Bismuth has puzzled scientists for nearly 20 years. Notably, the question of whether it is topological – that is, whether electrons behave differently on its surface than they do inside it – gets different answers depending on whether you ask a theorist or an experimentalist. Researchers in Japan now say they have found a way to resolve this conflict. A mechanism called surface relaxation, they report, may have masked or “blocked” bismuth’s true topological nature.

The classic way of describing topology is to compare objects that have a hole, such as a doughnut or a coffee mug, with objects that don’t, such as a muffin. Although we usually think of doughnuts as having more in common with muffins than with mugs – you can’t eat a mug – the fact that they have the same number of holes means the mug and doughnut share topological features that the muffin does not.

While no-one has ever wondered whether they can eat an electron, scientists have long been curious about whether materials conduct electricity. As it turns out, topology is one way of answering that question.

“Previously, people classified materials as metallic or insulating,” says Yuki Fuseya, a quantum solid state physicist at Kobe University. Beginning in the 2000s, however, Fuseya says scientists started focusing more on the topology of the electrons’ complex wavefunctions. This enriched our understanding of how materials behave, because wavefunctions with apparently different shapes can share important topological features.

For example, if the topology of certain wavefunctions on a material’s surface corresponds to that of apparently different wavefunctions within its bulk, the material may be insulating in its bulk, yet still able to conduct electricity on its surface. Materials with this property are known as topological insulators, and they have garnered a huge amount of interest due to the possibility of exploiting them in quantum computing, spintronics and magnetic devices.

Topological or not topological

While it’s not possible to measure the topology of wavefunctions directly, it is generally possible to detect whether a material supports certain surface states. This information can then be used to infer something about its bulk using the so-called bulk-edge state correspondence.

In bismuth, the existence of these surface states ought to indicate that the bulk material is topologically trivial. However, experiments have delivered conflicting information.

Fuseya was intrigued. “If you look at the history of solid-state physics, many physical phenomena were found firstly in bismuth,” he tells Physics World. Examples include diamagnetism, the Seebeck effect and the Shubnikov-de Haas effect, as well as phenomena related to the giant spin Hall effect and the potential for Turing patterns that Fuseya discovered himself. “That’s one of the reasons why I am so interested in bismuth,” he says.

Fuseya’s interest attracted colleagues with different specialisms. Using density functional theory, Rikako Yaguchi of the University of Electro-Communications in Tokyo calculated that layers of bismuth’s crystal lattice expand, or relax, by 3-6% towards the surface. According to Fuseya, this might not have seemed noteworthy. However, since the team was already looking at bismuth’s topological properties, another colleague, Kazuki Koie, went ahead and calculated how this lattice expansion changed the material’s surface wavefunction.

These calculations showed that the expansion is, in fact, significant. This is because bismuth is close to the topological transition point, where a change in parameters can flip the shape of the wavefunction and give topological properties to a material that was once topologically trivial. Consequently, the reason it is not possible to observe surface states indicating that bulk bismuth is topologically trivial is that the material is effectively different – and topologically non-trivial – on its surface.

Topological blocking

Although “very surprised” at first, Fuseya says that after examining the physics in more detail, they found the result “quite reasonable”. They are now looking for evidence of similar “topological blocking” in other materials near the transition point, such as lead telluride and tin telluride.

“It is remarkable that there are still big puzzles when trying to match data to the theoretical predictions,” says Titus Mangham Neupert, a theoretical physicist at the University of Zurich, Switzerland, who was not directly involved in the research. Since “so many compounds that made the headlines in topological physics” contain bismuth, Neupert says it will be interesting to re-evaluate existing experiments and conceive new ones. “In particular, the implication for higher-order topology could be tested,” he says.

Fuseya’s team is already studying how lattice relaxation might affect hinges where two surfaces come together. In doing so, they hope to understand why angle resolved photoemission spectroscopy (ARPES), which probes surfaces, yields results that contradict those from scanning tunnelling microscopy experiments, which probe hinges. “Maybe we can find a way to explain every experiment consistently,” Fuseya says. The insights they gain, he adds, might also be useful for topological engineering: by bending a material, scientists could alter its lattice constants, and thereby tailor its topological properties.

This aspect also interests Zeila Zanolli and Matthieu Verstraete of Utrecht University in the Netherlands. Though not involved in the current study, they had previously shown that free-standing two-dimensional bismuth (bismuthene) can take on several geometric structures in-plane – not all of which are topological – depending on the material’s strain, bonding coordination and directionality. The new work, they say, “opens the way to (computational) design of topological materials, playing with symmetries, strain and the substrate interface”.

The research is published in Physical Review B.

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By adapting their quantum twisting microscope to operate at cryogenic temperatures, researchers have made the first observations of a type of phonon that occurs in twisted bilayer graphene.  These “phasons” could have implications for the electron dynamics in these materials.

Graphene is a layer of carbon just one atom thick and it has range of fascinating and useful properties – as do bilayer and multilayer versions of graphene. Since 2018, condensed-matter physicists have been captivated by the intriguing electron behaviour in two layers of graphene that are rotated with respect to each other.

As the twist angle deviates from zero, the bilayer becomes a moiré superlattice. The emergence of this structure influences electronic properties of the material, which can transform from a semiconductor to a superconductor.

In 2023, researchers led by Shahal Ilani at the Weizmann Institute of Science in Israel developed a quantum twisting microscope to study these effects. Based on a scanning probe microscope with graphene on the substrate and folded over the tip such as to give it a flat end, the instrument allows precise control over the relative orientation between two graphene surfaces – in particular, the twist angle.

Strange metals

Now Ilani and an international team have operated the microscope at cryogenic temperatures for the first time. So far, their measurements support the current understanding of how electrons couple to phasons, which are specific modes of phonons (quantized lattice vibrations). Characterizing this coupling could help us understand “strange metals”, whose electrical resistance increases at lower temperatures – which is the opposite of normal metals.

There are different types of phonons, such as acoustic phonons where atoms within the same unit cell oscillate in phase with each other, and optical phonons where they oscillate out of phase. Phasons are phonons involving lattice oscillations in one layer that are out of phase or antisymmetric with oscillations in the layer above.

“This is the one that turns out to be very important for how the electrons behave between the layers because even a small relative displacement between the two layers affects how the electrons go from one layer to the other,” explains Weizmann’s John Birkbeck as he describes the role of phasons in twisted bilayer graphene materials.

For most phonons the coupling to electrons is weaker the lower the energy of the phonon mode. However for twisted bilayer materials, theory suggests that phason coupling to electrons increases as the twist between the two layers approaches alignment due to the antisymmetric motion of the two layers and the heightened sensitivity of interlayer tunnelling to small relative displacements.

Unique perspective

“There are not that many tools to see phonons, particularly in moiré systems” adds Birkbeck. This is where the quantum twisting microscope offers a unique perspective. Thanks to the atomically flat end of the tip, electrons can tunnel between the layer on the substrate and the layer on the tip whenever there is a matching state in terms of not just energy but also momentum too.

Where there is a momentum mismatch, tunnelling between tip and substrate is still possible by balancing the mismatch with the emission or absorption of a phonon. By operating at cryogenic temperatures, the researchers were able to get a measure of these momentum transactions and probe the electron phonon coupling too.

“What was interesting from this work is not only that we could image the phonon dispersion, but also we can quantify it,” says Birkbeck stressing the absolute nature of these quantified electron phonon coupling-strength measurements.

The measurements are the first observations of phasons in twisted bilayer graphene and reveal a strong increase in coupling as the layers approach alignment, as predicted by theory. However, the researchers were not able to study angles smaller than 6°. Below this angle the tunnelling resistance is so low that the contact resistance starts to warp readings, among other limiting factors.

Navigating without eyes

A certain amount of technical adjustment was needed to operate the tool at cryogenic temperatures, not least to “to navigate without eyes” because the team was not able to incorporate their usual optics with the cryogenic set up. The researchers hope that with further technical adjustments they will be able to use the quantum twisting microscope in cryogenic conditions at the magic angle of 1.1°, where superconductivity occurs.

Pablo Jarillo Herrero, who led the team at MIT that first reported superconductivity in twisted bilayer graphene in 2018 but was not involved in this research describes it as an “interesting study” adding, “I’m looking forward to seeing more interesting results from low temperature QTM research!”

Hector Ochoa De Eguileor Romillo at Columbia University in the US, who proposed a role for phason–electron interactions in these materials in 2019, but was also not involved in this research describes it as “a beautiful experiment”. He adds, “I think it is fair to say that this is the most exciting experimental technique of the last 15 years or so in condensed matter physics; new interesting data are surely coming.”

The research is described in Nature.

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