Tag: Bi-2212

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.

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.