Effects of surface undulations on asymmetric X-ray diffraction: a rocking-curve topography study

Very asymmetric crystal diffraction was obtained from a finely polished silicon crystal set to reflect in Bragg diffraction at grazing incidence for the 333 reflection. The angle of incidence to achieve Bragg diffraction was varied between 1.08 and 0.33° by changing the X-ray energy from 8.100 to 8.200 keV. The effects of surface undulations were identified, and the results are compared with dynamical X-ray diffraction calculations.

Abstract

The results are reported of an X-ray diffraction study of an Si crystal designed and fabricated for very asymmetric diffraction from the 333 reflection for X-ray energies of 8.100 and 8.200 keV. A crystal with an asymmetry angle of 46 ± 0.1° between the surface and the (111) planes was studied. The grazing angles of incidence were near 1.08 and 0.33° for these two energies, respectively. Features arising from surface undulations were not observed at 8.100 keV, but were observed at 8.200 keV. The results at 8.100 keV allow an alternative explanation based on strain near the surface to be ruled out. Topographic images were obtained as a function of rocking angle, and in the case of 8.200 keV the surface morphology is evident. The results are found to be in agreement with dynamical X-ray diffraction calculations made with the Takagi–Taupin equations specialized to a surface having convex or concave features, as reported in the accompanying paper [Macrander (2020). J. Appl. Cryst.53, 793–799].

1. Introduction  

Surface morphology is not incorporated into the classical X-ray dynamical diffraction theory of Ewald, Laue and Darwin (Darwin, 1914 ▸; Ewald, 1916 ▸, 1917 ▸; von Laue, 1931 ▸; Batterman & Cole, 1964 ▸; Zachariasen, 1967 ▸; James, 1982 ▸; Authier, 2001 ▸). The theory is rigorous in the sense that Maxwell’s equations are solved but is approximate in the sense that only diffraction from perfectly planar surfaces is treated. Surface morphology is commonly ignored when comparing experimental reflectivity data. The result has been that further development of classical dynamical theory that can account for a non-planar surface has not been pursued. In the present work we have addressed the possible effect of surface undulations on X-ray diffraction from a well polished single crystal of silicon. We report that undulations can indeed be ignored provided that the grazing angle of incidence is greater than about 1°. However, such surface morphology cannot be ignored if the grazing angle becomes much lower. We have imaged the undulations for angles of incidence near 0.33° for an asymmetric reflection. The measurements were made on beamline 1-BM of the Advanced Photon Source (Macrander et al., 2016 ▸) with a technique referred to as rocking-curve topography (RCTOPO), employing upstream conditioning crystal optics set to diffract sequentially (Stoupin et al., 2016 ▸).

2. Experimental arrangement  

The experimental arrangement is shown in Fig. 1 ▸. Here, the X-ray energy is set by a double crystal monochromator (DCM) configured to diffract vertically in the (+, −) arrangement. The third reflection is from a silicon crystal set to diffract from the 333 reflection in a symmetric geometry, that is, the angle of incidence equals the angle of exit. The crystal for this third reflection is referred to as a conditioning crystal and is configured to be in the (−, −) diffraction geometry in relation to the second crystal of the DCM. As a result, the angle–energy phase space of the beam after the conditioning crystal is constrained on the Dumond diagram on which the energy at diffraction is plotted versus angle. The angle–energy phase-space region has a divergence of 30 µrad and a bandpass of 0.22 eV.

Figure 1.

Experimental arrangement for the rocking-curve topography measurements

The fourth and final reflection is from the crystal under study, which is shown schematically in Fig. 2 ▸. This crystal was designed and fabricated to diffract from the 333 reflection at grazing incidence, as shown for the case of 8.200 keV in Fig. 3 ▸. The asymmetry angle was 46 ± 0.1°. Images were taken on a commercial CCD camera having 13 × 13 µm pixels (PIXIS-XF, Princeton Instruments, https://www.princetoninstruments.com/). The point-spread function for this camera is 40 µm. In applying the RCTOPO technique one assumes a one-to-one relationship between points in the image and locations on the crystal (Macrander et al., 2005 ▸). In the present measurements, the exit divergences were less than 1.5 µrad and the distance between the crystal and the camera was 0.3 m. The product of the rocking-curve width and this distance is 0.4 µm, which easily satisfies the requirement that the reflected beam does not ‘walk off’ an effective pixel, and we can safely make the one-to-one assumption over the width of the point-spread function. The result is that the spatial resolution of the topographic images that we report is  40 µm. As detailed below, the surface features were larger than 40 µm, with the result that these topographic images can be interpreted as images of the surface.

Figure 2.

Schematic drawing for the Si crystal, showing the 46 ± 0.1° angle between the surface and the (111) planes.

Figure 3.

The diffraction geometry for the Si(333) reflection with a 46 ± 0.1° asymmetry angle. The angle of incidence satisfying the Bragg condition is near 0.33°.

Not shown in Fig. 1 ▸ is a rotating disc of carbon, commonly referred to as a diffuser and placed just upstream of the conditioning crystal. On beamline 1-BM, the DCM is separated from the conditioning crystal by Be windows, and these windows were found to produce unwanted and spurious contrast features on the topographic images. This effect is well known (Snigirev et al., 1996 ▸). As reported by others (Cloetens et al., 1996 ▸; Irvine et al., 2010 ▸), a rotating diffuser disc can remove these contrast features. We inserted a rotating 1.5 mm thick disc of carbon-fibre composite, and we found the diffuser to be very effective in removing spurious contrast from the topographs. We observed that the images became quite clean, and the contrast features of interest could be ascribed to the crystal under study simply by moving the crystal and then noting that the features moved with the crystal.

3. Metrology with visible light  

The surface of the crystal was characterized with visible light using an interferometer, model ADE-Phase Shift MicroXAM RTS (https://www.aps.anl.gov/Optics/Optical-Metrology). The results are shown in Fig. 4 ▸. An average peak-to-valley height difference of 30 nm applies in Fig. 4 ▸. Although peak and valley features are of various sizes, we estimate a peak-to-peak distance of about 1 mm for the dominant features.

Figure 4.

Surface profile results obtained using metrology with visible light. The area shown is 1500 µm square. The average peak-to-valley height difference is 30 nm.

4. Results at 8.100 keV  

Rocking curves were measured for each pixel by collecting images over a wide rocking-angle range in steps of 5 µrad. Two of these topographic images are shown in Fig. 5 ▸. The images are for a region measuring 3.90 mm square. When the counts in all the pixels in this region are summed at each rocking angle, an overall average rocking curve is obtained. The rocking angle near the half-peak value both below and above the angle for the peak of this average rocking curve can then be determined. The images shown in Fig. 5 ▸ are for the corresponding topographs. Also shown in Fig. 5 ▸ are the counts in pixels lying along the dashed lines in each image. We note the absence of a correspondence between these two vertically crosscut data sets. When local rocking curves for the pixels at points in the topographs are compared they are not shifted in angle. That is, we find that rocking curves are aligned to within about 3 µrad, a value close to 5 µrad, which was the angular step size between images. The above results for 8.100 keV are key to the present finding that surface undulations can be imaged at 8.200 keV (see below) because they rule out an alternative explanation based on strain in the near-surface region. Since the extinction depth can be made quite small as the grazing angle is reduced, one can seemingly study strain close to the surface in a diffraction geometry such as the one reported here. However, as we show in the present work, the effect of surface morphology may then also need to be considered.

Figure 5.

Topographic images obtained at 8.100 keV, along with crosscut amplitudes along the dashed lines. Both topographs are 3900 µm square. The images were obtained at rocking angles near the half-peak intensity both below and above the peak in intensity.

5. Results at 8.200 keV  

Similar to what was done for the 8.100 keV data presented above, we show results obtained for 8.200 keV in Fig. 6 ▸. In contrast to the results at 8.100 keV, the crosscuts are now clearly correlated. Maxima for the high-angle case correspond to minima for the low-angle case. This would be the case if the rocking curves were shifted in angle from point to point, and this effect is depicted in Fig. 7 ▸, which shows the two rocking curves for the points A and B identified in Fig. 6 ▸. Point A is for a dark point in the topograph taken at a high angle, whereas point B is for a bright point at a high angle. The contrasts for these points are reversed at low angle. That this effect results from a shift in the rocking curve for point B relative to point A is illustrated in Fig. 7 ▸. We note that there is a clear distinction from the case at 8.100 keV where no such effect was observed.

Figure 6.

Topographic images obtained at 8.200 keV, along with crosscut amplitudes along the dashed lines. Both topographs are 3900 µm square. As for Fig. 5 ▸, the images were recorded near the half-peak intensity. Also shown are two points labelled A and B, for which rocking curves are shown in Fig. 7.

Figure 7.

Rocking curves for 8.200 keV. The data are for points A and B in Fig. 6 ▸ and reveal a shift in the rocking angle at peak intensity between these points. The contrast differences shown in Fig. 6 ▸ are for the same abscissa values, as indicated by the vertical lines. Below the peaks point A has the greater intensity, whereas above the peaks point B has the greater intensity. These results can be explained by Takagi–Taupin X-ray diffraction simulations, which show a similar shift in the rocking curve for a convex surface relative to a concave surface.

6. Comparison with dynamical diffraction simulations  

Dynamical X-ray diffraction simulations of very asymmetric diffraction from single crystals of silicon were made to accompany the present experimental rocking-curve topography study and are reported in a parallel paper (Macrander, 2020 ▸). The development of Uragami (1969 ▸) for Takagi–Taupin simulations was followed and applied to both convex and concave surface features. The theoretical development and applicable simulation results are reported in the accompanying paper. Shown in Fig. 8 ▸ are both the rocking-curve data for point A and the result of the simulation for a convex surface as detailed by Macrander (2020 ▸). We consider the agreement for the FWHM to be reasonable. Further evidence to support a comparison with the present data is presented in the accompanying paper. Effects on rocking curves were simulated and, in agreement with the experimental data, a shift between a convex and a concave surface was found at 8.200 keV, for which the incidence angles at the Bragg condition are near 0.33°. However, such a shift was absent at 8.100 keV, for which the incidence angles are near 1.08°. We conclude that the presently observed effect of surface features only occurs at quite small grazing angles.

Figure 8.

Rocking-curve data for point A in Fig. 6 ▸ and the result of Takagi–Taupin simulations for a convex surface (solid line).

7. Summary and conclusions  

We have found a good agreement between a topographic rocking-curve study of very asymmetric diffraction from a silicon crystal and the Takagi-–Taupin dynamical diffraction simulations reported by Macrander (2020 ▸). This agreement allows us to conclude that convex and concave surface features can be seen in Bragg diffraction. Furthermore, an alternative explanation for the observed shift in the rocking curve based on strain at or near the surface has been ruled out from data taken at an energy 100 eV lower. We note that this is a significant aspect of the present work, since the occurrence or absence of near-surface strain based on shifts in rocking curves would otherwise be misinterpreted.

Funding Statement

This work was funded by U.S. Department of Energy, Office of Basic Energy Sciences grant DE-AC-02-06CH11357.