Moiré Materials Magic

Over the past couple of decades, researchers have learned how to prepare two-dimensional materials — crystals that are only one atom or one molecular layer thick — and measure their properties. When two-dimensional materials that have a small difference in lattice constant or in orientation are overlaid, they form a moiré pattern in which the coordination between layers changes slowly on a long moiré length scale. If the host two-dimensional crystals are semiconductors or semimetals, electronic and lattice properties are then accurately described by effective quantum models that have the periodicity of the moiré pattern instead of that of the host crystal. In this way, it is now possible to create artificial two-dimensional crystals, or ‘moiré materials,’ with lattice constants that are about 100 times larger than those of their naturally occurring counterparts. The large lattice constants make it possible to use electrical gates, like those that turn transistors off and on, to vary the number of electrons per artificial atom — the parameter that is varied across the rows and columns of the periodic table — and in this way to achieve a modern-day version of the alchemist’s dream.

Like their atomic-scale counterparts, each moiré material has properties that flow from complex quantum dynamics and have to be teased out by careful research. New moiré materials have the potential to realize new fundamental physics and also the potential to be valuable for new applications. At present, only a few different classes of moiré materials have been realized in the laboratory. It is nevertheless clear from the example of graphene bilayers, in which the speed of electrons is reduced to zero at magic twist angles, that moiré materials can have surprising properties. This project will use advanced quantum theory to identify new host two-dimensional crystals from which moiré materials can be formed and to derive models that describe their behavior on the moiré length scale. It will use mathematics to deepen understanding of the magic twist angles and the quasiperiodic Schrodinger operators that are realized in moiré materials that have been subjected to external magnetic fields or have two or more distinct moiré patterns.

Eric Cancès is a professor at École des Ponts (Paris) who works on mathematical and numerical analysis of molecular simulation models. He has developed widely used algorithms for electronic structure calculations, molecular dynamics and multiscale approaches. He notably invented robust self-consistent field algorithms for density functional theory, efficient techniques to explore potential energy surfaces and implicit solvent models for quantum chemistry and molecular dynamics. Cancès contributed to the mathematical analysis of mean-field and correlated electronic structure models, proving the existence of solutions for the Kohn–Sham LDA and GW models and setting up mathematically sound frameworks to describe the electronic structures of various kinds of aperiodic materials: crystals with local defects, disordered materials, incommensurate structures such as multilayer 2D materials. Cancès was also a pioneer in the numerical analysis of electronic structure models. He established the first proof of convergence for self-consistent field algorithms, a mathematical justification of the diffusion Monte Carlo algorithm and optimal error estimators for Kohn–Sham and related models.

Svetlana Jitomirskaya is a mathematician at Georgia Tech/University of California, Irvine whose major focus is on the spectral theory of ergodic — most notably quasiperiodic — operators. In particular, she has obtained fundamental results on the key models that describe 2D crystals in magnetic fields, such as the Aubry–Andre (aka Harper’s/Hofstadter’s) model, also known in math literature as the ‘almost Mathieu operator,’ and high-contrast models of graphene. She is acknowledged as a pioneer and leader in nonperturbative analysis of small denominator problems in spectral theory. Jitomirskaya is a member of the American Academy of Arts and Sciences since 2018. She will be a 2022 ICM Plenary speaker, was a 2002 ICM speaker, a 2006 ICMP Plenary speaker, and has received AMS Satter Prize (2005) and APS & AIP Dannie Heineman Prize for Mathematical Physics (2020).

Efthimios Kaxiras is a professor of physics and of applied mathematics at Harvard University. He is the founding director of the Institute for Applied Computational Science and served as the director of the Initiative on Innovative Computing. He has also served in faculty appointments and in administrative positions in Switzerland (EPFL) and in Greece (University of Crete, University of Ioannina, FoRTH). His research interests encompass a wide range of topics in the physics of solids and fluids, with recent emphasis on materials for renewable energy, especially batteries and photovoltaics, and on the physics and applications of two-dimensional materials. He serves on the editorial board of several scientific journals, has published over 400 papers in refereed journals and several review articles and chapters in books, as well as a graduate textbook on the properties of solids. His group has developed several original methods for efficient simulations of solids using high-performance computing as well as multiscale approaches for the realistic modeling of materials. He holds several distinctions, including Fellow of the American Physical Society and Chartered Physicist and Fellow of the Institute of Physics (UK).

Lin Lin is an associate professor of mathematics at University of California, Berkeley focusing on the development of new numerical methods for complex first principles electronic structure theory calculations. He developed the PEXSI (pole expansion and selected inversion) method, a Green’s function–based fast algorithm that enables high efficiency first principle computation of quasi-2D quantum systems and has been used to perform ab initio simulation in systems with more than ten thousand atoms. The PEXSI method has been integrated into a number of community electronic structure software packages. He also developed the selected columns of density matrix (SCDM) method which has been integrated into the Wannier90 code, the adaptively compressed polarizability (ACP) method, which reduces the asymptotic scaling of large-scale phonon calculations, and semi-definite relaxation methods for improving the robustness of the dynamical mean-field theory and density-matrix embedding theory calculations.

Mitchell Luskin is a professor of mathematics at the University of Minnesota who has developed mathematical formulations and analysis to model and compute material properties with defects and microstructure. His early work gave numerical analysis and computational methods for defects in the molecular orientation of liquid crystals and for macroscopic properties of materials with fine scale sub-grid oscillation between symmetry-related states (frustration). Subsequent work proposed and analyzed hybrid atomistic-to-continuum coupling methods for materials with defects and developed and analyzed accelerated and thermostatted molecular dynamics methods. Luskin’s recent work has focused on the development of mathematical foundations and computational methods for 2D van der Waals heterostructures. He and his research group have developed real and momentum space methods using configuration space to model, compute and analyze the electronic properties (local density of states, density of states, Kubo conductivity) and mechanical properties (relaxation) of general incommensurate 2D multilayer heterostructures without supercell or continuum approximation. More recently, Luskin and Watson have given a rigorous proof of the existence of the first magic angle for the chiral model of bilayer graphene.

Allan MacDonald is a condensed matter theorist at University of Texas at Austin with a focus on new or ununderstood phenomena related to the quantum physics of interacting electrons in materials. Among other topics, he has made theoretical contributions to theories of the integer and fractional quantum Hall effects, spintronics in metals and semiconductors, topological Bloch bands, correlated electron-hole fluids and exciton and polariton condensation, and two-dimensional materials. In 2010, MacDonald predicted that it would be possible to realize strong correlation physics in graphene bilayers twisted to a magic relative orientation angle, foreshadowing the rise of twistronics and moiré materials. His recent work is focused on anticipating new physics in moiré superlattices, and on achieving a full understanding of the magic-angle graphene and transition-metal dichalcogenide systems. MacDonald is a fellow of the National Academy of Sciences and the American Academy of Arts and Sciences.

Angel Rubio is the director of the Theory Department of the Max Planck Institute for the Structure and Dynamics of Matter and a distinguished research scientist at the Simons Foundation’s Flatiron Institute (NY, USA). His research interests are rooted to the modeling and theory of electronic and structural properties of condensed matter as well as to the development of new theoretical tools to investigate the electronic response of materials, 2D materials, nanostructures and hybrid materials to external electromagnetic fields. He is acknowledged as pioneer and leader in the area of computational materials physics. In the last few years, he has pioneered the development of the theoretical framework of quantum electrodynamical density functional theory (QEDFT) that enables the ab initio modeling of strong light-matter interaction phenomena in materials, nanostructures and molecules, opening the new field of polaritonic chemistry. Rubio is a fellow of the American Physical Society and the American Association for the Advancement of Science, a member of the Academia Europaea, European Academy of Sciences and a foreign associate member of the National Academy of Sciences.

Maciej Zworski works on mathematical aspects of classical/quantum (particle/wave) correspondence, in particular on its manifestations in the theory of partial differential equations — that is, on microlocal analysis. His focus has been the study of scattering resonances which appear in different guises in mathematics and physics — from zeros of zeta functions to quasinormal modes of gravitational waves. His recent interests also include semiclassical problems in condensed matter physics such as the distributions of magic angles in twisted bilayers, where the small angle is the semiclassical parameter. He is involved in not-for-profit scientific publishing through his work at Mathematical Science Publishers. He received his Ph.D. from Massachusetts Institute of Technology in 1989 under the direction of Richard Melrose, and moved to University of California, Berkeley in 1998, after holding positions at Harvard, Johns Hopkins and the University of Toronto. He is a fellow of Royal Society of Canada and of the American Academy of Arts and Sciences.