Illustration of a one dimensional moire pattern that emerges on twisting a two-dimensional crystal

Illustration of a one dimensional moire pattern that emerges on twisting a two-dimensional crystal

Credit: Yves Kwan

Unusual new state of matter

Quantum materials
Rudolf Peierls Centre for Theoretical Physics

Writing in Nature today, 4 May 2022, a team of physicists including three members of Oxford’s Rudolf Peierls Centre for Theoretical Physics –  DPhil student Yves Kwan and Professors Shivaji Sondhi and Sid Parameswaran – report the emergence of an unusual new state of matter in a newly-explored moiré system.

One of the most exciting recent developments in the study of quantum materials is the ability to engineer ‘moiré superlattices’ (a term whose origins lie in the textile industry, see boxout) by layering two or more atomically thin two-dimensional crystalline layers with a slight mismatch in their orientation or inter-atomic spacing. Electrons moving through such a system experience a ‘moiré superlattice potential’ generated by the positively-charged ions of both layers. While this potential is approximately periodic in space, the scale over which the pattern repeats is much larger than the inter-atomic spacing. In certain cases, an electron moving in the moiré potential moves much slower than it would in an individual, isolated layer, so that its kinetic energy is much weaker than its interactions with other electrons. This allows strong correlations – cooperative behaviour of many interacting electrons, responsible for spectacular macroscopic quantum phenomena such as high-temperature superconductivity – to come into play.

In 2018, experimentalists at the Massachusetts Institute of Technology observed superconductivity and other strong correlation effects in the most famous moiré system: ‘twisted bilayer graphene’, made by twisting a pair of atomically thin sheets of graphene (two-dimensional carbon) to a ‘magic angle’ of around 1 degree. This prompted a surge of theoretical and experimental activity studying both twisted graphene and a host of other synthetic superlattices made by stacking a variety of other two-dimensional materials, in a quest to explore what new states of matter might emerge in this entirely new setting.

Groundbreaking experiments

The work of Kwan and Professors Sondhi and Parameswaran grew out of measurements of the electrical conductivity of twisted bilayers of tungsten ditelluride (WTe2), cooled to a few kelvin above absolute zero. These groundbreaking experiments were done in the group of Professor Sanfeng Wu, an Assistant Professor of Physics at Princeton University, and also involved collaborators at the Massachusetts Institute of Technology, Rice University, the Fraunhofer Institute in Hanau, Germany, and the National Institute for Materials Science in Tsukuba, Japan.

Electrons in an individual WTe2 layer move with comparable speeds in all directions in two dimensions. However, when the two layers are twisted at around 5 degrees, the resulting moiré effect produces a ‘stripy’ potential; electrons moving in this potential propagate relatively freely in the ‘easy’ direction along the stripes, but have great difficulty in moving in the ‘hard’ direction perpendicular to the stripes. This can be diagnosed by measuring the electrical resistance along the chains: at low temperatures, it is around a thousand times lower than that perpendicular to them. In contrast, measuring this anisotropy between the two directions in a single layer gives a factor of just 3 or so. Understanding how the moiré phenomenon drives this colossal enhancement of the anisotropy was a key puzzle for theory.

Key proof of principle

WTe2 is one of a family of materials known as transition metal dichalcogenides that can be isolated into 2D layers similarly to graphene. All other TMDs crystallise in a pattern akin to graphene, and have broadly similar physics, with some differences due to the fact that a single TMD layer is insulating rather than conducting. Unlike its counterparts, however, an isolated layer of WTe2 no longer resembles graphene. Therefore, the Oxford team had to combine a mix of pencil-and-paper calculations and detailed numerical simulations to come up with a theoretical model of twisted WTe2. This was a key proof of the principle that moiré physics could lead to essentially one-dimensional electronic transport consistent with that seen in Wu’s experiments.

'Luttinger liquid' electrical transport

Individual one-dimensional electronic systems have been synthesised before – one famous example is that of carbon nanotubes (essentially a rolled-up sheet of graphene), but there are several others. One-dimensional metals have the striking property that no matter how weak the interactions, they no longer behave in a way that resembles their non-interacting counterparts. This is in striking contrast to the situation for metals in higher dimensions, nearly all of which are ‘Fermi liquids’: a termed coined by Russian physicist Lev Landau in the 1950s, who argued that for many purposes ignoring interactions gives a pretty good description of most of their properties. A key signature of the breakdown of Fermi liquid theory in one dimension, in favour of what is now termed a ‘Luttinger liquid’, is to examine how easy it is to tunnel electrons into the one-dimensional wire as temperature is lowered. In a Fermi liquid the tunnelling rate remains non-zero even as the temperature is taken to zero, whereas in a repulsively-interacting Luttinger liquid it vanishes – essentially because electrons inside the wire resist the addition of the incoming electron trying to tunnel in. Certain types of electrical measurements are sensitive to this tunnelling. Luttinger liquid behaviour can be inferred by exploring how these measurements change with temperature, or by studying similar effects on changing an external voltage bias.

The picture described above is for a single one-dimensional wire. For an array of such wires, as in Wu’s twisted WTe2 samples, there is inevitably a small probability that electrons can jump between wires, and ultimately the physics becomes Fermi-liquid like at very low temperatures when the system can ‘sense’ that it is really two-dimensional.  Repeated efforts have attempted to identify scenarios where Luttinger-liquid behaviour persists down to zero temperature in such a two-dimensional ‘coupled wire’ setting, but the models where such physics seems theoretically possible are very difficult to realise experimentally.

New states of matter

Excitingly, besides the huge anisotropy, the second striking feature of Wu’s experiments is that electrical measurements along both easy and hard directions agree with theoretical predictions for Luttinger liquid tunnelling behaviour, down to extremely low temperatures. The standard moiré mechanism, which ignores the effect of interactions between electrons, would see the behaviour change to ‘normal’ Fermi liquid behaviour at much higher temperatures, indicating that electronic interactions are significant. Kwan, Sondhi, and Parameswaran were able to rule out many competing theoretical explanations, and explained how a Luttinger liquid interpretation could be consistent with the easy-vs-hard anisotropy. This analysis raises the tantalising possibility that this new system realises a long-sought example of a highly anisotropic, tuneable, two-dimensional ‘sliding Luttinger liquid’ (so named since electrons remain tied to each wire, they can in effect ‘slide’ past those in other wires). Kwan, who led the theoretical effort and is due to take up a postdoctoral fellowship at the Princeton Center for Theoretical Science later this year, said: ‘The emergence of Luttinger liquid physics here is remarkable as it expands, in a fundamentally new way, the dizzying array of fascinating phases that have been observed in moiré heterostructures. Our results only scratch the surface, and it is exciting to see the frontiers of correlated quasi-1D physics extended in twisted bilayer WTe2 and related systems.’

Many puzzles remain, but the exceptional tunability of the platform and the wealth of experimental probes mean that many of its properties can be investigated with a high level of detail – for instance, by attempting more direct tests such as tunnelling electrons from an atomic-scale ‘tip’ into the system. Apart from being interesting in its own right, a sliding Luttinger liquid could also seed a host of other interesting states of matter on applying a magnetic field, which introduces new possibilities for tunnelling between wires with the aid of interactions. All in all, the new work illustrates the rich new physics that can emerge in moiré materials, and the interplay of experiments, simulation, and fundamental theoretical ideas in probing the frontiers of this new branch of materials science.

Theoretical work on moiré systems at Oxford is supported by grants from the European Research Council, the UK Engineering and Physical Sciences Research Council, and the Flanders Science Foundation.

One-dimensional Luttinger liquids in a two-dimensional moiré lattice, Pengjie Wang et al, Nature 605, 4 May 2022

The moiré revolution

Over the past few years, an entirely new type of material system has taken the physics world by storm: ‘moiré heterostructures’, obtained by layering two-dimensional crystalline materials on top each other but with a small relative twist angle or difference in atomic spacing. The propagation of electrons through the periodic lattice of positive ions in each individual atomically-thin layer follows relatively well-understood rules written down by Felix Bloch in the 1930s. However, when two such layers are twisted relative to each other, electrons in one layer also experience forces due to ions in the other. This creates a ‘superlattice’ with a period that can be as much as a few hundred times bigger than the microscopic crystal, and can lead to dramatically different properties that can be tuned by adjusting the twist angle, or by applying relatively weak electric fields.

It has only become possible to engineer moiré superlattices for electrons relatively recently, but the underlying effect is familiar to anyone who has ever photographed or filmed a striped pattern. The French term moiré itself has origins in the textile trade and the English mohair, and refers to the rippled pattern produced when two sheets of fabric with slightly differently-spaced or rotated weaves are pressed together. Atomic-scale moiré patterns are far more challenging to create than their textile counterparts. The first requirement, the ability to reliably produce individual atomically-thin crystalline materials, was only reached in the mid-2000s when Andre Geim and Kostya Novoselov, both then at the University of Manchester, and Philip Kim, at Columbia University, managed to make flakes of graphene – pure carbon – by the breathtakingly simple expedient of peeling off single layers from graphite crystals using scotch tape. Geim and Novoselov received a Nobel Prize in 2010 for their work on graphene, but it took nearly a decade for physicists to develop the ability to manipulate individual layers with sufficient precision to produce twisted layers on demand. Andrea Young, now at the University of California, Santa Barbara, one of the first to accomplish this feat, likens it to ‘reusing Saran Wrap [cling film] — it gets wrinkled and it’s a mess,’ he said. ‘Now imagine that wrap was 30,000 times thinner — we struggled with that for quite a while!’ [1]

In parallel, Allan Macdonald and Rafi Bistritzer at the University of Texas developed ingenious techniques to precisely compute how electrons move through these moiré heterostructures, focusing on the special case when each layer is a single-atom-thick graphene sheet. In 2010, they predicted [2] that if one could engineer a twist of approximately 1 degree between the layers, the electrons would dramatically slow down, enhancing the effect of their interactions. Most strikingly, they suggested that in the right circumstances – temperatures of a few kelvin above absolute zero – these electrons could even become superconducting. This was spectacularly confirmed in the laboratory of Pablo Jarillo-Herrero at the Massachusetts institute of Technology [3,4] (where Young did his pioneering work as a postdoctoral researcher) and replicated in short order by a host of other groups around the world.

Several laboratories in the United States, Europe, and Asia have now perfected these ‘tear-and-stack’ techniques, and have begun exploring a dizzying array of heterostructures built not only from single- double- and triple-layer graphene, but also a host of other two-dimensional materials. Over the past two years, Oxford has established itself as a centre for theoretical investigations of moiré materials, with research activity focused in the Rudolf Peierls Centre for Theoretical Physics, and involving Professors Sid Parameswaran, Steve Simon, and Shivaji Sondhi, postdoctoral fellow Dr Nick Bultinck, graduate student Yves Kwan, and recent alumnus Dr Glenn Wagner, now a postdoc at the University of Zürich. Their collaboration has produced a string of recent papers exploring diverse aspects of moiré systems [5-9], notably the proposal of an entirely new type of ‘insulating spiral’ ordered state of electrons as a unified explanation for a host of intriguing phenomena in twisted bilayer graphene.

[2] R. Bistritzer, A.H. MacDonald PNAS 108, 12233 (2011). []
[3] Y. Cao, et al., Nature 556, 43 (2018). []
[4] Y. Cao, et al., Nature 556, 80 (2018). []
[5] Y.H. Kwan, Y. Hu, S.H. Simon, and S.A. Parameswaran, Phys. Rev. Lett. 126, 137601 (2021) []
[6] Y.H. Kwan, G. Wagner, N. Chakraborty, S.H. Simon, and S.A. Parameswaran, Phys. Rev. B 104, 115404 (2021). []
[7] Y.H. Kwan, G. Wagner, T. Soejima, M.P. Zaletel, S.H. Simon, S.A. Parameswaran, and N. Bultinck, Phys. Rev. X 11, 041063 (2021). []
[8] G. Wagner, Y.H. Kwan, N. Bultinck, S.H. Simon, and S.A. Parameswaran, Phys. Rev. Lett. 128, 156401. []
[9] Y.H. Kwan, G. Wagner, N. Bultinck, S.H. Simon, and S.A. Parameswaran, arXiv:2112.06936 (2021). []

Transitional metal dichalcogenides

WTe2 is one of a series of  transition metal dichalcogenides (TMDs), crystals built where each repeated unit contains a single transition metal atom such as tungsten or molybdenum and two ‘chalcogens’ atoms from Group 16 of the periodic table, such as sulphur (S), selenium (Se), or tellurium (Te). A single layer of most TMDs crystallises in a honeycomb pattern similar to that of graphene, but because it is insulating rather than conducting, the corresponding moiré problem is relatively simple to study by adapting an approach successfully developed to study twisted graphene layers.

Tungsten ditelluride is an intriguing exception to this rule: while the chemistry is the same as its cousins, the inter-atomic forces are sufficiently different that it crystallises in a completely different pattern. Worse still, there remains substantial debate over the properties of a single layer, posing a substantial challenge to giving a clear theoretical interpretation of Wu’s experiments. A definitive theoretical treatment of twisted WTe2 remains an important outstanding problem whose solution is being actively pursued at Oxford and elsewhere.