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X-Ray Diffraction Analysis of Residual Stress in Thin Polycrystalline Anatase Films and Elastic Anisotropy of Anatase

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Abstract

The importance of residual stress in anatase thin films for their photo-induced hydrophilicity was proved recently. Detailed X-ray diffraction (XRD) studies of residual stresses in titanium dioxide films are presented here. Measurements including multiple hkl reflections on several series of these films revealed the presence of tensile stresses in the films that were obtained by crystallization from amorphous state. Significant anisotropy of the strain was also found and compared with that of anatase, resulting from its theoretically calculated single-crystal elastic constants. The XRD data support the experimental evidence of the hypothesis that the [00l] axis is the elastically soft anatase direction, whereas the directions in the [h00] × [hk0] plane are elastically stiff. This is in agreement with the anisotropy predicted by single-crystal elastic constants that are obtained from ab-initio calculations. Residual stress analysis for materials with tetragonal symmetry is described and the theory is used to analyze the data. The anisotropy is very different from that for the rutile phase, and the experimental results agree well with the values calculated for anatase. A simplified method of XRD residual stress analysis in thin anatase films by total pattern fitting (TPF) is also presented. Tensile stresses are formed during the crystallization process and increase rapidly with reduced film thickness. They inhibit crystallization, which is then very slow in the thinnest films.

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Acknowledgments

This work was supported by the Academy of Sciences of the Czech Republic under Contract Nos. KAN400720701 and IAA101120803 and also as a part of the research plan MSM 0021620834 financed by the Ministry of Education of the Czech Republic.

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Authors and Affiliations

  1. Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University in Prague, 121 16, Praha 2, Czech Republic

    Z. Matěj (Postdoctoral Student), R. Kužel (Associate Professor and Head of the Department) & L. Nichtová (Postdoctoral Student)

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  1. Z. Matěj
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  2. R. Kužel
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Correspondence to R. Kužel.

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Manuscript submitted March 16, 2010.

Appendix A

Appendix A

Explicit formulas for conversion of the C ij stiffness elastic constants into the S ij compliance constants in the case of the 4/mmm Laue class can be found in Reference 44 or obtained directly from Eq. [4]:

$$ \begin{aligned} S_{11} & = \left( {C_{11} C_{33} - C_{13}^{2} } \right)/\left\{ {\left( {C_{11} - C_{12} } \right)\left[ {\left( {C_{11} + C_{12} } \right)C_{33} - 2C_{13}^{2} } \right]} \right\} \\ S_{12} & = \left( {C_{13}^{2} - C_{12} C_{33} } \right)/\left\{ {\left( {C_{11} - C_{12} } \right)\left[ {\left( {C_{11} + C_{12} } \right)C_{33} - 2C_{13}^{2} } \right]} \right\} \\ S_{33} & = \left( {C_{11} + C_{12} } \right)/\left[ {\left( {C_{11} + C_{12} } \right)C_{33} - 2C_{13}^{2} } \right] \\ S_{13} & = - C_{13} /\left[ {\left( {C_{11} + C_{12} } \right)C_{33} - 2C_{13}^{2} } \right] \\ S_{44} & = 1/C_{44} , \; S_{66} = 1/C_{66} \\ \end{aligned} $$
(A1)

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Matěj, Z., Kužel, R. & Nichtová, L. X-Ray Diffraction Analysis of Residual Stress in Thin Polycrystalline Anatase Films and Elastic Anisotropy of Anatase. Metall Mater Trans A 42, 3323–3332 (2011). https://doi.org/10.1007/s11661-010-0468-z

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