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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.