Active Learning Guided Computational Discovery of 2D Materials with Large Spin Hall Conductivity

Technical Summary: Active Learning Guided Computational Discovery of 2D Materials with Large Spin Hall Conductivity

Problem Statement
The discovery of two-dimensional (2D) materials with intrinsically high Spin Hall Conductivity (SHC) is critical for next-generation spintronic devices, which rely on the conversion of longitudinal charge currents into transverse spin currents via spin-orbit coupling (SOC). However, identifying such materials is hindered by the vast chemical space of 2D systems and the prohibitive computational cost of conventional first-principles methods. While high-throughput screening exists, it often relies on algorithmic simplifications that compromise accuracy, particularly in the construction of tight-binding (TB) models using Maximally Localized Wannier Functions (MLWFs). Errors in automated MLWF generation for dispersive band structures can significantly degrade SHC prediction accuracy.

Methodology
The authors developed an active learning framework that integrates Density Functional Theory (DFT), tight-binding modeling, and machine learning (ML) to efficiently navigate the chemical space of approximately 2,000 2D materials from the Materials Cloud Two-Dimensional Structure Database (MC2D).

1. Computational Workflow:

   • DFT: Electronic band structures (EBS) were calculated using Quantum ESPRESSO with GGA-PBE functionals and norm-conserving pseudopotentials.
   • MLWF & TB Modeling: To ensure accuracy, MLWFs were constructed manually (user-guided) from DFT wavefunctions using Wannier90, avoiding the errors associated with fully automated high-throughput generation. These MLWFs formed the basis for a minimal basis set TB Hamiltonian.
   • SHC Calculation: The Spin Hall Conductivity was computed using the Kubo formalism. Spin-orbit coupling (SOC) was introduced as an additional term to the Hamiltonian, with parameters tuned to match DFT results. The calculation was performed on a dense k-grid ($800 \times 800 \times 1$).

2. Active Learning Loop:

   • Initial Dataset: The process began with a diverse training set of 24 2D systems covering various chemical families and symmetries.
   • Feature Engineering: Models were trained using 158 features, including 132 elemental features from the matminer library (e.g., covalent radius, electronegativity, atomic weight) and 26 hand-crafted symmetry features (e.g., presence of inversion centers, mirror planes).
   • Model Selection: A symmetric log transformation was applied to the target SHC values due to their wide range. Among Ridge Regression, Support Vector Regression, and Gaussian Process Regression, Ridge Regression consistently yielded the lowest cross-validation error and was selected for screening.
   • Acquisition Strategy: The "Expected Improvement" (EI) acquisition function was used to balance exploration (high uncertainty) and exploitation (high predicted SHC). In each of three active learning loops, 5–7 candidates with high EI values were shortlisted for rigorous DFT-SOC+TB validation.
   • Iteration: Newly computed SHC values were added to the training set to retrain the model for the subsequent round.

Key Results

• Discovery of High-SHC Candidates: The active learning strategy successfully identified materials with significantly enhanced SHC. The top-performing candidate, Fe$_2$TeSe (discovered in Round 3), exhibited an SHC of 271.52 $\hbar/e\Omega^{-1}$.
• Performance Improvement: This value is approximately 23 times higher than the best candidate found in the initial random selection (AuSe, 11.7 $\hbar/e\Omega^{-1}$). The second-best candidate, K$_2$PtTe$_2$, showed a 16-fold increase over the initial best.
• Model Accuracy: While the ML model struggled to predict extreme outliers with perfect precision (a known challenge in active learning), it successfully guided the search toward regions of the chemical space with higher SHC values, as evidenced by the consistent rightward shift in SHC distributions across the three rounds.
• Chemical Insights: SHAP (SHapley Additive exPlanations) and correlation analyses revealed that high SHC is not solely determined by heavy elements or broken inversion symmetry. Key factors included:
  • Metallicity: 9 of the top 10 candidates were metallic.
  • Orbital Character: Dominance of $d$-orbitals near the Fermi energy ($E_F$), even from non-heavy transition metals (e.g., Fe, Mn, Cu).
  • Symmetry: The absence of roto-inversion symmetry was a strong indicator of high SHC. Conversely, the presence of multiple mirror planes or non-symmorphic space groups did not consistently correlate with high SHC in this dataset.
  • Covalent Radius: A large difference between the maximum and minimum covalent radii of constituent elements (Magpie feature) was the most influential predictor.

Significance and Claims
The authors claim that this work demonstrates the efficacy of combining active learning with accurate, physics-based models (DFT-SOC+TB) to accelerate materials discovery in large chemical spaces. The study highlights that:

1. Efficiency: Active learning can identify superior materials with significantly fewer computational resources than exhaustive screening.
2. Complexity of SHC: High SHC arises from a complex interplay of factors including metallic character, specific orbital symmetries near $E_F$, and specific symmetry breaking (lack of roto-inversion), rather than simple heuristics like the presence of heavy atoms alone.
3. Data Availability: The generated dataset and trained ML models are made publicly available to facilitate further research in 2D spintronics.
4. Validation: The top candidates, including Fe$_2$TeSe, are predicted to be dynamically stable and exfoliable, making them viable targets for future experimental investigation.

The paper concludes that while the ML model serves as a powerful screening tool, the integration of user-guided, high-fidelity TB modeling is essential for obtaining reliable SHC values, particularly for systems with complex band structures.