Amorphous topological insulators

Physicists from the Indian Institute of Science, Bengaluru, have found that topological insulators needn’t always be crystals – as has been thought.

In an amorphous material like glass, the constituent atoms are arranged randomly. Credit: Hans/pixabay

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.