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铁电材料在室温下具有可以在外加电场作用下改变方向的自发极化, 不同方向的极化在材料内部形成畴结构, 会对其物理特性和实际应用具有显著影响. 本文将最初用于微磁模拟的布朗方程引入铁电材料的大尺度模拟中, 研究其中可能出现的重要畴结构. 在以有效哈密顿量方法为基础推导出铁电材料中关于电偶极子的布朗方程后, 以
${\rm{BaTiO_3}}$ ,${\rm{PbTiO_3}}$ 块体和${\rm{SrTiO_3}}$ /${\rm{PbTiO_3}}$ /${\rm{SrTiO_3}}$ 夹心结构等钙钛矿铁电材料为研究对象, 验证了布朗方程的有效性并讨论了其中的多种畴结构, 如周期性条带状畴、涡旋型拓扑畴结构等, 并与相关实验结果进行了对比分析.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. 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 ${\rm{BaTiO_3}}$ bulk,${\rm{PbTiO_3}}$ bulk, and${\rm{SrTiO_3}}$ /${\rm{PbTiO_3}}$ /${\rm{SrTiO_3}}$ sandwiched structures. We demonstrate the reliability and feasibility of Brown's equation in ferroelectrics through the simulation of${\rm{BaTiO_3}}$ bulk and${\rm{PbTiO_3}}$ bulk, which are consistent with experiments. Then, using Brown's equation derived in our work, we obtain various domain structures in${\rm{SrTiO_3}}$ /${\rm{PbTiO_3}}$ /${\rm{SrTiO_3}}$ sandwiched structures, including periodic stripe domains and vortex domains. The simulation results are compared with related exprimental results.-
Keywords:
- topological domains /
- Brown’s equation /
- perovskites
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图 1 (a)—(c)是BTO系统模拟得到的三种偶极子极化构型, 宏观极化分别沿
$ \langle111\rangle $ ,$ \langle110\rangle $ ,$ \langle100\rangle $ 方向; (d)计算得到的所有样本中偶极子构型能量和宏观极化方向的关系, 横轴表示不同次计算获得的结果, 纵轴表示每个偶极子的平均能量(meV); (e)说明了图(a)—(c)中箭头颜色表示的偶极子极化方向, 不同颜色的箭头表明偶极子极化方向处于不同的象限. 下节PTO块体的构型也沿用了图(e)表示的颜色方向Fig. 1. (a)–(c) Three types of dipole polarization configurations in BTO simulation, where the macroscopic polarizations are along
$ \langle111\rangle $ ,$ \langle 110 \rangle $ ,$ \langle100\rangle $ , respectively; (d) energy (per dipole) with respect to the calculated configurations obtained from simulations; (e) dipole polarization direction indicating the colors used by the arrows in panels (a)–(c). The colors shown in panel (e) are also used for PTO bulk in the next section
图 2 PTO宏观极化沿
$ \langle 100\rangle $ 方向的偶极子构型Fig. 2. The dipole configuration of PTO bulk whose the macroscopic polarization is along
$ \langle 100\rangle $ 
图 3 (a) STO
$ _{6} $ /PTO$ _{8} $ /STO$ _{6} $ 体系模拟结果, 红色箭头和蓝色箭头分别表示极化沿$ \left[001\right] $ 正方向和负方向; (b) 图(a)中沿$ \left[100\right] $ 方向第28层偶极子平面投影图, 即图(a)黑色纵切面, 黄色圆表示涡旋畴结构Fig. 3. (a) Simulation results of STO
$ _{6} $ /PTO$ _{8} $ /STO$ _{6} $ system, where red arrows and blue arrows indicate that the polarization is along the positive and negative directions of$ \left[001\right] $ respectively; (b) projection of dipoles in the 28th$ (100) $ plane along the$ \left[100\right] $ direction (the black arrows in panel (a)). The yellow circles show the vortex domain structures
图 4 STO5/PTO20/STO5体系模拟结果 (a) STO/PTO/STO偶极子极化构型图, 为使图像更加清晰, 沿
$ \left[010\right] $ 方向每隔四层画一层, 红色箭头和蓝色箭头分别表示极化沿$ \left[001\right] $ 正方向和负方向; (b)图(a)中沿$ \left[010\right] $ 方向的第25层偶极子的纵切面投影图; (c)图(b)中黄色区域的放大图, 包括其沿$ \left[010\right] $ 方向的投影图和三维结构, 该涡旋畴沿$ \left[010\right] $ 方向呈柱状; (d)位于图(a)中沿$ [010] $ 方向的第13层偶极子纵切面底部; 在图(a)中, 图(b)—(d)表示结构的所在平面都用黑色箭头标注Fig. 4. Simulation results of STO
$ _{5} $ /PTO$ _{20} $ / STO$ _{5} $ system: (a) STO/PTO/STO dipole polarization configuration, drawing every 5th layers along the$ \left[010\right] $ direction in order to make the figure clearer; (b) the projected dipole pololarization of the 25th layer along the$ \left[010\right] $ direction; (c) detailed view of the yellow area in panel (b), the vortex domain is cylindrical along the$ \left[010\right] $ direction; (d) the dipole configuration of the 13th layer along the$ [010] $ direction. The planes shown in panels (b)–(d) are marked by black arrows in panel (a)
图 5 STO/PTO/STO模拟结果 (a) STO/PTO/STO偶极子极化构型示例(和图4(a)是同一个三维图), 沿[001]方向每隔三层画出一层中的偶极子, 同时画出PTO和STO交界层; (b) PTO中间层, 即沿
$ \left[001\right] $ 方向第15层的投影图; (c) PTO底部和STO交界层沿$ \left[001\right] $ 方向的投影图Fig. 5. (a) Dipole polarization configuration of STO/PTO/STO system, which is the configuration as Fig. 4(a), drawing every 4th layer along the
$ \left[001\right] $ direction in order to make the figure clearer, the interface layer between PTO and STO is also shown; (b) projection of the middle PTO layer along the$ \left[001\right] $ direction; (c) projection of the bottom interface PTO layer along$[001]$ 表 1 图1(a)—(c)所示构型中的近似沿
$ \langle 111\rangle $ 方向的偶极子数目Table 1. The number of unit cell dipoles that are approximately along
$ \langle 111\rangle $ direction for the configurations in Figs. 1(a)–(c), respectively宏观极化方向 $ \left\langle 111\right\rangle $ $ \left\langle 110\right\rangle $ $ \left\langle 100\right\rangle $ 近似沿$ \left\langle 111\right\rangle $方向的偶极子极化个数 1728 1536 1348 近似沿$ \left\langle 111\right\rangle $方向的偶极子极化占比 100% 89% 78% 
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[1] Seidel J, Ramesh R, Scott J, Catalan G 2012 Rev. Mod. Phys. 84 119
Google Scholar
[2] Nataf G F, Guennou M, Gregg J M, Meier D, Hlinka J, Salje E K H, Kreisel J 2020 Nat. Rev. Phys. 2 634
Google Scholar
[3] Scott J 2016 Ferroelectrics 503 117
Google Scholar
[4] 谭丛兵, 钟向丽, 王金斌 2020 物理学报 69 127702
Google Scholar
Tan C B, Zhong X L, Wang J B 2020 Acta Phys. Sin. 69 127702
Google Scholar
[5] Lu X M, Huang F Z, Zhu J S 2020 Acta Phys. Sin. 69
Google Scholar
[6] Tian G, Yang W, Chen D, Fan Z, Hou Z, Alexe M, Gao X 2019 Natl. Sci. Rev. 6 684
Google Scholar
[7] Das S, Tang Y L, Hong Z, Gonçalves M A P, McCarter M R, Klewe C, Nguyen K X, Gómez-Ortiz F, Shafer P, Arenholz E, Stoica V A, Hsu S L, Wang B, Ophus C, Liu J F, Nelson C T, Saremi S, Prasad B, Mei A B, Schlom D G, Íñiguez J, García-Fernández P, Muller D A, Chen L Q, Junquera J, Martin L W, Ramesh R 2019 Nature 568 368
Google Scholar
[8] Huang J, Tan P, Wang F, Li B 2022 Crystals 12 786
Google Scholar
[9] Das S, Ghosh A, McCarter M R, Hsu S L, Tang Y L, Damodaran A R, Ramesh R, Martin L W 2018 APL Mater. 6 100901
Google Scholar
[10] Seidel J 2016 Topological Structures in Ferroic Materials, volume 228 of Springer Series in Materials Science (Cham: Springer International Publishing)
[11] Lu L, Nahas Y, Liu M, Du H, Jiang Z, Ren S, Wang D, Jin L, Prokhorenko S, Jia C L, Bellaiche L 2018 Phys. Rev. Lett. 120 177601
Google Scholar
[12] Marton P, Rychetsky I, Hlinka J 2010 Phys. Rev. B 81 144125
Google Scholar
[13] Zhong W, Vanderbilt D, Rabe K M 1995 Phys. Rev. B 52 6301
Google Scholar
[14] Chen L Q 2008 J. Am. Ceram. Soc. 91 1835
Google Scholar
[15] Liu J, Chen W, Wang B, Zheng Y 2014 Materials 7 6502
Google Scholar
[16] Jiang Z, Xu B, Li F, Wang D, Jia C L 2015 Phys. Rev. B 91 014105
Google Scholar
[17] Wojdeł J C, Íñiguez J 2014 Phys. Rev. Lett. 112 247603
Google Scholar
[18] Zhong W, Vanderbilt D, Rabe K M 1994 Phys. Rev. Lett. 73 1861
Google Scholar
[19] Liu Y, Tang Y L, Zhu Y L, Wang W Y, Ma X L 2016 Adv. Mater. Interfaces 3 1600342
Google Scholar
[20] Brown W F 1941 Phys. Rev. 60 139
Google Scholar
[21] Brown W F 1940 Phys. Rev. 58 736
Google Scholar
[22] Aharoni A, et al. 2000 Introduction to the Theory of Ferromagnetism (Vol. 109) (Clarendon Press)
[23] García-Palacios J L, Lázaro F J 1998 Phys. Rev. B 58 14937
Google Scholar
[24] Kumar D, Adeyeye A O 2017 J. Phys. D: Appl. Phys. 50 343001
Google Scholar
[25] Zhu B, Lo C C H, Lee S J, Jiles D C 2001 J. Appl. Phys. 89 7009
Google Scholar
[26] Shen K, Tatara G, Wu M W 2011 Phys. Rev. B 83 085203
Google Scholar
[27] Li B L, Liu X P, Fang F, Zhu J L, Liu J M 2006 Phys. Rev. B 73 014107
Google Scholar
[28] Wang D, Liu J, Zhang J, Raza S, Chen X, Jia C L 2019 Comput. Mater. Sci. 162 314
Google Scholar
[29] Hong Z, Damodaran A R, Xue F, Hsu S L, Britson J, Yadav A K, Nelson C T, Wang J J, Scott J F, Martin L W, et al. 2017 Nano Lett. 17 2246
Google Scholar
[30] Chen L Q, Zhao Y 2022 Prog. Mater. Sci. 124 100868
Google Scholar
[31] Petković M, Petković L, Herceg ff 2010 J. Comput. Appl. Math. 233 1755
Google Scholar
[32] Sugiura H, Hasegawa T 2019 J. Comput. Appl. Math. 358 136
Google Scholar
[33] 黄云清, 舒适, 陈艳萍 2009 数值计算方法 (北京: 科学出版社)
Huang Y Q, Shu S, Chen Y P 2009 Numerical Computation Method (Beijing: Science Press) (in Chinese)
[34] Nishimatsu T, Iwamoto M, Kawazoe Y, Waghmare U V 2010 Phys. Rev. B 82 134106
Google Scholar
[35] Nishimatsu T, Aoyagi K, Kiguchi T, J Konno T, Kawazoe Y, Funakubo H, Kumar A, V Waghmare U 2012 J. Phys. Soc. Jpn. 81 124702
Google Scholar
[36] Nishimatsu T, Grünebohm A, Waghmare U V, Kubo M 2016 J. Phys. Soc. Jpn. 85 114714
Google Scholar
[37] Kwei G H, Lawson A C, Billinge S J L, Cheong S W 1993 J. Phys. Chem. 97 2368
Google Scholar
[38] Bersuker I B 1966 Phys. Lett. 20 589
Google Scholar
[39] Ravel B, Stern E A, Vedrinskii R I, Kraizman V 1998 Ferroelectrics 206 407
Google Scholar
[40] Liu J, Jin L, Jiang Z, Liu L, Himanen L, Wei J, Zhang N, Wang D, Jia C L 2018 J. Chem. Phys. 149 244122
Google Scholar
[41] Tinte S, Stachiotti M, Sepliarsky M, Migoni R, Rodriguez C 1999 J. Phys.: Condens. Matter 11 9679
Google Scholar
[42] Zhang F, Zhang J, Jing H, Li Z, Wang D, Jia C L 2022 Phys. Rev. B 105 024106
Google Scholar
[43] Tadmor E, Waghmare U, Smith G, Kaxiras E 2002 Acta Mater. 50 2989
Google Scholar
[44] Sani A, Hanfland M, Levy D 2002 J. Solid State Chem. 167 446
Google Scholar
[45] Abid A Y, Sun Y, Hou X, Tan C, Zhong X, Zhu R, Chen H, Qu K, Li Y, Wu M, et al. 2021 Nat. Commun. 12 1
Google Scholar
[46] Lemee N, Infante I C, Hubault C, Boulle A, Blanc N, Boudet N, Demange V, Karkut M G 2015 ACS Appl. Mater. Interfaces 7 19906
Google Scholar
[47] Pereiro M, Yudin D, Chico J, Etz C, Eriksson O, Bergman A 2014 Nat. Commun. 5 1
Google Scholar
[48] Sichuga D, Bellaiche L 2011 Phys. Rev. Lett. 106 196102
Google Scholar
[49] Kim J H, Lange F F 1999 J. Mater. Res. 14 1626
Google Scholar
[50] Zhang J, Wang Y J, Liu J, Xu J, Wang D, Wang L, Ma X L, Jia C L, Bellaiche L 2020 Phys. Rev. B 101 060103
Google Scholar
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