High-throughput discovery of high Curie point two-dimensional ferromagnetic materials

Recent demonstrations of ferromagnetism in monolayer CrI3 and Cr2Ge2Te6 at low temperature—and even room-temperature ferromagnetism reported for VSe2 and MnSe2—have energized research into 2D spintronic, valleytronic, sensing and memory materials. While the Mermin–Wagner theorem forbids long-range order in 2D isotropic Heisenberg systems, strong magnetocrystalline anisotropy in 2D materials can enable ferromagnetism. Numerous 2D ferromagnets have been computationally predicted, and large databases exist, but they lack Curie temperatures (Tc) due to the complex, manual nature of Tc estimation: enumerating ground/low-energy spin states, identifying exchange mechanisms, and mapping to spin models for Monte Carlo simulations. The reliability of Tc predictions based on simplified models (e.g., Ising) is questionable for materials with modest anisotropy. Building on a recent algorithm that efficiently predicts collinear ground/low-energy spin states in bulk materials, this work develops an automated first-principles-to-Monte Carlo pipeline to accurately predict Tc for 2D magnetic materials and to enable high-throughput discovery. The goal is to find practically relevant 2D ferromagnets with high Tc and to complement physics-based simulations with a data-driven ML model for rapid screening.

The paper situates itself within prior computational discovery of 2D ferromagnets and the development of databases (e.g., C2DB, 2DMatPedia). It highlights limitations of existing Tc estimates that often use Ising models, potentially overestimating Tc for materials with moderate anisotropy. It references the Mermin–Wagner theorem and the role of magnetocrystalline anisotropy in enabling 2D ferromagnetism, as well as the Goodenough–Kanamori rules for superexchange that govern expected FM/AFM behavior in bulk. The study builds on an algorithm capable of near-exhaustive, symmetry-based enumeration of collinear magnetic states to remove heuristics and better identify true ground and low-energy states. It also notes recent ML models predicting Tc for bulk materials and the scarcity of reliable 2D Tc data, motivating a tailored ML approach for 2D materials.

Automated workflow: Starting from a unit-cell structure, the pymatgen magnetism module generates symmetry-allowed FM and AFM spin configurations, removing heuristic bias. Structures are relaxed with spin-polarized DFT+U and collinear energies are computed. If the ground state is AFM, the material is discarded. Non-collinear DFT including spin–orbit coupling computes magnetocrystalline anisotropy energy (MAE) and the easy magnetization axis. Local magnetic moments (from Bader analysis) provide spin magnitudes. Heisenberg Hamiltonian: Collinear energies are fit to an anisotropic Heisenberg model including exchange couplings for up to 4 neighbor shells (J1–J4) and single-ion anisotropy constants (kx, ky, kz). Spin magnitudes S are derived from local moments. Fitting remedies (e.g., dropping the most unstable AFM and reducing neighbor order) are applied if the linear system is singular until a physically meaningful fit is found. Monte Carlo simulations: A large supercell (e.g., 50×50, thousands of spins) is constructed. Neighbors are mapped using a GPU-accelerated search. A just-in-time compiled (numba) Metropolis single-spin update MC code simulates temperature-dependent magnetization and susceptibility to extract Tc from the Heisenberg model. For materials with negligible in-plane anisotropy (XY magnets), Tc is estimated from an XY-model BKT relation: Tc = 0.89/(8kB) × (EAFM – EFM) per atom. Database search: The Computational 2D Materials Database (C2DB) was selected due to a large set of materials classified as FM (786), systematic coverage of layered materials and substitutions, and available stability/electronic properties. The pipeline reassesses FM vs AFM by exhaustive spin configuration enumeration. Machine learning: From 157 materials with computed Tc, features are generated from structures and compositions using automatminer/matminer. TPOT performs stochastic model and hyperparameter searches over extensive pipelines. For regression (Tc prediction), a pipeline using SelectPercentile, ZeroCount, and GradientBoostingRegressor is selected. For FM/AFM classification on 525 labeled samples, a pipeline using SelectPercentile, MaxAbsScaler, and ExtraTreesClassifier is chosen. The trained pipelines are evaluated on a separate test set of reported and database-sourced 2D materials with new structures and compositions. DFT setup and U values: Calculations use VASP (PAW, PBE-GGA), with GPU acceleration utilized. GGA+U is applied for correlated d-electrons. Cutoff energy 520 eV; dense k-point settings for static and MAE runs; stringent electronic and ionic convergence; large out-of-plane vacuum (>25 Å). Effective U values for common transition metals are taken from Materials Project calibrations; for others with low magnetization, material-averaged linear-response U values are determined and applied. For the 26 most promising high-Tc materials, material-specific U values are also computed via linear response for refined Tc estimates. Software engineering: The Python codebase uses numba JIT to accelerate MC kernels and GPU acceleration for neighbor mapping, enabling high-throughput runs on workstation-grade hardware (multi-GPU) with scalability to HPC. AIMD stability: Ab initio MD at 400 K (NVT, Nosé–Hoover) for 6.5 ps on large supercells (≥144 atoms) assesses structural stability of selected high-Tc candidates.

- Reclassification of C2DB FM entries: Of 786 materials labeled FM, 368 (~47%) are AFM upon exhaustive symmetry-based exploration and DFT evaluation. - Tc computed for 157 2D ferromagnets (including 12 XY magnets). These span >20 structural prototypes. - High-Tc discoveries: 26 materials have Tc > 400 K; 32 have Tc ≥ 300 K, indicating potential for practical devices. Some involve metals (Rh, Ru, Mo, W, Sc, Ti, Zr) typically showing low bulk magnetism but exhibit sizable 2D moments (≈0.59–3.96 μB/atom) and high Tc. - Multi-neighbor vs N1-only models: In most cases, Tc from only nearest-neighbor exchange (Tc′) is close to the multi-neighbor Tc (Tc_exact), except for prototypes with multiple metal layers (e.g., CH, GaSe, CdI2-like MXenes including Ti2CO2, Ti2CH2O2) and square/rectangular lattices (FeOCl-, FeSe-, GeS-, NiSe-type), where further-neighbor exchanges are critical. - Experimental validation: Predicted Tc for CrI3, Cr2Ge2Te6, and MnSe2 closely match experiments without manual parameter tuning. For T-phase VSe2, Tc ≈ 114.33 K is predicted; discrepancies with room-temperature experimental reports are attributed to substrate effects and in-plane magnetism consistent with experiments. - U-value refinement: Material-specific linear-response U often raises Tc; among 26 high-Tc candidates, all but MoIN_Pmmn remain near or above 400 K with refined U, and some show significant Tc enhancement. - ML performance: Tc regression pipeline attains average CV MSE ≈ 94.57 K^2 on training; generalization MSE on test ≈ 30,335.46 K^2, with errors attributed to unseen crystal chemistries/structures. FM/AFM classifier achieves ≈72.89% training accuracy and ≈73.17% on a 123-sample test set. - New candidates from ML: Identification of high-Tc materials such as CrO2_P4/mmm and ZnNi2O5; confirmation of high-Tc character in Cr3Te4 with Tc lower than prior Ising-model estimates but higher than bulk experimental values. - Goodenough–Kanamori violations: Multiple cases where the expected superexchange angle rule (180° → AFM, 90° → FM) fails, notably strong FM in planar CrO2_P4/mmm with 180° cation–anion–cation angle, likely due to high covalency and 2D bonding characteristics. - Thermal stability (AIMD 400 K, 6.5 ps): CrIN_Pmmm, RhCl2_C2/m, and ZnNi2O5_Pmmn maintain structure (with reduced crystallinity), whereas Mn2H2CO2_P-3m1 (MXene) melts after ~3 ps, indicating limited applicability despite high predicted Tc.

The study addresses the lack of Curie temperatures in 2D magnetic materials databases by delivering an automated, rigorous DFT-to-Heisenberg-MC pipeline capable of high-throughput Tc prediction. The reclassification of many supposed ferromagnets as antiferromagnets underscores the need for exhaustive, symmetry-driven magnetic configuration searches and consistent DFT settings. Close agreement with experimental Tc for benchmark materials validates the modeling choices, including the anisotropic Heisenberg model and MAE inclusion, avoiding overestimation seen with Ising approximations. The findings reveal that for many 2D systems, nearest-neighbor exchange suffices for Tc estimation, but multi-neighbor coupling is essential for structures with multiple metal layers or square/rectangular lattices where direct exchange and small N1–N2 separations matter. Observed violations of Goodenough–Kanamori rules suggest that 2D covalent bonding and reduced dimensionality can qualitatively alter exchange mechanisms compared to bulk expectations. The ML pipeline demonstrates that generalized chemical and structural features can capture aspects of the complex Tc-determining physics, but generalization suffers with limited and non-representative training data and structural novelty in the test set. Nonetheless, ML successfully flags additional high-Tc candidates, complementing physics-based screening. Thermal stability checks highlight that not all high-Tc predictions translate to viable free-standing monolayers, emphasizing the role of substrates and finite-temperature effects. Overall, the results enable prioritized experimental targeting of high-Tc 2D ferromagnets and provide methodological guidance on when simplified exchange models are adequate versus when higher-order interactions and material-specific U values are needed.

The work delivers an automated, GPU-accelerated DFT+U to anisotropic Heisenberg Monte Carlo pipeline that computes Curie temperatures for 2D magnetic materials in high throughput. Screening of C2DB and related sources yields 157 materials with computed Tc, including 26 with Tc > 400 K and 32 with Tc ≥ 300 K. Predictions align with experimental benchmarks and uncover cases where multi-neighbor interactions are essential and Goodenough–Kanamori rules break down in 2D. A complementary ML model, trained on 157 data points, identifies additional promising candidates and provides rapid estimates, though broader training data are needed for robust generalization. AIMD assessments suggest some high-Tc candidates retain structural integrity at 400 K, while others (e.g., certain MXenes) may be unsuitable as free-standing layers. Future directions include expanding the dataset for ML to improve generalization and potentially replace costly simulations, incorporating substrate/strain effects and charge density wave distortions into stability and magnetic property predictions, and systematically determining material-specific U values in a scalable way to refine Tc estimates in high-throughput workflows.

- Data and model scope: Only 157 materials yielded stable Tc estimates; 261 cases failed due to symmetry recognition issues, non-magnetism upon relaxation, DFT convergence problems, inability to stabilize AFM states, ill-conditioned Hamiltonian fits, or phase changes upon relaxation. - Model assumptions: The Heisenberg approach may not capture itinerant magnetism (e.g., Fe3GeTe2) or strong spin fluctuations beyond the classical treatment; XY model approximations are used when in-plane anisotropy is negligible. - U parameter sensitivity: Tc depends on Hubbard U; high-throughput uses element-level U approximations, while material-specific linear-response U is computationally expensive but can significantly change Tc. - Generalization of ML: Limited training size and structural novelty reduce predictive accuracy on unseen materials; feature sets may not fully capture subtle exchange and anisotropy physics. - Stability considerations: Some predicted high-Tc monolayers show dynamic or thermal instability as free-standing layers; real-device behavior may depend on substrates, strain, and possible CDW distortions. - Computational constraints: Despite acceleration, MC simulations and MAE calculations remain costly, especially with higher coordination and multi-neighbor interactions.