Machine Learning Prediction of Magnetic Proximity Effect in van der Waals Heterostructures: From Atoms to Moiré

TL;DR

A machine learning framework that efficiently predicts large-scale proximity-induced magnetism in van der Waals heterostructures in van der Waals heterostructures is introduced, overcoming the high computational cost of density functional theory (DFT).

Abstract

We introduce a machine learning framework that efficiently predicts large-scale proximity-induced magnetism in van der Waals heterostructures, overcoming the high computational cost of density functional theory (DFT). We apply it to graphene/\CGT, which exhibits a previously unrecognized dichotomy. Unlike the spin polarization at the Fermi level, which follows the pseudospin, the proximity-induced magnetic moments vary across carbon atoms, defying analytical modeling. To address this, we develop an ensemble-based regression model trained on DFT data and employ local environment descriptors to map the local ($\sim 2$\,nm$^2$) atomic-scale geometry to the carbon magnetic moments. Besides demonstrating locality, the model reveals rich magnetic moiré textures. Crucially, this method can be broadly applied to orbital and spin proximity effects that are highly sensitive to local atomic environments and are beyond analytical description.

Figures (19)

• Figure 1: ML workflow for modeling magnetic proximity effects in graphene/Cr$_2$Ge$_2$Te$_6$. (a) 3D and top view of a representative DFT simulation cell. Proximity to the 2D Ising ferromagnet Cr$_2$Ge$_2$Te$_6$ induces a magnetic moment in the carbon atoms. (b) We employ DFT to determine the proximitized magnetization of the graphene layer in a set of heterostructures with varying sizes, twist angles $\theta$, and interlayer distances $d_\mathrm{IL}$. Together with the structural/geometrical information encoded by the SOAP descriptor, the data set is then used to train an ML regression model to predict the proximitized magnetization in much larger structures, as indicated.
• Figure 2: The accuracy of the ML model is largely determined by the descriptor's cut-off radius $r_\mathrm{cut}$ which defines the local environment. The prediction error saturates at $r_\mathrm{cut}=6$ Å.
• Figure 3: Two types of proximity effects. Presented are DFT simulations for graphene/CGT at twist angle $\theta=8.948\degree$. (a) Sublattice-determined spin polarization $P(E_F)$ of the carbon atoms in the supercell. The spin polarization has only two values, locked to the pseudospin, being of type (i). The pseudospin character of $P(E_F)$ is further reflected in the calculated projected density of states (PDOS) around Fermi energy $E_F$ for the two indicated sublattices, shown in (c). The curves are not smooth due to computational limitations. (b) Distribution of the carbon magnetic moment. The values are widely spread, showing its type (ii) pseudospin-breaking character. (d) Band structure for spin-down bands with color-coded projections on Te states. Anti-crossings (gray circles) indicate resonant hybridization, affecting the local proximity magnetic moments, between Te valence states and graphene $p_z$ orbitals.
• Figure 4: Magnetic moiré patterns emerging in larger simulation cells for various twist angles $\theta$ as predicted by the ML model. The scale bar in the last panel holds for all panels. Due to the hexagonal shape of both material layers, the pattern repeats with a 60$\degree$ periodicity and is symmetric about $\theta=30\degree$. Thus, $\theta=20\degree$ corresponds to $\theta=40\degree$.
• Figure 5: Correlation between DFT-calculated and ML-predicted values when using our home-made descriptor (see text).