BiFeO₃（BFO）與 BaTiO₃（BTO）為重要之鐵電材料，廣泛應用於非揮發性記憶體、感測器及致動器等領域。近年來，BFO–BTO 固溶體與超晶格結構因兼具高極化特性與可調控之電性行為而受到廣泛關注。然而，第一原理計算受限於系統尺寸與計算成本，難以直接模擬大尺度鐵電行為，因此建立適用於 BFO–BTO 系統之可轉移勢能函數具有重要意義。
本研究利用密度泛函理論（Density Functional Theory, DFT）計算 BiFeO₃、BaTiO₃、0.5BaTiO₃–0.5BiFeO₃ 固溶體以及 BFO–BTO 超晶格結構之晶格常數、彈性常數及能量等性質，作為勢能函數擬合目標。接著採用 Buckingham potential 結合 Spring potential 建立核殼模型（Core–Shell Model），並利用粒子群最佳化法（Particle Swarm Optimization, PSO）結合梯度下降法（Gradient Descent, GD）進行參數擬合，以建立適用於 BFO–BTO 鐵電系統之可轉移核殼模型勢能函數。至於電荷決定使用波恩有效電荷還有Bader Charge 決定，其中Bader為殼核淨電荷，波恩有效電荷矩陣的對角線數值平均，另外氧的電荷取自BTO和BFO的平均。
本研究以 Buckingham potential 與 Spring potential 建立核殼模型勢能函數，其中 Buckingham potential 用於描述 Ba、Ti、O、Fe 與 Bi 五種原子間共 15 組短程交互作用，其 A、ρ 與 C 三項參數皆作為擬合變數；同時，五種原子之 core–shell Spring potential 參數 K₂ 與 K₄ 亦納入最佳化程序。最終共擬合 55 個勢能參數，以建立可同時描述 BiFeO₃、BaTiO₃、固溶體及超晶格結構之可轉移核殼模型勢能函數。
結果顯示，本研究建立之勢能函數可合理重現各結構之晶格常數、晶胞角度、彈性常數及相對能量趨勢，並成功維持 BFO、BTO、固溶體與超晶格結構間之相對穩定性。此外，BiFeO₃與BaTiO₃於分子動力學定溫下比較各個溫度各種相的能量時具有實驗上相轉變的趨勢，且Superlattice模型也具有相似於純相模型的趨勢，至於固溶體模型則是和實驗常溫下的相符。能重現鐵電材料之極化行為與 P–E 遲滯迴圈特徵。雙極三角波電場模擬結果顯示，BiFeO₃、BaTiO₃、固溶體及超晶格結構之殘餘極化分別約為 42.18 μC/cm²、8.10 μC/cm²、5.42–6.43 μC/cm² 及 10.41–12.93 μC/cm²，與單極方波電場測試結果具有良好一致性，顯示本研究建立之勢能函數具有良好的極化穩定性與再現性。
綜合以上結果，本研究成功建立一套可同時描述 BiFeO₃、BaTiO₃、固溶體及超晶格結構之可轉移核殼模型勢能函數，並驗證其在結構、力學及鐵電性質模擬上的適用性。所建立之勢能函數可作為後續 BFO–BTO 鐵電材料大尺度分子動力學模擬與性質研究之基礎。

BiFeO₃ (BFO) and BaTiO₃ (BTO) are important ferroelectric materials widely used in nonvolatile memories and actuators. In recent years, BFO–BTO solid solutions and superlattice structures have attracted considerable attention because of their high polarization and tunable electrical properties. However, first-principles calculations are limited by computational cost and accessible system size, making direct simulations of large-scale ferroelectric behavior challenging. Therefore, developing a transferable interatomic potential for BFO–BTO systems is essential.
In this study, density functional theory (DFT) calculations were performed for BiFeO₃, BaTiO₃, a 0.5BTO–0.5BFO solid solution, and BFO–BTO superlattice structures. Their lattice parameters, elastic constants, and relative energies were used as fitting targets. A core–shell model combining Buckingham and spring potentials was constructed. The potential parameters were optimized using particle swarm optimization (PSO), followed by gradient descent (GD) refinement, to establish a core–shell potential applicable to BFO–BTO ferroelectric systems.
The developed potential reproduces the lattice parameters, elastic constants, and relative energy trends of the structures. It also preserves the relative structural stability among BFO, BTO, the solid solution, and the superlattices. Molecular dynamics simulations further demonstrate that all structures remain stable under the investigated conditions. In addition, the model reproduces the polarization response and characteristic P–E hysteresis behavior of ferroelectric materials.
Overall, a transferable core–shell potential capable of simultaneously describing BiFeO₃, BaTiO₃, BFO–BTO solid solutions, and superlattice structures was successfully developed and validated in terms of structural, mechanical, and ferroelectric properties. The proposed potential provides a foundation for future large-scale molecular dynamics simulations and property investigations of BFO–BTO ferroelectric materials.

[1] P. B. Littlewood, “Physics of Ferroelectrics,” 27 Jan. (2002)
[2] Singh, Sushil & Maruyama, K. & Ishiwara, Hiroshi.. Frequency-dependent polarization in BiFeO3 thin films. Integrated Ferroelectrics. 98. 83-89.. (2010)
[3] Wang, Q.; Yan, H.-Z.; Zhao, X.; Wang, C.-M. Polymorphic Phase Transition and Piezoelectric Performance of BaTiO3-CaSnO3 Solid Solutions.Actuators,10, 129. (2021)
[4] T. Correia, M. Stewart, A. Ellmore, K. Albertsen, “Lead‐Free Ceramics with High Energy Density and Reduced Losses for High Temperature Applications,” Adv. Eng. Mater, 19, (2017)
[5] H. Pan, Y. Zeng, Y. Shen, Y.-H. Lin, J. Ma, L. Li, C.-W. Nan, “BiFeO3–SrTiO3 thin film as a new lead-free relaxor-ferroelectric capacitor with ultrahigh energy storage performance,” J. Mater. Chem. A, 5, 5920-5926, (2017)
[6] Y. Sun, H. Liu, F. Liu et al. “Dielectric and electrical energy storage properties of BiFeO3–BaTiO3–SrTiO3 ternary bulk ceramics.” J Mater Sci: Mater Electron, 32, 21188–21196, (2021)
[7] V. M. Goldschmidt, “Die Gesetze Der Krystallochemie,” Naturwissenschaften, 14, 477–485, (1926)
[8] Karpinsky, D.V., Eliseev, E.A., Xue, F. et al. Thermodynamic potential and phase diagram for multiferroic bismuth ferrite (BiFeO 3 ). npj Comput Mater 3, 20 (2017).
[9] Ying-Hao Chu, Lane W. Martin, Mikel B. Holcomb, Ramamoorthy Ramesh,"Controlling magnetism with multiferroics "Materials Today,Volume 10, Issue 10,Pages,16-23(2007)
[10] Fu. Desheng, Mitsuru Itoh. “Role of Ca Off-Centering in Tuning Ferroelectric Phase Transitions in Ba(Zr,Ti)O3System,” Ferroelectric Materials - Synthesis and Characterization, InTech, (2015.)
[11] Hemme, Pierre & Djemia, Philippe & Rovillain, Pauline & Gallais, Yann & Sacuto, Alain & Forget, Anne & Colson, Dorothee & Charron, Eric & Perrin, Bernard & Belliard, Laurent & Cazayous, Maximilien.. Elastic properties assessment in the multiferroic BiFeO3 by pump and probe method. 10.48550/arXiv.2102.05580. (2021)
[12] He Xiao-Kang et al Determination of elastic, piezoelectric, and dielectric constants of an R:BaTiO3 single crystal by Brillouin scattering Chinese Phys. B 21 067801(2012)
[13] Yongxing Wei, Changqing Jin, Yiming Zeng, Xiaotao Wang, Dong Gao, Xiaoli Wang,A coexistence of multi-relaxor states in 0.5BiFeO3–0.5BaTiO3,Ceramics International,Volume 43, Issue 18,Pages 17220-17224,ISSN 0272-8842, (2017)
[14] Zunger, AlexWei, S. H.Ferreira, L. G.Bernard, James E., Special quasirandom structures, Physical Review Letters, Volume 65, Issue 3, Pages 353-356,
[15] Wang, Xin & Yang, Jucai & Zhao, Erjun & Cao, Zhenzhu. Theoretical insights into the ultrahigh piezoelectric performance of (BiFeO3)n/(BaTiO3)n (n = 1–5) superlattices. Journal of Applied Physics. (2024)
[16] Hohenberg, Pierre, Walter, Kohn. "Inhomogeneous electron gas." Physical review 136.3B, B864, (1964)
[17] Walter, Kohn. and Lu Jeu Sham. "Self-consistent equations including exchange and correlation effects." Physical review 140.4A, A1133, (1965)
[18] P. E. Blochl, "Projector augmented-wave method." Physical Review B, 50(24), 17953–17979, Dec (1994)
[19] Richard Buckingham A. "The classical equation of state of gaseous helium, neon and argon." Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 168.933: 264-283, (1938)
[20] Paul. Peter. Ewald "Ewald summation." Ann. Phys 369.253, 1-2, (1921)
[21] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, S. J. Plimpton, “LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales,” Comp Phys Comm, 271, 10817, (2022)
[22] Loup. Verlet, "Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules." Physical review 159.1, 98, (1967)
[23] Shuichi Nosé, “A unified formulation of the constant temperature molecular dynamics methods”, J. Chem. Phys. 81, 511–519, (1984)
[24] Shuichi. Nosé, "A unified formulation of the constant temperature molecular dynamics methods." The Journal of chemical physics 81.1, 511-519, (1984)
[25] Anubhav. Jain, et al. "Commentary: The Materials Project: A materials genome approach to accelerating materials innovation." APL materials 1.1, (2013)
[26] Perdew, John P., Kieron Burke, and Matthias Ernzerhof. "Generalized gradient approximation made simple." Physical review letters, 77.18, 3865, (1996)
[27] R. F. Bader, and T. T. Nguyen-Dang. "Quantum theory of atoms in molecules–Dalton revisited." Advances in quantum chemistry. Vol. 14. Academic Press, 63-124, (1981)
[28] W. Tang, E. Sanville, and G. Henkelman “A grid-based Bader analysis algorithm without lattice bias,” J. Phys.: Condens. Matter 21, 084204, (2009)
[29] E. Sanville, S. D. Kenny, R. Smith, and G. Henkelman “An improved grid-based algorithm for Bader charge allocation,” J. Comp. Chem. 28, 899-908, (2007)
[30] G. Henkelman, A. Arnaldsson, and H. Jónsson, “A fast and robust algorithm for Bader decomposition of charge density,” Comput. Mater. Sci. 36, 354-360, (2006)
[31] M. Yu and D. R. Trinkle, “Accurate and efficient algorithm for Bader charge integration,” J. Chem. Phys. 134, 064111, (2011)
[32] Xie, G. & Xiong, Ying & Li, Baohua & Zhu, Yong & Li, Jiancheng & Gu, Xiaochen & Xiao, Yongguang & Tang, Minghua.. Radiation Damage Effects by Molecular Dynamics Simulation in BaTiO3 Ferroelectric Crystal. IEEE Transactions on Nuclear Science. 59. 1731-1737. 10.1109/TNS.2012.2201172. (2012)
[33] C. F. Wu, and J.H. Jean. "Constrained sintering of Bi2O3‐doped ZnO." International Journal of Ceramic Engineering & Science 1.3, 155-165, (2019)
[34] Erlebach, Andreas & Kurland, Heinz-Dieter & Grabow, Janet & Müller, Frank & Sierka, Marek. Structure evolution of nanoparticulate Fe2O3. Nanoscale. 7. 10.1039/C4NR06989G. (2014)
[35] Crawford, J and Jacobs, P. "Point defect energies for strontium titanate: A pair-potentials study." Journal of Solid State Chemistry, vol. 144, no. 2 , May. (1999)
[36] K. V. Mirskaya, "Combination rules for interatomic potential functions of Buckingham form." Tetrahedron, 29.5, 679-682, (1973)
[37] Hikaru Azuma, Tomohiro Ogawa, Shuji Ogata, Ryo Kobayashi, Masayuki Uranagase, Takahiro Tsuzuki, Frank Wendler,Unique temperature-dependence of polarization switching paths in ferroelectric BaTiO3: A molecular dynamics simulation study,Acta Materialia,Volume 296,(2025)
[38] Graf, Mónica & Sepliarsky, Marcelo & Tinte, Silvia & Stachiotti, Marcelo. Phase Transitions and Antiferroelectrivity in BiFeO3 from Atomic Level Simulations. Physical Review B. 90. (2014).
[39] J. Kennedy and R. Eberhart, "Particle swarm optimization," Proceedings of ICNN'95 - International Conference on Neural Networks, Perth, WA, Australia, , pp. 1942-1948 vol.4, doi: 10.1109/ICNN.1995.488968. (1995)
[40] Neaton, J. B., et al.. "First-principles study of spontaneous polarization in multiferroic BiFeO3." Physical Review B 71(1): 014113 (2005)
[41] Song, W., et al. "Influence of the Magnitude of Ferroelectric Domain Polarization on the Photochemical Reactivity of BaTiO3." ACS Applied Materials & Interfaces 10(48): 41450–41457. (2018).
[42] Sone, K., Naganuma, H., Miyazaki, T., Nakajima, T., & Okamura, S.. Crystal structures and electrical properties of epitaxial BiFeO3 thin films with (001), (110), and (111) orientations. Japanese Journal of Applied Physics, 49(9 PART 2), Article 09MB03.(2010).
[43] Shang, S. L., Sheng, G.,Wang, Y.,Chen, L. Q.,Liu, Z. K. Elastic properties of cubic and rhombohedral BiFeO3 from first-principles calculations, American Physical Society, Phys. Rev. B, 80.052102(2009)
[44] Wang, J.J. & Meng, F. & Ma, Xingqiao & Xu, M. & Chen, L.. (2010). Lattice, elastic, polarization, and electrostrictive properties of BaTiO3 from first-principles. Journal of Applied Physics. 108. 034107-034107. 10.1063/1.3462441.
[45]Upadhyay, Sanjay & Reddy, VARIMALLA & Nambakkat, Lakshmi. (2013). Study of (1−x) BaTiO3–x Ni0.5Zn0.5Fe2O4 (x = 5, 10 and 15%) magneto-electric ceramic composites. Journal of Asian Ceramic Societies. 1. 10.1016/j.jascer.2013.10.001.
[46] Lebeugle, D. & Colson, D. & Forget, A. & Viret, Michel. (2007). Very Large Spontaneous Electric Polarization in BiFeO3 Single Crystals at Room Temperature and Its Evolution Under Cycling Fields. Applied Physics Letters. 91. 10.1063/1.2753390.
[47] Grimme, S., Ehrlich, S. and Goerigk, L. (2011), Effect of the damping function in dispersion corrected density functional theory. J. Comput. Chem., 32: 1456-1465. https://doi.org/10.1002/jcc.21759
[48] Lee, M.H., Kim, D.J., Park, J.S., Kim, S.W., Song, T.K., Kim, M.-H., Kim, W.-J., Do, D. and Jeong, I.-K. High-Performance Lead-Free Piezoceramics with High Curie Temperatures. Adv. Mater., 27: 6976-6982. (2015),
[49] A study of particle swarm optimization particle trajectories F. van den Bergh, A.P. Engelbrecht , Information Sciences 176,937–971 (2006)