鐵電材料具有可與外加電場自由切換的自發電極化特性。作為一種典型的鐵電材料，鈦酸鋇(BaTiO₃)的自發極化被認為是鈦原子在封閉的氧八面體內偏離中心的結果，但其鐵電躍遷的詳細微觀性質一直是各種實驗和理論激烈研究的主題。鐵電躍遷的兩種描述模型—位移模型和有序–無序模型捕捉了一些實驗表征BaTiO₃的現象，而第一性原理模擬可以提供對相變本質的寶貴微觀理解。

計算機模擬材料的鐵電相變需要三個關鍵成分：描述原子和結構扭曲的能量響應的勢能面模型，在相關的有限溫度熱力學條件下采樣的自由能面，以及通過對樣本的平均來確定宏觀極化的單個配置極化。

Fig. 2 Spatial correlations of the unit-cell dipoles computed on a 5 × 5 × 5 supercell simulated at 250 K.

密度泛函理論(DFT)在探索BaTiO₃的勢能面及軟聲子等方面取得了成功，但有效的模型依賴于哈密頓量的顯示參數化。因此，為了對熱力學做出第一性原理的準確預測，最好使用一種無偏的、不可知的方法，而不以勢能面的形式進行任何先驗假設。

來自瑞士洛桑聯邦理工學院材料研究所計算科學與模擬實驗室的Lorenzo Gigli等，開發了一個現代的、通用的機器學習(ML)框架，來描述鈣鈦礦鐵電體的有限溫度和功能性質(介電響應)，并將其具體應用于BaTiO₃。

該框架在進行分子動力學時不需要在模擬規模和尺度間進行妥協。該框架基于原子間ML勢和極化矢量ML模型的組合，可同時預測鐵電材料的總能量、原子力和極化，以探索其復雜的、隨溫度變化的相圖，并預測其功能特性。

Fig. 5 Phases of BaTiO3 in CV space.

他們的方法可計算宏觀的可觀測量，如化學勢和介電磁化率，其精度相當于基礎DFT計算的理論水平，但計算成本要小得多。它適用于任何鈣鈦礦，甚至任何其他類型的鐵電材料，包括二維鐵電體等。

這項研究為理解和表征已知鐵電材料以及發現和設計具有改進性能的新候選化合物開辟了一條新途徑。該論文近期發表于npj Computational Materials 8: 209 (2022)。

Editorial Summaryd

Ferroelectric materials possess a spontaneous electric polarization that can be switched with an external electric field. The spontaneous polarization of the ferroelectric material —barium titanate (BaTiO₃) is thought to be the result of the titanium atom off-centering within the enclosing oxygen octahedron, but the detailed microscopic nature of the ferroelectric transition has been the subject of intense, ongoing research with a variety of experimental and theoretical techniques.?

The two models describing the ferroelectric transition, the displacive model and the order-disorder model, capture some of the phenomena characterized by experiments on BaTiO₃, experimentally observed in characterizing BaTiO₃, such as phonon softening at the transition temperatures—consistent with the displacive model—and diffuse X-ray scattering in all phases except the rhombohedral one—consistent with the order-disorder model—leading also to approaches combining the two models.

In this context, simulations—especially from first principles—can offer a precious microscopic understanding of the nature of the phase transitions. A computer simulation of the ferroelectric phase transition of any given material requires three key ingredients: first, a model of the potential energy surface (PES) that describes the energetic response to atomic and structural distortions, second, the free-energy surface (FES) sampled at the relevant, finite-temperature thermodynamic conditions, and third, the polarization of individual configurations that determines, through averaging over samples, the macroscopic polarization.?

Fig. 10 Temperature and pressure dependence of the imaginary part of the dielectric response spectrum, all computed in the?cubic phase on a 4 × 4 × 4 supercell.

Density—functional theory (DFT) calculations have been successful in exploring the PES of BaTiO₃ and soft phonons and so on, but effective models rely on the choice of an explicit parametrization of the Hamiltonian. Therefore, in order to make accurate first-principles predictions of the thermodynamics, it is desirable to use an unbiased, agnostic approach without any prior assumption in the form of the PES.?

Lorenzo Gigli at al. from the Laboratory of Computational Science and Modeling, Institute for Materials, école Polytechnique Fédérale de Lausanne, Switzerland, developed a modern, general machine learning (ML) framework to describe at once the finite-temperature and functional properties (dielectric response) of perovskite ferroelectrics and apply it specifically to model BaTiO₃.?

The framework does not need to compromise between simulation scale and scale when conducting molecular dynamics (MD). This framework, based on a combination of an interatomic ML potential and a vector ML model for the polarization, is used to simultaneously predict the total energy, atomic forces, and polarization of a ferroelectric material in order to explore its complex, temperature-dependent phase diagram as well as to predict its functional properties. This approach can be used to compute macroscopic observables —chemical potentials and dielectric susceptibilities, specifically—with an accuracy equivalent to that of the level of theory of the underlying DFT calculations, but at a much smaller computational cost. Moreover, it is applicable with only minor changes to any perovskite or even any other type of ferroelectric material, including 2D ferroelectrics.?

Fig. 13 Validation of the GAP.

The work opens the door for a new avenue of fruitful research into the understanding and characterization of known ferroelectric materials, as well as the discovery and design of new candidate compounds with improved industrially relevant properties.?Thisarticle was recently?published in?npj?Computational Materials?8:?209?(2022).

原文Abstract及其翻譯

Thermodynamics and dielectric response of BaTiO₃ by data-driven modeling (數據驅動模擬BaTiO₃的熱力學和介電響應)

Lorenzo Gigli,?Max Veit,?Michele Kotiuga,?Giovanni Pizzi,?Nicola Marzari?&?Michele Ceriotti

摘要第一性原理模擬鐵電材料是密度泛函理論的成功之一，也是許多發展的驅動力，這需要準確描述電子過程和熱力學平衡，它們驅動了自發對稱破缺和宏觀極化的出現。我們展示了一個集成機器學習模型的開發和應用，該模型描述了BaTiO₃相同的基礎結構、能量和功能特性，這是一種典型的鐵電學。該模型利用從頭計算作為參考，在時間和長度尺度上實現了對能量和極化準確而廉價的預測，這是直接從頭模擬無法實現的。這些預測使我們能夠評估鐵電躍遷的微觀機制。Ti離心態的有序–無序躍遷是鐵電躍遷的主要驅動因素，即使對稱性破缺和晶胞畸變之間的耦合決定了中間相、部分有序相的存在。此外，我們還深入地探索了BaTiO₃在其相圖上的靜態和動態行為，而不需要引入對鐵電躍遷的粗粒度描述。最后，我們應用極化模型，以完全從頭算的方式計算了材料的介電響應特性，再次再現了正確的定性實驗行為。