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Landau-Devonshire 自由能函数的高阶展开

Landau-Devonshire 自由能函数的高阶展开. 侯剑堃 2013 年 10 月 12 日.

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Landau-Devonshire 自由能函数的高阶展开

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  1. Landau-Devonshire自由能函数的高阶展开 侯剑堃 2013年10月12日

  2. To study the stable and metastable regions in temperature concentration phase diagrams in the perovskite-type ferroelectrics, we use Landau-Devonshire free energy function expressed in terms of the polarization components (p1,p2,p3), which are the transition parameters in the present case[1]:

  3. Landau-Devonshire free energy

  4. 相图

  5. MPB处存在单斜 • The MPB is known to appear in many perovskite-type solid solutions, and these materials with the concentration near MPB show excellent properties for various applications.[2] • Nohedal et al. found that a monoclinic phase appears between the tetragonal and rhombohedral phases in the MPB region, and they pointed out that the existence of the monoclinic phase can be pictured as providing a bridge between the T and R phases.[2]

  6. 研究L-D自由能[3] • 展开到四阶：有T和R相 • 展开到六阶：有T、R和O相，但是没有单斜 • 展开到八阶：有T、R、O三个相以及三个单斜相 （MA,MB,MC），但是MA的面积比较大,MC尤其是MB面积比较小；但是注意没有三斜 • 展开到十阶：依然没有三斜 • 展开到十二阶：拥有三斜

  7. 问题的提出 • 但是文献[3]中只是给出了展开至四、六、八、十、十二阶的结论，并没有描述展开的方法 • 最常见的截断次数为六阶，但是这样会无法体现单斜 • 为了研究单斜，我们应当展开至八阶（如果是研究三斜则应展开至十二阶） • 问题就是我们应当努力寻找高阶展开的方法

  8. 论文的检索

  9. 论文的主要引用模式 • （1）单纯的引用 • ①Devonshire理论在六阶和八阶（十二阶）时的结果不同 • ②引用文中的图 • ③引用一级相变或二级相变的结论 • （2）实验的引用 • ①通过实验验证文献[3]作者的猜想 • ②实验后得到数据与文献[3]模型相吻合 • （3）引用Devonshire理论中的单斜相(MA,MB,MC)

  10. 增加检索条件 • Devonshire & eighth

  11. 增加检索条件 • 78篇文献 • 成功下载58篇（学位论文有16篇） • 在剩下的42篇论文中 • 符合（1）单纯的引用的有33篇 • 符合（2）实验的引用的有2篇 • 符合（3）单斜的引用的有4篇 • 真正提到了8次展开的有3篇.[4-6]

  12. 参考文献[4]

  13. 参考文献[4] • B1 B2与温度有关 • 高次参数

  14. 参考文献[4] • 文中表明研究铁电相变时，自由能至少应该展开至八次而不是六次 • 然而Landau理论当中的某些系数是随温度改变的，最后也通过实验验证差距等等

  15. 参考文献[5] • Conventional wisdom suggests that for compositions close to the MPB, the piezoelectric coefficients maximize. • A number of relatively discoveries have driven a reassessment of our understanding of the MPB in PZT • (1)The unprecendented performance of complex perovskite single crystals; the mechanism appears to reply on a field-induced rotation. • (2)Monoclinic phase at the MPB in PZT

  16. 参考文献[5] • (3)The local atomic displacements in PZT appear to differ from those consistent with the macroscopic symmetry: for example, in the rhombohedral phase the lead cation displacements seem to be composed of local, short-range, random arrangements of tetragonal type displacements superimposed on the average, long range rhombohedral order.

  17. 参考文献[5] • Gibb’s free energy expansion(stress-free,8-order): • Ei: corresponding components of the applied electric field

  18. 参考文献[6] • In a narrow Ti concentration range(x=46-52%),the low temperature structure of PZT was found to be monoclinic MA • A similar MA structure has also been recently seen in PMN-PT below room temperature for x=35% • Another fine structure of the morphotropic phase boundary has been found in PZN-PT.

  19. 参考文献[6] • What is the most general phenomenological model of the phase diagram of a cubic system induced by a ferroelectric order parameter? • The results of the analysis of a simple twelfth-order model are presented in this paper.

  20. 参考文献[6] • 引入相互独立的J1 J2 J3，并通过他们展开成复杂式子

  21. 参考文献[6] • 之后提出两个要求： • （1）将式子简化； • （2）能够表现所有的相：C T O R MA MB MC Tri； • 对模型进行直接的数学研究非常复杂，文章中主要对PZT和PZN做研究

  22. 参考文献[6] • 一级相变： • 八个相中任意两个之间的一级相变都可以实现 • 二级相变： • 模型模拟关系图 和实际情况完全一样

  23. 参考文献 [1]Iwata M, Ishibashi Y. Coexistence States near the Morphotropic Phase Boundary: II. The Scaling of the Free Energy in PbZr_1-xTi_xO_3[J]. Japanese Journal of Applied Physics, 2013, 52(9): 09KF07. [2]Iwata M, Ishibashi Y. Coexistence States near the Morphotropic Phase Boundary[J]. Japanese Journal of Applied Physics, 2012, 51(9). [3]Vanderbilt D, Cohen M H. Monoclinic and triclinic phases in higher-order Devonshire theory[J]. Physical Review B, 2001, 63(9): 094108.

  24. 参考文献 [4] Sokolov A I. Fluctuations, higher order anharmonicities, and Landau expansion for barium titanate[J]. Physics of the Solid State, 2009, 51(2): 351-355. [5] Bell A J. Factors influencing the piezoelectric behaviour of PZT and other “morphotropic phase boundary” ferroelectrics[J]. Journal of materials science, 2006, 41(1): 13-25. [6]Sergienko I A, Gufan Y M, Urazhdin S. Phenomenological theory of phase transitions in highly piezoelectric perovskites[J]. Physical Review B, 2002, 65(14): 144104.

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