Machine learning makes twisted 2D materials simulable at a fraction of the cost

Twisting two sheets of 2D materials against each other creates so-called moiré patterns, and with them physical properties that the constituents lack. Understanding these properties requires quantum-mechanical simulation, which is extremely costly at moiré length scales. Researchers from the Max Planck Institute for the Structure and Dynamics of Matter (MPSD) and the University of York propose a machine learning framework that predicts the electronic structure of large twisted bilayers with first-principles accuracy. It captures extreme band-structure changes and ferroelectricity 10–100 times faster than direct density functional theory. The paper was recently published in the inaugural issue of PRX Intelligence.
 

Two-dimensional (2D) materials, for example graphene, are only one or a few atoms thick. Stack two such layers and twist one layer slightly against the other, and a new large-scale structure emerges: the atoms of the two layers line up differently from place to place, creating a repeating “moiré pattern” with a much longer period than the original lattice. This small twist can transform the materials’ properties significantly. For example, at a very specific “magic angle”, twisted bilayer graphene becomes a superconductor, carrying electricity with zero resistance. The choice of materials and twist angle offers high flexibility for realizing desired physical properties. However, attaining better control over these properties requires a deeper understanding that needs to be derived from simulations.

Density functional theory (DFT) is a widely used and successful computational method to simulate the electronic structure of many materials. Its computational cost usually scales cubically with the number of atoms per unit cell, and thus hits a prohibitive wall when simulating a large number of atoms, as in twisted bilayer materials. Zekun Lou, Alan Lewis, and Mariana Rossi have shown that a machine-learning (ML) tool called SALTED (Symmetry-Adapted Learning of Three-Dimensional Electron Densities), developed by Rossi's group and collaborators, can massively decrease this cost. SALTED is a “machine learning for density functional theory” (ML-DFT) method which learns the connection between structural geometries and electron densities. It predicts the electron densities of unseen structures from only atomic coordinates as input, then restarts the DFT calculation to obtain electronic properties, reducing computational cost. A trained SALTED model is 10 to 100 times faster than direct DFT for larger systems of this kind, while maintaining very high accuracy.

In this work, SALTED is applied to different 2D materials: it learns from a group of small untwisted bilayers that are cheap to calculate with DFT, then predicts the electron density of much larger twisted structures. An interesting finding of this study is the importance of embedding long-range physics into the SALTED model, something still rarely done in machine-learning architectures: most ML models only look at each atom’s immediate neighborhood and ignore effects from further away, such as charge rearrangements that span the whole moiré pattern. This proved essential to achieve a proper description of the band structure of more complex and polarizable 2D materials, such as hexagonal boron nitride (hBN) and molybdenum disulfide (MoS₂), whose electrons redistribute much more strongly than in graphene.

“As you twist the bilayers, the local stacking changes gradually across the whole pattern,” explains Zekun Lou. “This builds up a large-scale interlayer charge rearrangement or polarization, and only the long-range models can capture these effects.” In twisted hBN, for example, neighboring moiré regions carry opposite electric polarization, known as ferroelectricity. Mariana Rossi adds: “Machine learning models are usually evaluated by a single metric, such as the global density error, and that is not informative enough. A density error that is overall small but concentrated in one region can still distort the band structure significantly. Models should be validated explicitly against the properties you actually care about.”

This work advances the state of the art of ML-DFT in both model construction and evaluation, impacting many areas of materials research, including systematic screening of moiré materials, inverse design of moiré materials, and discovery of correlated quantum phases.