research papers
Study of Xray topography using the superBorrmann effect
^{a}Synchrotron Radiation Research Center, Hyogo Science and Technology Association, 14902 Kouto, Shingu, Tatsuno, Hyogo 6795165, Japan, and ^{b}Graduate School of Science, University of Hyogo, 321 Kouto, Kamigori, Hyogo 6781297, Japan
^{*}Correspondence email: tsusaka@sci.uhyogo.ac.jp
Xray topography exerting the superBorrmann effect has been performed using synchrotron radiation to display dislocation images with a highspeed and highresolution CMOS camera. Forwardtransmitted Xrays are positively employed instead of reflected Xrays to reveal dislocations in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes owing to the superBorrmann effect. Before the experiment, minimum values of the attenuation coefficients A_{min}^{P} for σ and π polarizations of the incident Xrays in the threebeam case are calculated. Results demonstrate that A_{min}^{P} for both polarizations are almost 20 times larger than those in the twobeam (usual Borrmann effect) case. The transmitted Xrays can be used to confirm the efficacy of taking topographs under the superBorrmann conditions, as well as under multiplediffraction conditions. Furthermore, superBorrmann topographs can be considered for relatively thick crystals, where a conventional Lang Xray topography technique is difficult to apply.
Keywords: topography; superBorrmann effect.
1. Introduction
The effect of ) if the Bragg condition is satisfied for the 111 and reflections simultaneously in the wideangle diagram of perfect germanium (Ge) crystals with thickness t = 0.8 mm and 1.2 mm with Cu Kα radiation. Enhanced intensity spots for the 111 and reflections appear at the Kossel line intersection of the T_{111} and traces of the reflected beams, respectively. Furthermore, enhanced intensity spots for the 111 and reflections appear on the R_{111} and traces of the transmitted (refracted in the strict sense) beams, respectively, and are symmetrical to the intersection point of the T_{111} and traces with respect to the respective reflecting planes. Calculation results and interpretation for the decrease in the was provided (Hildebrandt, 1966, 1967) for the enhanced spots in the threebeam case (/hkl means χ_{g–h} when g = 111 and h = ). Later, the theoretical understanding based on detailed calculation was advanced (Feldman & Post, 1972), and was confirmed experimentally (Uebach & Hildebrandt, 1969; Hildebrandt, 1978).
can be enhanced (Borrmann & Hartwig, 1965Other combinations of simultaneous reflections for threebeam cases such as (Umeno & Hildebrandt, 1975) and (Umeno, 1972) were investigated. The for fourbeam and sixbeam cases involving 220 reflections was found to be also enhanced (Joko & Fukuhara, 1967). Theoretical explanations were also discussed by Afanasev & Kohn (1975, 1976, 1977). This enhanced is called the `superBorrmann' effect (Lang, 1998).
In terms of the application of the superBorrmann effect to Xray topography to image lattice defects in crystals, such as dislocations, as well as conventional Xray topography, few reports have been published probably owing to a too low Xray source intensity and large Xray beam divergence to develop clear defect images for a wide visual field. However, since the availability of synchrotron radiation, Xray topography can be applied for imaging lattice defects in crystals by choosing the correct Xray wavelength. In addition, it was previously reported that topographs combined with a highspeed and highresolution CMOS camera taken by employing forwardtransmitted Xrays under multiple diffraction conditions (brightfield Xray topographs) can reveal dislocations without noticeable image deformations (Tsusaka et al., 2016, 2019).
Furthermore, as a major advantage, it is expected that the forwardtransmitted Xrays riding on the superBorrmann effect also reveal dislocations existing in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes such as (111) and where conventional Lang Xray topography is difficult to apply. Therefore, this study deals with Xray topography performed under threebeam multiplediffraction conditions for thick Ge crystals using synchrotron radiation.
2. threebeam case
Fig. 1 shows an example of threebeam multiple diffraction in of a perfect Ge crystal. Note that the beam is not drawn here considering that it passes symmetrical to the 111 beam with respective to the (100) plane of symmetry in order to simplify calculation of the absorption decreases controlling the superBorrmann effect. Figs. 1(a) and 1(b) demonstrate two cases of different energy of the incident Xrays for E = E_{1} and E = E_{2}, respectively, which is higher than E_{1}. The black dashed triangle in Fig. 1 comprises the original K_{o}, K_{111} and g_{111}, where K_{o} is the incident Xray wavevector, K_{111} is the 111reflected Xray wavevector and g_{111} = K_{111} − K_{o} is the diffraction vector of the 111 reflection lying on the same plane. Rectangles OPQR and PP′O′O represent projections on and (100), respectively. The lengths of sides , and are given as follows,
where ω is an elevation angle of K_{o} (or K_{111}) from the rectangle OPQR parallel to the entrance surface. It is clear that both and are independent of E; however, becomes larger when E increases, as observed in Figs. 1(a) and 1(b). Then, we put a unit vector of K_{o} as s_{o} and unit vectors of the polarization components of K_{o} as σ_{o} and π_{o}, for horizontal and vertical polarizations, respectively. σ_{o} lies in the base plane and is perpendicular to s_{o}. Therefore, π_{o} is also perpendicular to s_{o} and σ_{o}.
 Figure 1 of Ge, where only the 111 reflection is represented, considering the beam travels in a symmetrical direction with the plane of symmetry (100) at incident Xray energies of 
Next, we put , and in Fig. 1 as components of K_{111} in the s_{o}, σ_{o} and π_{o} directions, respectively. The magnitudes of these vectors were calculated as , and , respectively. The lengths of sides and were found to have the following values,
Then, we obtained = and = as a result for the plane parallel to the entrance surface.
Since is symmetrical with K_{111}, = ; however, the s_{o}, π_{o} components of and K_{111} are identical. Consequently, the refracted beam K_{o} and two reflected beams K_{111} and are summarized as follows,
3. Absorption coefficients in the threebeam diffraction cases
Since the calculation process for the Kα_{1} radiation has already been provided by Authier (2001), explanation of the calculation will be kept to a minimum. Based on the fundamental equations of Xray the projections of the electric displacements D in the threebeam case to the plane normal to K_{o} can be written for the three beams as follows,
in the threebeam case under CuIt is possible to rewrite the above equations by using excitation errors , , ,
Since the 200 reflection and reflectin are forbidden ( = = 0), the above equations are expressed as follows,
The two relations are obtained from the second and third lines of equation (3) shown above,
From D_{o[111]} and one can obtain D_{111[o]} and using a vector formula A × (B × C) = (A · C)B − (A · B)C in a similar way to that described by Authier (2001). By substituting D_{111[o]} and thus obtained in the first line of equation (3), we obtain a relation involving only D_{o}. If we decompose D_{o} into two components, , parallel to the plane of symmetry, and , perpendicular to the plane of symmetry,
The first line of equation (2), which is
can be replaced using two scalar values and , given as follows,
where A, B and C are the coefficients of , and , respectively,
Considering X can be separated into σ_{o} and π_{o},
Therefore, we can derive the determinant as follows,
and hence, inevitably,
Note that the components of K_{o}, K_{111} and are given by equation (1) and = . Therefore, we understand that B vanishes, and hence AC also vanishes.
From the above considerations, A (coefficient of ) and C (coefficient of ) should be null independently for σ_{o} and π_{o} polarizations, respectively. As a result, we obtain
for σ_{o} polarization and
for π_{o} polarization.
Considering χ_{o} is minimum when ξ_{o} = ξ_{111}, the subsequent values for ξ_{o} can be given as
for σ_{o} polarization and
for π_{o} polarization.
In the twobeam case,
When ξ_{o} = ξ_{111}, satisfying the Bragg condition exactly,
For a cubic crystal such as Ge, the F_{111} and Fourier component of the dielectric susceptibility χ_{111} for the 111 reflection are given as follows,
where f_{Ge} is the of Ge, r_{e} is the classical electron radius, λ is the wavelength of the Xrays, and V_{c} is the volume of a of Ge. F_{111_r} and F_{111_i} are the real and imaginary parts of the complex number F_{111}, respectively. The magnitudes of F_{111} and χ_{111} can be derived from the corresponding atomic scattering factors as follows,
Then,
The minimum value of the g, = threebeam case is given as follows,
in thewhere μ_{o} is the normal and χ_{111_i} and χ_{o_i} are imaginary parts of χ_{111} and χ_{o}, respectively.
The minimum t is the slab thickness and γ_{o} = n_{hkl} · s_{o} is a direction cosine of the incident Xray wavevector K_{o} (its unit vector is s_{o}) to n_{hkl}, the normal to the Xray entrance surface. In the present threebeam case, it is found from Fig. 2 that the direction cosine γ_{o} is expressed as
is = , wherefor K_{o} to .
However, the Xray energy in the present experimental case using synchrotron radiation was E = 10 keV and the Ge slab thickness was t = 0.05 cm with the (100) entrance surface. Because the lattice parameter of Ge is a = 0.56754 nm and the of the 111 reflection becomes = 7.2458° leading to = = 0.12613, we can derive γ_{o} for n_{hkl} = n_{001} from Fig. 2 as follows,
which corresponds to 31.608° as an angle between n_{001} and s_{o}. In this case, the polarization factors P for the σ_{o} and π_{o} components are introduced from equation (6) as
for σ_{o} polarization and
for π_{o} polarization
This makes it possible to calculate the effective μ_{e} and the minimum in the threebeam case for the Ge slab having the (001) entrance surface with thickness of 0.05 cm, as demonstrated in Table 1, by retrieving the data on the attenuation length from CXRO (https://henke.lbl.gov/optical_constants/atten2.html). Furthermore, it was observed that for both polarizations in the threebeam case are approximately 20 times the values in the twobeam case, due to which the phenomenon is called the superBorrmann effect.

4. Topography experiment using the superBorrmann effect for Ge crystals
To take Xray topographs with minimized image deformation, we employed forwardtransmitted (but refracted) Xrays that satisfied the Bragg conditions for the two {111} adjacent planes lying symmetrically with respect to the {100} plane of symmetry, as demonstrated in Fig. 3. An Xray diffraction goniometer with an Xray source of approximately 1.2 mm × 1.2 mm was used with 10 keV Xrays from the synchrotron radiation through a silicon doublecrystal monochromator at the BL24XU8 beamline of SPring8 (Tsusaka et al., 2001), similar to previously reported multiplebeam diffraction topography (Tsusaka et al., 2016, 2019). In order to avoid the harmonics of the incident synchrotron beam, the usual detuning treatment was carried out before carrying out the topography experiment.
Various interference patterns on diffracted and transmitted images with defect appearance were also studied using a coherent Xray beam under multiplediffraction conditions (Okitsu et al., 2003; Okitsu, 2003). However, in the present case, topographic images were taken directly by the forwardtransmitted Xray beam instead of the diffracted Xray beam using an Xray imaging detector (Hondoh et al., 1989). The detector comprises a 20 µmthick Gd_{3}Al_{2}Ga_{3}O_{12} (GAGG) scintillator, relay lens optics and a highspeed CMOS camera (Hamamatsu, C1144022CU). This detector resolved a 1 µm lineandspace pattern.
A Ge slab of dimensions 10 mm (width) × 14 mm (height) × 0.5 mm (thickness) and the (001) surface was prepared for the superBorrmann topography experiment. The slab was rotated in the clockwise direction around the [100] axis until bright spots corresponding to the reflections from two adjacent {111} planes, for example (111) and , could be recognized on a fluorescent sheet. It is clear that this multiple (nbeam) diffraction from a single crystal is not considered to be socalled umweganregung (Reninger, 1937a,b) but simply simultaneous excitation of the plural diffractions. After confirming the double fluorescent spots by the two 111 reflections on the sheet, the images formed by the forwardtransmitted beam were directly captured by the CMOS camera. As demonstrated in Fig. 3, an adjacent pair of {111} planes was selected by rotating the slab 90° clockwise around the normal to the (001) slab surface.
Figs. 4(a)–4(c) show a fluorescent spot from (a) the directly transmitted Xray beam denoted as `0', (b) the direct beam and the 111 reflected beam, and (c) the direct beam, the 111 reflected beam and the reflected beam. It can be easily noticed that the triple fluorescent spots in Fig. 4(c) are much brighter than those in Fig. 4(b), indicating the superBorrmann effect. The shining light on the righthand side of Fig. 4(c) is due to a specular reflection by the Ge crystal surface from the 111 reflection spot on the fluorescent sheet. After the triple fluorescent spots were recognized with nearly the same by sample rotation adjustment around [100] and [001], the topographic image formed by the transmitted beam was captured by the CMOS camera. During the usual Borrmann topography adjustment procedure, no clear dislocation images were recognized on the monitor.
Fig. 5 shows one of the topographs taken under the superBorrmann conditions shown in Fig. 4(c) using a pair of 111 and reflections without deformation correction. Considering the Xray source is approximately 1.2 mm × 1.2 mm in size, four shots of topographic images are pasted together to achieve a widearea topograph. Regarding the usual Borrmann topography (twobeam case), the images of the dislocations correspond to local lower transmitted intensities (in the forwardrefracted direction) or lower diffraction intensities (along the diffraction direction), since crystal imperfection can destroy the This is also true for the current superBorrmann topography (threebeam case), since double excitation of the 111 and reflections only enhances the i.e. black contrast on the camera monitor corresponds to the local lower diffraction intensity and white contrast corresponds to the local higher intensity, a phenomenon contradictory to that on negative film.
There are four combinations of two adjacent 111 reflections, i.e. 111 and reflections (called Atype), and reflections (Btype), and reflections (Ctype), and and 111 reflections (Dtype). Additionally, there are two combinations of diagonal 111 reflections, i.e. 111 and reflections (Etype), and and reflections (Ftype). Therefore, if one observes dislocation images disappearing only in an Atype topograph, the dislocation should have a Burgers vector of , considering this vector is commonly perpendicular to both [111] and . Similarly, from the invisibility rule g · b = 0, where g is the diffraction vector and b the dislocation Burgers vector, B, C and Dtype topographs do not include any images of the dislocations with Burgers vectors of, respectively, ( 1/2)[101], (1/2)[011] and . However, the combination of diagonal 111 reflections (E and Ftype) does not develop into the superBorrmann effect owing to the existence of χ_{220} instead of χ_{200} in equation (2). According to the partial lack of the superBorrmann conditions, Burgers vectors of all the dislocations cannot be determined completely by only observing the A, B, C and Dtype topographs. Nevertheless, we can conclude that the Burgers vector of the dislocations disappearing only on the Atype topograph should belong to . For example, some parts of A and Btype topographs are shown in Figs. 6(a) and 6(b), respectively, for the same part of the specimen. The dislocation configurations circled in red can be seen in both topographs.
5. Conclusions
In this study, we conducted synchrotron Xray topography exerting the superBorrmann effect for imaging dislocations using a CMOS camera. Forwardtransmitted Xrays can reveal dislocations in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes owing to the superBorrmann effect. SuperBorrmann topographs can be captured for relatively thick crystals, even when a conventional Lang Xray topography technique is difficult to apply.
Prior to the experiment, the minimum attenuation coefficients and for σ and πpolarizations, respectively, of the incident Xrays in the threebeam (superBorrmann) case were calculated. It was found that and were almost 20 times larger than those in the twobeam (usual Borrmann effect) case.
Although it is possible to determine Burgers vectors for some of the dislocations based on the invisibility criteria, it is difficult to finalize the Burgers vectors of most dislocations considering that the employment of a pair of diagonal {111} planes does not produce the superBorrmann effect.
In addition to the topographs taken by employing forwardtransmitted Xrays under multiplediffraction conditions (brightfield Xray topographs), the forwardtransmitted Xrays riding on the superBorrmann effect also reveal dislocations existing in comparatively thick crystals by simultaneously exciting a pair of adjacent {111} planes such as (111) and . Therefore, this study deals with Xray topography using synchrotron radiation performed under threebeam multiple diffraction conditions exerting the superBorrmann effect for thick Ge crystals. Future research will attempt to experimentally detect dislocation behaviors around the very initial growth stage in the necking parts of dislocationfree silicon crystals.
It was clarified that forwardtransmitted Xrays using synchrotron radiation can be used to confirm the efficacy for capturing topographs not only under usual multiple diffraction conditions but also under superBorrmann conditions.
Acknowledgements
The authors would like to thank Messrs Y. Itoh and Y. Namioka for help with the Xray topography experiments and Dr Umeno for discussions about the superBorrmann effect. The synchrotron radiation experiments were performed at BL24XU of SPring8 with approval from the Japan Synchrotron Radiation Research Institute (Proposal Nos. 2019B3202, 2020A3202, 2021A3202, 2021B3202).
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