X-ray Laue diffraction by sectioned multilayers. I. Pendellösung effect and rocking curves

Punegov, V.I.

doi:10.1107/S1600577521006408

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 28| Part 5| September 2021| Pages 1466-1475

https://doi.org/10.1107/S1600577521006408

X-ray Laue diffraction by sectioned multilayers. I. Pendellösung effect and rocking curves

Vasily I. Punegov ^a ^*

^aInstitute of Physics and Mathematics, Federal Research Center "Komi Scientific Center", The Ural Branch of the Russian Academy of Sciences, Syktyvkar 167982, Russian Federation
^*Correspondence e-mail: vpunegov@dm.komisc.ru

Edited by A. Momose, Tohoku University, Japan (Received 1 February 2021; accepted 20 June 2021; online 9 August 2021)

Using the Takagi–Taupin equations, X-ray Laue dynamical diffraction in flat and wedge multilayers is theoretically considered. Recurrence relations are obtained that describe Laue diffraction in structures that are inhomogeneous in depth. The influence of sectioned depth, imperfections and non-uniform distribution of the multilayer period on the Pendellösung effect and rocking curves is studied. Numerical simulation of Laue diffraction in multilayer structures W/Si and Mo/Si is carried out. It is shown that the determination of sectioned depths based on the period of the interference fringes of the experimental rocking curves of synchrotron radiation is not always correct.

Keywords: sectioned multilayers; dynamical X-ray Laue diffraction; recurrence relations; Pendellösung effect; rocking curves.

1. Introduction

Elements of multilayer X-ray optics are widely used in synchrotron radiation installations for transporting X-ray beams (Rack et al., 2010 ) and focusing radiation (Lyatun et al., 2020 ), in extreme-ultraviolet lithography (Chkhalo et al., 2017 ) and in astronomy (Tamura et al., 2018 ). Such multilayers mainly refer to grazing-incidence reflectors or, in terms of X-ray diffraction in crystals, to Bragg geometry (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ).

To focus hard X-rays, it was proposed to create multilayer Laue lenses (Maser et al., 2004 ), which, like Fresnel zone plates (Kagoshima & Takayama, 2019 ), transport radiation in transmission geometry. The fabrication of Laue lenses, which are depth-sectioned and thickness-graded multilayers, is a complex problem (Kang et al., 2007 ). Therefore, the first step in the study of Laue diffraction was presented by a synchrotron X-ray study of laterally bounded (sectioned) multilayers with a constant period (Kang et al., 2004 , 2005 ).

The dynamical theory of X-ray scattering in the Laue case is well developed for crystals, both for symmetric and asymmetric geometry (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ). Theoretically, asymmetric diffraction is also possible for sectioned multilayers; however, technologically this is a difficult problem associated with oblique sectioning of a multilayer structure.

Dynamical X-ray Laue diffraction in a multilayer differs compared with diffraction in grazing-incidence geometry. One of the main features is the pendulum (Pendellösung) effect (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ), when the intensity of the X-ray beam of the transmission wave is transmitted into the diffraction beam and then, with increasing depth, to the contrary, the intensity of the diffraction wave is transmitted into the transmission beam.

In the absence of an algorithm for simulating dynamical Laue diffraction, the analysis of experimental results using the interference-fringe spacing of the rocking curve was not entirely correct (Kang et al., 2005, 2007). Therefore, the present work is devoted to a consistent consideration of the Laue dynamical theory of diffraction in multilayers.

2. Dynamical Laue diffraction in a sectioned multilayer with a constant period

Multilayer Laue lenses are graded multilayers. Unfortunately, on the basis of the Takagi–Taupin equations (Takagi, 1962 , 1969 ; Taupin, 1964 ; Afanas'ev & Kohn, 1971 ; Kato, 1973 ), in the general case, it is impossible to obtain analytical solutions describing Laue diffraction in structures with period variation. Only some laws of changes in the period of the structure allow analytical solutions; in particular, such solutions were previously obtained for crystals with quadratic (Kolpakov & Punegov, 1985 ) and exponential (Andreev, 2001 ) displacement fields, and crystals with a transition layer (Chukhovskii & Khapachev, 1985 ; Kato, 1990 ). In most cases, one has to limit oneself to the numerical solution of the Takagi–Taupin equations in an oblique coordinate system (Punegov, 2020 ; Lomov et al., 2021 ), for example, using the `half-step derivative' method (Epelboin, 1985 ).

Therefore, let us consider X-ray dynamical Laue diffraction in a sectioned multilayer with a constant period d = d_t + d_b, where d_t is the thickness of the upper layer and d_b is the thickness of the lower layer (Fig. 1).

Figure 1
A schematic representation of Laue diffraction by a multilayer. L_x is the sectioned depth and d is the multilayer period.

A plane X-ray wave is incident on the left side of the sectioned multilayer at an angle θ = θ_B + ω, where ω is the deviation of the X-ray beam from the Bragg angle θ_B (Fig. 1). In contrast to the Ewald–Laue approach (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ), we will proceed from the one-dimensional Takagi–Taupin equations for the periodic structure, which in the Cartesian coordinate system have the form

$[\Bigg\{ \eqalign{ {(\partial / {\partial x)}}{E_0}(\eta, x) & = i{a_0}{E_0}(\eta, x)\, + i{a_{ - 1}}\,f\,{E_1}(\eta, x), \cr {(\partial / {\partial x})}{E_1}(\eta, x) &= i({a_0} + \eta){E_1}(\eta, x)\, + i{a_1}\,f\,{E_0}(\eta, x), } \eqno(1)]$

where E₀(η, x) and E₁(η, x) are the amplitudes of the transmission and diffraction X-ray waves, respectively, a₀ = $[\pi {\chi _0}/]$ $[(\lambda \cos {\theta _{\rm B}})]$ , a₁= $[C\pi {\chi _1}/]$ $[(\lambda \cos {\theta _{\rm B}})]$ , a₋₁ = a₁, $[\eta=]$ $[4\pi \sin ({\theta _{\rm B}})\, \omega /\lambda]$ is the angular parameter, λ is the wavelength of the X-ray radiation in vacuum, and C is the polarization factor. The Fourier coefficients of the X-ray polarizability for a structure with a two-layer period in the directions of transmission χ₀ and diffraction χ₁ are written as

$[{\chi _0} = {{{\chi _{\rm t}}{d_{\rm t}} + {\chi _{\rm b}}{d_{\rm b}}} \over d}, \qquad {\chi _1} = {{{\chi _{\rm t}} - {\chi _{\rm b}}} \over \pi }\sin \left({\pi {{{d_{\rm t}}} \over d}} \right). \eqno (2)]$

In relation (2), χ_t and χ_b are the Fourier coefficients of polarizabilities and thicknesses of the upper (t) and lower (b) layers of the period of the multilayer. X-ray polarizabilities of chemical elements are calculated using the tabular values of optical constants: χ_j = 2(δ_j + iβ_j), where $[{\delta _j}={r_0}({N_j} {\lambda^2}/{2\pi })]$ $[\times(\,{{Z_j} + \Delta {{f}^{\,\prime}_j}})]$ ; $[{\beta_j}= - {r_0}({{{N_j} {\lambda ^2}} / {2\pi }})]$ $[{\Delta {{f}^{\,\prime\prime}_j}}]$ ; j = t or b, which indicates the corresponding layer in the period of the multilayer; r₀ = e²/mc² is the classical radius of an electron, where e and m are the charge and mass of an electron, respectively; N_j is the atomic density; Z is the number of electrons in an atom; and $[\Delta{f}^{\,\prime}_j]$ and $[\Delta{f}^{\,\prime\prime}_j]$ are dispersion corrections to the atomic amplitude. Equation (1) contains an attenuation factor f, which describes the attenuation of X-ray reflection in a multilayer. This coefficient characterizes the disturbances in the periodic structure of multilayers during their creation. In particular, contribution to the attenuation factor is given by the roughness of the boundaries layers in the structure (Névot & Croce, 1980 ; de Boer, 1994 ), errors in layer thickness of the structure (Spiller & Rosenbluth, 1985 , 1986 ), diffusion blurring of boundaries between layers (presence of transition layers) (Stearns, 1989 ), random loss in a short-period multilayer of a heavy or light layer (Kopylets et al., 2019 ), local bending of layers during polishing (Kang et al., 2007), etc.

For example, accounting for interlayer roughness in Parratt's method of recurrence relations (Parratt, 1954 ) was considered using the Névot–Croce and Debye–Waller attenuation factors (Bushuev & Sutyrin, 2001 ; Kohn, 2003 ). Unfortunately, without an analysis of diffuse scattering, it is difficult to determine the contribution of each type of defect to the attenuation factor. Therefore, in our theoretical consideration of dynamical Laue diffraction, the attenuation factor will take on the value 0 ≤ f ≤ 1 without determining the specific types of defects.

We consider the model generally accepted in the dynamical Laue diffraction theory (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ), in which the front of a plane incident X-ray wave and the size of the multilayer are spatially unrestricted in the vertical direction (Fig. 1), hence the amplitudes of X-ray waves in equation (1) depend only on one horizontal coordinate x. Such an X-ray diffraction model can be considered if only the rocking curves are analyzed. In the case where the X-ray beam incident on the multilayer is spatially restricted, it is necessary to use the two-dimensional Takagi–Taupin equations (Punegov et al., 2017 ; Punegov & Karpov, 2021 ). This approach allows one to analyze X-ray reciprocal-space maps (RSMs).

The solution of coupled equation (1) for Laue diffraction can be obtained using the boundary conditions $[{E_0}(\eta, 0)]$ = $[E_0^{\,\rm in}]$ = 1 and E₁(η, 0) = 0. Applying these boundary conditions, we can obtain expressions for the amplitudes of the transmission E₀(η, x) and diffraction E₁(η, x) waves,

$[E_0^{}(\eta, x) = \exp (i\psi \,x) \left[ { \cos\left({{\xi \,x}\over{2}}\right) - i{{\eta}\over{\xi}} \sin\left( {{\xi\,x}\over{2}}\right) } \right] \eqno(3)]$

and

$[E_1^{}(\eta, x) = {{i2{a_1}\,f\exp (i\psi \,x)} \over \xi } \,\sin\left({{\xi}\,x\over{2}}\right), \eqno(4)]$

where ξ = −(η² + 4f²a₁a₋₁)^1/2 and ψ = a₀ + η/2.

2.1. Pendellösung effect in homogeneous sectioned multilayer

Let us first consider the pendulum (Pendellösung) effect under Laue diffraction conditions. If the exact Bragg condition is satisfied (η = 0) and taking into account a₁ = a₋₁, the intensity distributions of the transmission and diffraction X-ray waves can be written as

$[{I_0}(x) = \exp \left(- {\mu _0}\,x\right) \left[{\cos^2} \left(\,f\,{a^{\prime}_1}\,x\right) + {\rm{sin}}{{\rm{h}}^{\rm{2}}} \left(\,f\,{a^{\prime\prime}_1}x\right) \right] \eqno(5)]$

and

$[{I_1}(x) = \exp \left(- {\mu _0}\,x\right)\left[{\sin ^2} \left(\,f\,{a'_1}\,x\right) + {\sin}{{\rm{h}}^{\rm{2}}} \left(\,f\,{a^{\prime\prime}_1}x\right)\right], \eqno(6)]$

where $[{\mu _0}=]$ $[2\,{\mathop{\rm Im}\nolimits} ({a_0})]$ is the linear coefficient of absorption of X-rays in the multilayer. The dynamical coefficient in equation (1) is represented as a₁ = a′₁ + ia′′₁, where $[{a^{\prime}_1}=]$ $[C\pi \chi _1^{\rm r}/(\lambda \cos {\theta _{\rm B}})]$ is a real part and $[{a^{\prime\prime}_1}=]$ $[C\pi \chi _1^{\rm Im}/(\lambda \cos {\theta _{\rm B}})]$ is an imaginary part. Considering $[\chi _1^{\rm Im}\, \ll \,\chi _1^{\rm r}]$ and $[|{{\chi _1}}|=]$ $[\chi _1^{\rm r}[{1 + {{(\chi _1^{\rm Im}/\chi _1^{\rm r})}^2}}]^{1/2}]$ $[\simeq \chi _1^{\rm r}]$ , solutions (5) and (6) can be rewritten in a more visual form for attenuation Pendellösung oscillations as

$[{I_0}(x) = \exp (- {\mu _0}\,x)\cos ^2{{\pi \,x\,f}\over {l_{\rm Pen}}} \eqno(7)]$

and

$[{I_1}(x) = \exp (- {\mu _0}\,x)\sin ^2{{\pi \,x\,f}\over {l_{\rm Pen}}}. \eqno(8)]$

Expressions (7) and (8) clearly describe the Pendellösung effect of Laue diffraction when the exact Bragg condition is satisfied. The pendulum beat period $[l_{\rm Pen} = \lambda |{\cos {\theta _{\rm B}}}|]$ $[/(C| {{\chi _1}}|)]$ for a perfect multilayer depends on the Fourier polarizability coefficient χ₁, which characterizes the interaction of X-ray waves in a periodic medium.

Fig. 2 shows the intensity distributions over the depth of a perfect (f = 1) Mo/Si multilayer with the same thicknesses of alternating layers, d_Mo = d_Si = 3.5 nm, calculated using exact solutions (5) and (6), as well as using approximate solutions (7) and (8). In the numerical calculations, the synchrotron radiation wavelength λ = 0.1305 nm was used. The optical constants are presented in Tables 1 and 2. The total depth of the multilayer is L_x = 2l_Pen = 76.4 µm, which corresponds to two Pendellösung distances. At an exact Bragg angle (η = 0), the transmission intensity at a depth of x = 19.1 µm, which corresponds to half the Pendellösung distance, almost completely (corrected for photoelectric absorption) transforms into a diffraction wave. Furthermore, the reverse process is observed. The visible difference between the curves of the pendulum effect in the case of exact and approximate solutions is observed at large sectioned depths of the multilayer (Fig. 2).

Figure 2
Pendellösung oscillations in an Mo/Si multilayer for transmission (1 and 2) and diffraction (3 and 4) intensities. Curves 1 and 3 were calculated using solutions (5)

and (6)

. Curves 2 and 4 were obtained on the basis of solutions (7)

and (8)

Fig. 3 demonstrates Pendellösung oscillations for a perfect and imperfect Mo/Si multilayer. The attenuation factor for the defect structure is f = 0.8. It can be seen from the figure that the period of Pendellösung oscillations increases in the case of an imperfect multilayer. This is due to the fact that the presence of defects in the multilayer reduces the reflectivity of the periodic structure. A similar behavior of Pendellösung oscillations was observed in the case of dynamical Laue diffraction in a crystal with defects (Punegov & Pavlov, 1992 ).

Figure 3
Pendellösung oscillations in an imperfect (curves 1 and 3, attenuation factor is f = 0.8) and perfect (curves 2 and 4, f = 1) Mo/Si multilayer. Curves 1 and 2 are transmitted intensities, while curves 3 and 4 are diffraction intensities.

Solutions (3) and (4) make it possible to obtain the intensity distributions of the transmission I₀(x)= $[{| {{E_0}({\eta _\omega },x)}|^2}]$ and diffraction I₁(x)= $[{| {{E_1}({\eta _\omega },x)}|^2}]$ waves inside the multilayer, depending on the value of the angular parameter η (Fig. 4). So, for example, with an angular deviation of the incident X-ray wave from the Bragg angle by ω = 0.18 mrad, a decrease in the period of the Pendellösung oscillations is observed (Fig. 4). The transmission X-ray wave is not completely transmitted into the diffraction beam (Fig. 4, curve 3). Therefore, the diffraction intensity inside the multilayer is lower than at the exact Bragg angle (compare curves 3 and 4).

Figure 4
Pendellösung oscillations in an Mo/Si multilayer. Curves 1 and 3 show the calculated transmission and diffraction intensities for η = 0.16 µm⁻¹ (ω = 0.18 mrad), respectively. Curves 2 and 4 show them for η = 0.

If in equation (1) a₋₁ = 0, this means that there is no transfer of intensity from the diffraction beam back to the transmitted beam, solutions (3) and (4) are transformed into expressions corresponding to the kinematical approximation,

$[E_0^{}(\eta, x) = \exp (i{a_0}x) \eqno(9)]$

and

$[E_1^{}(\eta, x) = i{a_1}\,f{{\sin (\eta \,x/2)} \over {\eta /2}}\exp (i\psi \,x). \eqno(10)]$

In the kinematical approximation, for the intensities of the transmission and diffraction waves, we obtain

$[{I_0}(x) = \exp (- {\mu _0}\,x) \eqno(11)]$

and

$[{I_1}(x) = \exp (- {\mu _0}\,x)\,a_1^2\,{f^2}L(\eta \,x), \eqno(12)]$

where $[L(\eta \,x)=]$ $[{{{{\sin }^2}(\eta \,x/2)}/{{{(\eta /2)}^2}}}]$ is the Laue interference function (James, 1950 $[James, R. W. (1950). The Optical Principles of the Diffraction ofX-rays. London: G. Bell and Sons.]$ ). In the case of kinematical diffraction, the intensity of the transmitted wave, according to equation (11), exponentially attenuates due to photoelectric absorption in the multilayer. The intensity of the diffraction wave has a typical angular distribution of kinematical diffraction. Solutions (9)–(12) are valid for multilayers with a very small sectioned depth.

2.2. Rocking curves of a sectioned multilayer with constant period

In experimental works (Kang et al., 2004, 2005, 2007), rocking curves of Laue diffraction from sectioned multilayers are presented as angular dependences of q_x scans in reciprocal space. In our consideration, the deviation from the Bragg angle in solutions (3) and (4) is determined by the parameter η, which corresponds to the q_x scan in reciprocal space (Kang et al., 2005; Punegov et al., 2014 , 2016 ).

In Fig. 5, rocking curves of Mo/Si multilayers of different depths L_x are shown. In the case when the depth of the multilayer is equal to half the Pendellösung distance (L_x = l_Pen/2 = 19.1 µm), the intensity of the diffraction wave, according to Fig. 2, is maximum and is equal to $[{I_1}(\eta = 0)=]$ $[0.6\,I_0^{\,\rm in}]$ , where $[I_0^{\,\rm in}]$ is the intensity of the incident radiation. The intensity of the transmitted wave in the exact Bragg orientation is I₀(η = 0) ≃ 0 [Fig. 5(a)].

Figure 5
Rocking curves of Laue diffraction depending on the sectional depth L_x of an Mo/Si multilayer. (a) L_x = l_Pen/2, (b) L_x = 3l_Pen/4 and (c) L_x = l_Pen. Curve 1 is transmitted intensity, while curve 2 is diffracted intensity.

The rocking curves of the transmitted and diffracted intensities from an Mo/Si multilayer with a depth of L_x = l_Pen × 3/4 = 28.7 µm are shown in Fig. 5(b). The maximum of the diffracted intensity is $[{I_1}(\eta = 0)=]$ $[0.24\,I_0^{\,\rm in}]$ . In the case where L_x = l_Pen = 38.2 µm, the gap is observed in the profile of the rocking curve of the diffraction wave at the exact Bragg orientation [Fig. 5(c)] and the minimum of the diffracted intensity is $[{I_1}(\eta = 0)=]$ $[ 5\, \times {10^{ - 6}}\,I_0^{\,\rm in}]$ . This behavior of the rocking curve is typical for all periodic media in the case of Laue diffraction, when the depth of the sectioned structure is equal to the Pendellösung distance.

3. Dynamical Laue diffraction in an inhomogeneous sectioned multilayer: recurrence relations

In addition to flat multilayer Laue lenses, to improve focusing, it has been proposed to manufacture wedged multilayer Laue lenses (Yan et al., 2007 ; Conley et al., 2008 ), in which the period of the structure changes not only in thickness z but also along the depth x of the sectioned multilayer system. Equation (1) is written for a multilayer with a constant period and does not allow taking into account the change in the structure period along the x coordinate. A multilayer with wedge layers is a structure whose period varies along the x axis (Fig. 6). Since equation (1) describes diffraction by a structure with a constant period, we divide the wedge multilayer into vertical elementary sections, within which the period of the structure is constant (Fig. 6). We denote the depth of the vertical sections in the direction of the x axis as x^p − x^p−1 = l^p. Here x^p and x^p−1 are the coordinates of the left and right boundaries of the vertical elementary section with a number p, respectively, and l^p is the depth of this section, while the sections are numbered from left to right (p = 0, 1, 2,…, P). The superscript of all parameters corresponds to the elementary section number. Let the diffraction in the first vertical section be described by equation (1), then for the section with the number p instead of the parameter η in (1) one should write η^p = η + Δη^p, where Δη^p = 2π Δd^p/d² and Δd^p = d − d^p is the mismatch of the period of the pth vertical elementary section relative to the first vertical section with a multilayer period d¹ = d.

Figure 6
A schematic representation of X-ray Laue diffraction in a wedge multilayer.

X-ray diffraction in the pth elementary vertical section of the multilayer is described by a system of equations of the form

$[\Bigg\{ \eqalign{ {{\partial E_0^{\,p}(\eta, x)} /{\partial x}} & = i{a_0}\,E_0^{\,p}(\eta, x)\, + i{a_{ - 1}}\,{f^{\,p}}\,E_1^{\,p}(\eta, x), \cr {{\partial E_1^{\,p}(\eta, x)} / {\partial x}} &= i({a_0} + \eta + \Delta {\eta ^{\,p}})\,E_1^{\,p}(\eta, x)\, + i{a_1}\,{f^{\,p}}\,E_0^{\,p}(\eta, x). } \eqno (13)]$

Let $[E_0^{\,p - 1}(\eta, {x^{p - 1}})]$ and $[E_1^{\,p - 1}(\eta, {x^{p - 1}})]$ represent the amplitudes of the X-ray fields at the boundary (p − 1) of the pth vertical elementary section. Then, taking into account these boundary conditions, we can obtain solutions for the transmission and diffraction waves inside the elementary section with number p,

$[E_0^{\,p}(\eta, x) = B_1^{\,0}\exp \left[i\xi _1^{\,p}\left(x - {x^{\,p - 1}}\right)\right] - B_2^{\,0}\exp \left[i\xi _2^{\,p}\left(x - {x^{\,p - 1}}\right)\right] \eqno(14)]$

and

$[E_1^{\,p}(\eta, x) = B_1^{1}\exp \left[i\xi _1^{\,p}\left(x - {x^{\,p - 1}}\right)\right] - B_2^{1}\exp \left[i\xi _2^{\,p}\left(x - {x^{\,p - 1}}\right)\right], \eqno(15)]$

where

$[\xi _1^{\,p} = {{(2{a_0} + {\eta ^{\,p}}) - \left[{{{\left(\eta _{_{}}^{\,p}\right)}^2} + {{\left({\,f^{p}}\right)}^2}4{a_1}{a_{-1}}} \right]^{1/2} }\over 2},]$

$[\xi _2^{\,p} = {{(2{a_0} + {\eta ^{\,p}}) + \left[{{{\left(\eta _{_{}}^{\,p}\right)}^2} + {{\left({\,f^{p}}\right)}^2}4{a_1}{a_{-1}}} \right]^{1/2} } \over 2},]$

$[\xi _{}^{\,p} = \xi _1^{\,p} - \xi _2^{\,p} = -\left[{{{\left(\eta _{_{}}^{\, p}\right)}^2} + {{\left({\,f^{p}}\right)}^2}4{a_1}{a_{-1}}} \right]^{1/2},]$

$[\varphi _1^{0} = {{{a_0} - \xi _2^{\,p}} \over {\xi _{}^{\,p}}},]$

$[\varphi _2^{0} = {{{a_0} - \xi _1^{\,p}} \over {\xi ^{\,p}}},]$

$[\varphi _1^{1} = {{{a_0} + {\eta ^{\,p}} - \xi _2^{\,p}} \over {\xi ^{\,p}}},]$

$[\varphi _2^{1} = {{{a_0} + {\eta ^{\,p}} - \xi _1^{\,p}} \over {\xi ^{\,p}}},]$

$[B_{1,2}^{0} = \varphi _{1,2}^{0}E_0^{\,p - 1}(\eta, {x^{\,p - 1}}) + {{{a_{ - 1}}\,E_1^{\,p - 1}(\eta, {x^{p - 1}})} \over {\xi _{}^{p}}}]$

and

$[B_{1,2}^{1} = \varphi _{1,2}^{1}E_1^{\,p - 1}(\eta, {x^{\,p - 1}}) + {{{a_{ - 1}}\,E_0^{\,p - 1}(\eta, {x^{\,p - 1}})} \over {\xi ^{\,p}}}.]$

At the boundary between the pth and the (p + 1)th elementary vertical sections, the solutions for the amplitudes of X-ray waves have the form

$[E_0^{\,p}(\eta, {x^{p}}) = B_1^{\,0}\exp (i\xi _1^{\,p}{l^{\,p}}) - B_2^{\,0}\exp (i\xi _2^{\,p}{l^{\,p}}) \eqno(16)]$

and

$[E_1^{\,p}(\eta, {x^{p}}) = B_1^{1}\exp (i\xi _1^{\,p}{l^{\,p}}) - B_2^{1}\exp (i\xi _2^{\,p}{l^{\,p}}). \eqno(17)]$

The amplitudes $[E_{0}^{\,p}(\eta, {x^{p}})]$ and $[E_{1}^{\,p}(\eta, {x^{p}})]$ serve as the boundary conditions for diffraction in the (p + 1) elementary vertical section. Performing sequentially the procedure of recurrent calculations with the initial boundary conditions on the input face, $[E_0^{\,0}(\eta, {x^{0}}) = 1]$ and $[E_1^{\,0}(\eta, {x^{0}}) = 0]$ , we obtain the amplitudes of the transmitted $[E_0^{\,P}(\eta, {x^{P}})]$ and diffracted $[E_1^{\,P}(\eta, {x^{P}})]$ waves on the output face of the multilayer inhomogeneous in depth L_x, where $[{L_x} = \textstyle\sum_{p = 0}^P {{l^{\,p}}}]$ .

3.1. Pendellösung effect in the case of a wedge sectioned multilayer

To calculate the intensity distribution of the transmission and diffraction waves in the wedge multilayer (Fig. 6), we used the recurrence relations (16) and (17). We will consider cases of weak and strong linear period variation. In the first case, the period varies with the depth of the Mo/Si multilayer from 7.000 to 6.997 nm. In the case of a strong gradient, the period changes from 7.000 to 6.986 nm. Fig. 7 shows the depth distributions of intensities in Mo/Si multilayers with variations in the periods. In the case of a weak gradient of the period variation at the initial depth interval from 0 to 10 µm, the behavior of the transmission and diffraction intensities coincides with the Pendellösung effect of a multilayer with a constant period. Furthermore, due to the gradient of the period variation of the Mo/Si system, the classical Pendellösung effect is violated. As a result, the distance of the Pendellösung fringe decreases and the intensity of the transmission wave is not completely transferred to the diffraction beam [Fig. 7(a)]. For a strong gradient of the period variation of the Mo/Si multilayer, the Pendellösung profiles of the transmission and diffraction X-ray waves do not intersect. The period of the Pendellösung fringe decreases monotonically with the depth of the sectioned multilayer [Fig. 7(b)].

Figure 7
Distributions of transmitted (curve 1) and diffracted (curve 3) intensities inside an Mo/Si multilayer with weak (a) and strong (b) gradients of period variation. Curves 2 and 4 correspond to a perfect Mo/Si multilayer with a constant period. The number of elementary vertical sections for a wedge multilayer is P = 200.

The intensity distributions of the transmission and diffraction waves also depend on the lateral disturbance gradient of the multilayer structure. The Pendellösung effect depending on the weak and strong gradient of the attenuation factor is shown in Fig. 8. The attenuation factor varies linearly from 1 to 0.8 in the case of a weak gradient and from 1 to 0.6 in the case of a strong gradient. Since the perfection of the multilayer structure deteriorates along the depth of the Mo/Si multilayer, the period of Pendellösung oscillations in the direction of the x axis increases. We see that the stronger the attenuation-factor-variation gradient, the more visible the increase in the Pendellösung distance along the depth of the multilayer (Fig. 8).

Figure 8
Distributions of transmitted (curves 1 and 2) and diffracted (3 and 4) intensities inside an Mo/Si multilayer with weak (1 and 3) and strong (2 and 4) attenuation-factor gradients. The number of elementary vertical sections is P = 200.

3.2. Rocking curves of wedge sectioned multilayers

We calculated X-ray rocking curves of Mo/Si multilayers that are nonuniform over depth x (L_x = l_Pen = 38.2 µm) using the recurrent solutions (16) and (17). In Fig. 9, the rocking curves of transmission and diffraction waves with the strong variation gradient of the multilayer period for different numbers of fragmentations P of the structure depth into elementary vertical sections are shown. Starting from P = 30, an increase in the number of fragmentations into elementary sections does not change the rocking-curve profile.

Figure 9
Rocking curves of the transmitted (a) and diffracted (b) waves in the case of a strong period-variation gradient of Mo/Si multilayer for different numbers of fragmentations P into elementary vertical sections. Curve 1 corresponds to fragmentation P = 4, while curve 2 refers to the case where P = 30. L_x = l_Pen = 38.2 µm.

Fig. 10 demonstrates the rocking curves of wedge Mo/Si multilayers with weak and strong gradients of period variation along the depth of the sectioned structure. On the profile of the rocking curve of the diffracted wave in the case of the weak gradient, a gap is observed [Fig. 10(a)], which is less deep compared with the splitting of the rocking curve of a flat multilayer [Fig. 5(c)]. The rocking curve of the multilayer with the strong period-variation gradient has two symmetrical gaps [Fig. 10(b)].

Figure 10
Rocking curves of transmitted (curve 1) and diffracted (curve 2) waves from an Mo/Si multilayer with weak (a) and strong (b) gradients of period variation. The number of elementary vertical sections is P = 300, while the sectioned depth of the multilayer is L_x = l_Pen = 38.2 µm.

The influence of linearly varying attenuation factors along the depth of the sectional Mo/Si multilayer with a constant period d on the profiles of the rocking curves is shown in Fig. 11. The weak gradient of the attenuation-factor variation insignificantly affects the rocking curves [compare Fig. 11(a) with Fig. 5(c)]. On the other hand, a strong attenuation-factor gradient strongly changes the profile of the diffraction curve [Fig. 11(b)].

Figure 11
Rocking curves of transmitted (curve 1) and diffracted (curve 2) waves from an Mo/Si multilayer with weak (a) and strong (b) attenuation-factor gradients. The number of elementary vertical sections is P = 300, while the sectioned depth of the multilayer is L_x = l_Pen = 38.2 µm.

4. Determination of the depth of sectioned multilayers

An experimental study of Laue diffraction in sectioned W/Si and Mo/Si multilayers with hard synchrotron radiation energies (9.5 and 19.5 keV) was performed for sectioned depths from 2 to 17 µm (Kang et al., 2005). The period of the W/Si multilayer was 29 nm (sample A). The volume fraction of tungsten in the W/Si period was 60%. The second Mo/Si structure had a period of 7 nm with an equal proportion of Mo and Si (sample B). Since in the article (Kang et al., 2005) the results of diffraction were mainly discussed using synchrotron radiation with a wavelength of λ = 0.1305 nm (energy E = 9.5 keV), in this work numerical calculations were performed using this wavelength for structures A and B.

The optical constants of tungsten, molybdenum and silicon with respect to the wavelength λ = 0.1305 nm of synchrotron radiation were obtained using the X-ray Server computer program (Stepanov, 1997 , 2013 ; Stepanov & Forrest, 2008 ), and are shown in Table 1. Based on these data, the Fourier coefficients of X-ray polarizability, χ₀ and χ₁, for the case of Laue diffraction in W/Si and Mo/Si multilayers, were calculated (Table 2).

Table 1
Optical constants of silicon, molybdenum and tungsten at a synchrotron radiation wavelength λ = 0.1305 nm

Material	Density (gm cm⁻³)	Energy (keV)	Fourier coefficients of X-ray polarizability χ_j (× 10⁻⁵)
Si – amorphous	2.0	9.5	−0.219 + i 0.00089
Mo – amorphous	4.5	9.5	−1.783 + i 0.08958
W – amorphous	19.0	9.5	−6.279 + i 0.37728

Table 2
Fourier coefficients of X-ray polarizability of Mo/Si and W/Si multilayers at a synchrotron radiation wavelength λ = 0.1305 nm

Multilayer	d (nm)	λ (nm)	θ_B (mrad)	χ₀ (× 10⁻⁵)	χ₁ (× 10⁻⁵)
Mo/Si	7	0.1305	2.25	−1.465 + i 0.058	−0.341 + i 0.0267
W/Si	29	0.1305	9.32	−4.152 + i 0.228	−1.626 + i 0.107

The Pendellösung distance for the W/Si multilayer is more than four times less than the corresponding value for the Mo/Si structure ( $[l_{\rm Pen}^{\,\rm MoSi/Si} = \,38.2\,\rm\micro m]$ ). There are two options for determining the sectioned depth of multilayers L_x. The first approach is based on the fact that the depths of the sectioned multilayers L_x are determined by measuring the period of interference fringes on the rocking-curve profile. There are two drawbacks to this approach. First, this method is precisely realized within the kinematical approximation. Second, good angular resolution is needed in the measurement of the rocking curves. The second approach is based on the numerical simulation of the rocking curves for Laue diffraction from multilayers using dynamical diffraction solutions. Even with the absence of information on the structural perfection of multilayers, the second approach gives an insignificant error in numerical simulation and subsequent comparison with experimental measurements.

For a synchrotron radiation energy of 9.5 keV in the Laue diffraction experiment involving W/Si and Mo/Si multilayers, the angular distribution of the scattered intensity was investigated by scanning the scattering vector in the direction of the projection Q_x (Kang et al., 2005), which corresponds to the angular parameter η = Q_x in our consideration. The W/Si and Mo/Si samples were wedge shaped in the y direction perpendicular to the diffraction plane (x, z), which gave a change in the sectioned depth L_x. The rocking curves were calculated using solution (4) for the diffracted wave.

As in the experimental work (Kang et al., 2005), in the numerical simulation, we used the W/Si structure with a period d = d_W + d_Si = 29 nm, where d_W = 17.4 nm and d_Si = 11.6 nm. In the case of the Mo/Si multilayer, the period was d = d_Mo + d_Si = 7 nm with d_Mo = d_Si = 3.5 nm. The optical constants of these multilayers are presented in Table 2.

It is known that the rocking-curve profiles always have a gap when the sectional depth of the sample is close to the Pendellösung distance (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ). The calculated X-ray rocking curves of the sectioned W/Si and Mo/Si multilayers for different depths are shown in Fig. 12.

Figure 12
Calculated diffraction rocking curves in the Laue geometry from W/Si (a) and Mo/Si (b) multilayers with different sectioned depths L_x. The diffracted intensities are offset by factors of 10² for clarity.

For W/Si multilayers with depths L_x equal to 2.6, 4.4, 6.6 and 8.8 µm, the calculated rocking curves [Fig. 12(a)] are close to the experimental data (Kang et al., 2005), with the characteristic gap in the profile of the diffraction curve appearing for L_x = 8.8 µm ( $[l_{\rm Pen}^{\,\rm W/Si} = \,8.01\,\rm \micro m]$ ). For a relatively large depth L_x = 10.7 µm, the calculated rocking curve differs from the experimental diffraction curve (Kang et al., 2005).

The diffraction curves of the Mo/Si multilayers with small sectioned depths of 3.9, 6.3 and 8.6 µm [Fig. 12(b)] are consistent with experimental data (Kang et al., 2005). However, the calculated rocking curves for sectioned depths of 11.2 and 15.0 µm [Fig. 12(b)] differ significantly from the experimental results (Kang et al., 2005). For example, the experimental diffraction curve from an Mo/Si multilayer with a depth of 15.0 µm has a gap in the middle of the rocking-curve profile. Such a gap is only possible when the sectional depth L_x is close in value to the Pendellösung distance ( $[l_{\rm Pen}^{\,\rm MoSi/Si} = \,38.2\,\rm \micro m]$ ). Thus, analysis of experimental data to determine the sectioned depth $[L_x^{{\eta _0}} = \,\,2\pi /{\eta _0}]$ (Kang et al., 2005) using the value of the interference-fringe spacing η₀ is not always justified. For example, for the depth L_x = 15.0 µm and η₀ = 0.42 µm⁻¹, and for L_x = 38.2 µm and η₀ = 0.165 µm⁻¹.

The question then arises: why for small sectioned depths of multilayers are the calculated and experimental rocking curves close to each other, but for large L_x they are significantly different? Fig. 13 shows the calculated rocking curves of the Mo/Si multilayer structure for the sectioned depth L_x = 38.2 µm with different angular resolutions. We also show the influence of the instrumental function on X-ray diffraction from multilayers in the Laue geometry (curves 2, Fig. 13). In our calculations, we used a diffraction scheme containing a four-bounces Ge(220) monochromator and a three-bounces Ge(220) analyzer.

Figure 13
Calculated rocking curves (curve 1) from an Mo/Si multilayer with sectioned multilayer depth L_x = l_Pen = 38.2 µm. The diffraction curves (curve 2) are also presented, taking into account the instrumental function. The angular resolutions (a) are Δη = 0.02 µm⁻¹ and (b) Δη = 0.1 µm⁻¹, while the periods of the interference fringes are (a) η₀ = 0.1646 µm⁻¹ and (b) η₀ = 0.335 µm⁻¹.

In the case of a large depth L_x, interference oscillations are located close to each other; therefore, all oscillations are observed only at a high angular resolution Δη = 0.02 µm⁻¹ [Fig. 13(a)] and the period of oscillations is η₀ = 0.1646 µm⁻¹. In the case of a low resolution, for example, η₀ = 0.335 µm⁻¹, not all interference oscillations are registered; therefore, in this case, the distance between the oscillations is η₀ = 0.335 µm⁻¹ [Fig. 13(b)]. From this value, we can obtain the sectioned depth L_x = 18.8 µm, which is approximately two times less than in the case of high angular resolution. Consequently, the analysis of the experimental data of X-ray Laue diffraction to determine the sectional depth of multilayer structures is possible only under the condition of a high angular resolution of the rocking curves.

5. Concluding remarks

We have theoretically investigated in detail X-ray Laue diffraction in multilayers in the case of an incident plane wave. The influence of sectional depth, imperfections (defects) and non-uniform distribution of the period of multilayers on the Pendellösung effect and rocking curves has been shown. However, more complete information on the structure of multilayers is provided by the use of triple-axis diffractometry (Iida & Kohra, 1979 ; Punegov, 2015 ). Therefore, the next step will be to consider the Laue diffraction in the case of restricted X-ray beams (Punegov et al., 2017; Punegov & Karpov, 2021). This will make it possible to calculate RSMs in Laue geometry, as well as compare the calculated q_x and q_z sections of RSMs with experimental data (Kang et al., 2005).

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