Phase-field Modeling of Flexoelectric Contributions to Ferroelectricity

该奖项支持理论研究、计算建模和教育，旨在更好地理解铁电体，这是一种多功能材料，具有许多用途，包括执行器、传感器、内存存储和微机电系统。这些材料不仅能在外加电场作用下产生电信号或在外加机械应力场作用下产生宏观形状变形，而且能在外加机械应力作用下产生电信号或在外加电场作用下产生形状变形作用。人们对电信号、电场、均匀机械载荷、温度和均匀形状变形之间的耦合进行了广泛的研究，并且对均匀形状变形如何影响以“压电效应”为特征的铁电体电性能的基础科学有了相当好的理解。在这个项目中，PI将重点了解非均匀应力或形状变形如何通过“挠曲电效应”影响铁电体的多功能特性。我们将致力于开发有效的计算方法，并利用它们来模拟、预测和理解铁电材料在挠曲电效应影响下的内部、纳米尺度的非均匀结构和性质。计算研究将在众多实验小组、计算物理学家和应用数学家的密切合作下进行。在这个项目中获得的基本理解和开发的计算工具应该为开发可以利用所有材料中存在的挠曲电效应的材料系统提供指导。由于相变和微观结构演化的相场模拟已被纳入宾夕法尼亚州立大学的研究生课程，这项拟议的研究有望为材料研究生教育做出贡献。特别是，在NSF之前的支持下，为相场模型的一些应用开发了用户友好的图形界面。该软件已应用于两门研究生课程和一门本科生课程。此外，它还被用于计算热力学和相变动力学的夏季短期课程，这些课程提供给国家实验室的研究科学家，工业工程师以及学术界的教授和学生。这个项目将提供新的计算工具来说明如何通过挠曲电效应来改变材料的特性。PI将让本科生参与研究，参与宾夕法尼亚州立大学的一些项目，包括高级论文项目和少数民族本科生研究经验项目。技术概述铁电体是一类在其准电到铁电相变温度以下发生自发电极化的材料。在电场作用下，单晶的自发极化方向可以在晶体学上确定的取向之间重新定向。由于铁电跃迁时晶体结构的变化而产生的自发应变常常伴随着自发极化的出现。因此，铁电晶体的状态一般可以用极化和应变这两个量级的参数来宏观表征。正是有序参数、极化和应变以及热力学变量（如温度、应力和电场）之间的耦合导致了铁电晶体的多功能性，从介电、压电到热释电性质，从而在各种电子器件中得到了广泛的应用，包括电容器、致动器、非易失性存储器和微机电系统。虽然这些耦合的热力学已经很好地建立了，但阶参数及其梯度之间的耦合还不太清楚。该计划的主要目标是从根本上理解挠曲电效应的作用，极化和应变梯度之间的耦合在晶体的铁电性，畴结构，畴壁极化分布和畴切换中的作用。有足够的证据表明，挠曲电效应在宏观体系中是很小的，通常被忽略，但随着尺寸的减小，接近纳米结构，特别是在具有强介电性能的铁电材料中，挠曲电效应可能变得重要甚至占主导地位。PI计划采用结合中尺度弹性和静电理论的相场建模方法。这项建议的主要目标是：(1)建立包含挠性电贡献的铁电畴结构和开关的相场模型；(2)研究挠性电贡献是否能显著改变铁电畴壁的性质，并发现由挠性电效应诱导的潜在的新畴壁特征；(3)研究挠性电贡献对薄膜极化分布的作用，从而对畴结构的影响。(4)研究局部机械力作用下铁电薄膜的挠性电响应，探索铁电极化机械开关的可能性。本研究的目标是：(1)建立一个模拟铁电体挠性电响应的相场公式；(2)为理解挠性电效应在铁电性质（包括畴壁结构、极化分布和开关）中的作用做出重要贡献；(3)提出基于谱法求解涉及畴壁各向异性和挠性电的相场方程的先进数值算法。该项目将通过本科毕业论文和暑期研究培训研究生和本科生，从而促进人力资源发展。研究结果将通过档案出版物和会议、评论论文以及在讲习班和会议上的积极参与和演讲传播给广泛的受众。最后，PI将积极寻求与洛斯阿拉莫斯、阿贡、橡树岭等行业和国家实验室的合作，以及与北卡罗莱纳州立大学和宾夕法尼亚州立大学介电和压电中心（CDP）相关的行业成员的合作，为参与该项目的学生提供实习机会。

NONTECHNICAL SUMMARYThis award supports theoretical research and computational modeling, and education with an aim to better understand ferroelectrics, which are multifunctional materials that have many uses, including actuators, sensors, memory storage, and microelectromechanical systems. These materials can not only produce electric signals under an applied electric field or a macroscopic shape deformation under an applied mechanical stress field but can also produce electric signals in response to an applied mechanical stress or a shape deformation in response to an applied electric field. There have been extensive studies on the couplings among electric signals, electric fields, homogeneous mechanical loads, temperature, and homogeneous shape deformations, and the basic science of how a homogeneous shape deformation affects the electric properties of ferroelectrics, characterized by the "piezoelectric effect," is reasonably well understood. In this project, the PI will focus on understanding how an inhomogeneous stress or shape deformation affects the multifunctional properties of ferroelectrics, through the "flexoelectric effect." Effort will be devoted to developing efficient computational methods and employing them to model, predict, and understand internal, nanoscale inhomogeneous structures and properties of ferroelectric materials under the influence of the flexoelectric effect. The computational research will be carried out in close collaboration with numerous experimental groups, computational physicists, and applied mathematicians. The fundamental understanding achieved and the computational tools developed in this project should provide guidance to develop material systems that can exploit the flexoelectric effect that exists in all materials. The proposed research is expected to contribute to graduate education in materials as phase-field simulations of phase transformations and microstructure evolution are being incorporated into graduate courses at Penn State. In particular, user-friendly graphical interfaces for a number of applications of the phase-field models have been developed under prior NSF support. The software has been employed in two graduate courses and one undergraduate course. In addition, it has been used in summer short courses on computational thermodynamics and kinetics of phase transformations, which were offered to research scientists from national labs, engineers from industry, and professors and students from academia. This project will provide new computational tools to illustrate how materials properties may be modified through the flexoelectric effect. The PI will involve undergraduate students in the research by participating in a number of programs at Penn State including senior thesis projects and the Minority Undergraduate Research Experience program.TECHNICAL SUMMARYFerroelectrics are a class of materials in which a spontaneous electric polarization develops below their paraelectric to ferroelectric phase transition temperatures. The spontaneous polarization direction can be reoriented among crystallographically defined orientations in a single crystal by an electric field. Very often a spontaneous strain arising from the crystal structure change at the ferroelectric transition accompanies the appearance of spontaneous polarization. So, the state of a ferroelectric crystal can generally be characterized macroscopically by two order parameters, polarization and strain. It is the coupling between the order parameters, polarization and strain, and the thermodynamic variables such as temperature, stress, and electric field that leads to the multifunctionality of a ferroelectric crystal ranging from dielectric, piezoelectric to pyroelectric properties, and thus to many applications in a wide variety of electronic devices, including capacitors, actuators, nonvolatile memories, and microelectromechanical systems. Although the thermodynamics of these couplings has been well established, the coupling among order parameters and their gradients is much less well understood. The main goal of this proposed program is to fundamentally understand the role of the flexoelectric effect, the coupling between polarization and the gradient of strain in the ferroelectricity of a crystal, in domain structures, polarization distributions across domain walls, and domain switching. There is sufficient evidence that the flexoelectric effect, which is small and generally ignored in macroscopic systems, may become significant or even dominant with decreasing size approaching nanostructures, particularly in ferroelectric materials which exhibit strong dielectric properties. The PI plans to employ a phase-field modeling approach integrated with mesoscale elasticity and electrostatic theory. The main objectives of this proposal are: (1) to develop a phase-field model of ferroelectric domain structures and switching incorporating flexoelectric contributions, (2) to study whether the flexoelectric contribution can significantly modify the properties of a ferroelectric domain wall and to discover potentially new domain wall features induced by the flexoelectric effect, (3) to investigate the role of the flexoelectric contribution to the polarization distribution and thus to domain structure in thin films, and (4) to investigate the flexoelectric response of ferroelectric thin films under a local mechanical force and explore the possibility of mechanical switching of ferroelectric polarization. The proposed research is expected to: (1) yield a phase-field formulation for modeling flexoelectric response of ferroelectrics, (2) significantly contribute to the fundamental understanding of the roles of flexoelectric effect in ferroelectric properties including domain wall structures, polarization distribution, and switching, and (3) produce advanced numerical algorithms based on the spectral method for solving phase-field equations involving domain wall anisotropy and flexoelectricity. The project will contribute to human resource development by training both graduate and undergraduate students through undergraduate thesis and summer research. The research findings will be disseminated to a wide audience through archival publications and conferences, review papers, and active participation and lectures at workshops and conferences. Finally, the PI will actively pursue collaborations with industry and national labs such as Los Alamos, Argonne, Oak Ridge, and the industrial members associated with the Center for Dielectrics and Piezoelectrics (CDP) at North Carolina State University and Penn State to provide internship opportunities for students involved in the project.