to have a social platform to help people identify new moiré materials that no one else is working on in order to maximize the field’s overall progress.

Another possible way to speed up the exploration would be to automate the fabrication of moiré systems using advances in, for example, robotics and micro-manipulators. “I think it would be relatively straightforward,” Parkin said, “to build automated systems to exfoliate and put these exfoliated sheets one on top of each other at different angles and build the next, next generation structures that were discussed by Pablo yesterday.”

Yet another approach that had been discussed on the workshop’s first day, Parkin said, was to harness artificial intelligence and machine learning along with the “materials genome” project started about a decade ago to make predictions about which materials are likely to have which sorts of properties.

Another key issue from the workshop, he said, has been the question of just how exotic the properties of moiré materials are. “It’s fantastic that two sheets of carbon twisted at some angle shows so many interesting properties, particularly on gating,” he said. One of those properties is superconductivity. The electron pairing that leads to superconductivity is particularly strong in these materials, he noted, which raises the question of the mechanism that leads to superconductivity in magic-angle twisted bilayer graphene and other moiré materials. That question is still unanswered, but if the mechanism were known, it is possible that researchers could use that information to substantially increase the critical temperatures at which the materials become superconducting. There had been a mention on the workshop’s first day of a potential mechanism—one that involved skyrmions—but nothing further was said, and it might be useful to pursue that further, Parkin said. A related issue was the concept of topological superconductivity in twisted cuprate sheets. “It would be fantastic to experimentally demonstrate this,” he said.

Another exotic property that was discussed was orbital magnetism and the quantum anomalous Hall effect that has been observed in the twisted bilayer graphene on hexagonal boron nitride (hBN). Magnetism using just carbon materials is really exotic and was unexpected and not theoretically predicted, he said. One question is whether one might see orbital magnetism in some of the other twisted systems beyond graphene. Another question is what might orbital magnetism be used for.

An issue that had been discussed in the previous day’s discussion session was what keeps the two layers in magic-angle twisted bilayer graphene or other twisted bilayer systems from slipping back to a zero angle, which is energetically preferred. Are there some impurities—or “dirt,” as Parkin termed it—between the layers which are resisting the movement of the layers? And once there is a better understanding of the slipping of the two layers relative to one another, could devices be built that take advantage of that? Parkin mentioned, for instance, an oscillator in

which the two layers move back and forth a small amount relative to each other, thus switching from a moiré to a non-moiré configuration and back, over and over again. He mentioned in particular a 2018 paper in Science that described a device consisting of a layer of graphene on top of a layer of hBN in which the hBN layer could be freely rotated relative to the graphene to continuously vary the wavelength of the moiré pattern created by the two layers (Ribeiro-Palau et al. 2018).

Another important question to arise from the workshop’s discussions, Parkin said, was what other sorts of materials might be used for twistonics devices. Ramesh discussed correlated oxides, and Karunadasa spoke about halide perovskites, both of which are possibilities.

Size is another issue to consider, Parkin said. The moiré patterns that were described were on the order of about 10 nm, which in one sense is small but in another may be too large for certain appliications. In particular, he said, next-generation CMOS devices are expected to have a critical size about 2 nm, and in today’s devices the critical size is about 5–7 nm. Thus, he asked, is the moiré wavelength too big for certain types of applications, such as for use in quantum computing?

Speaking more generally about applications, Parkin said that novel fabrication methods may be needed for moiré devices. Are there natural moiré structures? Could thin film techniques be used, such as the directed growth of layers by ion beams, which has been used for liquid crystals?

Referring to something Jarillo-Herrero had said, Parkin pointed to the tunability of these systems as a major asset. Just by changing the number of electrons, for instance, one can tune the properties of one of these systems from being a superconductor to an insulator. In applications today, one of the goals is to reduce energy use, and since energy is proportional to voltage squared, the goal is to reduce the voltage. Ideally the devices would operate at 10 mV or so.

Finally, Parkin commented that it takes a long time from a materials discovery to application. For example, Kevlar was discovered in 1965, but it first came to market 17 years later—in 1982. In his own career, he said, he has been involved in several discoveries that eventually made their way into applications. One was the invention of the spin-valve device, which was implemented in magnetic disk drives at IBM, where he was working at the time, in less than a decade—a very fast development. By contrast, the type of random access memory his group proposed in 1995 took a quarter of a century to make it to market.

DISCUSSION

To open the discussion period, Parkin asked the presenters if there were any points they would like to make before he began asking questions. Karunadasa answered that she had a couple of points she would like to address. First, noting that there is a vast collection of two-dimensional materials to choose from when

designing a twisted bilayer structure, she said that it seems that if the goal is to get a flattening of the band at a particular twist angle, then “the simplest design principle that I can think of is that you want one of the bands to have states from both layers so that it is sensitive to the twist angle.” This would lead to a more rational approach to choosing which two-dimensional materials to combine to create these systems, she said.

Her second point, she said, was to reiterate the possibility of using organic molecules to bring the layers together. There are many two-dimensional that have organic sheets templated with organic groups, she explained, and perhaps by choosing the organic groups wisely it will be possible to dictate how the layers come together, including the distance between the sheets and perhaps even the twist angle, based on simply the length of the link. In response to a clarifying question from Parkin, Karunadasa explained further that she was envisioning two different two-dimensional sheets, each with its own organic molecules attached, with the organic molecules chosen so that when the sheets were brought together, they would interact and create the desired twist.

Workshop chair Aharon Kapitulnik followed up on that answer by asking for more details. “Is it possible to design the organic molecules to produce a known, well-defined twist between successive layers and then build, let’s say, a three-dimensional organo-metallic perovskite, but with a known twist, so you build it to be chiral? Is it possible?” Karunadasa answered that she believes it would be possible if one were to use organic molecules that are mostly rigid but that can twist at certain points.

Parkin followed up on that by referring to organic superconductors from a number of years ago in which the idea was to use long organic molecules to increase electron–electron coupling and thus strengthen the superconductivity of the materials. Could this have some application in the materials Karunadasa was talking about? Harunadasa said that such a thing might work with layers stacked with organic molecules between them and a certain twist angle between each of the layers, creating a spiraling material which, if doped, would create to a onedimensional conductor.

Next Ramesh offered two points of his own. The first was the question of whether it makes sense to think about moiré structures created by rotation around different axes than the 001 axis of rotation in twisted bilayer graphene. This choice of the axis of rotation is a degree of freedom that has not been discussed much, he said, and a different choice could lead to different options and properties. His second point was that it is important to think about the length of the innovation cycle for materials-based technologies. For some reason, he said, the number 18 years is like a magic number—a large number of innovations, from batteries to beer, take around 18 years from conception to being available for sale. So what would it take to reduce that by, say, a factor of four? The development cycle for software is more

like 3 years, so 18 years is not inevitable, but it would likely take doing a number of different things in parallel to decrease the materials innovation cycle by that much.

Franz then offered some comments on the role of theorists. From past experience, he said, it is clear that theorists are very effective in thinking of new things and predicting new phenomena in situations where the basic ingredients are relatively simple—like the case of graphene, for instance. It is a different issue, though, when the system is complicated, such as with the complex oxides and halides discussed earlier in the workshop. “I don’t know where to start modeling this type of a system,” he said. With that in mind, he commented that Ramesh’s idea of doing moiré structures in three dimensions was interesting, but “for this to be useful we need to develop some conceptual frameworks starting from some simple starting point.” One thing that came to mind, he said, was that one could start with Dirac and Weyl semi-metals because they are three-dimensional generalizations of graphene. “If you look at low energies, their spectrum is again simple,” he said. “There are point nodes that disperse linearly, but now this occurs in three dimensions rather than two. If we can devise some physical geometry where some moiré structure develops in three dimensions one could immediately start applying these tools that we’re familiar with.”

A second way in which theorists can be useful, he added, is explaining observed phenomena, although that can take time. It took 40 years for theorists to explain superconductivity in ordinary metals, he said, and theorists are still working to explain the high-temperature superconductors discovered more than three decades ago, although there have been other phenomena—such as the fractional quantum Hall effect—that theorists were able to explain in years rather than decades.

Parkin then asked Franz about his characterization of graphene as a simple system to study theoretically, pointing out that theorists cannot yet explain much of what has been observed in the magic-angle twisted bilayer graphene systems, such as orbital magnetism and superconductivity. Franz explained that the basic characteristics of these systems, such as flat bands and band folding, could be calculated and that such calculations told experimentalists such as Jarillo-Herrero where to look for some interesting phenomena. However, when one adds interactions to the system, it gets much more complicated, and even in a simple system such as twisted bilayer graphene, interactions are a problem for theorists. He predicted that theorists will eventually understand the graphene system with interactions, but it will require data from various experiments. “I doubt that anybody can calculate what the interacting phases are independent of any experiment,” he said. “That, I think, would be very difficult. But now that we have this input coming in from multiple groups, I think this understanding will come.”

Parkin then asked Franz whether the introduction of quantum computing systems would help theorists better predict the interactions in these systems, given that quantum computers would seem to be more consistent than existing computers

with the physical systems being modeled. Franz answered that he thinks quantum computing will help in some aspects, but certain areas—such as explaining superconductivity in the cuprates—may involve such complexities that even quantum computing will have diminished predictive powers. “I have great hopes for quantum computation and machine intelligence in the future,” he said, “but there will remain problems that are always challenging our ability to solve, especially on this complexity frontier.”

In response to a question from Parkin, Vishwanath then offered a brief explanation of a skyrmion-based theory he is working on to explain the superconductivity seen in magic-angle twisted bilayer graphene. “It is by no means the accepted mechanism,” he said, “but I think it has some interesting futures that seem to tie in with observations.” The first question to ask, he said, is, skyrmions of what? Think of an antiferromagnet on the lattice, spin up on the A sublattice and spin down on the B sublattice. (It is actually pseudo-spin, he noted, “but for our purposes this is sufficient.”) If there is a skyrmion of all these spins on the A sublattice, it will be followed by an anti-skyrmion on the B sublattice because of the antiferromagnetic coupling between the two. Because of special properties of the magic-angle twisted bilayer graphene, each skyrmion carries an electric charge of 2. “So if you have a finite density of electrons,” he said, “you end up getting a finite density of these fermions, and they could condense and give you a superconductor. And in simple models this works out.”

Next, Joe Checkelsky of the Massachusetts Institute of Technology and the moderator of the previous day’s discussion session, posed a question: What other communities should condensed matter physicists be reaching out to relative to this topic? Engineers? Mathematicians? Chemists? People from different areas use different language, and that can be helpful in understanding new phenomena. Ramesh suggested that it would be valuable to get organic chemists and device engineers involved. Jarillo-Herrero said that those communities are likely to get involved once physicists communicate their excitement about the materials and about the possibilities they represent. In that sense, communication with other communities will be vital, because once the chemists and engineers and others understand the opportunities, they will get involved.

Kapitulnik noted that most of the workshop’s focus had been on electronic/charge physics. So, he asked, what new opportunities do these moiré materials offer for spin physics or magnetism? Parkin answered that one interesting magnetic property in such materials might be long-range, periodic magnetism. “For example, if an electron goes through a periodic magnetic system it looks to the electron like some kind of a spin–orbit coupling effect,” he said. “So I think in this way, you could create artificial spin textures which could have interesting properties.”

Also, referring back to his previous comment about whether devices made from moiré materials might be impractically large to use in modern chip technology,

Parkin suggested that the issue might be sidestepped by moving to three dimensions. It is not clear how or whether that could be done, however.

In response to a question from Checkelsky about creating twisted layered materials that will not easily lose their twist when heated, Karunadasa said that the layered perovskite–non-perovskite heterostructure that she has built is stable, but that at this point she is not able to create it so that the layers are twisted relative to each other. Since they are created through a chemical self-assembly process in solution, the layers are fixed relative to one another. One approach would be to come up with a process by which the layers assemble in a twisted way, she said, but another would be to determine what is special about the moiré lattice—it is not the twist per se, but instead it is ultimately the electronic structure—and then come up with a way to capture that electronic structure in the heterostructure. “So, for example, I can think about engineering the material such that the two lattices have filled states of similar energy so that when they form a valence band, that valence band will have states from both materials,” she said. “Is that enough? That’s what I’m trying to figure out. What’s special about the moiré lattice that I can distill down to the electronic structure and then try to replicate that electronic structure in the heterostructure.”

Franz added that there is a vast world of interesting physics beyond just the electronic properties of these moiré materials. For example, it is not necessary to think about electrons when one is considering just the topological properties of moiré structures. “There’s been a whole field where people are exploring these various structures with topological properties for propagation of light, for propagation of acoustic waves, and all these phenomena,” he said. “I have a feeling that these moiré structures and wave propagation phenomena are the next frontier.”

Jarillo-Herrero reported that people have already made moiré and twisted phononic systems where they can slow down the phonons as well as photonic lattices where light can be slowed down. “People have built magic-angle acoustic graphene, they call it, and the same thing with photonic lattices where they can slow light because they can make a moiré structure where the velocity of light gets very, very small.

Referring to Karunadasa’s point about engineering the electronic structure in a lattice without a twist, Jarillo-Herrero said that people who work with cold atoms, because of the control they have over the system, have actually been able to make flat bands and magic-angle graphene without twisting the lattices. “All you need to engineer is the coupling,” he said “So the essence is not the twisting itself, the essence is the type of tunnel coupling you have between the two layers. So you can make the magic-angle graphene electronic structure without any twisting if you engineer the coupling between the cold-atom lattices.” Perhaps, he added, a similar thing could be accomplished in a chemical system.

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