimetre thick steel.
… Each LHC dipole contains coils of niobium-titanium superconducting cables, each of which contains stranded filaments a mere six microns thick – much smaller than a human hair. The LHC contains 1,200 tonnes of these remarkable filaments. If you unwrapped them, they would be long enough to encircle the orbit of Mars.
When operating, the dipoles need to be extremely cold, since they work only when the temperature is sufficiently low. The superconducting wires are maintained at 1.9 degrees above absolute zero … This temperature is even lower than the 2.7-degree cosmic microwave background radiation in outer space. The LHC tunnel houses the coldest extended region in the universe – at least that we know of. The magnets are known as cryodipoles to take into account their special refrigerated nature.
In addition to the impressive filament technology used for the magnets, the refrigeration (cryogenic) system is also an imposing accomplishment meriting its own superlatives. The system is in fact the world’s largest. Flowing helium maintains the extremely low temperature. A casing of approximately 97 metric tonnes of liquid helium surrounds the magnets to cool the cables. It is not ordinary helium gas, but helium with the necessary pressure to keep it in a superfluid phase. Superfluid helium is not subject to the viscosity of ordinary materials, so it can dissipate any heat produced in the dipole system with great efficiency: 10,000 metric tonnes of liquid nitrogen are first cooled, and this in turn cools the 130 metric tonnes of helium that circulate in the dipoles.

Featured image: A view of the experimental MgB2 transmission line at the LHC. Credit: CERN.

Superconductivity: From Feshbach to Fermi

(This post is continued from this one.)

After a bit of searching on Wikipedia, I found that the fundamental philosophical underpinnings of superconductivity were to be found in a statistical concept called the Feshbach resonance. If I had to teach superconductivity to those who only knew of the phenomenon superfluously, that’s where I’d begin. So.

Imagine a group of students who have gathered in a room to study together for a paper the next day. Usually, there is that one guy among them who will be hell-bent on gossiping more than studying, affecting the performance of the rest of the group. In fact, given sufficient time, the entire group’s interest will gradually shift in the direction of the gossip and away from its syllabus. The way to get the entire group back on track is to introduce a Feshbach resonance: cut the bond between the group’s interest and the entity causing the disruption. If done properly, the group will turn coherent in its interest and to focusing on studying for the paper.

In multi-body systems, such as a conductor harboring electrons, the presence of a Feshbach resonance renders an internal degree of freedom independent of those coordinates “along” which dissociation is most like to occur. And in a superconductor, a Feshbach resonance results in each electron pairing up with another (i.e., electron-vibrations are quelled by eliminating thermal excitation) owing to both being influenced by an attractive potential that arises out of the electron’s interaction with the vibrating lattice.

Feshbach resonance & BCS theory

For particulate considerations, the lattice-vibrations are quantized in the form of hypothetical particles called phonons. As for why the Feshbach resonance must occur the way it does in a superconductor: that is the conclusion, rather implication, of the BCS theory formulated in 1957 by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer.

The BCS theory essentially treats electrons like rebellious, teenage kids (I must be getting old). As negatively charged electrons pass through the crystal lattice, they draw the positively charged nuclei toward themselves, creating an increase in the positive charge density in their vicinity that attracts more electrons in turn. The resulting electrostatic pull is stronger near nuclei and very weak at larger distances. The BCS theory states that two electrons that would otherwise repel each other will pair up in the face of such a unifying electrostatic potential, howsoever weak it is.

This is something like rebellious teens who, in the face of a common enemy, will unite with each other no matter what the differences between them earlier were.

Since electrons are fermions, they bow down to Pauli’s exclusion principle, which states that no two fermions may occupy the same quantum state. As each quantum state is defined by some specific combination of state variables called quantum numbers, at least one quantum number must differ between the two co-paired electrons.

prof_wolfgang_pauli — Prof. Wolfgang Pauli (1900-1958)

In the case of superconductors, this is particle spin: the electrons in the member-pair will have opposite spins. Further, once such unions have been achieved between different pairs of electrons, each pair becomes indistinguishable from the other, even in principle. Imagine: they are all electron-pairs with two opposing spins but with the same values for all other quantum numbers. Each pair, called a Cooper pair, is just the same as the next!

Bose-Einstein condensates

This unification results in the sea of electrons displaying many properties normally associated with Bose-Einstein condensates (BECs). In a BEC, the particles that attain the state of indistinguishability are bosons (particles with integer spin), not fermions (particles with half-integer spin). The phenomenon occurs at temperatures close to absolute zero and in the presence of an external confining potential, such as an electric field.

Since bosons don’t follow Pauli’s exclusion principle, a major fraction of the indistinguishable entities in the condensate may and do occupy the same quantum state. This causes quantum mechanical effects to become apparent on a macroscopic scale.

By extension, the formulation and conclusions of the BCS theory, alongside its success in supporting associated phenomena, imply that superconductivity may be a quantum phenomenon manifesting in a macroscopic scale.

Note: If even one Cooper pair is “broken”, the superconducting state will be lost as the passage of electric current will be disrupted, and the condensate will dissolve into individual electrons, which means the energy required to break one Cooper pair is the same as the energy required to break the composition of the condensate. So thermal vibrations of the crystal lattice, usually weak, become insufficient to interrupt the flow of Cooper pairs, which is the flow of electrons.

The Meissner effect

In this context, the Meissner effect is simply an extrapolation of Lenz’s law but with zero electrical resistance.

Lenz’s law states that the electromotive force (EMF) because of a current in a conductor acts in a direction that always resists a change in the magnetic flux that causes the EMF. In the absence of resistance, the magnetic fields due to electric currents at the surface of a superconductor cancel all magnetic fields inside the bulk of the material, effectively pushing magnetic field lines of an external magnetic potential outward. However, the Meissner effect manifests only when the externally applied field is weaker than a certain critical threshold: if it is stronger, then the superconductor returns to its conducting state.

Now, there are a class of materials called Type II superconductors – as opposed to the Type I class described earlier – that only push some of the magnetic field outward, the rest remaining conserved inside the material in filaments while being surrounded by supercurrents. This state is called the vortex state, and its occurrence means the material can withstand much stronger magnetic fields and continue to remain superconducting while also exhibiting the hybrid Meissner effect.

Temperature & superconductivity

There are also a host of other effects that only superconductors can exhibit, including Cooper-pair tunneling, flux quantization, and the isotope effect, and it was by studying them that a strong relationship was observed between temperature and superconductivity in various forms.

Bardeen_Cooper_Schrieffer — (L to R) John Bardeen, Leon Cooper, and John Schrieffer

In fact, Bardeen, Cooper, and Schrieffer hit upon their eponymous theory after observing a band gap in the electronic spectra of superconductors. The electrons in any conductor can exist at specific energies, each well-defined. Electrons above a certain energy, usually in the valence band, become free to pass through the entire material instead of staying in motion around the nuclei, and are responsible for conduction.

The trio observed that upon cooling the material to closer and closer to absolute zero, there was a curious gap in the energies at which electrons could be found in the material at a particular temperature. This meant that, at that temperature, the electrons were jumping from existing at one energy to existing at some other lower energy. The observation indicated that some form of condensation was occurring. However, a BEC was ruled out because of Pauli’s exclusion principle. At the same time, a BEC-like state had to have been achieved by the electrons.

This temperature is called the transition temperature, and is the temperature below which a conductor transitions into its superconducting state, and Cooper pairs form, leading to the drop in the energy of each electron. Also, the differences in various properties of the material on either side of this threshold are also attributed to this temperature, including an important notion called the Fermi energy: it is the potential energy that any system possesses when all its thermal energy has been removed from it. This is a significant idea because it defines both the kind and amount of energy that a superconductor has to offer for an externally applied electric current.

enrico_fermi — Enrico Fermi, along with Paul Dirac, defined the Fermi-Dirac statistics that governs the behavior all identical particles that obey Pauli’s exclusion principle (i.e., fermions). Fermi level and Fermi energy are concepts named for him; however, as long as we’re discussing eponymy, Fermilab overshadows them all.

In simple terms, the density of various energy states of the electrons at the Fermi energy of a given material dictates the “breadth” of the band gap if the electron-phonon interaction energy were to be held fixed at some value: a direct proportionality. Thus, the value of the energy gap at absolute zero should be a fixed multiple of the value of the energy gap at the superconducting transition temperature (the multiplication factor was found to be 3.5 universally, irrespective of the material).

Similarly, because of the suppression of thermal excitation (because of the low temperature), the heat capacity of the material reduces drastically at low temperatures, and vanishes below the transition temperature. However, just before hitting zero at the threshold, the heat capacity balloons up to beyond its original value, and then pops. It was found that the ballooned value was always 2.5 times the material’s normal heat capacity value… again, universally, irrespective of the material!

The temperature-dependence of superconductors gains further importance with respect to applications and industrial deployment in the context of its possible occurring at higher temperatures. The low temperatures currently necessary eliminate thermal excitations, in the form of vibrations, of nuclei and almost entirely counter the possibility of electrons, or Cooper pairs, colliding into them.The low temperatures also assist in the flow of Cooper pairs as a superfluid apart from allowing for the energy of the superfluid being higher than the phononic energy of the lattice.

However, to achieve all these states in order to turn a conductor into a superconductor at a higher temperature, a more definitive theory of superconductivity is required. One that allows for the conception of superconductivity that requires only certain internal conditions to prevail while the ambient temperature soars. The 1986-discovery of high-temperature superconductors in ceramics by Bednorz and Muller was the turning point. It started to displace the BCS theory which, physicists realized, doesn’t contain the necessary mechanisms for superconductivity to manifest itself in ceramics – insulators at room temperature – at temperatures as high as 125 K.

A firmer description of superconductivity, therefore, still remains elusive. Its construction should not only pave the for one of the few phenomena that hardly appears in nature and natural processes to be fully understood, but also for its substitution against standard conductors that are responsible for lossy transmission and other such undesirable effects. After all, superconductors are the creation of humankind, and only by its hand while they ever be fully worked.

Stages of superconductivity: Researchers’ insight on ‘pseudogap’ an important advance (phys.org)

Getting started on superconductivity

After the hoopla surrounding and attention on particle physics subsided, I realized that I’d been riding a speeding wagon all the time. All I’d done is used the lead-up to (the search for the Higgs boson) and the climax itself to teach myself something. Now, it’s left me really excited! Learning about particle physics, I’ve come to understand, is not a single-track course: all the way from making theoretical predictions to having them experimentally verified, particle physics is an amalgamation of far-reaching advancements in a host of other subjects.

One such is superconductivity. Philosophically, it’s a state of existence so far removed from its naturally occurring one that it’s a veritable “freak”. It is common knowledge that everything that’s naturally occurring is equipped to resist change that energizes, to return whenever possible to a state of lower energy. Symmetry and surface tension are great examples of this tendency. Superconductivity, on the other hand, is the desistence of a system to resist the passage of an electric current through it. As a phenomenon that as yet doesn’t manifest in naturally occurring substances, I can’t really opine on its phenomenological “naturalness”.

In particle physics, superconductivity plays a significant role in building powerful particle accelerators. In the presence of a magnetic field, a charged particle moves in a curved trajectory through it because of the Lorentz force acting on it; this fact is used to guide the protons in the Large Hadron Collider (LHC) at CERN through a ring 27 km long. Because moving in a curved path involves acceleration, each “swing” around the ring happens faster than the last, eventually resulting in the particle traveling at close to the speed of light.

superconducting_quardupole_electromagnetic_LHC — A set of superconducting quadrupole-electromagnets installed at the LHC with the cryogenic cooling system visible in the background

In order to generate these extremely powerful magnetic fields – powerful because of the minuteness of each charge and the velocity required to be achieved – superconducting magnets are used that generate fields of the order of 20 T (to compare: the earth’s magnetic field is 25-60 μT, or close to 500,000-times weaker)! Furthermore, the direction of the magnetic field is also switched accordingly to achieve circular motion, to keep the particle from being swung off into the inner wall of the collider at any point!

To understand the role the phenomenon of superconductivity plays in building these magnets, let’s understand how electromagnets work. In a standard iron-core electromagnet, insulated wire is wound around an iron cylinder, and when a current is passed through the wire, a magnetic field is generated around the cross-section of the wire. Because of the coiling, though, the centre of the magnetic field passes through the axis of the cylinder, whose magnetic permeability magnifies the field by a factor of thousands, itself becoming magnetic.

When the current is turned off, the magnetic field instantaneously disappears. When the number of coils is increased, the strength of the magnetic field increases. When the strength of the current is increased, the strength of the magnetic field increases. However, beyond a point, the heat dissipated due to the wire’s electric resistance reduces the amount of current flowing through it, consequently resulting in a weakening of the core’s magnetic field over time.

It is Ohm’s law that establishes proportionality between voltage (V) and electric current (I), calling the proportionality-constant the material’s electrical resistance: R = V/I. To overcome heating due to resistance, resistance itself must be brought down to zero. According to Ohm’s law, this can be done either by passing a ridiculously large current through the wire or bringing the voltage across its ends down to zero. However, performing either of these changes on conventional conductors is impossible: how does one quickly pass a large volume of water through any pipe across which the pressure difference is miniscule?!

The solution to this unique problem, therefore, lay in a new class of materials that humankind had to prepare, a class of materials that could “instigate” an alternate form of electrical conduction such that an electrical current could pass through it in the absence of a voltage difference. In other words, the material should be able to carry large amounts of current without offering up any resistance to it. This class of materials came to be known as superconductors – after Heike Kamerlingh Onnes discovered the phenomenon in 1911.

In a conducting material, the electrons that essentially effect the flow of electric current could be thought of as a charged fluid flowing through and around an ionic 3D grid, an arrangement of positively charged nuclei that all together make up the crystal lattice. When a voltage-drop is established, the fluid begins to get excited and moves around, an action called conducting. However, the electrons constantly collide with the ions. The ions, then, absorb some of the energy of the current, start vibrating, and gradually dissipate it as heat. This manifests as the resistance. In a superconductor, however, the fluid exists as a superfluid, and flows such that the electrons never collide into the ions.

In (a classical understanding of) the superfluid state, each electron repels every other electron because of their charge likeness, and attracts the positively charged nuclei. As a result, the nucleus moves very slightly toward the electron, causing an equally slight distortion of the crystal lattice. Because of the newly increased positive-charge density in the vicinity, some more electrons are attracted by the nucleus.

This attraction, which, across the entirety of the lattice, can cause a long-range but weak “draw” of electrons, results in pairs of electrons overcoming their mutual hatred of each other and tending toward one nucleus (or the resultant charge-centre of some nuclei). Effectively, this is a pairing of electrons whose total energy was shown by Leon Cooper in 1956 to be lesser than the energy of the most energetic electron if it had existed unpaired in the material. Subsequently, these pairs came to be called Cooper pairs, and a fluid composed of Cooper pairs, a superfluid (thermodynamically, a superfluid is defined as a fluid that can flow without dissipating any energy).

Although the sea of electrons in the new superconducting class of materials could condense into a superfluid, the fluid itself can’t be expected to flow naturally. Earlier, the application of an electric current imparted enough energy to all the electrons in the metal (via a voltage difference) to move around and to scatter against nuclei to yield resistance. Now, however, upon Cooper-pairing, the superfluid had to be given an environment in which there’d be no vibrating nuclei. And so: enter cryogenics.

ILCTA_cryo — The International Linear Collider – Test Area’s (ILCTA) cryogenic refrigerator room

The thermal energy of a crystal lattice is given by E = kT, where ‘k’ is Boltzmann’s constant and T, the temperature. Demonstrably, to reduce the kinetic energy of all nuclei in the lattice to zero, the crystal itself had to be cooled to absolute zero (0 kelvin). This could be achieved by cryogenic cooling techniques. For instance, at the LHC, the superconducting magnets are electromagnets wherein the coiled wire is made of a superconducting material. When cooled to a really low temperature using a two-stage heat-exchanger composed of liquid helium jacketed with liquid nitrogen, the wires can carry extremely large amounts of current to generate very intense magnetic fields.

At the same time, however, if the energy of the superfluid itself surpassed the thermal energy of the lattice, then it could flow without the lattice having to be cooled down. Because the thermal energy is different for different crystals at different ambient temperatures, the challenge now lies in identifying materials that could permit superconductivity at temperatures approaching room-temperature. Now that would be (even more) exciting!

Graphene the Ubiquitous (astrohep.wordpress.com)

P.S. A lot of the related topics have not been covered in this post, such as the Meissner effect, electron-phonon interactions, properties of cuprates and lanthanides, and Mott insulators. They will be taken up in the future as they’re topics that require in-depth detailing, quite unlike this post which has been constructed as a superfluous introduction only.

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