Computation sorts thousands of 2D crystals for topology and twisting

In 2004, physicists isolated graphene, a sheet of carbon just one atom thick. It launched the study of two-dimensional (2D) materials: crystals a few atoms thick in which electrons are confined to a plane and can behave quite differently from those in ordinary solids. The family of known and predicted 2D crystals has since grown to thousands of members. Two recent studies show how computation can search that family for the most promising ones, the first for unusual electronic “topology”, the second for crystals well suited to twisting.

A hidden property of electrons

Electrons in a crystal can only have energies within allowed ranges called bands, separated by forbidden gaps. If a gap separates the filled electrons from the empty states, the material is an insulator. In a semimetal, filled and empty bands just touch, so a few electrons can move freely.

Two insulators can still differ at a deeper level, described by topology: a property of the whole set of electron states that cannot change smoothly, only by closing the gap or breaking a protecting symmetry. A doughnut and a coffee mug are topologically alike because each has one hole, and neither can become a sphere without tearing.

The idea took off with the quantum spin Hall effect. Proposed for graphene in 2005, it proved unobservable there because the effect is far too weak. A 2006 proposal for thin layers of mercury telluride was confirmed in 2007. In such a topological insulator the interior blocks current, yet the edges conduct, with electrons of opposite spin travelling in opposite directions, which protects them against many kinds of imperfection.

Symmetry-based methods known as topological quantum chemistry in 2017 made it possible to decide whether a computed electronic structure is topological. Applied to three-dimensional crystals, they pointed to thousands of candidates. Flat crystals lagged behind, with only about 400 known topological or related monolayers.

The first study, “Two-dimensional Topological Quantum Chemistry and Catalog of Topological Materials” ¹, adapts these methods to 2D crystals and applies them to 8,872 entries in two computational databases. It finds 4,073 with nontrivial topology or a related property called obstruction, including 905 topological insulators, 2,129 topological semimetals and 1,003 obstructed atomic insulators, plus 36 close variants. In an obstructed insulator the electrons are centred on empty positions between the atoms rather than on the atoms themselves. The bulk looks ordinary, but cutting the crystal along certain lines produces edge states, and small flakes can carry fractions of an electron’s charge at their corners. The study also calculates edge states for selected materials. Its catalogue includes over 100 monolayers already made in laboratories, but most entries are computer predictions whose properties remain to be tested.

Twisting layers

Stack two atomic sheets and rotate one slightly, and the mismatch creates a large interference pattern, called a moiré pattern, much like two overlapping window screens. Electrons feel this longer repeating pattern, and under the right conditions their bands become nearly flat. Electrons in a flat band have little energy of motion, so their mutual repulsion dominates and strongly correlated behaviour can emerge.

In 2011, calculations predicted that two graphene sheets twisted by about 1.1 degrees, a “magic angle”, would have such flat bands. In 2018, experiments at that angle revealed insulating states caused by electron repulsion, and superconductivity. Twisted layers of other materials, such as MoTe₂ and WSe₂, have since shown many correlated phases, though the mechanisms are still debated.

Most work has focused on these few materials, because most crystals have electronic structures too complicated to predict once twisted. The second study, “2D theoretically twistable material database” ², therefore defines a material as theoretically twistable if its low-energy electrons admit a simple description. In a metal, the relevant electrons must sit at isolated points, as in graphene. In an insulator, the top of the filled band or the bottom of the empty band must stand clear of other bands. Insulators must also have gaps between 0.2 and 3 electronvolts, because larger gaps make electrical contacts difficult. The search was limited to crystals with at most 12 atoms in the repeating cell and four chemical elements.

Starting from the first study’s database, it identifies 61 semimetals and 1,568 insulators. It ranks them with a score and sorts them by lattice shape (about half are hexagonal), by where in momentum space the relevant electrons sit, and by the strength of spin-orbit coupling, the link between an electron’s spin and its motion. These choices shape the moiré pattern, which can resemble honeycomb, kagome or square lattices. Square-lattice moiré systems have been proposed but not yet realized, and could help simulate models relevant to cuprate superconductors.

Calculations on twisted bilayers of SnSe₂, SnS₂, ZrS₂, HfS₂ and Sc₂CCl₂ at a twist of about 7 degrees show isolated flat bands, as the database predicts. Experiments connect the theory to the laboratory. Bulk crystals of SnSe₂, SnS₂, HfS₂, GaTe and ZrNCl were grown. Single layers of several candidates, including SnSe₂, HfSe₂, ZrS₂ and WS₂, were peeled off with a gold-assisted technique that yields larger, cleaner flakes. A dual-gated two-layer SnSe₂ device was built for electrical measurements, and a twisted two-layer HfS₂ device was used for scanning tunnelling measurements near absolute zero.

The direction is clear. Instead of discovering materials by trial and error, it is becoming possible to search systematically for crystals with chosen properties, then test them layer by layer.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research paper/s.