||Domain growth and Dielectric properties of Normal Ferroelectrics and Relaxors : A combined Ab initio and Monte Carlo Simulation Method
||Department of Materials Science and Engineering
Monte Carlo simulation
polar nano regions(PNRs)
Random-field和奈米極化區域(polar nano regions)的產生一直被認為是影響弛滯體各項性質的關鍵，然而其成因和機制至今仍未有定論。為探究一般鐵電材料和弛滯體中，random-field的有無對於其電域形貌及介電性質之影響，本研究分別採用第一原理及蒙地卡羅法兩種方法進行探討，利用第一原理及Nudged Elastic Band理論計算Pb0.5Zr0.5TiO3及Sr0.5Ba0.5Nb2O6之電域壁能障，然而由於NEB計算中，原子之實際擴散路徑並非本研究採用之直線路徑，因此其結果並未被採用。蒙地卡羅法模擬則利用Ising Model探討一般鐵電材料和弛滯體在不同電場條件下之電域形貌和介電性質，並探討電域形貌和介電性質之關聯性；材料的介電常數和該材料容不容易極化成正相關，越容易極化的材料其介電常數越高；對於鐵電材料或弛滯體等材料其電極化量主要的貢獻為晶格極化，因此在交流電場下，於每一電場週期內，晶格偶極反轉的個數越多，則介電常數會越大。模擬結果顯示偶極反轉的數量在溫度較高、電場較強或較低頻率狀態下較多，因此在這些條件下其介電常數也較高。由於弛滯體中的random-field效應，使其在低溫下便具有許多可反轉的偶極，因此其介電常數較一般鐵電材料高，介電常數實部峰值溫度Tm也較一般鐵電材料低。
The random-field and the generation of polar nano regions (PNRs) are considered as the main effect of the weird relaxors properties. However, there is still no conclusion about their generation mechanism. In order to investigate the random-field effect on domain morphology and dielectric properties in normal ferroelectrics and relaxors, we adopted Ab initio and Monte Carlo simulation, respectively. We calculated the domain wall energy barriers of Pb0.5Zr0.5TiO3 and Sr0.5Ba0.5Nb2O6 by Ab initio and Nudged Elastic Band. However, the actual atom diffusion path is not a linear path that we set, so the result was not accepted. On Monte Carlo part, we use Ising model to research the domain morphology and dielectric properties under different a.c. field for both materials and attempt to establish the corelation between domain morphology and dielectric properties. The susceptibility of the materials is positive correlated to their polarization ability. The easier the material polarized, the higher the susceptibility. For normal ferroelectric and relaxors, the lattice polarization contributes the highest proportion to their total polarizations. Thus under a.c. field, the susceptibility would rise as the flipping dipoles increase in the field cycle. The simulation results show that the quantities of flipping dipoles would be greater under higher temperature、stronger field amplitude or lower field frequency conditions so that the susceptibility would also be higher. Due to the random-field effect in relaxors, they possess more flipping dipoles under lower temperature conditions, so the susceptibility of relaxors is higher than normal ferroelectrics and Tm (the temperature where the peak of real part susceptibility tale place) is lower than normal ferroelectrics.
第一章 緒論 1
第二章 文獻回顧 3
2.2.1 鐵電性 3
2.2.2 極化機制與電域 4
2.3.1 鍶鋇鈮結構 8
2.3.2 弛滯體特性 10
2.3.3 奈米極化區域(PNRs) 11
2.4.1 直流電場下的移動模型 13
2.4.2 交流電場下的移動模型 14
2.4.3 介電時效化 16
2.4.4 鬆弛時間與介電常數 18
2.4.5 Cole-Cole plot模型 19
2.4.6 Dielectric Dispersion現象 21
2.4.7 Ising Model模擬回顧 23
第三章 模擬基礎理論回顧 24
3.1.1 密度泛函理論 24
3.1.2 Hohenberg-Kohn定理 24
3.1.3 Kohn-Sham方法 28
3.1.4 局部密度近似法 30
3.1.5 Kohn-Sham方程式與自洽場計算 31
3.2.1 Ising Model 34
3.2.2 Metropolis演算法 34
第四章 實驗設計與分析方法 38
4.1.1 第一原理計算設置 38
4.1.2 Ising Model設置 40
4.1.2 Random-field的建立 41
4.3.1 介電分析 43
4.3.2 電域面積變化分析 44
第五章 結果與討論 45
Part I 第一原理法計算 45
5.1 PZT結構計算 45
5.1.1 收斂驗證 45
5.1.2 最佳化結果 46
5.1.3 Born effective charge和極化量計算 47
5.2 SBN結構計算 48
5.2.1 收斂驗證 51
5.2.2 最佳化結果 51
5.2.2 Born effective charge和極化量計算 52
5.3 NEB位能障計算 53
5.4 第一原理計算總結 56
Part II 蒙地卡羅法計算 57
5.4.1 一般鐵電材料中之溫度效應 57
5.4.2 弛滯體中之溫度效應 63
5.5.1 一般鐵電材料中之電場強度效應 68
5.5.2 弛滯體中的電場強度效應 76
5.6.1 一般鐵電材料中的頻率效應 82
5.6.2 弛滯體中的頻率效應 88
第六章 結論與未來展望 93
第七章 參考文獻 95
 B. Meyer and D. Vanderbilt, "Ab initio study of ferroelectric domain walls in PbTiO3," Physical Review B, vol. 65, Mar 2002.
 T. Tybell, P. Paruch, T. Giamarchi, and J. M. Triscone, "Domain wall creep in epitaxial ferroelectric Pb(Zr0.2Ti0.8)O-3 thin films," Physical Review Letters, vol. 89, Aug 2002.
 M. S. Kim, P. Wang, J. H. Lee, J. J. Kim, H. Y. Lee, and S. H. Cho, "Site occupancy and dielectric characteristics of strontium barium niobate ceramics: Sr/Ba ratio dependence," Japanese Journal of Applied Physics Part 1-Regular Papers Short Notes & Review Papers, vol. 41, pp. 7042-7047, Nov 2002.
 J. A. Luna-Lopez, J. Portelles, O. Raymond, and J. M. Siqueiros, "Structural study of Sr0.3Ba0.7Nb2O6 and La0.030Sr0.255Ba0.700Nb2O6 ceramic systems," Materials Chemistry and Physics, vol. 118, pp. 341-348, Dec 2009.
 S. Podlozhenov, H. A. Graetsch, J. Schneider, M. Ulex, M. Wohlecke, and K. Betzler, "Structure of strontium barium niobate SrxBa1-xNb2O6 (SBN) in the composition range 0.32 <= x <= 0.82," Acta Crystallographica Section B-Structural Science, vol. 62, pp. 960-965, Dec 2006.
 V. V. Shvartsman, J. Dec, S. Miga, Tadeuszlukasiewicz, and W. Kleemann, "Ferroelectric Domains in SrxBa1-xNb2O6 Single Crystals (0.40.75)," Ferroelectrics, vol. 376, pp. 197-204, 2008.
 P. B. Jamieson, S. C. Abrahams, and J. L. Bernstein, "Ferroelectric Tungsten Bronze-Type Crystal Structures. I. Barium Strontium Niobate Ba[sub 0.27]Sr[sub 0.75]Nb[sub 2]O[sub 5.78]," The Journal of Chemical Physics, vol. 48, pp. 5048-5057, 1968.
 A. M. Glass, "INVESTIGATION OF ELECTRICAL PROPERTIES OF SR1-XBAXNB2O6 WITH SPECIAL REFERENCE TO PYROELECTRIC DETECTION," Journal of Applied Physics, vol. 40, pp. 4699-&, 1969.
 L. A. Bursill and P. J. Lin, "CHAOTIC STATES OBSERVED IN STRONTIUM BARIUM NIOBATE," Philosophical Magazine B-Physics of Condensed Matter Statistical Mechanics Electronic Optical and Magnetic Properties, vol. 54, pp. 157-170, Aug 1986.
 S. Lee, J. A. Bock, S. Trolier-McKinstry, and C. A. Randall, "Ferroelectric-thermoelectricity and Mott transition of ferroelectric oxides with high electronic conductivity," Journal of the European Ceramic Society, vol. 32, pp. 3971-3988, Dec 2012.
 W. Kleemann, "The relaxor enigma - charge disorder and random fields in ferroelectrics," Journal of Materials Science, vol. 41, pp. 129-136, Jan 2006.
 W. Kleemann, "Universal domain wall dynamics in disordered ferroic materials," in Annual Review of Materials Research. vol. 37, ed Palo Alto: Annual Reviews, 2007, pp. 415-448.
 A. A. Bokov and Z. G. Ye, "Recent progress in relaxor ferroelectrics with perovskite structure," Journal of Materials Science, vol. 41, pp. 31-52, Jan 2006.
 L. E. Cross, "RELAXOR FERROELECTRICS," Ferroelectrics, vol. 76, pp. 241-267, 1987.
 G. Burns and F. H. Dacol, "GLASSY POLARIZATION BEHAVIOR IN FERROELECTRIC COMPOUNDS PB(MG1/3NB2/3)O3 AND PB(ZN1/3NB2/3)O3," Solid State Communications, vol. 48, pp. 853-856, 1983.
 A. Naberezhnov, S. Vakhrushev, B. Dorner, D. Strauch, and H. Moudden, "Inelastic neutron scattering study of the relaxor ferroelectric PbMg1/3Nb2/3O3 at high temperatures," European Physical Journal B, vol. 11, pp. 13-20, Sep 1999.
 S. B. Vakhrushev, B. E. Kvyatkovsky, A. A. Naberezhnov, N. M. Okuneva, and B. P. Toperverg, "GLASSY PHENOMENA IN DISORDERED PEROVSKITE-LIKE CRYSTALS," Ferroelectrics, vol. 90, pp. 173-176, 1989.
 Y. Imry and S. Ma, "RANDOM-FIELD INSTABILITY OF ORDERED STATE OF CONTINUOUS SYMMETRY," Physical Review Letters, vol. 35, pp. 1399-1401, 1975.
 V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, "Nanopolar structure in Sr(x)Ba(1-x)Nb(2)O(6) single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy," Physical Review B, vol. 77, Feb 2008.
 T. T. Fang and T. Y. Chiu, "Polarization dynamics of polar nano-regions in Sr0.5Ba0.5Nb2O6 doped with combinations of Ce and Cr," Acta Materialia, vol. 59, pp. 1692-1699, Feb 2011.
 P. Chauve, T. Giamarchi, and P. Le Doussal, "Creep and depinning in disordered media," Physical Review B, vol. 62, pp. 6241-6267, Sep 2000.
 T. Nattermann, S. Stepanow, L. H. Tang, and H. Leschhorn, "DYNAMICS OF INTERFACE DEPINNING IN A DISORDERED MEDIUM," Journal De Physique Ii, vol. 2, pp. 1483-1488, Aug 1992.
 L. Roters, A. Hucht, S. Lubeck, U. Nowak, and K. D. Usadel, "Depinning transition and thermal fluctuations in the random-field Ising model," Physical Review E, vol. 60, pp. 5202-5207, Nov 1999.
 S. F. Edwards and D. R. Wilkinson, "THE SURFACE STATISTICS OF A GRANULAR AGGREGATE," Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 381, pp. 17-31, 1982.
 T. Nattermann, V. Pokrovsky, and V. M. Vinokur, "Hysteretic dynamics of domain walls at finite temperatures," Physical Review Letters, vol. 87, Nov 2001.
 R. Blinc, J. Dolinsek, A. Gregorovic, B. Zalar, C. Filipic, Z. Kutnjak, A. Levstik, and R. Pirc, "Local polarization distribution and Edwards-Anderson order parameter of relaxor ferroelectrics," Physical Review Letters, vol. 83, pp. 424-427, Jul 1999.
 J. P. Bouchaud, P. Doussineau, T. de Lacerda-Aroso, and A. Levelut, "Frequency dependence of aging, rejuvenation and memory in a disordered ferroelectric," European Physical Journal B, vol. 21, pp. 335-340, Jun 2001.
 L. K. Chao, E. V. Colla, M. B. Weissman, and D. D. Viehland, "Aging and slow dynamics in SrxBa1-xNb2O6," Physical Review B, vol. 72, Oct 2005.
 V. V. Shvartsman and W. Kleemann, "Evolution of nanodomains in the uniaxial relaxor Sr0.61Ba0.39Nb2O6 : Ce," Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control, vol. 53, pp. 2275-2279, Dec 2006.
 A. R. V. Hippel, Dielectric and Waves vol. Ch31.
 K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics I. Alternating current characteristics," Journal of Chemical Physics, vol. 9, pp. 341-351, Apr 1941.
 J. AK, "Dielectric Relaxation In Solids," ed, 1983.
 S. K. I. Rychetsky, V. Porokhonskyy, A. Pashkin, M. Savinov, V. Bovtun, J. Petzelt,M. Kosec, and M.Dressel,, "Frequency-independent dielectric losses (1/ f noise) in PLZT relaxors at low temperatures," Journal of Physics: Condensed Matter, vol. 15, pp. 6017-6030, 2003.
 X. Y. J. W. Zhai, M. Z. Wu, and L. Y. Zhang, "Preparation and microwave characterization of PbTiO3 ceramic and powder," Journal of Physics D: Applied Physics, vol. 34, pp. 1413-1416, 2001.
 J. G. P. a. W. Jackson, "The measurement of the dielectric properties of high-permittivity materials at centimetre wavelengths," Proceedings of the Institution of Electrical Engineers, vol. 96, pp. 383-389, 1949.
 G. Arlt, U. Bottger, and S. Witte, "DIELECTRIC-DISPERSION OF FERROELECTRIC CERAMICS AND SINGLE-CRYSTALS AT MICROWAVE-FREQUENCIES," Annalen Der Physik, vol. 3, pp. 578-588, 1994.
 U. Bottger and G. Arlt, "DIELECTRIC MICROWAVE DISPERSION IN PZT CERAMICS," Ferroelectrics, vol. 127, pp. 95-100, 1992.
 A. F. Devonshire, "THEORY OF BARIUM TITANATE .2," Philosophical Magazine, vol. 42, pp. 1065-1079, 1951.
 V. Mueller, Y. Shchur, and H. Beige, "Logarithmic domain-wall dispersion," Ferroelectrics, vol. 269, pp. 201-206, 2002.
 Y. Park, "Low-frequency-dispersion of Pb(Fe1/2Nb1/2)O-3 single crystal in the region of its paraelectric ferroelectric phase transition," Solid State Communications, vol. 113, pp. 379-383, 2000.
 L. Jin, V. Porokhonskyy, and D. Damjanovic, "Domain wall contributions in Pb(Zr,Ti)O-3 ceramics at morphotropic phase boundary: A study of dielectric dispersion," Applied Physics Letters, vol. 96, Jun 2010.
 W. Kleemann, J. Dec, S. Miga, T. Woike, and R. Pankrath, "Non-Debye domain-wall-induced dielectric response in Sr0.61-xCexBa0.39Nb2O6," Physical Review B, vol. 65, Jun 2002.
 Z. Q. Wu, W. H. Duan, Y. Wang, B. L. Gu, and X. W. Zhang, "Effect of defect-induced internal field on the aging of relaxors," Physical Review B, vol. 67, Feb 2003.
 Y. F. Zhang, C. L. Wang, H. C. Li, M. L. Zhao, and H. L. Zhang, "Simulation on dielectric susceptibility and domain evolution of relaxor ferroelectrics," Physica B-Condensed Matter, vol. 403, pp. 2822-2827, Aug 2008.
 Y. Laosiritaworn, K. Kanchiang, A. Ngamjarurojana, R. Yimnirun, R. Y. Guo, and A. S. Bhalla, "The Debye Dielectric Behavior of Mixed Normal and Relaxor-Ferroelectrics: Monte Carlo Investigation," Ferroelectrics, vol. 401, pp. 239-245, 2010.
 Y. H. Shin, I. Grinberg, I. W. Chen, and A. M. Rappe, "Nucleation and growth mechanism of ferroelectric domain-wall motion," Nature, vol. 449, pp. 881-U7, Oct 2007.
 Y. Laosiritaworn, K. Kanchiang, R. Yimnirun, R. Y. Guo, and A. S. Bhalla, "Monte Carlo Investigation of Mixed Normal and Relaxor Ferroelectrics," Ferroelectrics, vol. 382, pp. 28-35, 2009.
 D. Bolten, U. Bottger, and R. Waser, "Frequency and temperature dependence of the relative permittivity in ferroelectrics: Monte-Carlo simulation study," Journal of Applied Physics, vol. 93, pp. 2890-2894, Mar 2003.
 R. H. Dong, B. Zheng, and N. J. Zhou, "Creep motion of a domain wall in the two-dimensional random-field Ising model with a driving field," Epl, vol. 98, May 2012.
 Y. Tomita, T. Kato, and K. Hirota, "Monte Carlo Study of Relaxor Systems: A Minimum Model of Pb(In1/2Nb1/2)O-3," Journal of the Physical Society of Japan, vol. 79, Feb 2010.
 Y. F. Zhang, C. L. Wang, M. L. Zhao, J. C. Li, and R. Z. Zhang, "Potts-Ising model for simulation of polarization switching in polycrystalline ferroelectrics," Chinese Physics B, vol. 18, pp. 1665-1668, Apr 2009.
 T. Malcherek, "The ferroelectric properties of Cd2Nb2O7: a Monte Carlo simulation study," Journal of Applied Crystallography, vol. 44, pp. 585-594, Jun 2011.
 E. Ising, "Report on the theory of ferromagnetism," Zeitschrift Fur Physik, vol. 31, pp. 253-258, Feb-Apr 1925.
 R. D. Kingsmith and D. Vanderbilt, "THEORY OF POLARIZATION OF CRYSTALLINE SOLIDS," Physical Review B, vol. 47, pp. 1651-1654, Jan 1993.
 X. Duan, W. Luo, W. Wu, and J. S. Yuan, "Dielectric response of ferroelectric relaxors," Solid State Communications, vol. 114, pp. 597-600, 2000.
 C. C. Su, B. Vugmeister, and A. G. Khachaturyan, "Dielectric properties of material with random off-center defects: Monte Carlo simulation of relaxor ferroelectrics," Journal of Applied Physics, vol. 90, pp. 6345-6356, Dec 2001.
 Z. G. Wu and H. Krakauer, "First-principles calculations of piezoelectricity and polarization rotation in Pb(Zr0.5Ti0.5)O-3," Physical Review B, vol. 68, Jul 2003.
 B. Noheda, J. A. Gonzalo, L. E. Cross, R. Guo, S. E. Park, D. E. Cox, and G. Shirane, "Tetragonal-to-monoclinic phase transition in a ferroelectric perovskite: The structure of PbZr0.52Ti0.48O3," Physical Review B, vol. 61, pp. 8687-8695, Apr 2000.
 Y. Okuno, K. Kawato, M. Suzuki, A. Harada, and T. Oguchi, "Comparisons of piezoelectricities of lead zirconate titanate for various phases by a first-principles method," Japanese Journal of Applied Physics Part 1-Regular Papers Brief Communications & Review Papers, vol. 46, pp. 5199-5204, Aug 2007.
 G. Saghi-Szabo, R. E. Cohen, and H. Krakauer, "First-principles study of piezoelectricity in tetragonal PbTiO3 and PbZr1/2Ti1/2O3," Physical Review B, vol. 59, pp. 12771-12776, May 1999.
 L. B. Kong, J. Ma, H. T. Huang, W. Zhu, and O. K. Tan, "Lead zirconate titanate ceramics derived from oxide mixture treated by a high-energy ball milling process," Materials Letters, vol. 50, pp. 129-133, Aug 2001.
 F. Xu, S. Trolier-McKinstry, W. Ren, B. M. Xu, Z. L. Xie, and K. J. Hemker, "Domain wall motion and its contribution to the dielectric and piezoelectric properties of lead zirconate titanate films," Journal of Applied Physics, vol. 89, pp. 1336-1348, Jan 2001.
 H. D. Chen, K. R. Udayakumar, C. J. Gaskey, L. E. Cross, J. J. Bernstein, and L. C. Niles, "Fabrication and electrical properties of lead zirconate titanate thick films," Journal of the American Ceramic Society, vol. 79, pp. 2189-2192, Aug 1996.
 D. A. Barrow, T. E. Petroff, R. P. Tandon, and M. Sayer, "Characterization of thick lead zirconate titanate films fabricated using a new sol gel based process," Journal of Applied Physics, vol. 81, pp. 876-881, Jan 1997.
 T. T. Fang, H. Y. Chung, and C. H. Lee, "Defects, Structure Changes, and the Effect of Random Fields on the Displacement of Off-Center Ions in Sr0.5Ba0.5Nb2O6 Doped with Combinations of Ce and Cr," Journal of the American Ceramic Society, vol. 93, pp. 2339-2345, Aug 2010.
 S. B. Deshpande, H. S. Potdar, P. D. Godbole, and S. K. Date, "PREPARATION AND FERROELECTRIC PROPERTIES OF SBN/50 CERAMICS," Journal of the American Ceramic Society, vol. 75, pp. 2581-2585, Sep 1992.
 J. Zhang, G. S. Wang, F. Gao, C. L. Mao, F. Cao, and X. L. Dong, "Influence of Sr/Ba ratio on the dielectric, ferroelectric and pyroelectric properties of strontium barium niobate ceramics," Ceramics International, vol. 39, pp. 1971-1976, 2013.
 A. E. Glazounov, A. K. Tagantsev, and A. J. Bell, "Evidence for domain-type dynamics in the ergodic phase of the PbMg1/3Nb2/3O3 relaxor ferroelectric," Physical Review B, vol. 53, pp. 11281-11284, May 1996.
 S. M. Huang, C. D. Feng, L. D. Chen, and Q. Wang, "Relaxor behavior of Sr1-xBaxBi2Nb2O9 ceramics," Journal of the American Ceramic Society, vol. 89, pp. 328-331, Jan 2006.