Topology of chalcogen chains

In this project, we showed that the electronic states in elemental tellurium and selenium crystals have a nontrivial topology, which could explain recent experimental observations.

Selenium and Tellurium are elemental chalcogen crystals that consist of weakly coupled chains spiraling around a hexagonal array of helical axes. Because of the weak coupling between these chains, we can describe the electronic properties of this materials class with the help of a one-dimensional model taking into account only electrons from the three outer p orbitals of the chalcogen atoms. Also, these chains have a periodic pattern: every three atoms, the pattern repeats. They therefore realize a period-3 chain. Interestingly, this number coincides with the number of p orbitals in the model, which is not a coincidence as explained below.

The electronic bands of these materials have an energy gap separating valence and conduction electrons energetically. For this reason, they are so called band insulators. However, our calculations revealed that there are additional electronic states inside this energy gap, which are energetically detached from the valence and conduction bands. We found that they correspond to electrons localized at the two ends of the chains. A closer inspection of the electronic states of the energy bands showed us that these extra end electrons have a topological origin: they are the consequence of the nontrivial topology of the material’s valence band electrons.

Nontrivial topology means that the valence electrons are twisted in some sense rendering the material a topological insulator. This is in contrast to normal (or trivial) band insulators whose valence electrons do not feature such an unusual twist. We formulated and calculated the corresponding topological invariant and found it to be nonzero, which means that the chalcogen chains are indeed topological. We further found that the orbitals of the chalcogen chains form special types of dimers, which is similar to what happens in polyacetylene where carbon atoms are connected by an alternating pattern of double and single bonds: a strong-weak pattern. The situation in the chalcogen chain is a little more complicated: here, we have a period-three pattern of the form strong-weak-weak for each of the three p orbitals. In addition, the strong bond forms at different locations for each type of p orbital. As a consequence, no matter how we terminate the chalcogen chains, we will always cut through one of the dimers. This ensures the presence of the unbound end electrons and is a direct consequence of the material’s nontrivial topology.

Our work could explain recent experimental findings of electronic surface states on crystals of elemental selenium and tellurium.

A. Kłosiński, W. Brzezicki, A. Lau, C. E. Agrapidis, A. M. Oleś, J. van Wezel, and K. Wohlfeld,
Topology of chalcogen chains,
Physical Review B 107, 125123 (2023), arXiv:2212.04299.