Accelerated Discovery of Materials with Extreme Work Functions through Uncertainty-Aware Multi-Fidelity Screening

Technical Summary: Accelerated Discovery of Materials with Extreme Work Functions through Uncertainty-Aware Multi-Fidelity Screening

Problem Statement
The work function ($\Phi$) is a critical electronic property governing electron exchange at interfaces, essential for technologies ranging from thermionic emission and field emission to catalysis and organic electronics. Materials with extreme work functions—specifically low values ($\Phi < 2.5$ eV) for efficient electron emission and high values ($\Phi > 6.0$ eV) for effective hole transport or electron blocking—are highly sought after. However, predicting work functions is inherently complex due to crystallographic anisotropy, surface relaxation, reconstruction, and the influence of adsorbates. While Density Functional Theory (DFT) is the standard for calculation, it is computationally expensive for high-throughput screening of millions of candidates. Conversely, existing Machine Learning (ML) models, such as Random Forest (RF) regressors, offer speed but often lack reliable uncertainty quantification and domain of applicability (DoA) assessment, leading to potentially misleading confidence in predictions for unseen materials or extreme values.

Methodology
The authors developed a scalable, multi-fidelity screening workflow integrating an augmented ML model with computational simulations to screen $\approx$5.5 million compounds from the GNoME and Alexandria databases.

1. Augmented Random Forest Model:

   • Base Model: Built upon a previously published RF model trained on 58,332 unrelaxed DFT-calculated work functions.
   • Uncertainty Calibration: The raw uncertainty estimates (standard deviation of tree predictions) were calibrated using a calibrated bootstrap method. This process linearly rescales the uncertainty to ensure the resulting $z$-scores follow a standard normal distribution, aligning the predicted error bars with observed true errors.
   • Domain of Applicability (DoA): A Kernel Density Estimation (KDE) model was fitted to the training feature space to quantify the dissimilarity of new candidates. A dissimilarity score ($d$) was calculated; predictions with $d > 0.97$ were flagged as out-of-domain (OD) and filtered out to prevent extrapolation errors.

2. Multi-Fidelity Screening Workflow:

   • Initial Filtering: From GNoME and Alexandria, compounds were filtered for non-radioactive elements, thermodynamic stability ($E_{hull} \le 50$ meV/atom), and metallic character (using a graph neural network to predict HSE-equivalent band gaps, excluding those with $E_{gap} > 0$).
   • Surface Enumeration: Low-index surfaces ((100), (001), (010), (110), (101), (111)) were enumerated for $\approx$355,000 bulk compounds, generating $\approx$11 million unique surface terminations.
   • ML Screening: The augmented RF model predicted work functions, calibrated uncertainties, and DoA scores. Only in-domain (ID) predictions ($d \le 0.97$) were retained.
   • Surface Relaxation: Approximately 20,000 top-ranked surfaces (extreme low and high $\Phi$) underwent geometry relaxation using the universal Machine Learning Interatomic Potential (U-MLIP) M3GNet. This step was crucial as surface relaxation significantly alters work function values.
   • Targeted DFT: Final high-fidelity validation was performed using DFT (VASP, PBE functional) on the most promising relaxed surfaces to confirm work function values.

Key Results

• Model Performance: The augmented RF model achieved an $R^2$ of 0.995 and MAE of 0.032 eV on the full dataset. Cross-validation demonstrated strong generalization to unseen materials (MAE $\approx$ 0.12 eV), though extrapolation to values outside the training distribution (e.g., $<2$ eV or $>10$ eV) remained challenging without DoA filtering.
• Uncertainty and DoA: Calibration significantly improved the statistical consistency of uncertainty estimates, bringing the standard deviation of $z$-scores from 0.676 to 0.999. The DoA filter ($d \le 0.97$) successfully separated reliable predictions from unreliable ones, reducing the miscalibration area and ensuring that follow-up resources were focused on high-confidence candidates.
• Screening Outcomes: The workflow identified:
  • 209 surfaces with extreme low work functions ($\Phi_{DFT} < 2.0$ eV), corresponding to 136 unique materials.
  • 227 surfaces with extreme high work functions ($\Phi_{DFT} > 6.0$ eV), corresponding to 172 unique materials.
• Chemical Trends:
  • Low $\Phi$: Surfaces terminated by alkali metals, alkaline-earth metals, and lanthanides consistently exhibited low work functions. Notably, lanthanide-rich terminations were strongly associated with extremely low values.
  • High $\Phi$: Surfaces containing metalloids, phosphorus, sulfur, or selenium at the top layer correlated with exceptionally high work functions.
  • Feature Importance: SHAP analysis revealed that the minimum electronegativity of the top layer elements was the dominant predictor, followed by Mendeleev number and first ionization energy.

Significance and Claims
The paper claims to demonstrate a scalable strategy that leverages uncertainty-aware ML and multi-fidelity computational efforts to accelerate the discovery of materials with extreme work functions. By integrating uncertainty calibration and DoA assessment, the authors provide a framework that enhances prediction robustness and filters out unreliable extrapolations, which is critical for autonomous materials discovery. The study highlights that surface relaxation is a non-negotiable step in screening workflows, as it can shift work function values by hundreds of meV. Furthermore, the work uncovers both established chemical principles (e.g., alkali termination lowering $\Phi$) and less conventional motifs (e.g., lanthanide-rich or phosphorus-terminated surfaces), offering a data-driven foundation for designing materials for advanced electronic, energy-conversion, and catalytic applications. The authors emphasize that while the current study focused on metallic compounds, the methodology provides a pathway for future work on semiconductors and insulators.