Meissner effect – Close Read

At last, physicists report finding the ‘fourth sign’ of superconductivity

Using an advanced investigative technique, researchers at Stanford University have found that cuprate superconductors – which become superconducting at higher temperatures than their better-known conventional counterparts – transition into this exotic state in a different way. The discovery provides new insights into the way cuprate superconductors work and eases the path to discovering a room-temperature superconductor one day.

A superconductor is a material that can transport an electric current with zero resistance. The most well-known and also better understood superconductors are certain metallic alloys. They transition from their ‘normal’ resistive state to the superconducting state when their temperature is brought to a very low value, typically a few degrees above absolute zero.

The theory that explains the microscopic changes that occur as the material transitions is called Bardeen-Cooper-Schrieffer (BCS) theory. As the material crosses its threshold temperature, called the critical temperature, BCS theory predicts four signatures of superconductivity. If these four signatures occur, we can be sure that the material has become superconducting.

First, the material’s resistivity collapses and its electrons begin to flow without any resistance through the bulk – the electronic effect.

Second, the material expels all magnetic fields within its bulk – the magnetic (a.k.a. Meissner) effect.

A magnet levitating above a high-temperature superconductor, thanks to the Meissner effect. Credit: Mai-Linh Doan/Wikimedia Commons, CC BY-SA 3.0

Third, the amount of heat required to excite electrons to an arbitrarily higher energy is called the electronic specific heat. This number is lower for superconducting electrons than for non-superconducting electrons – but it increases as the material is warmed, only to drop abruptly to the non-superconducting value at the critical temperature. This is the effect on the material’s thermodynamic behaviour.

Fourth, while the energies of the electrons in the non-superconducting state have a variety of values, in the superconducting state some energy levels become unattainable. This shows up as a gap in a chart mapping the energy values. This is the spectroscopic effect. (The prefix ‘spectro-‘ refers to anything that can assume a continuous series of values, on a spectrum.)

Conventional superconductors are called so simply because scientists discovered them first and they defined the convention: among other things, they transition from their non-superconducting to superconducting states at very low temperature. Their unconventional counterparts are the high-temperature superconductors, which were discovered in the late 1980s and which transition at temperatures greater than 77 K. And when they do, physicists have thus far observed the corresponding electronic, magnetic and thermodynamic effects – but not the spectroscopic one.

A new study, published on January 26, 2022, has offered to complete this record. And in so doing, the researchers have uncovered new information about how these materials transition into their superconducting states: it is not the way low-temperature superconductors do.

The research team, at Stanford, reportedly did this by studying the thermodynamic effect and connecting it to the material’s spectroscopic effect.

The deeper problem with zeroing in on the spectroscopic effect in high-temperature superconductors is that an electron energy gap shows up before the transition, when the material is not yet a superconductor, and persists into the superconducting phase.

First, recall that at the critical temperature, the electronic specific heat stops increasing and drops suddenly to the non-superconducting value. The specific heat is directly related to the amount of entropy in the system (energy in the system that can’t be harnessed to perform work). The entropy is in turn related to the spectral function – an equation that dictates which energy states the electrons can and can’t occupy. So by studying changes in the specific heat, the researchers can understand the spectroscopic effect.

Second, to study the specific heat, the researchers used a technique called angle-resolved photo-emission spectroscopy (ARPES). These are big words but they have a simple meaning. Photo-emission spectroscopy refers to a technique in which energy-loaded photons are shot into a target material, where they knock out those electrons that they have the energy for. Based on the energies of the electrons knocked out, their position and their momenta, scientists can piece together the properties of the electrons inside the material.

ARPES takes this a step further by also recording the angle at which the electrons are knocked out of the material. This provides an insight into another property of the superconductor. Specifically, another way in which cuprates differ from conventional superconductors is the way in which the electrons pair up. In the latter, the pairs break rotational symmetry, such that the energy required to break up the pair is not equal in all directions.

This affects the way the thermodynamic and spectral effects look in the data. For example, photons fired at certain angles will knock out more electrons from the material than photons incoming at other angles.

The angle-specific measurements of the specific-heat coefficient (y-axis) versus the temperature (x-axis). Credit: https://doi.org/10.1038/s41586-021-04251-2

Taking all this into account, the researchers reported that a cuprate superconductor called Bi-2212 (bismuth strontium calcium copper oxide) transitions to becoming a superconductor in two steps – unlike the single-step transition of low-temperature superconductors.

According to BCS theory, the electrons in a conventional superconductor are encouraged to overcome their mutual repulsion and bind to each other in pairs when two conditions are met: the material’s lattice – the grid of atomic nuclei – has a vibrational energy of a certain frequency and the material’s temperature is lowered. These electron pairs then move around the material like a fluid of zero viscosity, thus giving rise to superconductivity.

The Stanford team found that in Bi-2212, the electrons pair up with each other at around 120 K, but condense into the fluid-like state only at around 77 K. The former gives rise to an energy gap – i.e. the spectroscopic effect – even as the superconducting behaviour itself arises only at the 77-K mark, when the pairs condense.

A small sample of Bi-2212 The side is 1 mm long. Credit: James Slezak, Cornell Laboratory of Atomic and Solid State Physics, CC BY-SA 3.0

There are two distinct feats here: finding the spectroscopic effect and finding the two-step transition. Both – but the first more so – were the product of technological advancements. The researchers obtained their Bi-2212 samples, created with specific chemical compositions so as to help analyse the ARPES data, from their collaborators in Japan, and then studied it with two instruments capable of performing ARPES studies at Stanford: an ultraviolet laser and the Synchrotron Radiation Lightsource.

Makoto Hashimoto, a physicist at Stanford and one of the study’s authors, said in a press statement: “Recent improvements in the overall performance of those instruments were an important factor in obtaining these high-quality results. They allowed us to measure the energy of the ejected electrons with more precision, stability and consistency.”

The second finding, of the two-step transition, is important foremost because it is new knowledge of the way cuprate superconductors ‘work’ and because it tells physicists that they will have to achieve two things – instead of just one, as in the case of conventional, low-temperature superconductors – if they want to recreate the same effects in a different material.

As Zhi-Xun Shen, the researcher who led the study at Stanford, told Physics World, “This knowledge will ultimately help us make better superconductors in the future.”

Featured image: A schematic illustration of an ARPES setup. On the left is the head-on view of the manipulator holding the sample and at the centre is the side-on view. On the right is an electron energy analyser. Credit: Ponor/Wikimedia Commons, CC BY-SA 4.0.

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.

A stinky superconductor

The next time you smell a whiff of rot in your morning’s eggs, you might not want to throw them away. Instead, you might do better to realise what you’re smelling could be a superconductor (under the right conditions) that’s, incidentally, riled up the scientific community.

The source of excitement is a paper published in Nature on August 17, penned by a group of German scientists, describing an experiment in which the compound hydrogen sulphide conducts electricity with zero resistance under a pressure of 90 gigapascals (about 888,231-times the atmospheric pressure) – when it turns into a metal – and at a temperature of 203.5 kelvin, about -70.5° C. The discovery makes it an unexpected high-temperature superconductor, doubly so for becoming one under conditions physicists don’t find too esoteric.

The tag of ‘high-temperature’ may be unfit for something operating at -70.5° C, but in superconductivity, -70.5° C approaches summer in the Atacama. When the phenomenon was first discovered – by the Dutch physicist Heike Kamerlingh Onnes in 1911 – it required the liquid metal mercury to be cooled to 4.2 kelvin, about -269° C. What happened in those conditions was explained by an American trio with a theory of superconductivity in 1957.

The explanation lies in quantum mechanics, where all particles have a characteristic ‘spin’ number. And QM allows all those particles with integer spin (0, 1, 2, …) to – in some conditions – cohere into one bigger ‘particle’ with enough energy of itself to avoid being disturbed by things like friction or atomic vibrations*. Electrons, however, have half-integer (1/2) spin, so can’t slip into this state. In 1957, John Bardeen, Leon Cooper and Robert Schrieffer proposed that at very low temperatures – like 4 K – the electrons in a metal interact with the positively charged latticework of atoms around them to pair up with each other. These electronic pairs are called Cooper pairs, kept twinned by vibrations of the lattice. The pair’s total spin is 1, allowing all of them to condense into one cohesive sea of electrons that then flows through the metal unhindered.

The BCS theory soon became a ‘conventional’ theory of superconductivity, able to explain the behaviour of many metals cooled to cryogenic temperatures. The German team’s hydrogen sulphide system is also one such conventional scenario – in which the gas had to compressed to form a metal before its superconducting abilities were teased out.

The team, led by Mikhail Eremets and Alexander Drozdov from the Max Planck Institute for Chemistry in Mainz, first made its claims last year, that under heavy pressure hydrogen sulphide becomes sulphur hydride (H₂S → H₃S), which in turn is a superconductor. At the time their experiment showed only one of two typical properties of a superconducting system, however: that its electrical resistance vanished at 190 K, higher than the previous record of 164 K.

Their August 17 paper reports that the second property has since been observed, too: that pressurised hydrogen sulphide doesn’t allow any external magnetic field to penetrate beyond its surface. This effect, called the Meissner effect, is observed only in superconductors. For Eremets, Drozdov et al, this is the full monty: a superconductor functioning at temperatures that actually exist on Earth. But for the broader scientific community, the paper marks the frenzied beginning of a new wave of experiments in the field.

Given the profundity of the findings – of a hydrogen-based high-temperature superconductor – they won’t enter the canon just yet but will require independent verification from other teams. A report by Edwin Cartlidge in Nature already notes five other teams around the world working on replicating the discovery. If and when they succeed, the implications will be wide-ranging – for physics as well as historical traditions of physical chemistry.

The BCS theory of superconductivity provided a precise mechanism of action that allowed scientists to predict the critical temperature (T_c) – below which a material becomes superconducting – of all materials that abided by the theory. Nonetheless, by 1957, the highest T_c reached had been 10 K despite scientists’ best efforts; so great was their frustration that in 1972, Philip Warren Anderson and Marvin Cohen predicted that there could be a natural limit at 30 K.

However, just a few years earlier – in 1968 – two physicists, Neil Ashcroft and Vitaly Ginzburg, refusing to subscribe to a natural limit on the critical temperature, proposed that the T_c could be very high in substances in which the vibrations of the atomic latticework surrounding the electrons was pretty energetic. Such vigour is typically found in the lighter elements like hydrogen and helium. Thus, the Ashcroft-Ginzburg idea effectively set the theoretical precedent for Eremets and Drozdov’s work.

But between the late 1960s and 2014, when hydrogen sulphide entered the fray of experiments, two discoveries threw the BCS theory off kilter. In 1986, scientists discovered cuprates, a class of copper’s compounds that were superconductors at 133 K (at 164 K under pressure) but didn’t function according to the BCS theory. Thus, they came to be called unconventional superconductors. The second discovery was of another class of unconventional superconductors, this time in compounds of iron and arsenic called pnictides, in 2008. The highest T_c among them was less than that of the cuprates. And because cuprates under pressure could muster a T_c of 164 K, scientists pinned their hopes on them of breaching the room-temperature barrier, and worked on developing an unconventional theory of superconductivity.

But for those choosing to persevere with the conventional order of things, there was a brief flicker of hope in 2001 with the discovery of magnesium diboride superconductors: they had a T_c of 39 K, an important but not very substantial improvement on previous records among conventional materials.

The work of Eremets & Drozdov was also indirectly assisted by a group of Chinese researchers in 2014, who were able to anticipate hydrogen sulphide’s superconducting abilities using the conventional BCS theory. According to them, hydrogen sulphide would become a metal under the application of 111 gigapascals of pressure, with a T_c between 191 K and 204 K. And once it survives independent experimental scrutiny intact, the Chinese theoretical work will prove valuable as scientists confront their next big challenge: pressure.

The ultimate fantasy would be to have a T_c is in the range of ambient temperatures. Imagine leagues of superconducting cables radiating out from coal-choked power plants, a gigawatt of power transmitted for a gigawatt of power produced**, or maglev trains running on superconducting tracks at lower costs and currents, or the thousands of superconducting electromagnets around the LHC that won’t have to be supercooled using jackets of liquid helium. Sadly, that Eremets & Drozdov have (probably) achieved a T_c of 203.5 K doesn’t mean that the engineering is accessible or affordable. In fact, what allowed them to fetch 203.5 K is what the barrier is for the tech to be ubiquitously used, making their feat an antecedence of possibilities rather than a demonstration itself.

It wasn’t possible until the 1970s to achieve pressures of a few gigapascals in the lab, and similar processes today are confined to industrial purposes. A portable device that’d sustain that pressure across large areas is difficult to build – yet that’s when metallic sulphur hydride shows itself. In their experiment, Eremets and Drozdov packed a cold mass of hydrogen sulphide against a stainless steel gasket using some insulating material like teflon, and then sandwiched the pellet between two diamond anvils that pressurised it. The diameter of the entire apparatus was a little more than a 100 micrometers across. Moreover, they also note in their paper that the ‘loading’ of the hydrogen sulphide between the anvils needs to be done at a low temperature – before pressurisation – so that the gas doesn’t decompose before the superconducting can begin.

These are impractical conditions if hydrogen sulphide cables have to be handled by a crew of non-specialists and in conditions nowhere near controllable enough as the insides of a small steel gasket. As an alternative, should independent verification of the Eremets & Drozdov experiment happen, scientists will use it as a validation of the Chinese theorists’ calculations and extend that to fashion a material more suited to their purposes.

*The foundation for this section of QM was laid by Satyendra Nath Bose, and later expanded by Albert Einstein to become the Bose-Einstein statistics.

**But not a gigawatt of power consumed, thanks to power thefts to the tune of Rs.2.52 lakh crore.

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)

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