A tale of vortices, skyrmions, paths and shapes

There are many types of superconductors. Some of them can be explained by an early theory of superconductivity called Bardeen-Cooper-Schrieffer (BCS) theory.

In these materials, vibrations in the atomic lattice force the electrons in the material to overcome their mutual repulsion and team up in pairs, if the material’s temperature is below a particular threshold (very low). These pairs of electrons, called Cooper pairs, have some properties that individual electrons can’t have. One of them is that all Cooper pairs together form an exotic state of matter called a Bose-Einstein condensate, which can flow through the material with much less resistance than individuals electrons experience. This is the gist of BCS theory.

When the Cooper pairs are involved in the transmission of an electric current through the material, the material is an electrical superconductor.

Some of the properties of the two electrons in each Cooper pair can influence the overall superconductivity itself. One of them is the orbital angular momentum, which is an intrinsic property of all particles. If both electrons have equal orbital angular momentum but are pointing in different directions, the relative orbital angular momentum is 0. Such materials are called s-wave superconductors.

Sometimes, in s-wave superconductors, some of the electric current – or supercurrent – starts flowing in a vortex within the material. If these vortices can be coupled with a magnetic structure called a skyrmion, physicists believe they can give rise to some new behaviour previously not seen in materials, some of them with important applications in quantum computing. Coupling here implies that a change in the properties of the vortex should induce changes in the skyrmion, and vice versa.

However, physicists have had a tough time creating a vortex-skyrmion coupling that they can control. As Gustav Bihlmayer, a staff scientist at the Jülich Research Centre, Germany, wrote for APS Physics, “experimental studies of these systems are still rare. Both parts” of the structures bearing these features “must stay within specific ranges of temperature and magnetic-field strength to realise the desired … phase, and the length scales of skyrmions and vortices must be similar in order to study their coupling.”

In a new paper, a research team from Nanyang Technical University, Singapore, has reported that they have achieved just such a coupling: they created a skyrmion in a chiral magnet and used it to induce the formation of a supercurrent vortex in an s-wave superconductor. In their observations, they found this coupling to be stable and controllable – important attributes to have if the setup is to find practical application.

A chiral magnet is a material whose internal magnetic field “typically” has a spiral or swirling pattern. A supercurrent vortex in an electrical superconductor is analogous to a skyrmion in a chiral magnet; a skyrmion is a “knot of twisting magnetic field lines” (source).

The researchers sandwiched an s-wave superconductor and a chiral magnet together. When the magnetic field of a skyrmion in the chiral magnet interacted with the superconductor at the interface, it induced a spin-polarised supercurrent (i.e. the participating electrons’ spin are aligned along a certain direction). This phenomenon is called the Rashba-Edelstein effect, and it essentially converts electric charge to electron spin and vice versa. To do so, the effect requires the two materials to be in contact and depends among other things on properties of the skyrmion’s magnetic field.

There’s another mechanism of interaction in which the chiral magnet and the superconductor don’t have to be in touch, and which the researchers successfully attempted to recreate. They preferred this mechanism, called stray-field coupling, to demonstrate a skyrmion-vortex system for a variety of practical reasons. For example, the chiral magnet is placed in an external magnetic field during the experiment. Taking the Rashba-Edelstein route means to achieve “stable skyrmions at low temperatures in thin films”, the field needs to be stronger than 1 T. (Earth’s magnetic field measures 25-65 µT.) Such a field could damage the s-wave superconductor.

For the stray-field coupling mechanism, the researchers inserted an insulator between the chiral magnet and the superconductor. Then, when they applied a small magnetic field, Bihlmayer wrote, the field “nucleated” skyrmions in the structure. “Stray magnetic fields from the skyrmions [then] induced vortices in the [superconducting] film, which were observed with scanning tunnelling spectroscopy.”


Experiments like this one reside at the cutting edge of modern condensed-matter physics. A lot of their complexity resides in scientists being able to closely control the conditions in which different quantum effects play out, using similarly advanced tools and techniques to understand what could be going on inside the materials, and to pick the right combination of materials to use.

For example, the heterostructure the physicists used to manifest the stray-field coupling mechanism had the following composition, from top to bottom:

  • Platinum, 2 nm (layer thickness)
  • Niobium, 25 nm
  • Magnesium oxide, 5 nm
  • Platinum, 2 nm

The next four layers are repeated 10 times in this order:

  • Platinum, 1 nm
  • Cobalt, 0.5 nm
  • Iron, 0.5 nm
  • Iridium, 1 nm

Back to the overall stack:

  • Platinum, 10 nm
  • Tantalum, 2 nm
  • Silicon dioxide (substrate)

The first three make up the superconductor, the magnesium oxide is the insulator, and the rest (except the substrate) make up the chiral magnet.

It’s possible to erect a stack like this through trial and error, with no deeper understanding dictating the choice of materials. But when the universe of possibilities – of elements, compounds and alloys, their shapes and dimensions, and ambient conditions in which they interact – is so vast, the exercise could take many decades. But here we are, at a time when scientists have explored various properties of materials and their interactions, and are able to engineer novel behaviours into existence, blurring the line between discovery and invention. Even in the absence of applications, such observations are nothing short of fascinating.

Applications aren’t wanting, however.


quasiparticle is a packet of energy that behaves like a particle in a specific context even though it isn’t actually one. For example, the proton is a quasiparticle because it’s really a clump of smaller particles (quarks and gluons) that together behave in a fixed, predictable way. A phonon is a quasiparticle that represents some vibrational (or sound) energy being transmitted through a material. A magnon is a quasiparticle that represents some magnetic energy being transmitted through a material.

On the other hand, an electron is said to be a particle, not a quasiparticle – as are neutrinos, photons, Higgs bosons, etc.

Now and then physicists abstract packets of energy as particles in order to simplify their calculations.

(Aside: I’m aware of the blurred line between particles and quasiparticles. For a technical but – if you’re prepared to Google a few things – fascinating interview with condensed-matter physicist Vijay Shenoy on this topic, see here.)

We understand how these quasiparticles behave in three-dimensional space – the space we ourselves occupy. Their properties are likely to change if we study them in lower or higher dimensions. (Even if directly studying them in such conditions is hard, we know their behaviour will change because the theory describing their behaviour predicts it.) But there is one quasiparticle that exists in two dimensions, and is quite different in a strange way from the others. They are called anyons.

Say you have two electrons in an atom orbiting the nucleus. If you exchanged their positions with each other, the measurable properties of the atom will stay the same. If you swapped the electrons once more to bring them back to their original positions, the properties will still remain unchanged. However, if you switched the positions of two anyons in a quantum system, something about the system will change. More broadly, if you started with a bunch of anyons in a system and successively exchanged their positions until they had a specific final arrangement, the system’s properties will have changed differently depending on the sequence of exchanges.

This is called path dependency, and anyons are unique in possessing this property. In technical language, anyons are non-Abelian quasiparticles. They’re interesting for many reasons, but one application stands out. Quantum computers are devices that use the quantum mechanical properties of particles, or quasiparticles, to execute logical decisions (the same way ‘classical’ computers use semiconductors). Anyons’ path dependency is useful here. Arranging anyons in one sequence to achieve a final arrangement can be mapped to one piece of information (e.g. 1), and arranging anyons by a different sequence to achieve the same final arrangement can be mapped to different information (e.g. 0). This way, what information can be encoded depends on the availability of different paths to a common final state.

In addition, an important issue with existing quantum computers is that they are too fragile: even a slight interaction with the environment can cause the devices to malfunction. Using anyons for the qubits could overcome this problem because the information stored doesn’t depend on the qubits’ existing states but the paths that they have taken there. So as long as the paths have been executed properly, environmental interactions that may disturb the anyons’ final states won’t matter.

However, creating such anyons isn’t easy.

Now, recall that s-wave superconductors are characterised by the relative orbital angular momentum of electrons in the Cooper pairs being 0 (i.e. equal but in opposite directions). In some other materials, it’s possible that the relative value is 1. These are the p-wave superconductors. And at the centre of a supercurrent vortex in a p-wave superconductor, physicists expect to find non-Abelian anyons.

So the ability to create and manipulate these vortices in superconductors, as well as, more broadly, explore and understand how magnet-superconductor heterostructures work, is bound to be handy.


The Nanyang team’s paper calls the vortices and skyrmions “topological excitations”. An ‘excitation’ here is an accumulation of energy in a system over and above what the system has in its ground state. Ergo, it’s excited. A topological excitation refers to energy manifested in changes to the system’s topology.

On this subject, one of my favourite bits of science is topological phase transitions.

I usually don’t quote from Wikipedia but communicating condensed-matter physics is exacting. According to Wikipedia, “topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending”. For example, no matter how much you squeeze or stretch a donut (without breaking it), it’s going to be a ring with one hole. Going one step further, your coffee mug and a donut are topologically similar: they’re both objects with one hole.

I also don’t like the Nobel Prizes but some of the research that they spotlight is nonetheless awe-inspiring. In 2016, the prize was awarded to Duncan Haldane, John Kosterlitz and David Thouless for “theoretical discoveries of topological phase transitions and topological phases of matter”.

David Thouless in 1995. Credit: Mary Levin/University of Washington

Quoting myself from 2016:

There are four popularly known phases of matter: plasma, gas, liquid and solid. If you cooled plasma, its phase would transit to that of a gas; if you cooled gases, you’d get a liquid; if you cooled liquids, you’d get a solid. If you kept cooling a solid until you were almost at absolute zero, you’d find substances behaving strangely because, suddenly, quantum mechanical effects show up. These phases of matter are broadly called quantum phases. And their phase transitions are different from when plasma becomes a gas, a gas becomes a liquid, and so on.

A Kosterlitz-Thouless transition describes a type of quantum phase transition. A substance in the quantum phase, like all substances, tries to possess as low energy as possible. When it gains some extra energy, it sheds it. And how it sheds it depends on what the laws of physics allow. Kosterlitz and Thouless found that, at times, the surface of a flat quantum phase – like the surface of liquid helium – develops vortices, akin to a flattened tornado. These vortices always formed in pairs, so the surface always had an even number of vortices. And at very low temperatures, the vortices were always tightly coupled: they remained close to each other even when they moved across the surface.

The bigger discovery came next. When Kosterlitz and Thouless raised the temperature of the surface, the vortices moved apart and moved around freely, as if they no longer belonged to each other. In terms of thermodynamics alone, the vortices being alone or together wouldn’t depend on the temperature, so something else was at play. The duo had found a kind of phase transition – because it did involve a change in temperature – that didn’t change the substance itself but only a topological shift in how it behaved. In other words, the substance was able to shed energy by coupling the vortices.

Reality is so wonderfully weird. It’s also curious that some concepts that seemed significant when I was learning science in school (like invention versus discovery) and in college (like particle versus quasiparticle) – concepts that seemed meaningful and necessary to understand what was really going on – don’t really matter in the larger scheme of things.

Amorphous topological insulators

A topological insulator is a material that conducts electricity only on its surface. Everything below, through the bulk of the material, is an insulator. An overly simplified way to understand this is in terms of the energies and momenta of the electrons in the material.

The electrons that an atom can spare to share with other atoms – and so form chemical bonds – are called valence electrons. In a metal, these electrons can have various momenta, but unless they have a sufficient amount of energy, they’re going to stay near their host atoms – i.e. within the valence band. If they do have energies over a certain threshold, then they can graduate from the valence band to the conduction band, flowing throw the metal and conducting electricity.

In a topological insulator, the energy gap between the valence band and the conduction band is occupied by certain ‘states’ that represent the material’s surface. The electrons in these states aren’t part of the valence band but they’re not part of the conduction band either, and can’t flow throw the entire bulk.

The electrons within these states, i.e. on the surface, display a unique property. Their spins (on their own axis) are coupled strongly with their motion around their host atoms. As a result, theirs spins become aligned perpendicularly to their momentum, the direction in which they can carry electric charge. Such coupling staves off an energy-dissipation process called Umklapp scattering, allowing them to conduct electricity. Detailed observations have shown that the spin-momentum coupling necessary to achieve this is present only in a few-nanometre-thick layer on the surface.

If you’re talking about this with a physicist, she will likely tell you at this point about time-reversal symmetry. It is a symmetry of nature that is said to (usually) ‘protect’ a topological insulator’s unique surface states.

There are many fundamental symmetries in nature. In particle physics, if a force acts similarly on left- and right-handed particles, it is said to preserve parity (P) symmetry. If the dynamics of the force are similar when it is acting against positively and negatively charged particles, then charge conjugation (C) symmetry is said to be preserved. Now, if you videotaped the force acting on a particle and then played the recording backwards, the force must be seen to be acting the way it would if the video was played the other way. At least if it did it would be preserving time-reversal (T) symmetry.

Physicists have known some phenomena that break C and P symmetry simultaneously. T symmetry is broken continuously by the second law of thermodynamics: if you videographed the entropy of a universe and then played it backwards, entropy will be seen to be reducing. However, CPT symmetries – all together – cannot be broken (we think).

Anyway, the surface states of a topological insulator are protected by T symmetry. This is because the electrons’ wave-functions, the mathematical equations that describe some of the particles’ properties, do not ‘flip’ going backwards in time. As a result, a topological insulator cannot lose its surface states unless it undergoes some sort of transformation that breaks time-reversal symmetry. (One example of such a transformation is a phase transition.)

This laboured foreword is necessary – at least IMO – to understand what it is that scientists look for when they’re looking for topological insulators among all the materials that we have been, and will be able, to synthesise. It seems they’re looking for materials that have surface states, with spin-momentum coupling, that are protected by T symmetry.


Physicists from the Indian Institute of Science, Bengaluru, have found that topological insulators needn’t always be crystals – as has been thought. Instead, using a computer simulation, Adhip Agarwala and Vijay Shenoy, of the institute’s physics department, have shown that a kind of glass also behaves as a topological insulator.

The band theory described earlier is usually described with crystals in mind, wherein the material’s atoms are arranged in a well-defined pattern. This allows physicists to determine, with some amount of certainty, as to how the atoms’ electrons interact and give rise to the material’s topological states. In an amorphous material like glass, on the other hand, the constituent atoms are arranged randomly. How then can something as well-organised as a surface with spin-momentum coupling be possible on it?

As Michael Schirber wrote in Physics magazine,

In their study, [Agarwala and Shenoy] assume a box with a large number of lattice sites arranged randomly. Each site can host electrons in one of several energy levels, and electrons can hop between neighboring sites. The authors tuned parameters, such as the lattice density and the spacing of energy levels, and found that the modeled materials could exhibit symmetry-protected surface currents in certain cases. The results suggest that topological insulators could be made by creating glasses with strong spin-orbit coupling or by randomly placing atoms of other elements inside a normal insulator.

The duo’s paper was published in the journal Physical Review Letters on June 8. The arXiv preprint is available to read here. The latter concludes,

The possibility of topological phases in a completely random system opens up several avenues both from experimental and theoretical perspectives. Our results suggest some new routes to the laboratory realization of topological phases. First, two dimensional systems can be made by choosing an insulating surface on which suitable [atoms or molecules] with appropriate orbitals are deposited at random (note that this process will require far less control than conventional layered materials). The electronic states of these motifs will then [interact in a certain way] to produce the required topological phase. Second is the possibility of creating three dimensional systems starting from a suitable large band gap trivial insulator. The idea then is to place “impurity atoms”, again with suitable orbitals and “friendly” chemistry with the host… The [interaction] of the impurity orbitals would again produce a topological insulating state in the impurity bands under favourable conditions.

Agarwala/Shenoy also suggest that “In realistic systems the temperature scales over which one will see the topological physics … may be low”, although this is not unusual. However, they don’t suggest which amorphous materials could be suitable topological insulators.

Thanks to penflip.com and its nonexistent autosave function, I had to write the first half of this article twice. Not the sort of thing I can forgive easily, less so since I’m loving everything else about it.