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Influence of asymmetrical angle on crystal lattice strain analysis using Voigt-function method

Zhu Jie Ji Meng Ma Shuang

Influence of asymmetrical angle on crystal lattice strain analysis using Voigt-function method

Zhu Jie, Ji Meng, Ma Shuang
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  • The Voigt function provides a rapid and easy method of explaining the breadths of diffraction profiles, and it defines two main broadening types: the domain size and strain component. The latter is caused by lattice imperfection (dislocation and different defects). Thus, diffraction can be used to measure crystal strain with very high precision and accuracy. However, each of all the crystals used in the present study has asymmetrical angle due to the processes of cutting grinding and polishing. This deviation angle is the angle between the considered lattice plane and crystal surface. The crystal with asymmetrical angle also satisfies Bragg's law but with different incident angle and reflected one. In the following, we investigate the crystal strain as a function of asymmetrical angle to evaluate the lattice distortion in detail. The single crystal silicon samples with different asymmetrical angles (in a range from 0.008 to 5.306) are prepared in this experiment. The lattice plane is (111). After grinding and polishing, the surface and subsurface damage are almost wiped off to remove internal stress which comes from cracks and grain refinement. Only broadening from lattice strain depends on the nature of imperfection, and the shape of crystallite can be left. It is convenient to acquire the full width at half maximum (FWHM) and integral breadth of diffraction curve by high resolution X-ray diffraction technique. Using the Voigt function method, diffraction line is characterized by all three parameters of the half-width integral breadth and form factor. The crystal lattice strains are calculated by analyzing the experimental line profile composed of Cauchy and Gaussian parts. Simulation of coherence diffraction of asymmetric crystal silicon is achieved by ray tracing code SHADOW. Both the theoretical calculation and experimental results show that if asymmetrical angle reaches 0.749, the half-width and integral breadth of diffraction curve change obviously compared with the situation where asymmetrical angle reaches 0.008. This is why the calculation error of crystal strain will be beyond 5% by the Voigt function method no matter whether we use theoretical value or experimental data. It is shown that the precise crystal cut is extremely important for device application. And this conclusion will also be helpful in other crystal studies by using X-ray diffraction parameters.
      Corresponding author: Zhu Jie, jzhu008@tongji.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11575127) and the National Key RD Program of China (Grant No. 2017YFA0403304).
    [1]

    Zaprazny Z, Korytar D, Siffalovic P, Jergel M, Demydenko M, Mikulik P, Dobrocka E, Ferrari C, Vagovic P, Mikloska M 2014 Advances in X-Ray/EUV Optics and Components IX 9207 920701Y

    [2]

    Guigay J P, Ferrero C 2016 Acta Cryst. A 72 489

    [3]

    Yang D R, Fan R X, Yao H N 1994 Mater. Sci. Eng. 12 33 (in Chinese) [杨德仁, 樊瑞新, 姚鸿年 1994 材料科学与工程 12 33]

    [4]

    Zhao B H, Chen D L 1991 J. Zhejiang Univ. -Sci. A 25 538 (in Chinese) [赵炳辉, 陈立登 1991 浙江大学学报 25 538]

    [5]

    Zhu N C, Li R S, Chen J Y, Xu S S 1990 Acta Phys. Sin. 39 770 (in Chinese) [朱南昌, 李润身, 陈京一, 许顺生 1990 物理学报 39 770]

    [6]

    Cembali F, Fabbri R, Servidori M, Zani A 1992 J. Appl. Cryst. 25 424

    [7]

    Huang J Y, E J C, Huang J W, Sun T, Fezzaa K, Xu S L, Luo S N 2016 Acta Mater. 114 136

    [8]

    Sun Y, Wang S L, Gu Q T, Xu X G, Ding J X, Liu W J, Liu G X, Zhu S J 2012 Acta Phys. Sin. 61 210203 (in Chinese) [孙云, 王圣来, 顾庆天, 许心光, 丁健旭, 刘文洁, 刘光霞, 朱胜军 2012 物理学报 61 210203]

    [9]

    Dickey E C, Drivid V P, Hubbard C R 1997 J. Am. Ceram. Soc. 80 2773

    [10]

    Yoshiike T, Fujii N, Kozaki S 1997 J. Appl. Phys. 36 5764

    [11]

    Ward Iii A, Hendricks R W 1997 Proceding 5th International Conference Residual Stresses Lynkping, Sweden, June 16-18, 1997 p1054

    [12]

    Suzuki H, Akita K, Misawa H 2003 Jpn. Soc. Appl. Phys. 42 2876

    [13]

    Langford J I 1978 J. Appl. Cryst. 11 10

    [14]

    Macherauch E, Wohlfahrt H, Wolfstieg U 1973 Hart. Tech. Mitt. 28 201

    [15]

    de Kerjser Th H, Langford J I, Mittemeijer E J, Vogels A B P 1982 J. Appl. Cryst. 15 308

    [16]

    Chang M, Xu S L 1993 Acta Phys. Sin. 42 446 (in Chinese) [常明, 许守廉 1993 物理学报 42 446]

    [17]

    Chang M, Xing J H, Xu S L 1994 Mater. Sci. Technol. 2 21 (in Chinese) [常明, 邢金华, 许守廉 1994 材料科学与工艺 2 21]

    [18]

    Xu S S, Feng D 1987 X Ray Diffraction Topography (Beijing: Science Press) p168 (in Chinese) [许顺生, 冯端 1987 X射线衍衬貌相学 (北京: 科学出版社)第168页]

    [19]

    Fan Q C 2012 Mater. Sci. 2 106 (in Chinese) [范群成 2012 材料科学 2 106]

    [20]

    Chen H F, Liu K J 2006 J. Changshu Institute Technol. 20 39 (in Chinese) [陈惠芬, 刘克家 2006 常熟理工学院学报 20 39]

    [21]

    Wang C C, Fang Q H, Chen J B, Liu Y W, Jin T 2016 Int. J. Adv. Manuf. Technol. 83 937

    [22]

    Buchwald R, Frohlich K, Wurzner S, Lehmann T, Sunder K, Moller H J 2013 Energy Procedia 38 901

    [23]

    Fukumori T, Futagami K, Kuroki K 2004 Jpn. J. Appl. Phys. 43 8331

    [24]

    Zhang Y X 2006 Ph. D. Dissertation (Dalian: Dalian University of Technology) (in Chinese) [张银霞 2006 博士学位论文(大连: 大连理工大学)]

    [25]

    Langford J I, Wilson A J C 1978 J. Appl. Cryst. 11 102

  • [1]

    Zaprazny Z, Korytar D, Siffalovic P, Jergel M, Demydenko M, Mikulik P, Dobrocka E, Ferrari C, Vagovic P, Mikloska M 2014 Advances in X-Ray/EUV Optics and Components IX 9207 920701Y

    [2]

    Guigay J P, Ferrero C 2016 Acta Cryst. A 72 489

    [3]

    Yang D R, Fan R X, Yao H N 1994 Mater. Sci. Eng. 12 33 (in Chinese) [杨德仁, 樊瑞新, 姚鸿年 1994 材料科学与工程 12 33]

    [4]

    Zhao B H, Chen D L 1991 J. Zhejiang Univ. -Sci. A 25 538 (in Chinese) [赵炳辉, 陈立登 1991 浙江大学学报 25 538]

    [5]

    Zhu N C, Li R S, Chen J Y, Xu S S 1990 Acta Phys. Sin. 39 770 (in Chinese) [朱南昌, 李润身, 陈京一, 许顺生 1990 物理学报 39 770]

    [6]

    Cembali F, Fabbri R, Servidori M, Zani A 1992 J. Appl. Cryst. 25 424

    [7]

    Huang J Y, E J C, Huang J W, Sun T, Fezzaa K, Xu S L, Luo S N 2016 Acta Mater. 114 136

    [8]

    Sun Y, Wang S L, Gu Q T, Xu X G, Ding J X, Liu W J, Liu G X, Zhu S J 2012 Acta Phys. Sin. 61 210203 (in Chinese) [孙云, 王圣来, 顾庆天, 许心光, 丁健旭, 刘文洁, 刘光霞, 朱胜军 2012 物理学报 61 210203]

    [9]

    Dickey E C, Drivid V P, Hubbard C R 1997 J. Am. Ceram. Soc. 80 2773

    [10]

    Yoshiike T, Fujii N, Kozaki S 1997 J. Appl. Phys. 36 5764

    [11]

    Ward Iii A, Hendricks R W 1997 Proceding 5th International Conference Residual Stresses Lynkping, Sweden, June 16-18, 1997 p1054

    [12]

    Suzuki H, Akita K, Misawa H 2003 Jpn. Soc. Appl. Phys. 42 2876

    [13]

    Langford J I 1978 J. Appl. Cryst. 11 10

    [14]

    Macherauch E, Wohlfahrt H, Wolfstieg U 1973 Hart. Tech. Mitt. 28 201

    [15]

    de Kerjser Th H, Langford J I, Mittemeijer E J, Vogels A B P 1982 J. Appl. Cryst. 15 308

    [16]

    Chang M, Xu S L 1993 Acta Phys. Sin. 42 446 (in Chinese) [常明, 许守廉 1993 物理学报 42 446]

    [17]

    Chang M, Xing J H, Xu S L 1994 Mater. Sci. Technol. 2 21 (in Chinese) [常明, 邢金华, 许守廉 1994 材料科学与工艺 2 21]

    [18]

    Xu S S, Feng D 1987 X Ray Diffraction Topography (Beijing: Science Press) p168 (in Chinese) [许顺生, 冯端 1987 X射线衍衬貌相学 (北京: 科学出版社)第168页]

    [19]

    Fan Q C 2012 Mater. Sci. 2 106 (in Chinese) [范群成 2012 材料科学 2 106]

    [20]

    Chen H F, Liu K J 2006 J. Changshu Institute Technol. 20 39 (in Chinese) [陈惠芬, 刘克家 2006 常熟理工学院学报 20 39]

    [21]

    Wang C C, Fang Q H, Chen J B, Liu Y W, Jin T 2016 Int. J. Adv. Manuf. Technol. 83 937

    [22]

    Buchwald R, Frohlich K, Wurzner S, Lehmann T, Sunder K, Moller H J 2013 Energy Procedia 38 901

    [23]

    Fukumori T, Futagami K, Kuroki K 2004 Jpn. J. Appl. Phys. 43 8331

    [24]

    Zhang Y X 2006 Ph. D. Dissertation (Dalian: Dalian University of Technology) (in Chinese) [张银霞 2006 博士学位论文(大连: 大连理工大学)]

    [25]

    Langford J I, Wilson A J C 1978 J. Appl. Cryst. 11 102

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  • Received Date:  15 September 2017
  • Accepted Date:  16 November 2017
  • Published Online:  05 February 2018

Influence of asymmetrical angle on crystal lattice strain analysis using Voigt-function method

    Corresponding author: Zhu Jie, jzhu008@tongji.edu.cn
  • 1. Key Laboratory of Advanced Micro-Structured Materials MOE, Institute of Precision Optical Engineering, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11575127) and the National Key RD Program of China (Grant No. 2017YFA0403304).

Abstract: The Voigt function provides a rapid and easy method of explaining the breadths of diffraction profiles, and it defines two main broadening types: the domain size and strain component. The latter is caused by lattice imperfection (dislocation and different defects). Thus, diffraction can be used to measure crystal strain with very high precision and accuracy. However, each of all the crystals used in the present study has asymmetrical angle due to the processes of cutting grinding and polishing. This deviation angle is the angle between the considered lattice plane and crystal surface. The crystal with asymmetrical angle also satisfies Bragg's law but with different incident angle and reflected one. In the following, we investigate the crystal strain as a function of asymmetrical angle to evaluate the lattice distortion in detail. The single crystal silicon samples with different asymmetrical angles (in a range from 0.008 to 5.306) are prepared in this experiment. The lattice plane is (111). After grinding and polishing, the surface and subsurface damage are almost wiped off to remove internal stress which comes from cracks and grain refinement. Only broadening from lattice strain depends on the nature of imperfection, and the shape of crystallite can be left. It is convenient to acquire the full width at half maximum (FWHM) and integral breadth of diffraction curve by high resolution X-ray diffraction technique. Using the Voigt function method, diffraction line is characterized by all three parameters of the half-width integral breadth and form factor. The crystal lattice strains are calculated by analyzing the experimental line profile composed of Cauchy and Gaussian parts. Simulation of coherence diffraction of asymmetric crystal silicon is achieved by ray tracing code SHADOW. Both the theoretical calculation and experimental results show that if asymmetrical angle reaches 0.749, the half-width and integral breadth of diffraction curve change obviously compared with the situation where asymmetrical angle reaches 0.008. This is why the calculation error of crystal strain will be beyond 5% by the Voigt function method no matter whether we use theoretical value or experimental data. It is shown that the precise crystal cut is extremely important for device application. And this conclusion will also be helpful in other crystal studies by using X-ray diffraction parameters.

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