Machine learning the Hubbard U parameter in DFT+U using Bayesian optimization
Source abstract
Abstract
Abstract Within density functional theory (DFT), adding a Hubbard U correction can mitigate some of the deficiencies of local and semi-local exchange-correlation functionals, while maintaining computational efficiency. However, the accuracy of DFT+U largely depends on the chosen Hubbard U values. We propose an approach to determining the optimal U parameters for a given material by machine learning. The Bayesian optimization (BO) algorithm is used with an objective function formulated to reproduce the band structures produced by more accurate hybrid functionals. This approach is demonstrated for transition metal oxides, europium chalcogenides, and narrow-gap semiconductors. The band structures obtained using the BO U values are in agreement with hybrid functional results. Additionally, comparison to the linear response (LR) approach to determining U demonstrates that the BO method is superior.
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Research context
OpenAlex topics: Machine Learning in Materials Science · Semiconductor materials and interfaces · Inorganic Chemistry and Materials
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References (62)
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1. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).10.1103/PhysRevLett.77.3865 ↗
2. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple [phys. rev. lett. 77, 3865 (1996)]. Phys. Rev. Lett. 78, 1396–1396 (1997).10.1103/PhysRevLett.78.1396 ↗
3. Curtarolo, S. et al. The high-throughput highway to computational materials design. Nat. Mater. 12, 191–201 (2013).10.1038/nmat3568 ↗
4. Jain, A. et al. The Materials Project: a materials genome approach to accelerating materials innovation. APL Mater. 1, 011002 (2013).10.1063/1.4812323 ↗
5. Saal, J. E., Kirklin, S., Aykol, M., Meredig, B. & Wolverton, C. Materials design and discovery with high-throughput density functional theory: The open quantum materials database (oqmd). JOM 65, 1501–1509 (2013).10.1007/s11837-013-0755-4 ↗
6. Hüfner, S. Electronic structure of NiO and related 3d-transition-metal compounds. Adv. Phys. 43, 183–356 (1994).10.1080/00018739400101495 ↗
7. Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J. & Sutton, A. P. Electron-energy-loss spectra and the structural stability of nickel oxide: An lsda+u study. Phys. Rev. B 57, 1505–1509 (1998).10.1103/PhysRevB.57.1505 ↗
8. Perdew, J. P. Climbing the ladder of density functional approximations. MRS Bull. 38, 743–750 (2013).10.1557/mrs.2013.178 ↗
9. Becke, A. D. A new mixing of Hartree-Fock and local density-functional theories. J. Chem. Phys. 98, 1372–1377 (1993).10.1063/1.464304 ↗
10. Heyd, J. & Scuseria, G. E. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 118, 8207–8215 (2003).10.1063/1.1564060 ↗
11. Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: hybrid functionals based on a screened coulomb potential [J. CHEM. Phys. 118, 8207 (2003)]. J. Chem. Phys. 124, 219906 (2006).10.1063/1.2204597 ↗
12. Anisimov, V. I., Zaanen, J. & Andersen, O. K. Band theory and Mott insulators: Hubbard U instead of Stoner I. Phys. Rev. B 44, 943–954 (1991).10.1103/PhysRevB.44.943 ↗
13. Yu, K. & Carter, E. A. Communication: comparing ab initio methods of obtaining effective U parameters for closed-shell materials. J. Chem. Phys. 140, 121105 (2014).10.1063/1.4869718 ↗
14. Cococcioni, M. & de Gironcoli, S. Linear response approach to the calculation of the effective interaction parameters in the LDA+U method. Phys. Rev. B 71, 035105 (2005).10.1103/PhysRevB.71.035105 ↗
15. Kulik, H. J., Cococcioni, M., Scherlis, D. A. & Marzari, N. Density functional theory in transition-metal chemistry: a self-consistent hubbard u approach. Phys. Rev. Lett. 97, 103001 (2006).10.1103/PhysRevLett.97.103001 ↗
16. Mosey, N. J. & Carter, E. A. Ab initio evaluation of coulomb and exchange parameters for DFT+U calculations. Phys. Rev. B 76, 155123 (2007).10.1103/PhysRevB.76.155123 ↗
17. Mosey, N. J., Liao, P. & Carter, E. A. Rotationally invariant ab initio evaluation of Coulomb and exchange parameters for DFT+U calculations. J. Chem. Phys. 129, 14103 (2008).10.1063/1.2943142 ↗
18. Aryasetiawan, F., Karlsson, K., Jepsen, O. & Schönberger, U. Calculations of Hubbard U from first-principles. Phys. Rev. B 74, 125106 (2006).10.1103/PhysRevB.74.125106 ↗
19. Miyake, T. & Aryasetiawan, F. Screened coulomb interaction in the maximally localized wannier basis. Phys. Rev. B 77, 085122 (2008).10.1103/PhysRevB.77.085122 ↗
20. Şaşíoğlu, E., Friedrich, C. & Blügel, S. Effective coulomb interaction in transition metals from constrained random-phase approximation. Phys. Rev. B 83, 121101 (2011).10.1103/PhysRevB.83.121101 ↗
21. Frazier, P. I. A tutorial on bayesian optimization. arXiv https://arxiv.org/abs/1807.02811 (2018).
22. Brochu, E., Cora, V. M., & De Freitas, N. A tutorial on bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. arXiv preprint at https://arxiv.org/abs/1012.2599 (2010).
23. Snoek, J., Larochelle, H. & Adams, R. P. Practical bayesian optimization of machine learning algorithms. In Advances In Neural Information Processing Systems. 2951–2959 (Neural Information Processing Systems Foundation, Inc., 2012).
24. Huhn, W. P. & Blum, V. One-hundred-three compound band-structure benchmark of post-self-consistent spin-orbit coupling treatments in density functional theory. Phys. Rev. Mater. 1, 033803 (2017).10.1103/PhysRevMaterials.1.033803 ↗
25. Topsakal, M. & Wentzcovitch, R. M. Accurate projected augmented wave (PAW) datasets for rare-earth elements (RE=La-Lu). Comput. Mater. Sci. 95, 263–270 (2014).10.1016/j.commatsci.2014.07.030 ↗
26. Zhang, G.-X., Reilly, A. M., Tkatchenko, A. & Scheffler, M. Performance of various density-functional approximations for cohesive properties of 64 bulk solids. New J. Phys. 20, 063020 (2018).10.1088/1367-2630/aac7f0 ↗
27. Garza, A. J. & Scuseria, G. E. Predicting band gaps with hybrid density functionals. J. Phys. Chem. Lett. 7, 4165–4170 (2016).10.1021/acs.jpclett.6b01807 ↗
28. Friedrich, C., Betzinger, M., Schlipf, M., Blügel, S. & Schindlmayr, A. Hybrid functionals and gw approximation in the flapw method. J. Phys. Condens. Matter 24, 293201 (2012).10.1088/0953-8984/24/29/293201 ↗
29. Williams, C. K. I. & Rasmussen, C. E. Gaussian Processes For Machine Learning. Vol. 2 (MIT press Cambridge, 2006).
30. Kick, M., Reuter, K. & Oberhofer, H. Intricacies of DFT+U, not only in a numeric atom centered orbital framework. J. Chem. Theory Comput. 15, 1705–1718 (2019).10.1021/acs.jctc.8b01211 ↗
31. Ye, L.-H., Luo, N., Peng, L.-M., Weinert, M. & Freeman, A. J. Dielectric constant of NiO and LDA+U. Phys. Rev. B 87, 075115 (2013).10.1103/PhysRevB.87.075115 ↗
32. Pask, J. E., Singh, D. J., Mazin, I. I., Hellberg, C. S. & Kortus, J. Structural, electronic, and magnetic properties of MnO. Phys. Rev. B 64, 024403 (2001).10.1103/PhysRevB.64.024403 ↗
33. Deng, H.-X., Li, J., Li, S.-S., Xia, J.-B., Walsh, A. & Wei, S.-H. Origin of antiferromagnetism in CoO: a density functional theory study. Appl. Phys. Lett. 96, 162508 (2010).10.1063/1.3402772 ↗
34. Dufek, P., Blaha, P., Sliwko, V. & Schwarz, K. Generalized-gradient-approximation description of band splittings in transition-metal oxides and fluorides. Phys. Rev. B 49, 10170 (1994).10.1103/PhysRevB.49.10170 ↗
35. Hüfner, S., Osterwalder, J., Riesterer, T. & Hulliger, F. Photoemission and inverse photoemission spectroscopy of NiO. Solid State Commun. 52, 793–796 (1984).10.1016/0038-1098(84)90007-3 ↗
36. Sawatzky, G. A. & Allen, J. W. Magnitude and origin of the band gap in NiO. Phys. Rev. Lett. 53, 2339 (1984).10.1103/PhysRevLett.53.2339 ↗
37. Himmetoglu, B. & Wentzcovitch, R. M. First-principles study of electronic and structural properties of CuO. Phys. Rev. B 84, 115108 (2011).10.1103/PhysRevB.84.115108 ↗
38. Jiang, H., Gomez-Abal, R. I., Rinke, P. & Scheffler, M. Localized and Itinerant states in lanthanide oxides united by GW@LDA+U. Phys. Rev. Lett. 102, 126403 (2009).10.1103/PhysRevLett.102.126403 ↗
39. Jiang, H., Gomez-Abal, R. I., Rinke, P. & Scheffler, M. First-principles modeling of localized d states with the GW@LDA+U approach. Phys. Rev. B 82, 045108 (2010).10.1103/PhysRevB.82.045108 ↗
40. Wachter, P. The optical electrical and magnetic properties of the europium chalcogenides and the rare earth pnictides. Crit. Rev. Solid State Mater. Sci. 3, 189–241 (1972).10.1080/10408437208244865 ↗
41. Ghosh, D. B., De, M. & De, S. K. Electronic structure and magneto-optical properties of magnetic semiconductors: Europium monochalcogenides. Phys. Rev. B 70, 115211 (2004).10.1103/PhysRevB.70.115211 ↗
42. Larson, P. & Lambrecht, W. R. L. Electronic structure and magnetism of europium chalcogenides in comparison with gadolinium nitride. J. Phys. Condens. Matter 18, 11333–11345 (2006).10.1088/0953-8984/18/49/024 ↗
43. Shi, S. Q. et al. Electronic structure and magnetism of EuX (X = O, S, Se and Te): A first-principles investigation. EPL 83, 4–9 (2008).
44. Schlipf, M., Betzinger, M., Ležaić, M., Friedrich, C. & Blügel, S. Structural, electronic, and magnetic properties of the europium chalcogenides: a hybrid-functional DFT study. Phys. Rev. B 88, 94433 (2013).10.1103/PhysRevB.88.094433 ↗
45. Ingle, N. J. C. & Elfimov, I. S. Influence of epitaxial strain on the ferromagnetic semiconductor EuO: first-principles calculations. Phys. Rev. B 77, 121202 (2008).10.1103/PhysRevB.77.121202 ↗
46. An, J. M., Barabash, S. V., Ozolins, V., van Schilfgaarde, M. & Belashchenko, K. D. First-principles study of phase stability of Gd-doped EuO and EuS. Phys. Rev. B 83, 064105 (2011).10.1103/PhysRevB.83.064105 ↗
47. Kim, Y.-S., Hummer, K. & Kresse, G. Accurate band structures and effective masses for InP, InAs, and InSb using hybrid functionals. Phys. Rev. B 80, 035203 (2009).10.1103/PhysRevB.80.035203 ↗
48. Massidda, S. et al. Structural and electronic properties of narrow-band-gap semiconductors: InP, InAs, and InSb. Phys. Rev. B 41, 12079–12085 (1990).10.1103/PhysRevB.41.12079 ↗
49. Soluyanov, A. A. et al. Optimizing spin-orbit splittings in InSb Majorana nanowires. Phys. Rev. B 93, 115317 (2016).10.1103/PhysRevB.93.115317 ↗
50. Malyi, O. I., Dalpian, G. M., Zhao, X.-G., Wang, Z. & Zunger, A. Realization of predicted exotic materials: the burden of proof. Mater. Today 32, 35–45 (2019).10.1016/j.mattod.2019.08.003 ↗
51. Vurgaftman, I., Meyer, J. R. & Ram-Mohan, L. R. Band parameters for III-V compound semiconductors and their alloys. J. Appl. Phys. 89, 5815–5875 (2001).10.1063/1.1368156 ↗
52. Madelung, O., Rössler, U. & Schulz, M. Group IV Elements, IV-IV and III-V Compounds. Part b - Electronic, Transport, Optical and Other Properties. Landolt-Börnstein - Group III Condensed Matter, 41A1β (Heidelberg Springer, Berlin, 2002).10.1007/b80447 ↗
53. Micnas, R., Ranninger, J. & Robaszkiewicz, S. Superconductivity in narrow-band systems with local nonretarded attractive interactions. Rev. Mod. Phys. 62, 113–171 (1990).10.1103/RevModPhys.62.113 ↗
54. Hase, I. & Yanagisawa, T. Madelung energy of the valence-skipping compound BaBiO3. Phys. Rev. B 76, 174103 (2007).10.1103/PhysRevB.76.174103 ↗
55. Nakamura, H., Hayashi, N., Nakai, N., Okumura, M. & Machida, M. First-principle electronic structure calculations for magnetic moment in iron-based superconductors: an LSDA+ negative U study. Phys. C Supercond. 469, 908–911 (2009).10.1016/j.physc.2009.05.096 ↗
56. Persson, C. & Mirbt, S. Improved electronic structure and optical properties of sp-hybridized semiconductors using LDA+U SIC. Braz. J. Phys. 36, 286–290 (2006).10.1590/S0103-97332006000300014 ↗
57. Cococcioni, M. The LDA+U approach: a simple Hubbard correction for correlated ground states. Correlated Electrons: From Models to Materials Modeling and Simulation. Vol. 2, Ch. 4 (Verlag des Forschungszentrum Jülich, Jülich, Germany, 2012).
58. Kresse, G. Ab initio molecular dynamics for liquid metals. J. Non Cryst. Solids 192–193, 222–229 (1995).10.1016/0022-3093(95)00355-X ↗
59. Rolland, A., Pouget, L., Brunel, M., Loas, G. & Alouini, M. Bridging the THz to RF gap by four-wave mixing in a highly nonlinear fiber. In Proc. 2013 Conference on Lasers and Electro-Optics, CLEO 2013 59, 11–19 (2013).
60. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).10.1103/PhysRevB.50.17953 ↗
61. Steiner, S., Khmelevskyi, S., Marsmann, M. & Kresse, G. Calculation of the magnetic anisotropy with projected-augmented-wave methodology and the case study of disordered Fe1−xCox alloys. Phys. Rev. B 93, 224425 (2016).10.1103/PhysRevB.93.224425 ↗
62. Yang, S., Wu, C. & Marom, N. Topological properties of SnSe/EuS and SnTe/CaTe interfaces. Phys. Rev. Mater. 4, 034203 (2020).10.1103/PhysRevMaterials.4.034203 ↗