Correlated diffuse x-ray scattering from periodically nanostructured surfaces

V. Soltwisch, A. Haase, J. Wernecke, J. Probst, M. Schoengen, S. Burger, M. Krumrey, and F. Scholze

Phys. Rev. B 94, 035419 – Published 13 July 2016

Abstract

Laterally periodic nanostructures were investigated with grazing incidence small angle x-ray scattering. To support an improved reconstruction of nanostructured surface geometries, we investigated the origin of the contributions to the diffuse scattering pattern which is correlated to the surface roughness. Resonant diffuse scattering leads to a palmlike structure of intensity sheets. Dynamic scattering generates the so-called Yoneda band caused by a resonant scatter enhancement at the critical angle of total reflection and higher-order Yoneda bands originating from a subsequent diffraction of the Yoneda enhanced scattering at the grating. Our explanations are supported by modeling using a solver for the time-harmonic Maxwell's equations based on the finite-element method.

Received 22 October 2015
Revised 3 May 2016

DOI:https://doi.org/10.1103/PhysRevB.94.035419

Physics Subject Headings (PhySH)

Nanostructures

Finite-element method Small-angle x-ray scattering X-ray diffuse scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. Soltwisch^1,*, A. Haase¹, J. Wernecke¹, J. Probst², M. Schoengen², S. Burger³, M. Krumrey¹, and F. Scholze¹

¹Physikalisch-Technische Bundesanstalt (PTB), Abbestr. 2-12, 10587 Berlin, Germany
²Helmholtz-Zentrum Berlin (HZB), Albert-Einstein-Str. 15, 12489 Berlin, Germany
³Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany

^*Victor.Soltwisch@ptb.de

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Vol. 94, Iss. 3 — 15 July 2016

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Images

Figure 1
(a) SEM image of a silicon grating sample similar to grating No. 5 (see table in Fig. 3). (b) Finite-element model employed for the simulations of above grating sample. (c) Scattering of a collimated x-ray beam under grazing incidence from a lamellar grating oriented parallel $(φ = 0^{\circ})$ with respect to the incident beam.
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Figure 2
Calculated electric field intensities at the vacuum interface for incident angles $α_{i}$ around the critical angle of silicon. (a) The dashed black line represents the surface field enhancement at the critical angle for a flat Si substrate. (b) The substrate was capped with an additional layer with reduced mass density (42%) and a thickness of 100 nm. (c) The field intensity for a grating with rectangular lines at oblique azimuthal incidence $(φ = 2^{\circ})$ .
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Figure 3
Electric field intensity at the vacuum interface normalized to the incident plane wave calculated for $α_{i}$ below the critical angle of silicon for a rectangular grating with a line height of 50 nm (a) and 100 nm (b) as a function of structure density. The angular scale is normalized to $α_{c}$ . Blue circles in both figures mark the experimentally determined position $α_{f}$ of the lower Yoneda band boundary. The numerical values for the simulated angles $α_{i}$ and measured $α_{f}$ are summarized in the table. (c), (d) Measured Yoneda band characteristic obtained from grating No. 4 and grating No. 3.
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Figure 4
GISAXS pattern from grating No. 4 obtained at 9.5 keV and $α_{i} = 0 . 5^{\circ}, φ = 0^{\circ}$ with logarithmic gray scale. The range of the scale was adjusted to optimize the visibility of the diffuse scattering background.
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Figure 5
(a) Central area under an azimuthal rotation $(E = 8$ keV, $α_{i} = 0 . 8^{\circ}, φ = 0 . 5^{\circ})$ of the grating No. 4. The diffraction cone and higher order Yoneda bands follow this rotation, in contrast to the RDS sheets. Red dots illustrate the expected reciprocal space locations for the higher orders of the Yoneda band. (b) Magnified view of figure (a) around the higher-order Yoneda bands.
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Figure 6
Magnified view of the diffuse scattering pattern obtained from (a) grating No. $5 (E = 6$ keV, $α_{i} = 0 . 1^{\circ}, φ = 0^{\circ})$ and (c) grating No. $4 (E = 5$ keV, $α_{i} = 0 . 15^{\circ}, φ = 0^{\circ})$ around the higher-order Yoneda bands. (b), (d), (e) Computation of the electric field intensities for both gratings (see text). The insets show the rectangular grating model including corner rounding (b), (d) and a trapezoidal model with $80^{\circ}$ sidewall angle (e).
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