Brown’s equation for the simulation of complex domain structures in ferroelectrics
Authors: Ouyang Xin-Jian , Zhang Zi-Yang , Zhang Feng , Zhang Jia-Le , Wang Da-Wei ,   School of Microelectronics, Faculty of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Published: December 2022

<sec>Ferroelectric material possesses spontaneous polarization at room temperature, which can be switched by an external electric field. The diverse domain structures within ferroelectric materials, consisting of polarizations in various directions, often significantly affect their physical properties and practical applications. Numerical simulations can aid in comprehending and validating the complex domains observed in experiments. They can also provide guidance for controlling such structures. One popular method for finding dipole configurations is to create an energy model and employ it in Monte-Carlo simulations to find dipole configuration. However, since these simulations usually reaches the ground state of the system (the state with the lowest global energy), they often miss the dipole configurations of interest, such as topological domain structures, which are usually metastable.</sec><sec>Here, in order to simulate complex domain, we introduce Brown's equation, which is originally used for micromagnetic simulation, into the large-scale simulation of ferroelectric materials. Using the effective Hamiltonian as the energy model, we derive the Brown's equations with respect to the electric dipoles in ferroelectric materials, and invesitgate perovskites such as<inline-formula><tex-math id="M6">\begin{document}${\rm{BaTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M6.png"/></alternatives></inline-formula> bulk, <inline-formula><tex-math id="M7">\begin{document}${\rm{PbTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M7.png"/></alternatives></inline-formula> bulk, and <inline-formula><tex-math id="M8">\begin{document}${\rm{SrTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M8.png"/></alternatives></inline-formula>/<inline-formula><tex-math id="M9">\begin{document}${\rm{PbTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M9.png"/></alternatives></inline-formula>/<inline-formula><tex-math id="M10">\begin{document}${\rm{SrTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M10.png"/></alternatives></inline-formula> sandwiched structures. We demonstrate the reliability and feasibility of Brown's equation in ferroelectrics through the simulation of <inline-formula><tex-math id="M11">\begin{document}${\rm{BaTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M11.png"/></alternatives></inline-formula> bulk and <inline-formula><tex-math id="M12">\begin{document}${\rm{PbTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M12.png"/></alternatives></inline-formula> bulk, which are consistent with experiments. Then, using Brown's equation derived in our work, we obtain various domain structures in <inline-formula><tex-math id="M13">\begin{document}${\rm{SrTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M13.png"/></alternatives></inline-formula>/<inline-formula><tex-math id="M14">\begin{document}${\rm{PbTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M14.png"/></alternatives></inline-formula>/<inline-formula><tex-math id="M15">\begin{document}${\rm{SrTiO_3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222150_M15.png"/></alternatives></inline-formula> sandwiched structures, including periodic stripe domains and vortex domains. The simulation results are compared with related exprimental results.</sec>