Predicting the Thermal Behavior of Semiconductor Defects with Equivariant Neural Networks

TL;DR

This work addresses the high cost of predicting temperature-dependent electronic properties of defects in semiconductors by introducing a unified, active-learning framework that couples equivariant MACE-based ML force fields for finite-temperature atomic dynamics with DeepH-E3 Hamiltonian learning for electronic structure. The approach is demonstrated on GaAs defects, achieving near-DFT accuracy for both structural dynamics and electronic structure while delivering substantial speedups, and enabling robust predictions of the temperature evolution of band gaps and defect levels. By preserving physical symmetries and leveraging active learning, the method maintains transferability across defect types, temperatures, and system sizes, offering a scalable path to explore defect physics in complex materials. The results point to practical applications in optoelectronics and photovoltaics and lay groundwork for extending to charged defects and higher-level electronic structure methods.

Abstract

The presence of defects strongly influences semiconductor behavior. However, predicting the electronic properties of defective materials at finite temperatures remains computationally expensive even with density functional theory due to the large number of atoms in the simulation cell and the multitude of thermally accessible configurations. Here, we present a neural network-based framework to investigate the electronic properties of defective semiconductors at finite temperatures efficiently. We develop an active learning approach that integrates two advanced equivariant graph neural networks: MACE for atomic energies and forces and DeepH-E3 for the electronic Hamiltonian. Focusing on representative point defects in GaAs, we demonstrate computational accuracy comparable to density functional theory at a fraction of the computational cost, predicting the temperature-dependent band gap of defective GaAs directly from larger scale molecular dynamics trajectories with an accuracy of few tens of meV. Our results highlight the potential of equivariant neural networks for accurate atomic-scale predictions in complex, dynamically evolving materials.

Figures (4)

• Figure 1: MPNN-based workflow for defect predictions.a. A preliminary MACE-based MLFF is trained on randomly displaced defective structures and used to run short MD simulations for further training. b. An accurate MLFF is obtained via active learning. MD-generated configurations are stored in a pool set, where force uncertainties from an ensemble of MACE models are used to select samples for DFT labeling. These are added to the labeled set for iterative retraining. Longer MD simulations are run and enter the pool set of Hamiltonian learning. c. DeepH-E3 models for Hamiltonian learning are also trained in an active learning scheme. Configurations in the pool set, which show high uncertainty in Hamiltonian matrix elements across an ensemble of DeepH models, are selected and ladded to the labeled set for iterative retraining. d. The final MLFF and Hamiltonian model are used together for predicting temperature temperature-dependent electronic properties of defects in semiconductors.
• Figure 2: Results of the active learning and accuracy of the MACE model in predicting atomic forces and dynamics in defective bulk GaAs.a. Evolution of the mean absolute error (MAE) in energies (blue solid line) and forces (orange dashed line) during successive rounds of active learning. Zero in the $x$-axis represents the model trained from randomly displaced configurations. b. Parity plot of MACE and DFT forces components for five types of defects: As interstitial (Asi), Ga interstitial (Gai), As vacancy (VAs), Ga vacancy (VGa), and As antisite (AsGa). Each data point represents a force component in a test structure, color-coded by defect type. The dashed line indicates perfect agreement. c. Vibrational density of states (VDOS) and d. radial distribution function, $g(r)$, calculated from AIMD (blue solid line) and MACE-predicted trajectories (orange dashed line), for bulk GaAs with the AsGa defect.
• Figure 3: Active learning and benchmarking of DeepH model for predicting the electronic structure in defective GaAs.a. Mean absolute error (MAE) of Hamiltonian matrix elements (blue solid line) and electronic eigenvalues for 30 bands around the band gap (orange dashed line) predicted by DeepH in comparison to DFT as a function of training set size. b, c. Electronic band structures of the neutral point defects Asi (panel b) and AsGa (panel c) obtained from DFT (red solid lines) and the DeepH model (blue dots). The data were obtained considering a random snapshot taken from a 432-atom supercell MD simulation with the MACE MLFF at 100 K.
• Figure 4: Temperature-dependent electronic properties of pristine and defective GaAs.a, b, c. Band gap of pristine GaAs (panel a) as well as as a function of temperature, b. Band gap of GaAs containing Asi and c. defect level energy, reported with respect to the valence band maximum, of Asi in GaAs as a function of temperature, computed by DFT (blue) and DeepH (orange). d. DeepH predicted defect level energy and broadening as a function of temperatures, where the conduction band minimum (CBM) is set to zero. The level broadening is defined as the standard deviation of the defect levels.