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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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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]:

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Landau-Devonshire free energy

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相图

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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]

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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

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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

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论文的检索

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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)

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增加检索条件Devonshire & eighth

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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]

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参考文献 [4]

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参考文献 [4]B1 B2 与温度有关

高次参数

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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

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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

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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.

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

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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.

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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.

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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

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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

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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

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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.

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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.