Chalcogen Impurity Barriers in 2D Systems via Semi-Empirical/Machine Learning Modeling: A Survey over 4000 Materials

1. Problem Statement

The characterization of two-dimensional (2D) materials for applications in catalysis, sensing, and surface functionalization requires identifying structures with low energy barriers for impurity adsorption and migration. While Density Functional Theory (DFT) is the gold standard for calculating these properties, its high computational cost makes it infeasible for screening large datasets (e.g., the Computational 2D Materials Database, C2DB, which contains thousands of materials). There is a critical need for a scalable, data-driven framework that can rapidly estimate adsorption energy barriers for a vast number of 2D systems without sacrificing physical interpretability.

2. Methodology

The authors propose a hybrid workflow integrating semi-empirical quantum mechanics with advanced machine learning (ML) and interpretability protocols. The process involves five sequential steps:

• Data Acquisition: The study utilizes 4,036 distinct 2D materials from the C2DB.
• Semi-Empirical Calculations (EHM):
  • Instead of performing time-consuming geometry optimizations, the authors employ the Extended Hückel Method (EHM) using the YAeHMOP software.
  • Phenomenological Equilibrium Distance ($d_{eq}$): To bypass geometry optimization, $d_{eq}$ is estimated using a simple effective expression based on covalent radii: $d_{eq} = 2 r_{cov} (1 + \delta)$, where $\delta \approx 0.3$. This was validated against DFT results for alkali metals on graphene.
  • Energy Profiles: For each material, impurities (Sulfur, Selenium, and Tellurium) are displaced along three in-plane trajectories ($x$, $y$, and $xy$ diagonal) over ~3 Å. The average energy barrier ($\bar{E}_b$) is calculated from the maximum energy of these three paths.
• Descriptor Extraction:
  • Physicochemical descriptors are generated using the Matminer library and direct C2DB data.
  • The "Full Set of Descriptors" (FSD) includes formation energy, bandgap, thickness, and atomic averages (valence electrons, electronegativity, atomic number, radius, molar mass).
• Machine Learning Modeling:
  • Four supervised learning algorithms were trained and compared: Linear Regression, Multilayer Perceptron (MLP), Decision Tree, and XGBoost (Extreme Gradient Boosting).
  • Hyperparameters for XGBoost were optimized using the Optuna library.
  • The target variable is the average energy barrier ($\bar{E}_b$).
• Interpretability Analysis:
  • The SHAP (SHapley Additive exPlanations) method was applied to the best-performing model (XGBoost) to quantify feature importance and uncover physical correlations.
  • Independent validation was performed using Pearson correlation matrices and K-means clustering.

3. Key Contributions

• Scalable Framework: Demonstrated that combining semi-empirical methods (EHM) with ML allows for the rapid screening of >4,000 2D materials, a scale unattainable with pure DFT.
• Novel Phenomenological Approach: Introduced a simplified method to estimate equilibrium distances without geometry optimization, significantly reducing computational overhead while maintaining reasonable accuracy for screening purposes.
• Comprehensive Chalcogen Survey: Provided the first large-scale comparative study of S, Se, and Te impurity migration barriers across a diverse set of 2D materials.
• Interpretability in Materials Science: Successfully linked "black-box" ML predictions to physical mechanisms (e.g., electronic properties, surface topology) using SHAP, moving beyond mere prediction to physical insight.

4. Results

A. Benchmarking and Trends

• Graphene Benchmark: The EHM results for S, Se, and Te on graphene matched known DFT trends, correctly predicting the order of barriers: S > Se > Te.
  • S: Highest barriers (~1.0 eV), showing strong anisotropy.
  • Se: Intermediate barriers (~0.5 eV), reduced anisotropy.
  • Te: Lowest barriers (~0.23 eV), quasi-isotropic behavior due to larger radius and polarizability.
• Database Statistics: Of the 4,036 materials, ~3,150 had barriers < 5.0 eV. The mean barriers were 1.32 eV (S), 1.59 eV (Se), and 1.03 eV (Te).

B. Machine Learning Performance

• Model Comparison: XGBoost significantly outperformed Linear Regression, MLP, and Decision Trees.
  • For the 0–2.0 eV range, XGBoost achieved an $R^2$ of 0.528 (testing) with a Mean Absolute Error (MAE) of 0.251 eV.
  • Linear Regression and Decision Trees suffered from poor generalization (overfitting) and low $R^2$ values (< 0.43).
• Energy Window Optimization: Restricting the dataset to lower energy barriers improved accuracy:
  • For barriers ≤ 1.0 eV (1,495 materials), the test RMSE dropped significantly, and the train-test gap narrowed, indicating better model stability for low-barrier candidates.
  • Tellurium (Te) showed the most robust generalization, while Selenium (Se) showed the largest error reduction when narrowing the energy window.

C. Interpretability (SHAP Analysis)

• Dominant Descriptors: For all three impurities, the top predictors were consistently average valence number, electronegativity, and atomic number.
• Impurity-Specific Insights:
  • Sulfur (S): Highly sensitive to surface geometry and coordination (e.g., coordination number 1). Lower valence/atomic number and higher electronegativity correlate with lower barriers.
  • Selenium (Se): Shows high sensitivity to local geometric disorder (e.g., Steinhardt bond orientation parameters) and surface thickness.
  • Tellurium (Te): Primarily driven by average electronic properties rather than local structural variations, consistent with its weaker, more delocalized interactions.
• Validation: Pearson correlation and K-means clustering confirmed the SHAP findings, validating that thinner, denser structures generally reduce migration barriers.

5. Significance

This work establishes a powerful, scalable pipeline for the inverse design of 2D materials. By proving that semi-empirical methods coupled with interpretable ML can accurately predict adsorption dynamics, the authors provide a tool to:

1. Accelerate Discovery: Rapidly identify promising 2D candidates for catalysis and sensing from massive databases.
2. Guide Experiments: Offer physical insights (via SHAP) into why certain materials exhibit low barriers, guiding the selection of specific structural features (e.g., specific coordination environments or thickness).
3. Bridge Scales: Demonstrate a viable path from low-cost screening to high-fidelity DFT validation, where only the top candidates from the ML filter require expensive ab initio calculations.

The study concludes that while the current framework is not a replacement for ab initio calculations, it serves as an essential, high-throughput filter that balances speed, accuracy, and physical interpretability.