Since its beginnings roughly a century ago, research in condensed-matter physics has focused mainly on the study of bulk crystals. These are solid-state materials in which electrons are confined to move among atoms arranged in a periodic array. More recently, however, researchers have started to look at artificial engineered lattices that mimic the properties of these crystals. Two well-known examples of such systems are photonic crystals, in which a periodic wave guide is used to trap light in a periodic array, and cold-atom traps, in which a lattice generated by lasers is used to trap atoms in a periodic array of light.

“In our work, we have made a new type of engineered lattice, in which we arranged topological insulator interface Dirac fermion states in a periodic array to produce an emergent electronic band structure,” explains lead author of the study, graduate student Ilya Belopolski. Topological insulators are naturally-occurring materials that are insulating in the bulk but can conduct electricity on their surface via special Dirac-like surface electronic states. These surface states are said to be topologically protected because they are robust to disorder in the crystal thanks to the fascinating topological properties of the electronic wavefunctions of the bulk material. In the new experiment, the researchers figured out a way to arrange copies of these topological Dirac-like electronic states in a periodic array, forming a new type of condensed-matter lattice.

“Our quantum-matter lattice allows us to engineer band structures by tuning the hybridization between topological interface states,” says Belopolski. “At the same time, it is still a true electron system, so it can be used to study electron transport or develop new devices and electrical components.”

A stacked quantum-matter lattice

The quantum-matter lattice made by the team has a layer-by-layer pattern and consists of two materials – the 3D topological insulator (TI), Bi2Se3, which is well known for its simple topological band structure, and a 3D “trivial” or normal insulator (NI) made by doping Bi2Se3 with indium.

“We used a technique called molecular-beam epitaxy to deposit alternating layers of the TI and NI on a substrate, creating a stacked quantum-matter lattice. Each layer is less than 10 nm or 50 atomic layers thick and to measure our lattice we used angle-resolved photoemission spectroscopy (ARPES), which is a powerful technique that allows us to directly observe the behaviour of electrons in a crystal,” says team leader M. Zahid Hasan.

Interfaces host protected Dirac fermion surface states

“Using ARPES, we studied the top layer of the heterostructure and found that we could control the electronic structure on the top surface by adjusting the thickness of the TI or NI layers or the level of doping in the NI layer.”

“At each interface within the stack there is a TI on one side and a NI on the other side,” adds Hasan. Such an interface must host a protected surface state taking the form of a Dirac cone (a feature in the band structure where the conduction and valence bands meet at a single point).

“Since our heterostructure has alternating TI and NI layers throughout, each interface in the lattice hosts such a topological Dirac cone interface state,” he tells “And, since the layers within the stack are very thin, there is a certain probability of electrons quantum tunnelling from layer to layer. The Dirac-cone interface states thus hybridize with one another, producing a new, emergent band structure built up from the TI interface states.

“By building up a heterostructure of TIs and NIs, we form a system in which the interfaces between layers act as atomic lattice sites and the Dirac-cone interface states as electron orbitals. By tuning the chemical composition or thickness of layers, we can precisely control the hybridization amplitude of nearby interface states, allowing us to engineer the emergent band structure.”

Better understanding 1D topological phases

According to the researchers, the new results could help them better understand 1D topological phases. “The results remind us in fact of the famous Su–Schrieffer–Heeger (SSH) model, which is the prototypical theoretical model for topological phases of matter,” explains Hasan. “The SSH model is a 1D model for the hydrocarbon molecule polyacetylene, in which, depending on the hybridization amplitude chosen, there either exists or does not exist a topological end mode on the 0D boundary of the 1D chain. So, the SSH model shows two phases, a topological phase with an end mode and a trivial phase without an end mode.

“In our work, we see an analogous result, where the Dirac-cone surface state on the top surface (the end mode) is or is not protected depending on the way hybridization amplitudes are chosen in the bulk of the heterostructure. Our work, in a sense, represents the first realization of this foundational model of TIs, upon which many new devices could be built up. It is rather like the first set of ‘Lego’ pieces with which we could design or fabricate many other topological states of matter.”

Magnetic Weyl semimetals and superconducting topological lattices?

The researchers, reporting their work in Science Advances DOI: 10.1126/sciadv.1501692, say that they could also go back to treating their emergent band structures as 3D systems and ask: what new topological phases might we be able to engineer?

So, what next? “One idea that naturally comes to mind is to introduce magnetism into our heterostructure to make some novel time-reversal symmetry-breaking topological phases,” says Hasan. “In particular, our system may provide a way to make a magnetic Weyl semimetal, which has never been realized in an experiment. Doing this would be an important direction for our field. In the same way, we might be able to make a superconducting topological lattice.”