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## Numerical extraction of electric field distribution from thermal pulse method based on Monte Carlo simulation

Liang Ming-Hui, Zheng Fei-Hu, An Zhen-Lian, Zhang Ye-Wen
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• #### 摘要

热脉冲法是测量聚合物介质薄膜空间电荷分布的有效方法之一, 其数据的分析涉及第一类Fredholm积分方程, 只能采用合适的数值计算方法进行求解, 而Monte Carlo法是近年来提出的数值求解该方程的方法之一. 本文尝试使用Monte Carlo法在频域内实现热脉冲数据的分析, 通过一系列模拟计算讨论Monte Carlo法的分析效果. 计算结果表明: Monte Carlo法可实现对热脉冲法实验数据的有效分析, 提取被测薄膜内的电场分布, 而且计算的电场分布在整个样品厚度上都与真实分布较好地符合, 可有效地弥补尺度变换法只在样品表面附近获得较高准确度的缺陷. 该方法的局限性在于计算结果存在一定的振荡, 且在噪声和数据误差的影响下, 其准确性很大程度上依赖于奇异值分解过程中容差的选择, 在应用的方便程度方面还有待进一步提升.

#### Abstract

Thermal-pulse method is a powerful tool for measuring space charge distributions in polymer films. The data analysis for thermal-pulse method involves the Fredholm integral equation of the first kind, which requires an appropriate numerical procedure to obtain a solution. Various numerical techniques, including scale transformation and regulation method, are proposed. Of those numerical methods, the scale transformation (ST) is the simplest and the most widely used method. However, it presents a high spatial resolution only near the sample surface. Monte Carlo (MC) method is one of the recently proposed ways to solve the equation numerically and has been successfully applied to the analysis of laser intensity modulation method data, which also involves the Fredholm integral equation of the first kind. In this paper we attempt to analyze thermal-pulse data in frequency domain with the MC method and discuss its effectiveness based on some numerical simulations. The simulation results indicate that the electric field profiles can be effectively extracted by the MC method. The computed profiles by the MC method consist well with the supposed distributions in the entire thickness of the sample, while the profiles reconstructed by the ST method fit very well to the supposed one at the vicinity of the target surface and distort sharply along the direction of the thermal pulse propagation in the sample bulk. On the other hand, the oscillations in the computed results by the MC method could deteriorate its accuracy in this study. The influence of noise level on the analysis based on the MC method is also tested by the use of the simulated data. The results show that the computed profiles would become more fluctuant as the noise level increases. This problem can be solved by selecting a larger value of tolerance during the singular value decomposition procedure. Thus, the value of tolerance is considered to be one of the key parameters in this algorithm, which is actually hard to determine. Additionally, the experimental data obtained from a polypropylene film under applied electric field are analyzed to illustrate the feasibility of MC method to be applied to the thermal-pulse experimental data. The results also show that the spatial accuracy by the MC method in the entire sample thickness is higher than by the ST method, which verifies that the MC method is more suitable for detecting the electric field distribution in the deep bulk of the sample. Owing to noise and error, the accuracy of MC calculation depends on the chosen tolerance value, which is now considered to be an obstacle in applying this method to the practical thermal-pulse measurement.

#### 作者及机构信息

###### 通信作者: 郑飞虎, feihuzheng@tongji.edu.cn
• 基金项目: 国家自然科学基金(批准号: 51277133, 51477119, 51477118)和教育部博士点基金(批准号: 20130072110046)资助的课题.

#### Authors and contacts

###### Corresponding author: Zheng Fei-Hu, feihuzheng@tongji.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 51277133, 51477119, 51477118) and the Ph. D. Programs Foundation of Ministry of Education of China (Grant No. 20130072110046).

#### 参考文献

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#### 施引文献

•  [1] Zheng F H, Lin C, Liu C D, An Z L, Lei Q Q, Zhang Y W 2012 Appl. Phys. Lett. 101 172904 [2] Takada T, Sakai T 1983 IEEE Trans. Dielec. El. In. 18 619 [3] Laurenceau P, Dreyfus G, Lewiner J 1977 Phys. Rev. Lett. 38 46 [4] Zheng F H, Zhang Y W, Wu C S, Li J X, Xia Z F 2003 Acta Phys. Sin. 52 1137 (in Chinese) [郑飞虎, 张冶文, 吴长顺, 李吉晓, 夏钟福 2003 物理学报 52 1137] [5] Collins R E 1977 Rev. Sci. Instrum. 48 83 [6] Lang S B, Das-Gupta D K 1986 J. Appl. Phys. 59 2151 [7] Zhou Y X, Wang N H, Wang Y S, Sun Q H, Liang X D, Guan Z C 2008 Trans. China Electrotech. Soc. 23 16 (in Chinese) [周远翔, 王宁华, 王云杉, 孙清华, 梁曦东, 关志成 2008 电工技术学报 23 16] [8] Zheng F H, Liu C D, Lin C, An Z L, Lei Q Q, Zhang Y W 2013 Meas. Sci. Technol. 24 065603 [9] Mellinger A, Singh R, Gerhard-Multhaupt R 2005 Rev. Sci. Instrum. 76 013903 [10] Ploss B, Emmerich R, Bauer S 1992 J. Appl. Phys. 72 5363 [11] Petre A, Marty-Dessus D, Berquez L, Franceschi J L 2004 Jpn. J. Appl. Phys. 43 2572 [12] Tuncer E, Lang S B 2005 Appl. Phys. Lett. 86 071107 [13] DeReggi A S, Guttman C M, Mopsik F I, Davis G T, Broadhurst M G 1978 Phys. Rev. Lett. 40 413 [14] Bauer S, Ploss B 1991 Ferroelectrics 118 363 [15] Bauer S 1993 Phys. Rev. B 47 11049 [16] Tuncer E, Gubanski S M 2001 IEEE Trans. Dielec. El. Inl. 8 310 [17] Lang S B, Fleming R 2009 IEEE Trans. Dielec. El. In. 16 809 [18] Ploss B 1994 Ferroelectrics 156 345
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##### 出版历程
• 收稿日期:  2015-12-06
• 修回日期:  2016-01-12
• 刊出日期:  2016-04-05

## 基于Monte Carlo的热脉冲法数据分析

• 1. 同济大学电气工程系, 上海 201804
• ###### 通信作者: 郑飞虎, feihuzheng@tongji.edu.cn
基金项目: 国家自然科学基金(批准号: 51277133, 51477119, 51477118)和教育部博士点基金(批准号: 20130072110046)资助的课题.

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