IT WAS March 2018. The atmosphere at the at the Los Angeles Convention Center was highly charged. The session had been moved to the atrium to accommodate the crowds, but people still had to cram onto the balconies to get a view of the action.
Rumours had it that , a physicist at the Massachusetts Institute of Technology, had something momentous to report. He and his colleagues had been experimenting with graphene, sheets of carbon just a single atom thick that are peeled from the graphite found in pencil lead. Graphene was already celebrated for its various promising electronic properties, and much more besides.
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Here, Jarillo-Herrero showed that if you stacked two graphene sheets and twisted, or rotated, one relative to the other at certain 鈥渕agic angles鈥, you could , where electric current barely flows, or a , where current flows with zero resistance. It was a staggering trick, and potentially hugely significant because superconductivity holds promise for applications ranging from quantum computing to nuclear fusion.
Researchers have since used twisted graphene to generate all manner of exotic quantum effects, including 鈥渜uasiparticles鈥 that can manifest as magnetic vortices and otherwise exhibit bizarre electronic properties. 鈥淲hat鈥檚 exciting about these systems is that they hold huge potential for surprise,鈥 says at Harvard University.
Even more exciting is that we have barely started on this journey. Now, by inserting more sheets of graphene or swapping in sheets of other materials to produce similar effects, we are delving deeper into the wild new physics lurking inside two-dimensional materials.
Studies of these skewed, layered substances offer a new way to investigate the fundamental nature of materials, and more specifically how the arrangement of atoms affects their properties. Whether a material conducts electric current or not stems from how the electrons in its atoms are distributed around their nuclei. Broadly speaking, the distributions of electrons on neighbouring atoms overlap to form extended 鈥渂ands鈥 throughout the material. In a conducting material, the electrons with the most energy occupy a band that has room for lots of other electrons, so they are mobile: apply a voltage and they can flow as an electric current between the electrodes. In an insulator, the highest energy band is, like the other bands at lower energies, totally filled with electrons. So the electrons, like a crowd in a packed room, have no freedom to move around.
In graphene, the carbon atoms are linked into a honeycomb-like hexagonal lattice and there is a band where electrons are free to roam throughout this, making them extraordinarily mobile 鈥 a promising quality for high-speed electronics. In fact, in pristine graphene, where there are no imperfections in the honeycomb lattice arrangement, , as if they have no mass at all.
But if you take two layers of graphene and twist one of them with respect to the other, you can change the way electrons move. The two hexagonal grids move in and out of alignment as you cross the lattice, creating a kind of 鈥渟uperlattice鈥 that repeats on a much larger scale. The same effect has been popular in 鈥渕oire鈥 textiles, where a pattern larger than the cloth mesh emerges from the play of light on the skewed layers of fabric. For electrons in graphene, their behaviour is no longer just influenced by the carbon atom lattice, but also by the moire lattice, which affects how easily they move between layers. The result is that conducting electrons in this twisted bilayer graphene can slow down drastically, changing their behaviour.
鈥淲hen electrons have lots of kinetic energy, when they move very fast, they barely have time to interact,鈥 says Jarillo-Herrero. But this changes as they slow down in twisted bilayer graphene, he adds. The strong interactions mean that the electron motions become very sensitive to, and dependent on, one another. In technical terms, they become highly correlated 鈥 and here is where things start to get interesting, because correlated electrons are capable of otherwise impossible feats.
Electrons team up
Take superconductivity. In conventional superconductors, the interactions between electrons result in them teaming up into 鈥淐ooper pairs鈥. The laws of quantum mechanics restricting the number of electrons that can share the same properties 鈥 energy, position and so on 鈥 don鈥檛 apply to these Cooper pairs. Thus they can gather en masse and race around, unimpeded by the lattice atoms, with no resistance.
In fact, a Cooper pair is an example of a quasiparticle: a collective state of many electrons that acts as if it is a new type of particle.
So if you鈥檙e looking for funky electronic properties, you want your electrons to be correlated. And if you want strong correlations, flat materials are your best bet. In three dimensions, the electrons have more ways of moving out of one another鈥檚 reach. In two dimensions, on the other hand, and particularly in sheet-like conductors such as graphene, the electrons are more likely to come together to perform their tricks.
at Rutgers University in New Jersey and her colleagues had already glimpsed magic-angle effects when they saw that odd things happened to the energy levels of electrons in samples where one graphene sheet lay atop another. The effect was particularly pronounced when one layer was rotated with respect to the other by around 1 degree. As other groups reported similar phenomena, theorists began to wonder what was going on. Among them were Allan MacDonald at the University of Texas in Austin and his colleague , now at Tel Aviv University in Israel. They and found that the velocity of the electrons would fall to zero at certain twist angles. 鈥淭he electrons just stop,鈥 says MacDonald. 鈥淭his was a complete surprise to us.鈥
The largest of these magic angles was around 1.16 degrees, which is still tiny and therefore demands a devilishly delicate level of control over the orientation of the microscopic graphene layers. But Jarillo-Herrero could see that this twist angle had potential as an unprecedented new 鈥渢uning knob鈥 for electronic properties, so he decided to give it a shot.
In 2016, his team found the of electron bands with very small kinetic energies, equivalent to the low velocities MacDonald and Bistritzer had predicted. Pressing on, the researchers searched for a type of non-conducting state called a Mott insulator that they thought might arise from strongly correlated electrons.
Sure enough, in 2018, Jarillo-Herrero鈥檚 team too 鈥 but also something more intriguing. If the researchers altered the applied voltage to fine-tune how many electrons were available to carry a current, . As with all superconducting materials, this behaviour only emerged at very low temperatures 鈥 below 1.7 kelvin, which is less than 2 degrees from absolute zero. No one had predicted this, says Jarillo-Herrero.
Superconductivity raises hopes
Naturally, researchers quickly flocked to study the exotic electronic effects produced by magic-angle graphene. They were energised not only by the prospect of discovering new fundamental physics, but also because superconductors are in demand. They can be used in quantum bits in quantum computers, which exploit the strange laws of quantum physics to speed up certain calculations, as well as in technologies that use strong magnetic fields, such as MRI machines and nuclear fusion reactors.
One way to produce a magnetic field is by running a current through a coiled wire. Ramp up the currents using superconducting wires and much larger magnetic fields are possible. But the need to cryogenically cool superconducting wires makes them hard to work with. Which explains why some people were intrigued by the possibility that the superconducting behaviour in magic-angle graphene might offer a way in to finally understanding why certain copper-based compounds called layered cuprates . First reported nearly 40 years ago, the behaviour has foxed the field ever since.
鈥淲e do not know yet if understanding magic-angle graphene will help us understand the origin of superconductivity in cuprates,鈥 says Jarillo-Herrero. Both the cuprates and the magic-angle graphene are layered materials and share other characteristics, he says, but they have many differences too. 鈥淢y intuition is that it will [help], but it鈥檚 too early to tell.鈥
In any case, there are plenty of intriguing phenomena to discover with magic-angle graphene. For example, it can be made ferromagnetic, like iron. Ferromagnetism results from a quantum property of electrons called spin. In ferromagnetic materials, all the electrons鈥 spins align. In 2019, by manipulating the electron bands of magic-angle graphene, David Goldhaber Gordon at the Stanford Institute for Materials and Energy Sciences in California and his colleagues were able to for the first time. A controllable ferromagnet that can be switched on and off might be useful in a type of electronic technology called spintronics, where information is encoded in the spin of electrons rather than in pulses of electrical current.
A quasiparticle hunting ground
Magic-angle graphene has also proved fertile ground for the discovery of new and exotic quasiparticles, including those that can have fractional charges. The charge of an electron is a fundamental unit 鈥 no free particle can have any smaller charge than this. But electrons can behave as if they have a fractional charge in a phenomenon called the fractional quantum Hall effect. While such fractionally charged quasiparticles generally behave as though they are isolated from one another, in magic-angle graphene they can line up into their own quasiparticle lattice, known as a .
More than just a scientific curiosity, these fractional quasiparticles command practical interest because they bear a striking resemblance to 鈥渁nyons鈥 鈥 a hypothetical quasiparticle keenly sought for quantum computing.
In the standard model of particle physics, all fundamental particles fall into one of two classes: fermions, such as electrons, and bosons, like photons. Quasiparticles generally follow the same dichotomy. Cooper pairs, for example, are bosons. But the anyon, should it exist, would be something between a boson and a fermion. One particular kind of anyon has been proposed as a quantum bit that would be resistant to the random flips in state that cause errors in quantum computation, which is currently a key obstacle to making good on the promise of quantum computers.
The discoveries keep coming. In March 2021, Ashvin Vishwanath at Harvard University and his colleagues developed a theory of superconductivity in twisted graphene based on quasiparticles called skyrmions, which manifest as magnetic vortices. Such skyrmions were reported in December by researchers at Princeton University. Then, earlier this year, Jarillo-Herrero discovered superconductivity in , a result replicated independently at Harvard University. Jarillo-Herrero鈥檚 team has since shown that the property exists even in four and five-layer systems.
鈥淲e have been able to realise pretty much all the phases of condensed-matter physics, all in a few years and by combining very simple materials,鈥 says Jarillo-Herrero. 鈥淚t鈥檚 quite extraordinary when you think about it.鈥
The feat is all the more impressive when you consider what it takes to produce magic-angle graphene in the first place. Typically, Jarillo-Herrero鈥檚 group takes a flake of hexagonal boron nitride (hBN) 鈥 so named because it has the same honeycomb structure as graphene 鈥 that is about 10 millionths to 30 millionths of a millimetre thick. They use this as a kind of sticky tape for stripping a single layer of graphene from graphite. Then they can pull off a second layer, rotated slightly relative to the first by manually reorienting the hBN flake, before adding the components needed to take electrical measurements.
The trouble is that reproducing these structures is excruciatingly difficult. 鈥淭he physical properties of a moire system can change with minuscule changes in twist angle,鈥 says Jarillo-Herrero. Yacoby also admits that if asked to make two identical devices, 鈥淚鈥檇 be struggling to do it鈥. In addition, he says, these systems might not be stable. 鈥淭here鈥檚 a lot of strain and distortion, and atoms move to find the most comfortable position.鈥
There is, however, an alternative that gets around these problems, because it doesn鈥檛 rely on magic angles at all. Two sheets of different materials with slightly different spacings between the atoms can also create a moire lattice. The so-called transition-metal dichalcogenides (TMDs), such as tungsten disulphide and tungsten diselenide, also form hexagonally bonded layers and offer many permutations in which the two (or more) layers are made from different materials. For tungsten disulphide and tungsten diselenide, for instance, there is a 4 per cent mismatch in their atomic spacing, producing a moire pattern that repeats every 8 nanometres.
at Cornell University in Ithaca, New York, says that because TMD bilayers aren鈥檛 reliant on magic angles and therefore aren鈥檛 sensitive to small variations in twist angle, experiments on them are much more reproducible. He and his colleague , also at Cornell, have teamed up with MacDonald to show that a moire system made from tungsten disulphide and tungsten diselenide can be used to explore all kinds of correlated electron behaviour by acting as of the most popular model used to describe them: .
Named after physicist John Hubbard, who proposed it in 1963, the Hubbard model breaks down electron energies into just two contributing factors 鈥 their kinetic energy and the energy of their interactions. It has been a popular go-to for investigating Mott insulators, superconductors 鈥 especially the cuprates 鈥 and ordered magnetic systems. Yet, for all its beguiling simplicity, it is so difficult to work with mathematically that an exact solution to the equations in the model still only exists for one-dimensional systems.
The good news is that 鈥渢hese moire materials are just about a perfect mapping of [the Hubbard model],鈥 says MacDonald. By altering the voltage applied to their TMD bilayer sample, Mak and Shan could make a range of material behaviours predicted by the model, such as the transition between a ferromagnetic state and an antiferromagnetic one, where the atomic spins alternate in orientation. Instead of a complicated calculation, they could deduce what the model predicts by experiment.
MacDonald says we can鈥檛 be completely confident what these moire systems will reveal. Even so, he is cautiously optimistic about the progress we can expect when it comes to understanding magic-angle graphene superconductivity. 鈥淭his progress will have implications for understanding high-temperature superconductivity,鈥 he says.
We should expect some surprises. Jarillo-Herrero鈥檚 initial discovery of superconductivity in these systems came completely out of the blue. And despite the progress made in the years since, he insists that 鈥渨e have barely scratched the surface of the many hundreds of possible moire systems we can build, with very different constituents, geometries and complexity鈥. These are our first steps into an uncharted landscape of possibilities.
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