Predicting temperature-dependent optoelectronic properties of semiconductor defects with equivariant neural networks
Source abstract
Abstract
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.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Machine Learning in Materials Science · Quantum many-body systems · Advanced Physical and Chemical Molecular Interactions
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (64)
Show source-provided references
1. Pantelides, S. T. The electronic structure of impurities and other point defects in semiconductors. Rev. Mod. Phys. 50, 797–858 (1978).10.1103/RevModPhys.50.797 ↗
2. Yu, P. Y. & Cardona, M. Graduate Texts in Physics. In Fundamentals of Semiconductors: Physics and Materials Properties (Springer Berlin Heidelberg, 2010).10.1007/978-3-642-00710-1 ↗
3. Sze, S. M., Li, Y. & Ng, K. K. Physics of Semiconductor Devices (John Wiley & Sons, 2021).
4. Park, J. S., Kim, S., Xie, Z. & Walsh, A. Point defect engineering in thin-film solar cells. Nat. Rev. Mater. 3, 194–210 (2018).10.1038/s41578-018-0026-7 ↗
5. Kirchartz, T. et al. The state of the art in photovoltaic materials and device research. Nat. Rev. Mater. 10, 335–354 (2025).10.1038/s41578-025-00784-4 ↗
6. Freysoldt, C. et al. First-principles calculations for point defects in solids. Rev. Mod. Phys. 86, 253–305 (2014).10.1103/RevModPhys.86.253 ↗
7. Puska, M. J., Pöykkö, S., Pesola, M. & Nieminen, R. M. Convergence of supercell calculations for point defects in semiconductors: vacancy in silicon. Phys. Rev. B 58, 1318–1325 (1998).10.1103/PhysRevB.58.1318 ↗
8. Dove, M. T. Introduction to Lattice Dynamics 1st edn (Cambridge University Press, 1993).10.1017/CBO9780511619885 ↗
9. Youssef, M. & Yildiz, B. Intrinsic point-defect equilibria in tetragonal ZrO2: density functional theory analysis with finite-temperature effects. Phys. Rev. B 86, 144109 (2012).10.1103/PhysRevB.86.144109 ↗
10. Glensk, A., Grabowski, B., Hickel, T. & Neugebauer, J. Breakdown of the Arrhenius Law in describing vacancy formation energies: the importance of local anharmonicity revealed by ab initio thermodynamics. Phys. Rev. X 4, 011018 (2014).
11. Qiao, S. et al. Temperature effect on charge-state transition levels of defects in semiconductors. Phys. Rev. B 105, 115201 (2022).10.1103/PhysRevB.105.115201 ↗
12. Cohen, A. V., Egger, D. A., Rappe, A. M. & Kronik, L. Breakdown of the static picture of defect energetics in halide perovskites: the case of the Br vacancy in CsPbBr3. J. Phys. Chem. Lett. 10, 4490–4498 (2019).10.1021/acs.jpclett.9b01855 ↗
13. Wang, B. et al. Electron-volt fluctuation of defect levels in metal halide perovskites on a 100 ps time scale. J. Phys. Chem. Lett. 13, 5946–5952 (2022).10.1021/acs.jpclett.2c01452 ↗
14. Lodeiro, L. et al. Methodological issues in first-principle calculations of CH3NH3PbI3 perovskite surfaces: quantum confinement and thermal motion. ACS Omega 5, 29477–29491 (2020).10.1021/acsomega.0c04420 ↗
15. Delgado, F. P., Simões, F., Kronik, L., Kaiser, W. & Egger, D. A. Machine-learning force fields reveal shallow electronic states on dynamic halide perovskite surfaces. ACS Energy Lett. 10, 3367–3374 (2025).10.1021/acsenergylett.5c01519 ↗
16. Mosquera-Lois, I. & Walsh, A. Dynamic vacancy levels in CsPbCl3 obey equilibrium defect thermodynamics. PRX Energy 4, 043008 (2025).10.1103/dxmb-8s96 ↗
17. Gilmer, G. H., Diaz de la Rubia, T., Stock, D. M. & Jaraiz, M. Diffusion and interactions of point defects in silicon: molecular dynamics simulations. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 102, 247–255 (1995).10.1016/0168-583X(95)80150-K ↗
18. Grabowski, B., Ismer, L., Hickel, T. & Neugebauer, J. Ab initio up to the melting point: Anharmonicity and vacancies in aluminum. Phys. Rev. B 79, 134106 (2009).10.1103/PhysRevB.79.134106 ↗
19. Leyssale, J.-M. & Vignoles, G. L. A large-scale molecular dynamics study of the divacancy defect in graphene. J. Phys. Chem. C. 118, 8200–8216 (2014).10.1021/jp501028n ↗
20. Hizhnyakov, V., Haas, M., Shelkan, A. & Klopov, M. Theory and molecular dynamics simulations of intrinsic localized modes and defect formation in solids. Phys. Scr. 89, 044003 (2014).10.1088/0031-8949/89/04/044003 ↗
21. Butler, K. T., Davies, D. W., Cartwright, H., Isayev, O. & Walsh, A. Machine learning for molecular and materials science. Nature 559, 547–555 (2018).10.1038/s41586-018-0337-2 ↗
22. Schmidt, J., Marques, M. R. G., Botti, S. & Marques, M. A. L. Recent advances and applications of machine learning in solid-state materials science. npj Comput. Mater. 5, 1–36 (2019).10.1038/s41524-019-0221-0 ↗
23. Himanen, L., Geurts, A., Foster, A. S. & Rinke, P. Data-driven materials science: status, challenges, and perspectives. Adv. Sci. 6, 1900808 (2019).10.1002/advs.201900808 ↗
24. Unke, O. T. et al. Machine learning force fields. Chem. Rev. 121, 10142–10186 (2021).10.1021/acs.chemrev.0c01111 ↗
25. Li, Z. et al. Machine learning in concrete science: applications, challenges, and best practices. npj Comput. Mater. 8, 1–17 (2022).10.1038/s41524-022-00810-x ↗
26. Kulik, H. J. et al. Roadmap on machine learning in electronic structure. Electro Struct. 4, 023004 (2022).10.1088/2516-1075/ac572f ↗
27. Goryaeva, A. M. et al. Efficient and transferable machine learning potentials for the simulation of crystal defects in bcc Fe and W. Phys. Rev. Mater. 5, 103803 (2021).10.1103/PhysRevMaterials.5.103803 ↗
28. Schattauer, C., Todorović, M., Ghosh, K., Rinke, P. & Libisch, F. Machine learning sparse tight-binding parameters for defects. npj Comput. Mater. 8, 1–11 (2022).10.1038/s41524-022-00791-x ↗
29. Pols, M., Brouwers, V., Calero, S. & Tao, S. How fast do defects migrate in halide perovskites: Insights from on-the-fly machine-learned force fields. Chem. Commun. 59, 4660–4663 (2023).10.1039/D3CC00953J ↗
30. Shi, T. et al. An investigation of self-interstitial diffusion in α-zirconium by an on-the-fly machine learning force field. AIP Adv. 14, 055010 (2024).10.1063/5.0211883 ↗
31. Mosquera-Lois, I., Kavanagh, S. R., Ganose, A. M. & Walsh, A. Machine-learning structural reconstructions for accelerated point defect calculations. npj Comput Mater. 10, 1–9 (2024).10.1038/s41524-024-01303-9 ↗
32. Mosquera-Lois, I., Klarbring, J. & Walsh, A. Point defect formation at finite temperatures with machine learning force fields. Chem. Sci. 16, 8878–8888 (2025).10.1039/D4SC08582E ↗
33. Mohanty, S., Wang, Y. & Cai, W. Generalizability of graph neural network force fields for predicting solid-state properties. Adv. Theory Simul. 8, 2401058 (2025).10.1002/adts.202401058 ↗
34. Ma, Y. et al. Transferable machine learning approach for predicting electronic structures of charged defects. Appl. Phys. Lett. 126, 044103 (2025).10.1063/5.0242683 ↗
35. Geiger, M. & Smidt, T. E3nn: Euclidean Neural Networks. Preprint at https://doi.org/10.48550/arXiv.2207.09453 (2022).10.48550/arXiv.2207.09453 ↗
36. Batzner, S., Musaelian, A. & Kozinsky, B. Advancing molecular simulation with equivariant interatomic potentials. Nat. Rev. Phys. 5, 437–438 (2023).10.1038/s42254-023-00615-x ↗
37. Corso, G., Stark, H., Jegelka, S., Jaakkola, T. & Barzilay, R. Graph neural networks. Nat. Rev. Methods Prim. 4, 1–13 (2024).
38. Batzner, S. et al. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat. Commun. 13, 2453 (2022).10.1038/s41467-022-29939-5 ↗
39. Batatia, I. et al. The design space of E(3)-equivariant atom-centred interatomic potentials. Nat. Mach. Intell. 7, 56–67 (2025).10.1038/s42256-024-00956-x ↗
40. Musaelian, A. et al. Learning local equivariant representations for large-scale atomistic dynamics. Nat. Commun. 14, 579 (2023).10.1038/s41467-023-36329-y ↗
41. Batatia, I. et al. The Design Space of E(3)-equivariant atom-centered interatomic potentials. Nat. Mach. Intell. 7, 56–67 (2022).10.1038/s42256-024-00956-x ↗
42. Batatia, I., Kovacs, D. P., Simm, G., Ortner, C. & Csanyi, G. MACE: higher order equivariant message passing neural networks for fast and accurate force fields. Adv. Neural Inf. Process. Syst. 35, 11423–11436 (2022).10.52202/068431-0830 ↗
43. Li, H. et al. Deep-learning density functional theory Hamiltonian for efficient ab initio electronic-structure calculation. Nat. Comput. Sci. 2, 367–377 (2022).10.1038/s43588-022-00265-6 ↗
44. Gong, X. et al. General framework for E(3)-equivariant neural network representation of density functional theory Hamiltonian. Nat. Commun. 14, 2848 (2023).10.1038/s41467-023-38468-8 ↗
45. Wang, Y. et al. DeepH-2: enhancing deep-learning electronic structure via an equivariant local-coordinate transformer. Preprint at https://doi.org/10.48550/arXiv.2401.17015 (2024).10.48550/arXiv.2401.17015 ↗
46. Zhong, Y., Yu, H., Su, M., Gong, X. & Xiang, H. Transferable equivariant graph neural networks for the Hamiltonians of molecules and solids. npj Comput. Mater. 9, 1–13 (2023).10.1038/s41524-023-01130-4 ↗
47. Yu, H., Xu, Z., Qian, X., Qian, X. & Ji, S. Efficient and equivariant graph networks for predicting quantum Hamiltonian. International Conference on Machine Learning (PMLR, 2023).
48. Zhouyin, Z. et al. Learning local equivariant representations for quantum operators. The Thirteenth International Conference on Learning Representations (2025).
49. Qian, C., Vitartas, V., Kermode, J. & Maurer, R. J. Equivariant electronic Hamiltonian prediction with many-body message passing. npj Comput. Mater (2026).10.1038/s41524-026-02020-1 ↗
50. Gu, Q. et al. Deep learning tight-binding approach for large-scale electronic simulations at finite temperatures with ab initio accuracy. Nat. Commun. 15, 6772 (2024).10.1038/s41467-024-51006-4 ↗
51. Schwade, M., Zhang, S., Vonhoff, F., Delgado, F. P. & Egger, D. A. Physics-informed Hamiltonian learning for large-scale optoelectronicproperty prediction. Nat. Commun. 17, 2652 (2026).10.1038/s41467-026-70865-7 ↗
52. Batatia, I. et al. A foundation model for atomistic materials chemistry. J. Chem. Phys. 163, 184110 (2025).
53. Homm, H., Laakso, J. & Rinke, P. Efficient dataset generation for machine learning halide perovskite alloys. Phys. Rev. Mater. 9, 053802 (2025).10.1103/PhysRevMaterials.9.053802 ↗
54. Bhatia, N., Rinke, P. & Krejčí, O. Leveraging active learning-enhanced machine-learned interatomic potential for efficient infrared spectra prediction. npj Comput. Mater. 11, 324 (2025).10.1038/s41524-025-01827-8 ↗
55. von Bardeleben, H. J., Stiévenard, D., Deresmes, D., Huber, A. & Bourgoin, J. C. Identification of a defect in a semiconductor: EL2 in GaAs. Phys. Rev. B 34, 7192–7202 (1986).10.1103/PhysRevB.34.7192 ↗
56. Meyer, B. K., Hofmann, D. M., Niklas, J. R. & Spaeth, J.-M. Arsenic antisite defect As Ga and EL 2 in GaAs. Phys. Rev. B 36, 1332–1335 (1987).10.1103/PhysRevB.36.1332 ↗
57. Dabrowski, J. & Scheffler, M. Theoretical evidence for an optically inducible structural transition of the isolated as antisite in GaAs: identification and explanation of EL 2? Phys. Rev. Lett. 60, 2183–2186 (1988).10.1103/PhysRevLett.60.2183 ↗
58. Thompson, A. P. et al. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comput. Phys. Commun. 271, 108171 (2022).10.1016/j.cpc.2021.108171 ↗
59. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).10.1103/PhysRevB.54.11169 ↗
60. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).10.1103/PhysRevB.50.17953 ↗
61. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).10.1103/PhysRevLett.77.3865 ↗
62. Ozaki, T. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B 67, 155108 (2003).10.1103/PhysRevB.67.155108 ↗
63. Ozaki, T. & Kino, H. Numerical atomic basis orbitals from H to Kr. Phys. Rev. B 69, 195113 (2004).10.1103/PhysRevB.69.195113 ↗
64. Morrison, I., Bylander, D. M. & Kleinman, L. Nonlocal Hermitian norm-conserving Vanderbilt pseudopotential. Phys. Rev. B 47, 6728–6731 (1993).10.1103/PhysRevB.47.6728 ↗