Study of X-ray topography using the super-Borrmann effect

Matsui, J.; Takatsu, K.; Tsusaka, Y.

doi:10.1107/S1600577522007779

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 29| Part 5| September 2022| Pages 1251-1257

https://doi.org/10.1107/S1600577522007779

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Study of X-ray topography using the super-Borrmann effect

J. Matsui,^a K. Takatsu ^b and Y. Tsusaka ^b ^*

^aSynchrotron Radiation Research Center, Hyogo Science and Technology Association, 1-490-2 Kouto, Shingu, Tatsuno, Hyogo 679-5165, Japan, and ^bGraduate School of Science, University of Hyogo, 3-2-1 Kouto, Kamigori, Hyogo 678-1297, Japan
^*Correspondence e-mail: tsusaka@sci.u-hyogo.ac.jp

Edited by Y. Amemiya, University of Tokyo, Japan (Received 19 May 2022; accepted 2 August 2022; online 17 August 2022)

X-ray topography exerting the super-Borrmann effect has been performed using synchrotron radiation to display dislocation images with a high-speed and high-resolution CMOS camera. Forward-transmitted X-rays are positively employed instead of reflected X-rays to reveal dislocations in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes owing to the super-Borrmann effect. Before the experiment, minimum values of the attenuation coefficients A_min^P for σ and π polarizations of the incident X-rays in the three-beam case are calculated. Results demonstrate that A_min^P for both polarizations are almost 20 times larger than those in the two-beam (usual Borrmann effect) case. The transmitted X-rays can be used to confirm the efficacy of taking topographs under the super-Borrmann conditions, as well as under multiple-diffraction conditions. Furthermore, super-Borrmann topographs can be considered for relatively thick crystals, where a conventional Lang X-ray topography technique is difficult to apply.

Keywords: topography; super-Borrmann effect.

1. Introduction

The effect of anomalous transmission can be enhanced (Borrmann & Hartwig, 1965 ) if the Bragg condition is satisfied for the 111 and $[\bar111]$ reflections simultaneously in the wide-angle 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 $[\bar111]$ reflections appear at the Kossel line intersection of the T₁₁₁ and $[{T_{\bar111}}]$ traces of the reflected beams, respectively. Furthermore, enhanced intensity spots for the 111 and $[\bar111]$ reflections appear on the R₁₁₁ and $[{R_{\bar111}}]$ traces of the transmitted (refracted in the strict sense) beams, respectively, and are symmetrical to the intersection point of the T₁₁₁ and $[{T_{\bar111}}]$ traces with respect to the respective reflecting planes. Calculation results and interpretation for the decrease in the absorption coefficient was provided (Hildebrandt, 1966 , 1967 ) for the enhanced spots in the $[111,\bar111/200]$ three-beam case (/hkl means χ_g–h when g = 111 and h = $[\bar111]$ ). Later, the theoretical understanding based on detailed calculation was advanced (Feldman & Post, 1972 ), and was confirmed experimentally (Uebach & Hildebrandt, 1969 ; Hildebrandt, 1978 ).

Other combinations of simultaneous reflections for three-beam cases such as $[111,1\bar{1}\bar{1}/022]$ (Umeno & Hildebrandt, 1975 ) and $[220,\bar{2}0\bar{2}/422]$ (Umeno, 1972 ) were investigated. The Borrmann effect for four-beam and six-beam 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 Borrmann effect is called the `super-Borrmann' effect (Lang, 1998 ).

In terms of the application of the super-Borrmann effect to X-ray topography to image lattice defects in crystals, such as dislocations, as well as conventional X-ray topography, few reports have been published probably owing to a too low X-ray source intensity and large X-ray beam divergence to develop clear defect images for a wide visual field. However, since the availability of synchrotron radiation, X-ray topography can be applied for imaging lattice defects in crystals by choosing the correct X-ray wavelength. In addition, it was previously reported that topographs combined with a high-speed and high-resolution CMOS camera taken by employing forward-transmitted X-rays under multiple diffraction conditions (bright-field X-ray topographs) can reveal dislocations without noticeable image deformations (Tsusaka et al., 2016 , 2019 ).

Furthermore, as a major advantage, it is expected that the forward-transmitted X-rays riding on the super-Borrmann effect also reveal dislocations existing in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes such as (111) and $[({\bar111})]$ where conventional Lang X-ray topography is difficult to apply. Therefore, this study deals with X-ray topography performed under three-beam multiple-diffraction conditions for thick Ge crystals using synchrotron radiation.

2. $[111,\bar{1}11/200]$ three-beam case

Fig. 1 shows an example of $[111,\bar111/200]$ three-beam multiple diffraction in reciprocal space of a perfect Ge crystal. Note that the $[\bar111]$ 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 super-Borrmann effect. Figs. 1(a) and 1(b) demonstrate two cases of different energy of the incident X-rays for E = E₁ and E = E₂, respectively, which is higher than E₁. The black dashed triangle in Fig. 1 comprises the original K_o, K₁₁₁ and g₁₁₁, where K_o is the incident X-ray wavevector, K₁₁₁ is the 111-reflected X-ray wavevector and g₁₁₁ = K₁₁₁ − K_o is the diffraction vector of the 111 reflection lying on the same plane. Rectangles OPQR and PP′O′O represent projections on $[(01\bar1)]$ and (100), respectively. The lengths of sides $[ \overline{PQ}]$ , $[\overline{OP}]$ and $[\overline{PP^{\,\prime}}]$ are given as follows,

$[\eqalign{ \overline{PQ} & = \left| \overline{OQ}\right| \cos\varphi = {{1}\over{\sqrt{3}}}\left|{{\bf{g}}}_{111}\right| = {{2}\over{\sqrt{3}}}\,k\sin\theta, \cr \overline{OP} & = \left| \overline{OQ}\right|\sin\varphi = {{\sqrt{2}}\over{\sqrt{3}}}\left|{{\bf{g}}}_{111}\right| = {{\sqrt{8}}\over{\sqrt{3}}}\,k\sin\theta, \cr \overline {{{PP^{\,\prime}}}} & = \overline {{{X}}{{{L}}_{\rm{o}}}} = k\sin \omega, }]$

where ω is an elevation angle of K_o (or K₁₁₁) from the rectangle OPQR parallel to the $[\left({0\bar11}\right)]$ entrance surface. It is clear that both $[\overline{{{PQ}}}]$ and $[\overline {{{OP}}}]$ are independent of E; however, $[\overline{{{PP^{\,\prime}}}}]$ 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 $[\left({01\bar1}\right)]$ base plane and is perpendicular to s_o. Therefore, π_o is also perpendicular to s_o and σ_o.

Figure 1
Schematic of $[111,\bar111/200]$ three-beam multiple diffraction in reciprocal space of Ge, where only the 111 reflection is represented, considering the $[\bar111]$ beam travels in a symmetrical direction with the plane of symmetry (100) at incident X-ray energies of E = E₁ in (a) and E = E₂ (>E₁) in (b) with rectangles OPQR and PP′O′O representing projections on the $[\left({01\bar1}\right)]$ and (100) planes, respectively. $[\left({01\bar1}\right)]$ : entrance surface; K_o: incident X-ray wavevector; K₁₁₁: 111-reflected X-ray wavevector; g₁₁₁: diffraction vector of the 111 reflection.

Next, we put $[{\bf{K}}_{111[0]}^{\,s}]$ , $[{\bf{K}}_{111[0]}^{\,\sigma}]$ and $[{\bf{K}}_{111[0]}^{\,\pi}]$ in Fig. 1 as components of K₁₁₁ in the s_o, σ_o and π_o directions, respectively. The magnitudes of these vectors were calculated as $[k\cos 2\theta]$ , $[({{2}/{\sqrt{3}}})\,k\sin\theta]$ and $[2k\sin\theta\,[(2/3)-\sin^2\theta]^{1/2}]$ , respectively. The lengths of sides $[\overline{{{OX}}}]$ and $[\overline{{{XP}}}]$ were found to have the following values,

$[\eqalign{\overline{OX} & = \overline{QX} = \overline{QL_{\rm{o}}}\cos\omega = k\cos\omega \cr&= {{\sqrt{3}}\over{\sqrt{8}}} \left(2k\sin\theta\right) = {{\sqrt{3}}\over{\sqrt{2}}}\,k\sin\theta,}]$

$[\eqalign{\overline{XP} & = \overline{OP}-\overline{OX} = \left({{\sqrt{2}}\over{\sqrt{3}}}-{{\sqrt{3}}\over{\sqrt{8}}}\right) \,\left(2k\sin\theta\right) \cr& = {{1}\over{\sqrt{24}}}\, \left(2k\sin\theta\right) = {{1}\over{3}}\,\overline{OX}.}]$

Then, we obtained $[\cos\omega]$ = $[({{\sqrt{3}}/\!{\sqrt{2}}})\sin\theta]$ and $[\sin\omega]$ = $[[1-(3/2)\sin^2\theta]^{1/2}]$ as a result for the plane parallel to the $[(0\bar11)]$ entrance surface.

Since $[{{\bf{K}}_{\bar111}}]$ is symmetrical with K₁₁₁, $[|{\bf{K}}_{\bar111[0]}^{\,\sigma}|]$ = $[-|{\bf{K}}_{111[0]}^{\,\sigma}|]$ ; however, the s_o, π_o components of $[{{\bf{K}}_{\bar111}}]$ and K₁₁₁ are identical. Consequently, the refracted beam K_o and two reflected beams K₁₁₁ and $[{{\bf{K}}_{\bar111}}]$ are summarized as follows,

$[\eqalignno{&\left(\,\matrix{{\bf{K}}_{\rm{o}} \cr {\bf{K}}_{111} \cr {\bf{K}}_{\bar{1}11}}\,\right) = \cr& \,\,\,\,\,\,\,k\left(\,\matrix{1 & 0 & 0 \cr \cos2\theta & -{{2}\over{\sqrt{3}}}\sin\theta & 2\sin\theta\left(\,{{2}\over{3}}-\sin^2\theta\right)^{1/2} \cr \cos2\theta & {{2}\over{\sqrt{3}}}\sin\theta & 2\sin\theta\left(\,{{2}\over{3}}-\sin^2\theta\right)^{1/2}}\,\right) \left(\,\matrix{{\bf{s}}_{\rm{o}} \cr \boldsigma_{\rm{o}} \cr \boldpi_{\rm{o}}}\,\right). &(1)}]$

3. Absorption coefficients in the three-beam diffraction cases

Since the calculation process for the absorption coefficient in the three-beam case under Cu Kα₁ radiation has already been provided by Authier (2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction, pp. 225-248. Oxford University Press.]$ ), explanation of the calculation will be kept to a minimum. Based on the fundamental equations of X-ray dynamical theory, the projections of the electric displacements D in the three-beam case to the plane normal to K_o can be written for the three beams as follows,

$[{{{\bf{K}}_{\rm{o}}^{2} - {k^2}} \over {{k^2}}}\,{{\bf{D}}_{\rm{o}}} = {\chi_{\rm{o}}}{{\bf{D}}_{\rm{o}}} + {\chi _{\bar 1\bar 1\bar 1}}{{\bf{D}}_{111\left [{\rm{o}} \right]}} + {\chi _{1\bar 1\bar 1}}{{\bf{D}}_{\bar111\left [{\rm{o}} \right]}},]$

$[{{{\bf{K}}_{111}^2 - {k^2}} \over {{k^2}}}\,{{\bf{D}}_{111}} = {\chi_{\rm{o}}}{{\bf{D}}_{111}} + {\chi _{111}}{{\bf{D}}_{{\rm{o}}\left [{111} \right]}} + {\chi _{200}}{{\bf{D}}_{\bar111\left [{111} \right]}},]$

$[{{{\bf{K}}_{\bar111}^2 - {k^2}} \over {{k^2}}}\,{{\bf{D}}_{\bar111}} = {\chi_{\rm{o}}}{{\bf{D}}_{\bar111}} + {\chi _{\bar111}}{{\bf{D}}_{{\rm{o}}\left [{\bar111} \right]}} + {\chi _{\bar 200}}{{\bf{D}}_{\left [{\bar111} \right]}}.]$

It is possible to rewrite the above equations by using excitation errors $[{\xi}_{{\rm{o}}}]$ , $[{\xi}_{111}]$ , $[{\xi}_{\bar{1}11}]$ ,

$[\matrix{ {2\xi}_{\rm{o}}{\bf{D}}_{\rm{o}} & -k{\chi}_{\bar{1}\bar{1}\bar{1}}{\bf{D}}_{111[{\rm{o}}]} & -k{\chi}_{1\bar{1}\bar{1}}{\bf{D}}_{\bar{1}11[{\rm{o}}]} & = 0, \cr -k{\chi}_{111}{\bf{D}}_{{\rm{o}}[111]} & +{2\xi}_{111}{\bf{D}}_{111} & -k{\chi}_{200}{\bf{D}}_{\bar{1}11[111]} &= 0, \cr -k{\chi}_{\bar{1}11}{\bf{D}}_{{\rm{o}} [\bar{1}11]} & -k{\chi}_{\bar{2}00}{\bf{D}}_{111[\bar{1}11]} & +{2{\xi}_{\bar{1}11}{\bf{D}}}_{\bar{1}11} & = 0. } \eqno(2)]$

Since the 200 reflection and $[\bar200]$ reflectin are forbidden ( $[{\chi_{200}}]$ = $[{\chi _{\bar 200}}]$ = 0), the above equations are expressed as follows,

$[\matrix{ {2\xi}_{\rm{o}}{\bf{D}}_{\rm{o}} & -k{\chi}_{\bar{1}\bar{1}\bar{1}}{\bf{D}}_{111[{\rm{o}}]} & -k{\chi}_{1\bar{1}\bar{1}}{\bf{D}}_{\bar{1}11[{\rm{o}}]} & = 0, \cr -k{\chi}_{111}{\bf{D}}_{{\rm{o}}[111]}& +{2\xi}_{111}{\bf{D}}_{111} & + 0 & = 0, \cr -k{\chi}_{\bar{1}11}{\bf{D}}_{{\rm{o}} [\bar{1}11]} & +0& +{2{\xi}_{\bar{1}11}{\bf{D}}}_{\bar{1}11}& = 0. } \eqno(3)]$

The two relations are obtained from the second and third lines of equation (3) shown above,

$[{\bf{D}}_{111} = {{k{\chi}_{111}}\over{{2\xi}_{111}}}{\bf{D}}_{{\rm{o}}\left[111\right]} = {{k{\chi}_{111}}\over{{2\xi}_{111}}}\left({\bf{D}}_{\rm{o}}-{{{\bf{K}}_{111}\cdot {\bf{D}}_{\rm{o}}}\over{{{K}_{111}}^{2}}}{\bf{K}}_{111}\right), ]$

$[{\bf{D}}_{\bar{1}11} = {{ k{\chi}_{\bar{1}11} }\over{ {2\xi}_{\bar{1}11} }} \,{\bf{D}}_{{\rm{o}}[\bar{1}11]} = {{k{\chi}_{\bar{1}11}}\over{{2\xi}_{\bar{1}11}}}\left({\bf{D}}_{\rm{o}}-{{{\bf{K}}_{\bar{1}11}\cdot {\bf{D}}_{\rm{o}}}\over{{\bf{K}}_{\bar{1}11}^{\,2}}}{\bf{K}}_{\bar{1}11}\right).]$

From D_o[111] and $[{{\bf{D}}_{{\rm{o}}[{\bar111}]}}]$ one can obtain D_111[o] and $[{{\bf{D}}_{\bar111\left [{\rm{o}} \right]}}]$ using a vector formula A × (B × C) = (A · C)B − (A · B)C in a similar way to that described by Authier (2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction, pp. 225-248. Oxford University Press.]$ ). By substituting D_111[o] and $[{{\bf{D}}_{\bar111\left [{\rm{o}} \right]}}]$ 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, $[D_{\rm{o}}^\sigma]$ , parallel to the plane of symmetry, and $[D_{\rm{o}}^\pi]$ , perpendicular to the plane of symmetry,

$[{\bf{D}}_{\rm{o}} = {D}_{\rm{o}}^{\sigma}{{\boldsigma}}_{\rm{o}}+{D}_{\rm{o}}^{\pi} {{\boldpi}}_{\rm{o}}.]$

The first line of equation (2), which is

$[{\bf{X}} = 2{\xi _{\rm{o}}}{{\bf{D}}_{\rm{o}}} - k{\chi _{\bar 1\bar 1\bar 1}}{{\bf{D}}_{111\left [{\rm{o}} \right]}} - k{\chi _{1\bar 1\bar 1}}{{\bf{D}}_{\bar111\left [{\rm{o}} \right]}} = 0,]$

can be replaced using two scalar values $[D_{\rm{o}}^\sigma]$ and $[D_{\rm{o}}^\pi]$ , given as follows,

$[{\bf X} = A{D}_{\rm{o}}^{\sigma } {{\boldsigma}}_{\rm{o}}+B\left({D}_{\rm{o}}^{\sigma}{ {\boldpi}}_{\rm{o}}+{D}_{\rm{o}}^{\pi } {{\boldsigma}}_{\rm{o}}\right)+C{D}_{\rm{o}}^{\pi }{ {\boldpi}}_{\rm{o}} = 0, \eqno(4)]$

where A, B and C are the coefficients of $[{D}_{\rm{o}}^{\sigma} {{\boldsigma }}_{\rm{o}}]$ , $[{D}_{\rm{o}}^{\sigma}{{\boldpi}}_{\rm{o}}+{D}_{\rm{o}}^{\pi} {{\boldsigma}}_{\rm{o}}]$ and $[{D}_{\rm{o}}^{\pi}{{\boldpi}}_{\rm{o}}]$ , respectively,

$[A = 8{\xi _{\rm{o}}}{\xi _{111}}{\xi _{\bar111}} - {k^2}{\chi _{111}}{\chi _{\bar 1\bar 1\bar 1}}\left({1 - {4 \over 3}{{\sin }^2}\theta } \right)\left({2{\xi _{111}} + 2{\xi _{\bar111}}} \right),]$

$[B = {k^2}{\chi _{111}}{\chi _{\bar 1\bar 1\bar 1}}\left[{{4 \over {\sqrt 3 }}\,\sin^2\theta\, \left({{2 \over 3} - {{\sin }^2}\theta } \right)^{1/2} } \right]\left({2{\xi _{111}} - 2{\xi _{\bar111}}} \right),]$

$[\eqalign{C = {}& 8\,{\xi _{\rm{o}}}{\xi _{111}}{\xi _{\bar111}} - {k^2}{\chi _{111}}{\chi _{\bar 1\bar 1\bar 1}}\left [{1 - 4\sin^2\theta\, \left({{2 \over 3} - {\sin^2}\theta } \right)} \right] \cr &\times\left({2{\xi _{111}} + 2{\xi_{\bar111}}}\right).}]$

Considering X can be separated into σ_o and π_o,

$[{\bf X} = \left(AD_{\rm{o}}^{\sigma} + B{D}_{\rm{o}}^{\pi}\right)\,{{\boldsigma}}_{\rm{o}} + \left(B{D}_{\rm{o}}^{\sigma} + C{D}_{\rm{o}}^{\pi}\right)\,{{\boldpi}}_{\rm{o}} = 0. \eqno(5)]$

Therefore, we can derive the determinant as follows,

$[\left(\matrix{A & B \cr B & C \cr}\right) \left(\matrix{D_{\rm{o}}^\sigma \cr D_{\rm{o}}^\pi \cr}\right) = \left(\matrix{0 \cr 0 \cr}\right),]$

and hence, inevitably,

$[\det \left(\matrix{A & B \cr B & C \cr}\right) = AC-B^2 = 0. ]$

Note that the components of K_o, K₁₁₁ and $[{{\bf{K}}_{\bar111}}]$ are given by equation (1) and $[{\xi_{111}}]$ = $[{\xi_{\bar111}}]$ . Therefore, we understand that B vanishes, and hence AC also vanishes.

From the above considerations, A (coefficient of $[{D}_{\rm{o}}^{\sigma} {{\boldsigma}}_{\rm{o}}]$ ) and C (coefficient of $[{D}_{\rm{o}}^{\pi}{{\boldpi}}_{\rm{o}}]$ ) should be null independently for σ_o and π_o polarizations, respectively. As a result, we obtain

$[{\xi _{\rm{o}}}{\xi_{111}} = {1\over2}{k^2}{\chi_{111}}{\chi_{\bar1\bar1\bar1}}\left({1-{4\over3}\sin^2\theta} \right)]$

for σ_o polarization and

$[{\xi _{\rm{o}}}{\xi _{111}} = {1 \over 2}{k^2}{\chi _{111}}{\chi _{\bar 1\bar 1\bar 1}}\left [{1 - 4{{\sin }^2}\theta\, \left({{2 \over 3} - {{\sin }^2}\theta } \right)} \right]]$

for π_o polarization.

Considering χ_o is minimum when |ξ_o| = |ξ₁₁₁|, the subsequent values for ξ_o can be given as

$[{\xi}_{\rm{o}} = -{{1}\over{\sqrt{2}}}\,k\,\left({{\chi}_{111}{\chi}_{\bar{1}\bar{1}\bar{1}} } \right)^{1/2}\left({1-{{4}\over{3}}\sin^2\theta} \right)^{1/2} \eqno(6a)]$

for σ_o polarization and

$[{\xi}_{\rm{o}} = -{{1}\over{\sqrt{2}}}\,k\,\left({{\chi}_{111}{\chi}_{\bar{1}\bar{1}\bar{1}}}\right)^{1/2} \left[ {1-4\sin^2\theta\,\left({{2}\over{3}}-\sin^2\theta\right)} \right]^{1/2} \eqno(6b)]$

for π_o polarization.

In the two-beam case,

$[{\xi_{\rm{o}}}{\xi _{111}} = \xi _{\rm{o}}^{\,2} = {1 \over 4}{k^2}\left({P{\chi _{111}}} \right)\left({P{\chi _{\bar 1\bar 1\bar 1}}} \right). ]$

When |ξ_o| = |ξ₁₁₁|, satisfying the Bragg condition exactly,

$[{\xi _{\rm{o}}} = - {1 \over 2}\,kP\left({{\chi _{111}}{\chi _{\bar 1\bar 1\bar 1}}} \right)^{1/2}. ]$

For a cubic crystal such as Ge, the structure factor F₁₁₁ and Fourier component of the dielectric susceptibility χ₁₁₁ for the 111 reflection are given as follows,

$[{F}_{111} = 4\,{f}_{\!\rm{Ge}}\left(1+i\right), \qquad {\chi}_{111} = -{{{r}_{\rm{e}}{\lambda}^{2}}\over{\pi{V}_{\rm{c}}}} \left({F}_{111\_{\,\rm{r}}}+i{F}_{111\_{\,\rm{i}}}\right),]$

where f_Ge is the atomic scattering factor of Ge, r_e is the classical electron radius, λ is the wavelength of the X-rays, and V_c is the volume of a unit cell of Ge. F_{111_r} and F_{111_i} are the real and imaginary parts of the complex number F₁₁₁, respectively. The magnitudes of F₁₁₁ and χ₁₁₁ can be derived from the corresponding atomic scattering factors as follows,

$[\left| {{F_{111}}} \right| = 4\left({{1^2} + {{\left|\,i\,\right|}^2}} \right)^{1/2} \left|\,f\,\right| = 4\sqrt 2\, \left| {{\,f_{111\_\,{\rm{r}}}} + {\,f_{111\_\,{\rm{i}}}}} \right|,]$

$[\left| {{\chi _{111}}} \right| = - {{{r_{\rm{e}}}{\lambda^2}} \over {\pi{V_{\rm{c}}}}}\left({4\sqrt 2\, \left|\, {{f_{111\_\,{\rm{r}}}} + \,{f_{111\_\,{\rm{i}}}}} \right|} \right). ]$

Then,

$[{{\left|{{\chi _{111\_\,{\rm{i}}}}}\right|} \over {\left|{{\chi_{\rm{oi}}}}\right|}} = {{4\sqrt2\,{f_{111\_\,{\rm{i}}}}}\over{8\,{f_{{\rm{o}}\_\,{\rm{i}}}}}} = {1\over{\sqrt2}}.]$

The minimum value of the absorption coefficient in the g, = $[111,\bar{1}11/200]$ three-beam case is given as follows,

$[{\mu_{\rm{e}}} = {\mu_{\rm{o}}}\left({1-P\,{{\left|{{\chi_{111\_\,{\rm{i}}}}} \right|} \over {\left|{{\chi_{{\rm{o}}\_\,{\rm{i}}}}}\right|}}}\right), \eqno(7)]$

where μ_o is the normal absorption coefficient and χ_{111_i} and χ_{o_i} are imaginary parts of χ₁₁₁ and χ_o, respectively.

The minimum attenuation coefficient is $[A_{{\rm{min}}}^P]$ = $[\exp[{-{\mu_{\rm{e}}}({t/{{\gamma_{\rm{o}}}}})}]]$ , where t is the slab thickness and γ_o = n_hkl · s_o is a direction cosine of the incident X-ray wavevector K_o (its unit vector is s_o) to n_hkl, the normal to the X-ray entrance surface. In the $[111,\bar{1}11/200]$ present three-beam case, it is found from Fig. 2 that the direction cosine γ_o is expressed as

$[{\gamma}_{\rm{o}} = \cos\left(\,{{\pi}\over{2}}-\omega\right) = \sin\omega = {{\left|\overline{XL_{\rm{o}}}\right|}\over{k}} = \left({1-{{3}\over{2}}{\sin}^{2}\theta}\right)^{1/2}, \eqno(8)]$

for K_o to $[{{\bf{n}}_{0\bar11}}]$ .

Figure 2
Schematic of the relation between K_o, the wavevector of the incident X-rays, and n₀₀₁, the (001) surface normal, or $[{{\bf{n}}_{0\bar11}}]$ , the $[\left({0\bar11}\right)]$ surface normal. Angles $[({\pi/2})-{{\omega}}]$ and $[{{\omega}}-({\pi/4})]$ correspond to direction cosines γ_o of K_o to n₀₀₁ and $[{{\bf{n}}_{0\bar11}}]$ , respectively.

However, the X-ray 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 Bragg angle of the 111 reflection becomes $[{\theta_{111}}]$ = 7.2458° leading to $[\sin{\theta}_{111}]$ = $[1.23984/[2\left(a/\sqrt{3}\right)E\,]]$ = 0.12613, we can derive γ_o for n_hkl = n₀₀₁ from Fig. 2 as follows,

$[\eqalign{ {\gamma}_{\rm{o}} & = \cos\left(\omega-{{\pi}\over{4}}\right) = {{1}\over{\sqrt{2}}}\left(\cos\omega + \sin\omega \right) \cr& = {{1}\over{\sqrt{2}}}\left[\sqrt{\,{{3}\over{2}}\,}\sin\theta \,+ \left({1-{{3}\over{2}}\sin^2\theta}\right)^{1/2}\right] = 0.85165, }]$

which corresponds to 31.608° as an angle between n₀₀₁ and s_o. In this case, the polarization factors P for the σ_o and π_o components are introduced from equation (6) as

$[P = \sqrt2\, \left(1-{4\over3}\sin^2\theta_{111}\right)^{1/2} \eqno(9a)]$

for σ_o polarization and

$[P = \sqrt2\, \left[1-4\sin^2\theta_{111} \left({2\over3}-\sin^2\theta_{111}\right)\right]^{1/2} \eqno(9b)]$

for π_o polarization

This makes it possible to calculate the effective absorption coefficient μ_e and the minimum attenuation coefficient $[A_{{\rm{min}}}^P]$ in the $[111,\bar{1}11/200]$ three-beam 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 $[A_{{\rm{min}}}^P]$ for both polarizations in the three-beam case are approximately 20 times the values in the two-beam case, due to which the phenomenon is called the super-Borrmann effect.

Table 1
μ_o, μ_e, $[A_{{\rm{min}}}^P]$ for σ_o and π_o polarizations in the $[111,\bar{1}11/200]$ three-beam reflection case for a Ge slab (thickness: 0.05 cm; X-ray entrance surface: (100); X-ray energy: 10 keV)

Polarization	Linear absorption coefficient μ_o	Polarization component P	Effective absorption coefficient μ_e	Minimum attenuation coefficient $[A_{{\rm{min}}}^P]$ = $[\exp\left(-\mu_{\rm{e}}t/\gamma_{\rm{o}}\right)]$
σ_o	192.213	1.380	4.638	0.672
π_o	192.213	1.349	8.878	0.594

4. Topography experiment using the super-Borrmann effect for Ge crystals

To take X-ray topographs with minimized image deformation, we employed forward-transmitted (but refracted) X-rays 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 X-ray diffraction goniometer with an X-ray source of approximately 1.2 mm × 1.2 mm was used with 10 keV X-rays from the synchrotron radiation through a silicon double-crystal monochromator at the BL24XU8 beamline of SPring-8 (Tsusaka et al., 2001 ), similar to previously reported multiple-beam 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.

Figure 3
Incident X-rays K_o and reflected X-rays K₁₁₁ and $[{{\bf{K}}_{\bar111}}]$ simultaneously satisfying the Bragg conditions for the 111 and $[\bar111]$ reflections, respectively, where sample rotations around [001] and [100] are performed, respectively, to select a pair of {111} planes and for Bragg condition adjustment to satisfy the individual three-beam diffraction condition.

Various interference patterns on diffracted and transmitted images with defect appearance were also studied using a coherent X-ray beam under multiple-diffraction conditions (Okitsu et al., 2003 ; Okitsu, 2003 ). However, in the present case, topographic images were taken directly by the forward-transmitted X-ray beam instead of the diffracted X-ray beam using an X-ray imaging detector (Hondoh et al., 1989 ). The detector comprises a 20 µm-thick Gd₃Al₂Ga₃O₁₂ (GAGG) scintillator, relay lens optics and a high-speed CMOS camera (Hamamatsu, C11440-22CU). This detector resolved a 1 µm line-and-space pattern.

A Ge slab of dimensions 10 mm (width) × 14 mm (height) × 0.5 mm (thickness) and the (001) surface was prepared for the super-Borrmann 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 $[\left({\bar111}\right)]$ , could be recognized on a fluorescent sheet. It is clear that this multiple (n-beam) diffraction from a single crystal is not considered to be so-called 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 forward-transmitted 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 X-ray beam denoted as `0', (b) the direct beam and the 111 reflected beam, and (c) the direct beam, the 111 reflected beam and the $[\bar111]$ 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 super-Borrmann effect. The shining light on the right-hand 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 brightness 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.

Figure 4
Reflection spots on a fluorescent sheet. (a) Directly transmitted X-ray beam denoted as `0', (b) direct beam and 111 reflected beam (suggesting the usual Borrmann case) and (c) direct beam and 111 and $[\bar111]$ reflected beams (suggesting the super-Borrmann case).

Fig. 5 shows one of the topographs taken under the super-Borrmann conditions shown in Fig. 4(c) using a pair of 111 and $[\bar111]$ reflections without deformation correction. Considering the X-ray source is approximately 1.2 mm × 1.2 mm in size, four shots of topographic images are pasted together to achieve a wide-area topograph. Regarding the usual Borrmann topography (two-beam case), the images of the dislocations correspond to local lower transmitted intensities (in the forward-refracted direction) or lower diffraction intensities (along the diffraction direction), since crystal imperfection can destroy the Borrmann effect. This is also true for the current super-Borrmann topography (three-beam case), since double excitation of the 111 and $[\bar111]$ reflections only enhances the Borrmann effect, 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.

Figure 5
X-ray topograph of a germanium slab taken by simultaneous 111 and $[\bar111]$ reflections without deformation correction. Four shots of images are pasted together to obtain a wide-area topograph.

There are four combinations of two adjacent 111 reflections, i.e. 111 and $[\bar111]$ reflections (called A-type), $[\bar111]$ and $[\bar1\bar11]$ reflections (B-type), $[\bar1\bar11]$ and $[1\bar11]$ reflections (C-type), and $[1\bar11]$ and 111 reflections (D-type). Additionally, there are two combinations of diagonal 111 reflections, i.e. 111 and $[\bar1\bar11]$ reflections (E-type), and $[\bar111]$ and $[1\bar11]$ reflections (F-type). Therefore, if one observes dislocation images disappearing only in an A-type topograph, the dislocation should have a Burgers vector of $[({1/2})\left[{01\bar1}\right]]$ , considering this vector is commonly perpendicular to both [111] and $[\left[{\bar111}\right]]$ . Similarly, from the invisibility rule g · b = 0, where g is the diffraction vector and b the dislocation Burgers vector, B-, C- and D-type topographs do not include any images of the dislocations with Burgers vectors of, respectively, ( 1/2)[101], (1/2)[011] and $[({1/2})\left[{\bar101}\right]]$ . However, the combination of diagonal 111 reflections (E- and F-type) does not develop into the super-Borrmann effect owing to the existence of χ₂₂₀ instead of χ₂₀₀ in equation (2). According to the partial lack of the super-Borrmann conditions, Burgers vectors of all the dislocations cannot be determined completely by only observing the A-, B-, C- and D-type topographs. Nevertheless, we can conclude that the Burgers vector of the dislocations disappearing only on the A-type topograph should belong to $[({1/2})\left[{01\bar1}\right]]$ . For example, some parts of A- and B-type 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.

Figure 6
X-ray topographs of a germanium slab taken by (a) simultaneous 111 and $[\bar111]$ reflections and (b) simultaneous $[\bar111]$ and $[\bar1\bar11]$ reflections with deformation correction where dislocation configurations circled in red in (a) and (b) correspond to each other.

5. Conclusions

In this study, we conducted synchrotron X-ray topography exerting the super-Borrmann effect for imaging dislocations using a CMOS camera. Forward-transmitted X-rays can reveal dislocations in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes owing to the super-Borrmann effect. Super-Borrmann topographs can be captured for relatively thick crystals, even when a conventional Lang X-ray topography technique is difficult to apply.

Prior to the experiment, the minimum attenuation coefficients $[A_{{\rm{min}}}^{{\sigma}}]$ and $[A_{{\rm{min}}}^{{\pi}}]$ for σ- and π-polarizations, respectively, of the incident X-rays in the three-beam (super-Borrmann) case were calculated. It was found that $[A_{{\rm{min}}}^{{\sigma}}]$ and $[A_{{\rm{min}}}^{{\pi}}]$ were almost 20 times larger than those in the two-beam (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 super-Borrmann effect.

In addition to the topographs taken by employing forward-transmitted X-rays under multiple-diffraction conditions (bright-field X-ray topographs), the forward-transmitted X-rays riding on the super-Borrmann effect also reveal dislocations existing in comparatively thick crystals by simultaneously exciting a pair of adjacent {111} planes such as (111) and $[\left({\bar111}\right)]$ . Therefore, this study deals with X-ray topography using synchrotron radiation performed under three-beam multiple diffraction conditions exerting the super-Borrmann 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 dislocation-free silicon crystals.

It was clarified that forward-transmitted X-rays using synchrotron radiation can be used to confirm the efficacy for capturing topographs not only under usual multiple diffraction conditions but also under super-Borrmann conditions.

Acknowledgements

The authors would like to thank Messrs Y. Itoh and Y. Namioka for help with the X-ray topography experiments and Dr Umeno for discussions about the super-Borrmann effect. The synchrotron radiation experiments were performed at BL24XU of SPring-8 with approval from the Japan Synchrotron Radiation Research Institute (Proposal Nos. 2019B3202, 2020A3202, 2021A3202, 2021B3202).

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JOURNAL OF
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Volume 29| Part 5| September 2022| Pages 1251-1257

https://doi.org/10.1107/S1600577522007779

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