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Dynamical Theory of X-Ray Diffraction in a Crystal with a Surface Grating of Another Material

  • ATOMS, MOLECULES, OPTICS
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Abstract

A general theory is developed for dynamical X-ray diffraction in a crystal on the surface of which a lateral periodic structure of thin-film lines (strips) of another material is formed. On the basis of the model of edge forces, the fields of elastic lattice displacements in the substrate are calculated that arise as a result of formation of a lateral surface grating (SG). With the use of the formalism of diffraction of spatially restricted X-ray beams, solutions are obtained for the amplitudes of X-ray waves reflected from a crystal with an SG whose chemical composition differs from the composition of the substrate. A numerical simulation is carried out of X-ray diffraction in a silicon substrate with SGs of tungsten and SiO2 oxide. It is shown that the angular distributions of the scattering intensity by a silicon crystal with tungsten and oxide lines of identical size differ significantly, and the physical nature of such difference is established.

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Funding

This work was supported in part by the Ural Branch of the Russian Academy of Sciences within the Fundamental Research Program (project no. 18-10-2-23) and by the Russian Foundation for Basic Research (project no. 17-02-00090-a).

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Authors and Affiliations

  1. Institute of Physics and Mathematics, Komi Research Center, Ural Branch, Russian Academy of Sciences, 167982, Syktyvkar, Russia

    V. I. Punegov

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  1. V. I. Punegov
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Correspondence to V. I. Punegov.

Additional information

Translated by I. Nikitin

APPENDIX

APPENDIX

The elastic deformations arising in the substrate can be calculated with the use of relations (33):

$$\begin{gathered} {{\varepsilon }_{{xx}}} = \frac{1}{{{{E}_{s}}}}[{{\sigma }_{{xx}}} - {{\nu }_{s}}{{\sigma }_{{zz}}}] \\ = - \frac{{2{{F}_{f}}}}{{\pi {{E}_{s}}}}\left( {\frac{{{{x}^{3}}}}{{{{{({{x}^{2}} + {{z}^{2}})}}^{2}}}} - {{\nu }_{s}}\frac{{x{{z}^{2}}}}{{{{{({{x}^{2}} + {{z}^{2}})}}^{2}}}}} \right), \\ \end{gathered} $$
((A.1))
$$\begin{gathered} {{\varepsilon }_{{zz}}} = \frac{1}{{{{E}_{s}}}}[{{\sigma }_{{zz}}} - {{\nu }_{s}}{{\sigma }_{{xx}}}] \\ = - \frac{{2{{F}_{f}}}}{{\pi {{E}_{s}}}}\left( {\frac{{x{{z}^{2}}}}{{{{{({{x}^{2}} + {{z}^{2}})}}^{2}}}} - {{\nu }_{s}}\frac{{{{x}^{3}}}}{{{{{({{x}^{2}} + {{z}^{2}})}}^{2}}}}} \right). \\ \end{gathered} $$
((A.2))

Knowing the relation between the strains and the fields of lattice displacements, εxx = dux/dx and εzz = duz/dx, we obtain

$${{u}_{x}} = - \frac{{2{{F}_{f}}}}{{\pi {{E}_{s}}}}\left( {\frac{{{{x}^{2}} + {{\nu }_{s}}{{z}^{2}}}}{{2({{x}^{2}} + {{z}^{2}})}} + \frac{1}{2}\ln {\text{|}}{{x}^{2}} + {{z}^{2}}{\text{|}}} \right),$$
((A.3))
$${{u}_{z}}\, = \,\frac{{2{{F}_{f}}}}{{\pi {{E}_{s}}}}\left( {(1\, + \,{{\nu }_{s}})\frac{{zx}}{{2({{x}^{2}}\, + \,{{z}^{2}})}} - (1\, - \,{{\nu }_{s}})\frac{1}{2}{\text{arctan}}\frac{z}{x}} \right).$$
((A.4))

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Punegov, V.I. Dynamical Theory of X-Ray Diffraction in a Crystal with a Surface Grating of Another Material. J. Exp. Theor. Phys. 129, 197–209 (2019). https://doi.org/10.1134/S1063776119070185

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  • DOI: https://doi.org/10.1134/S1063776119070185

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