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Volume 70 
Part 1 
Pages 64-71  
January 2014  

Received 11 October 2013
Accepted 6 November 2013
Online 20 December 2013

Darwin's approach to X-ray diffraction on lateral crystalline structures

aKomi Research Center, Ural Division, Russian Academy of Sciences, 167982, Syktyvkar, Russian Federation,bSchool of Science and Technology, University of New England, NSW 2351, Australia, and cSchool of Physics, Monash University, VIC 3800, Australia
Correspondence e-mail: vpunegov@dm.komisc.ru

Darwin's dynamical theory of X-ray diffraction is extended to the case of lateral (i.e., having a finite length in the lateral direction) crystalline structures. This approach allows one to calculate rocking curves as well as reciprocal-space maps for lateral crystalline structures having a rectangular cross section. Numerical modelling is performed for these structures with different lateral sizes. It is shown that the kinematical approximation is valid for thick crystalline structures having a small length in the lateral direction.

1. Introduction

There are several approaches to X-ray dynamical diffraction by crystals (see, e.g., Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]; Pietsch et al., 2004[Pietsch, U., Holý, V. & Baumbach, T. (2004). High Resolution X-ray Scattering: From Thin Films to Lateral Nanostructures, 2nd ed. New York: Springer-Verlag.] and references therein). Among them Darwin's diffraction theory (Darwin, 1914a[Darwin, C. G. (1914a). Philos. Mag. 27, 315-333.],b[Darwin, C. G. (1914b). Philos. Mag. 27, 675-691.]), based on recurrence relations, presents arguably the most simple and transparent way to describe X-ray and neutron dynamical diffraction by crystals. However, Darwin's theory (in its original formulation) has more restrictions in comparison with Laue-Ewald's theory (see, e.g., Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]); therefore it was not widely used. Borie (1966[Borie, B. (1966). Acta Cryst. 21, 470-472.]) first used Darwin's recurrence relations to describe the Borrmann effect. Using Borie's approach, Bezirganian & Navasardian (1969[Bezirganian, P. H. & Navasardian, M. A. (1969). Izv. Akad. Nauk Arm. SSR Fiz. 3, 269-274.]) showed that the Borrmann effect depends on the lateral size of crystals. Darwin's theory was also used to investigate (i) diffraction in the asymmetric Laue geometry (Borie, 1967[Borie, B. (1967). Acta Cryst. 23, 210-216.]; Kuznetsov & Fofanov, 1970[Kuznetsov, A. V. & Fofanov, A. D. (1970). Sov. Phys. J. 13, 1269-1274.]), (ii) multiwave diffraction (Kuznetsov & Fofanov, 1972[Kuznetsov, A. V. & Fofanov, A. D. (1972). Sov. Phys. J. 15, 559-563.]; Ignatovich, 1992[Ignatovich, V. K. (1992). Sov. Phys. Crystallogr. 37, 588-595.]) and (iii) properties of the dispersion surface (Borie, 1967[Borie, B. (1967). Acta Cryst. 23, 210-216.]). The recurrence relations for the amplitudes of transmitted and reflected waves were also employed to investigate neutron diffraction by polyatomic crystals (Ignatovich, 1990[Ignatovich, V. K. (1990). Sov. Phys. JETP, 70, 913-917.]) and scattering of visible light in liquid crystals (Chandrasekhar & Rao, 1968[Chandrasekhar, S. & Rao, K. N. S. (1968). Acta Cryst. A24, 445-451.]). Darwin's recurrence relations were also used in multilayer materials optics (Dub & Litzman, 1999[Dub, P. & Litzman, O. (1999). Acta Cryst. A55, 613-620.]).

The crystal truncation rod (CTR) method (Robinson, 1986[Robinson, I. K. (1986). Phys. Rev. B, 33, 3830-3836.]) was originally developed within the framework of kinematical diffraction for characterization of surface layers. Darwin's dynamical diffraction approach was implemented in the CTR technique in Caticha (1994[Caticha, A. (1994). Phys. Rev. B, 49, 33-38.]), Nakatani & Takahashi (1994[Nakatani, S. & Takahashi, T. (1994). Surf. Sci. 311, 433-439.]), Takahashi & Nakatani (1995[Takahashi, T. & Nakatani, S. (1995). Surf. Sci. 326, 347-360.]), Durbin & Follis (1995[Durbin, S. & Follis, G. (1995). Phys. Rev. B, 51, 10127-10133.]) and Takahashi et al. (2000[Takahashi, T., Yashiro, W., Takahasi, M., Kusano, S., Zhang, X. & Ando, M. (2000). Phys. Rev. B, 62, 3630-3638.]).

Yashiro & Takahashi (2000[Yashiro, W. & Takahashi, T. (2000). Acta Cryst. A56, 163-167.]) analysed the reflection and transmission coefficients of a single atomic plane for an arbitrary two-dimensional Bravais lattice. Their results (Yashiro & Takahashi, 2000[Yashiro, W. & Takahashi, T. (2000). Acta Cryst. A56, 163-167.]) are similar to those obtained by Borie (1967[Borie, B. (1967). Acta Cryst. 23, 210-216.]); however, they differ from ones obtained by Durbin (1995[Durbin, S. M. (1995). Acta Cryst. A51, 258-268.]). Later, on the basis of Yashiro & Takahashi (2000[Yashiro, W. & Takahashi, T. (2000). Acta Cryst. A56, 163-167.]) a variant of Darwin's theory for grazing-incidence geometry was developed (Yashiro et al., 2001[Yashiro, W., Ito, Y., Takahasi, M. & Takahashi, T. (2001). Surf. Sci. 490, 394-408.]).

Prins (1930[Prins, J. A. (1930). Z. Phys. 63, 477-493.]) analysed the effects of refraction and absorption for semi-infinite crystals. An exact solution of Darwin's recurrence equations for a crystal with an arbitrary number of reflecting planes was obtained by Perkins & Knight (1984[Perkins, R. T. & Knight, L. V. (1984). Acta Cryst. A40, 617-619.]) with the use of the Chebyshev polynomials (Abramowitz & Stegun, 1972[Abramowitz, M. & Stegun, I. A. (1972). Editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications.]). Chen & Bhattacharya (1993[Chen, Y. C. & Bhattacharya, P. K. (1993). J. Appl. Phys. 73, 7389-7394.]) showed that the Darwin recurrence relations are identical to Takagi's differential equations (Takagi, 1962[Takagi, S. (1962). Acta Cryst. 15, 1311-1312.]) as one goes from an array of discrete atomic planes to a continual model of medium.

Most crystalline structures are non-ideal, e.g., they contain defects. Statistical dynamical diffraction theory (in the case of a plane-wave illumination) (Bushuev, 1989[Bushuev, V. A. (1989). Sov. Phys. Solid State, 31, 1877-1882.]; Punegov, 1990[Punegov, V. I. (1990). Sov. Phys. Crystallogr. 35, 336-340.], 1991[Punegov, V. I. (1991). Sov. Phys. Solid. State, 33, 136-140.], 1993[Punegov, V. I. (1993). Phys. Status Solidi, 136, 9-19.]; Pavlov & Punegov, 1998[Pavlov, K. M. & Punegov, V. I. (1998). Acta Cryst. A54, 214-218.], 2000[Pavlov, K. M. & Punegov, V. I. (2000). Acta Cryst. A56, 227-234.]) using Takagi's equations (Takagi, 1962[Takagi, S. (1962). Acta Cryst. 15, 1311-1312.]) is one possible way to describe X-ray dynamical diffraction in such structures. However, Darwin's theory (Darwin, 1914a[Darwin, C. G. (1914a). Philos. Mag. 27, 315-333.],b[Darwin, C. G. (1914b). Philos. Mag. 27, 675-691.]), which was originally developed for an ideal crystal, can also be modified for use with non-ideal crystals. For instance, Chung & Durbin (1999[Chung, J.-S. & Durbin, S. M. (1999). Acta Cryst. A55, 14-19.]) considered the thermal vibration effect. Li et al. (1997[Li, M., Ress, H., Gerhard, T., Landwehr, G., Cui, S. F. & Mai, Z. H. (1997). J. Appl. Phys. 81, 2143-2147.]) examined the influence of statistically distributed defects. X-ray diffraction is particularly sensitive to deformation of the crystalline lattice, which is another variant of deviations from an ideal crystalline structure. First attempts at modelling X-ray diffraction by a crystal with a linear lattice parameter variation using Darwin's approach were done by Fitzgerald & Darlington (1976[Fitzgerald, W. J. & Darlington, C. N. W. (1976). Acta Cryst. A32, 671-672.]). Their numerical results were confirmed later by analytical solutions (Kolpakov & Punegov, 1985[Kolpakov, A. V. & Punegov, V. I. (1985). Solid State Commun. 54, 573-578.]; Punegov & Vishnjakov, 1995[Punegov, V. I. & Vishnjakov, Y. V. (1995). J. Phys. D Appl. Phys. 28, A184-A188.]). Prudnikov obtained analytical solutions for the cases of crystals distorted by surface acoustic waves (Prudnikov, 1998[Prudnikov, I. R. (1998). Acta Cryst. A54, 1034-1036.]) and non-ideal heterostructures (Prudnikov, 2000[Prudnikov, I. R. (2000). Phys. Status Solidi, 217, 725-735.]).

Recurrence relations (similar to Darwin's equations) were obtained for multilayer structures. Therewith the reflection and transmission coefficients for a single atomic plane were replaced by the appropriate coefficients for crystalline layers (Belyaev & Kolpakov, 1983[Belyaev, Y. N. & Kolpakov, A. V. (1983). Phys. Status Solidi, 76, 641-646.]; Bartels et al., 1986[Bartels, W. J., Hornstra, J. & Lobeek, D. J. W. (1986). Acta Cryst. A42, 539-545.]). This approach was further extended to multiwave diffraction (Ladanov & Punegov, 1989[Ladanov, A. V. & Punegov, V. I. (1989). Twelfth European Crystallographic Meeting, Moscow, USSR, August 20-29, 1989. Collected Abstracts Vol. 3, p. 137.]; Punegov, 1993[Punegov, V. I. (1993). Phys. Status Solidi, 136, 9-19.]), highly asymmetrical diffraction geometry (Punegov & Ladanov, 1989b[Punegov, V. I. & Ladanov, A. V. (1989b). Twelfth European Crystallographic Meeting, Moscow, USSR. August 20-29, 1989. Collected Abstracts Vol. 3, p. 93.], 1990[Punegov, V. I. & Ladanov, A. V. (1990). Poverkhnost, 4, 45-50.]) and glancing geometry (Punegov & Ladanov, 1989a[Punegov, V. I. & Ladanov, A. V. (1989a). Sov. Phys. Tech. Phys. 34, 1351-1352.]).

All the above-mentioned Darwin recurrence relations approaches were obtained for planar structures with atomic planes, which are assumed to be infinitely large in the lateral direction. In recent years there has been a renewed interest in X-ray diffraction on lateral structures (see, e.g., Kaganer & Belov, 2012[Kaganer, V. M. & Belov, A. Y. (2012). Phys. Rev. B, 85, 125402.]; Minkevich et al., 2011[Minkevich, A. A., Fohtung, E., Slobodskyy, T., Riotte, M., Grigoriev, D., Metzger, T., Irvine, A. C., Novák, V., Holý, V. & Baumbach, T. (2011). EPL, 94, 66001.]; Lee et al., 2006[Lee, K., Yi, H., Park, W., Kim, Y. K. & Baik, S. (2006). J. Appl. Phys. 100, 051615.] and references therein). Different dynamical diffraction approaches (Olekhnovich & Olekhnovich, 1978[Olekhnovich, N. M. & Olekhnovich, A. I. (1978). Acta Cryst. A34, 321-326.]; Thorkildsen & Larsen, 1999b[Thorkildsen, G. & Larsen, H. B. (1999b). Acta Cryst. A55, 840-854.]; Kolosov & Punegov, 2005[Kolosov, S. I. & Punegov, V. I. (2005). Crystallogr. Rep. 50, 357-362.]) using Takagi's equations were employed to calculate rocking curves from ideal (i.e. non-deformed) crystals with rectangular cross section. Kinematical diffraction theory was used to simulate X-ray diffraction on deformed crystals having a trapezium cross-sectional shape (Punegov et al., 2006[Punegov, V. I., Kolosov, S. I. & Pavlov, K. M. (2006). Tech. Phys. Lett. 32, 809-812.]; Punegov & Kolosov, 2007[Punegov, V. I. & Kolosov, S. I. (2007). Crystallogr. Rep. 52, 191-198.]) or an arbitrary cross-sectional shape (Punegov et al., 2007[Punegov, V. I., Maksimov, A. I., Kolosov, S. I. & Pavlov, K. M. (2007). Tech. Phys. Lett. 33, 125-127.]).

However, using Takagi's equations to simulate dynamical X-ray diffraction on lateral crystalline structures (Becker, 1977[Becker, P. (1977). Acta Cryst. A33, 243-249.]; Becker & Dunstetter, 1984[Becker, P. & Dunstetter, F. (1984). Acta Cryst. A40, 241-251.]; Olekhnovich & Olekhnovich, 1980[Olekhnovich, N. M. & Olekhnovich, A. I. (1980). Acta Cryst. A36, 22-27.]; Saldin, 1982[Saldin, D. K. (1982). Acta Cryst. A38, 425-432.]; Chukhovskii et al., 1998[Chukhovskii, F. N., Hupe, A., Rossmanith, E. & Schmidt, H. (1998). Acta Cryst. A54, 191-198.]; Thorkildsen & Larsen, 1999a[Thorkildsen, G. & Larsen, H. B. (1999a). Acta Cryst. A55, 1-13.]) is a time-consuming procedure that hinders the use of dynamical diffraction to solve inverse problems for such structures. It is timely to explore possibilities offered by simple algebraic Darwin recurrence relations in application to lateral crystalline structures. This will extend the original one-dimensional Darwin approach to the two-dimensional case in both the Fourier space (reciprocal-space maps, RSMs) and real space (lateral crystalline structures).

The purpose of this paper is to extend Darwin's approach to X-ray dynamical diffraction by lateral crystalline structures. In particular, we demonstrate how our new approach can be used to simulate RSMs for lateral plane-parallel crystalline structures of different sizes and thicknesses.

2. Darwin's diffraction on a plane-parallel crystal

Before proceeding to obtain a new approach to X-ray diffraction by lateral plane-parallel crystalline structures, we provide a short review of Darwin's approach in the case of plane-parallel crystals. Darwin's approach considers a crystal as a combination of atomic planes with a distance d between those planes. Unlike Laue's theory (see, e.g., Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]), the Darwin model of crystals assumes that all electron density is placed on those atomic planes. The amplitude reflection coefficients, q and [\bar q], and transmission coefficient, 1 - iq0, of an atomic plane can be calculated using Fresnel diffraction (Borie, 1967[Borie, B. (1967). Acta Cryst. 23, 210-216.]; Yashiro & Takahashi, 2000[Yashiro, W. & Takahashi, T. (2000). Acta Cryst. A56, 163-167.]). On the other hand, these transmission and reflection coefficients are expressible (Chen & Bhattacharya, 1993[Chen, Y. C. & Bhattacharya, P. K. (1993). J. Appl. Phys. 73, 7389-7394.]) in terms of the Fourier coefficients of dielectric susceptibility (polarizability) [{\chi _g} = ] [- {r_0}{\lambda ^2}{F_g}/(\pi {V_{\rm c}})], where Fg is the structure factor ([g = 0,\,\,h,\,\bar h]), [lambda] is the wavelength, Vc is the volume of the elementary cell, r0 = e2/(mc2) is the classical electron radius, c is the speed of light in vacuum, and e and m are electron charge and electron mass, respectively.

Thus, the appropriate coefficients in the Darwin recurrence relations can be written as [{q_0} = \pi d{\chi _0}/(\lambda \,\sin {\theta _{\rm B}})], q = [ \,\pi d{\chi _h}/(\lambda \,\sin {\theta _{\rm B}})] and [\bar q = \,\pi d{\chi _{ - h}}/(\lambda \,\sin {\theta _{\rm B}})], where [{\theta _{\rm B}}] is the Bragg angle. In this paper we consider a symmetrical coplanar Bragg diffraction case for [sigma]-polarization. The equations can be extended for [pi]-polarization by incorporating [\cos ({2{\theta _{\rm B}}} )] into [{\chi _{ - h,h}}].

Let us consider a plane-parallel crystal having a finite thickness of Lz = dN, where N is the number of reflecting atomic planes. The angle between the wavevector of an incident plane wave and the crystal surface is [\theta = {\theta _{\rm B}} + \Delta \theta], where [\Delta \theta] is a small deviation from the Bragg angle.

Then the transmitted, Tn, and reflected, Sn, wave amplitudes for the nth atomic plane can be written using the following recurrence relations (Darwin, 1914b[Darwin, C. G. (1914b). Philos. Mag. 27, 675-691.]; Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]):

[\eqalignno{{T_n} &= (1 - i{q_0})\exp(i\varphi ){T_{n - 1}} - i\bar q \exp(i2\varphi )S_n,&\cr {S_n} &= (1 - i{q_0})\exp(i\varphi ){S_{n + 1}} - iqT_n.&(1)\cr}]

Here, Tn - 1, Sn + 1 are the wave amplitudes for the (n - 1)th and (n + 1)th atomic planes, respectively. The additional phase shift, [\varphi = (2\pi d/\lambda)\sin \theta], is caused by the propagation of the wavefield between the atomic planes. Note that we use a definition of the phase shift without the minus sign (cf. Darwin, 1914b[Darwin, C. G. (1914b). Philos. Mag. 27, 675-691.]; Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]) because we use the following definition for plane waves, [\exp [i({\bf k} \cdot {\bf r} - \omega t)]], instead of [\exp [ - i({\bf k} \cdot {\bf r} - \omega t )]] used by other authors. As the angular deviation [\Delta \theta] is small, the phase shift [\varphi] can be written in the following form: [\varphi = (2\pi d/\lambda)({\sin {\theta _{\rm B}} + \cos {\theta _{\rm B}}\Delta \theta } )]. Taking into account the boundary conditions T0 = 1 and SN = 0, the analytical solutions of equation (1)[link] can be presented as follows (Punegov, 1992[Punegov, V. I. (1992). Sov. Tech. Phys. Lett. 18, 120-122.]):

[\eqalignno{{S_n} &= - B{{u_2^Nu_1^n - u_1^Nu_2^n} \over {(Au_1^{} - 1)u_2^N - (Au_2^{} - 1)u_1^N}},&\cr {T_n} &= {{(Au_1^{} - 1)u_2^Nu_1^n - (Au_2^{} - 1)u_1^Nu_2^n} \over {(Au_1^{} - 1)u_2^N - (Au_2^{} - 1)u_1^N}}, &(2)\cr}]

where [{u_{1,2}} = \hat x \pm ({\hat x}^2-1)^{1/2}],

[\hat x = {{1 + {{(1 - i{q_0})}^2}\exp(i2\varphi) + q\bar q\exp(i2\varphi )} \over {2(1 - i{q_0})\exp(i\varphi )}},]

[A = (1 - i{q_0})\exp(i\varphi )] and B = - iq. From equation (1)[link] we can get the amplitude reflection, S0, and transmission coefficient, TN, of the entire crystal:

[\eqalignno{{S_0} &= - B{{u_2^N - u_1^N} \over {(Au_1^{} - 1)u_2^N - (Au_2^{} - 1)u_1^N}},&\cr {T_N} &= A{{u_1^{} - u_2^{}} \over {(Au_1^{} - 1)u_2^N - (Au_2^{} - 1)u_1^N}}. &(3)\cr}]

Taking into account that u1nu2n = 1, equation (3)[link] can be further transformed into

[\eqalignno{{S_0} &= {{\, - iq} \over {1 - (1 - i{q_0})\exp(i\varphi )\sin \left[{(N - 1)\arccos \hat x} \right]/\sin \left({N\arccos \hat x} \right)}}, &\cr {T_N} &= {{(1 - i{q_0})\exp(i\varphi )(1 - {\hat x}^2)^{1/2} } \over {\sin \left({N\arccos \hat x} \right) - (1 - i{q_0})\exp(i\varphi )\sin \left[{(N - 1)\arccos \hat x} \right]}}. &\cr &&(4)\cr}]

This result is identical to equations (12) and (13) in Perkins & Knight (1984[Perkins, R. T. & Knight, L. V. (1984). Acta Cryst. A40, 617-619.]).

Equation (4)[link] can also be rewritten in a more compact form (cf. Vardanyan et al., 1985[Vardanyan, D. M., Manoukyan, H. M. & Petrosyan, H. M. (1985). Acta Cryst. A41, 212-217.]):

[\eqalignno{{S_0} &= \left( {{q \over {\bar q}}}\right)^{1/2} {{\sin \left({N\chi } \right)} \over {\sin \left({N\chi + \psi } \right)}}\exp (- i\varphi),&\cr T_N &= {{\sin (\psi)} \over {\sin \left({N\chi + \psi } \right)}}\exp (- iN\varphi), &(5)\cr}]

where

[\hat y ={{1 - {{(1 - i{q_0})}^2}\exp(i2\varphi ) - q\bar q\exp(i2\varphi )} \over {2q\bar q\exp(i\varphi )}},]

[\chi = \arccos (\hat x)] and [\psi = \arccos (\hat y)].

3. Darwin's diffraction on a lateral plane-parallel crystalline structure

Now we use Darwin's methodology of recurrence relations to describe X-ray diffraction on a lateral crystalline structure having width of Lx and thickness of Lz (see Fig. 1[link]). The origin is on a line that is the intersection of two planes, namely the left vertical face and top surface of the structure. The x axis and z axis form a diffraction plane so that the x axis is the intersection of the diffraction plane and the top surface of the structure, and the z axis is the intersection of the diffraction plane and the left vertical face of the structure. Such a shape of the structure is similar, for instance, to the shape of a single quantum wire. We restrict ourselves to the case of a symmetric coplanar Bragg diffraction. The angle between the wavevector of an incident plane wave and the x axis is [\theta] (see Fig. 1[link]).

[Figure 1]
Figure 1
The scheme of Darwin's X-ray diffraction on a lateral crystalline structure with a rectangular cross section.

Let us consider an X-ray beam that goes through the origin. Then this beam travels [d/\sin \theta] before being reflected by the next atomic plane. The projection of this distance (i.e., [d/\sin \theta]) on the lateral direction is [\Delta x = d\,\cot \theta]. We can use [\Delta x] as a step size along the x axis to indicate the positions [{x_m} = m\,\Delta x] (m is an integer) where this beam will be partially transmitted to the next atomic plane or partially reflected.

Let Tnm and Snm be the amplitudes of the transmitted and reflected waves, respectively, upstream of the (m; n) node of a two-dimensional rectangular lattice. Here, both m and n are integers, where m corresponds to the node's number in the horizontal (lateral) direction, and n in the vertical direction (see Fig. 2[link]). The total number of nodes, Mx + 1 and Nz + 1, along the x and z axes, respectively, is determined by the structure width, [{L_x} = {M_x}\Delta x], and thickness, Lz = Nzd.

[Figure 2]
Figure 2
The transmitted, Tij, and reflected, Sij, waves in the Darwin approach.

Taking into account dynamical interactions of waves inside the structure, we obtain the following recurrence relations between the reflected, S, and transmitted, T, beams:

[\eqalignno{T_{n + 1}^m &= a\,\,T_n^{m - 1} + {b_1}S_n^{m - 1},&\cr S_n^m &= a\,S_{n + 1}^{m - 1} + {b_2}T_{n + 1}^{m - 1}, &(6)\cr}]

where [a = (1 - i{q_0})\exp (i{\varphi _l})], [{b_1} = - i\bar q\exp (i{\varphi _l})] and b2 = [- iq\exp (i{\varphi _l})]. It should be noted that in equation (6)[link] we use the expressions for the constants q0, q and [\bar q] obtained for an infinitely large crystal, which is, obviously, an approximation. Potentially, this method can be extended by way of replacing these three constants by three functions depending on the x and z coordinates. This will allow one to describe local non-homogeneity in the lateral crystalline structures. The phase shift [{\varphi _l} = ({2\pi d} )/({\lambda \sin {\theta _{\rm B}}} )] is an additional phase shift occurring when the wave propagates from one node to another.

Plane-parallel crystals are infinitely large in the lateral direction. Therefore their diffracted intensity distribution shape is a delta-like function depending on the vertical coordinate in Fourier space. Unlike the plane-parallel crystals, the lateral plane-parallel crystalline structures produce a diffracted intensity distribution that has a more complex shape. That is, their diffracted intensity distribution (i.e., RSM) depends on both the vertical and horizontal coordinates in Fourier space.

Let us now assume that the angle between the wavevector k of the incident plane wave and the x axis is [{\theta _1} = \,{\theta _{\rm B}} + \Delta {\theta _1}] (see Fig. 3[link]). The reflected wave is registered in the direction of the wavevector [{\bf k'}]. The angle between [{\bf k'}] and the x axis is [{\theta _2} = {\theta _{\rm B}} + \Delta {\theta _2}]. Both the wavevectors, k and [{\bf k'}], lie in the diffraction plane and [| {\bf k} | = | {\bf k'} | = k = 2\pi /\lambda]. We consider the case when the angular deviations [\Delta {\theta _1}] and [\Delta {\theta _2}] are small. The deviation of the scattering vector [{\bf Q} = {\bf k'} - {\bf k}] from the reciprocal-lattice vector h is defined by the vector q (see Fig. 3[link]). The appropriate projections of the vector q are

[\eqalignno{{q_x} &= k\sin {\theta _{\rm B}}(\Delta {\theta _1} - \Delta {\theta _2}),&\cr {q_z} &= - k\cos {\theta _{\rm B}}(\Delta {\theta _1} + \Delta {\theta _2}). &(7)\cr}]

We can also rewrite these relations as [\Delta {\theta _{1,2}} =] [\pm{(2k\cos {\theta _{\rm B}})^{ - 1}}({q_x}\cot {\theta _{\rm B}} \mp {q_z}).]

[Figure 3]
Figure 3
The wavevectors k, [{\bf k'}] of the incident and reflected waves, respectively. h is the vector of the reciprocal lattice. The deviation of the scattering vector, [{\bf Q} = {\bf k'} - {\bf k}], from the reciprocal-lattice vector h is defined by the vector q. The angular deviations of k and [{\bf k'}] from the Bragg angle position are described by [\Delta {\theta _1}] and [\Delta {\theta _2}], respectively.

Traditionally, the angular deviations of the sample and analyser crystal in the so-called triple-crystal scheme (Iida & Kohra, 1979[Iida, A. & Kohra, K. (1979). Phys. Status Solidi, 51, 533-542.]) are defined by [\omega] and [\varepsilon], respectively. Then using [\Delta {\theta _1} = \omega] and [\Delta {\theta _2} = \varepsilon - \omega] we can rewrite equation (7)[link] in the following form, which explicitly connects the position in reciprocal space and the experimentally measured angular parameters [\omega] and [\varepsilon]:

[{q_x} = k\sin {\theta _{\rm B}}(2\omega - \varepsilon), ]

[{q_z} = - k\cos {\theta _{\rm B}}\varepsilon. ]

The entire phase shift consists of two components: the phase shift of the transmitted wave and the phase shift of the reflected wave. These additional phase shifts are defined via optical path differences. We choose the origin (see Figs. 1[link] and 4[link]) as a reference point in our calculation of the additional phase shits. For X-rays incident on the left vertical face of the structure (x = 0) the optical path difference increases along the z direction as [z\sin {\theta _1}] (see Fig. 4[link]). The phase shift at the node positions zn = nd along the z direction is [\varphi _{z,{\rm in}}^n =] [ (2\pi /\lambda)nd\sin {\theta _1}]. Thus the boundary conditions at the left face of the structure (x = 0) are [T_n^0 = \exp (i\varphi _{z,{\rm in}}^n)] and Sn0 = 0, where [n = 0,1,2,\ldots,{N_z} - 1].

[Figure 4]
Figure 4
Optical path differences [{\Delta _1}=x\,\cos {\theta _1} + z\sin {\theta _1}] and [{\Delta _2}= - x\,\cos {\theta _2}] [ +\, z\sin {\theta _2}] for an arbitrary point (x,z) in a lateral crystalline structure. Note that the angular deviations are not to scale.

For X-rays incident on the top surface of the structure (z = 0) the optical path difference increases along the x direction as [m\Delta x\cos {\theta _1}]. Therefore, the phase shift of the incident wave in both the lateral and vertical directions depends on [{\theta _1}]. The appropriate boundary condition at the top surface of the structure can be written as [T_0^m = \exp (i\varphi _{x,{\rm in}}^m)], where [\varphi _{x,{\rm in}}^m =] [ (2\pi /\lambda)m\Delta x\cos {\theta _1}] and [m = 1,2,\ldots, {M_x}].

Considering that the exiting X-ray wave emerges from the top surface and the right vertical face of the structure (see Fig. 1[link]), the boundary conditions for the reflected wave S at the bottom surface and the left vertical face of the structure are SNzm = 0 and Sn0 = 0, respectively.

In accordance with the model (see Fig. 2[link]), we use a rectangular  Mx ×Nz lattice having a fixed distance between nodes to describe the dynamical diffraction process.

The simulation procedure based on equation (6)[link] consists of the external and internal cycles. The external cycle (from left to right) starts with the first column (m = 1) and goes up to the last column (m = Mx). Note that the left vertical face of the structure corresponds to the 0th column. The internal cycle (from top to bottom) starts with the first reflecting plane (n = 1) and goes up to the last one (n = Nz). Note that the top surface of the structure corresponds to the 0th plane. For instance, for the first column we use the external boundary conditions to calculate amplitudes of the reflected and transmitted waves at each node of the column [\{ S_n^1\}\! =\! (S_0^1,S_1^1,S_2^1,\ldots, S_{{N_z} - 1}^1)] and [\{ T_n^1\} \!= \!(T_1^1,T_2^1,T_3^1,\ldots, T_{{N_z}}^1)], respectively. Then substituting the amplitudes calculated for the first column in equation (6)[link] we can calculate the amplitudes for the second column, namely { Sn2} and { Tn2}, and so on. At the end of the external cycle we obtain arrays of the reflected and transmitted amplitudes, namely { Snm} and { Tnm}, at each node of the two-dimensional lattice. Note that according to the boundary conditions we apply for the amplitudes the following phase factors: [\exp ({i\varphi _{x,{\rm in}}^m} )], where [\varphi _{x,{\rm in}}^m = (2\pi /\lambda)m\Delta x\cos {\theta _1}], when we do calculations for the mth column, and [\exp({i\varphi _{z,{\rm in}}^n})], where [\varphi _{z,{\rm in}}^n = (2\pi /\lambda)nd\sin {\theta _1}], when we do calculations for the nth plane.

Hence, using equation (6)[link] and the boundary conditions for the entering wavefield we are able to calculate the reflection amplitudes, [\{ S_n^m({\theta _1})\}], at each node. To calculate the entire reflection amplitude we have to take into account the additional phase shifts, defined by the direction of the vector [{\bf k'}] or the angle [{\theta _2}].

The amplitude reflection coefficient, [S({\theta _1},{\theta _2})], of the lateral plane-parallel crystalline structure is a sum of the amplitudes of waves exiting the top surface of the structure, namely [\{ S_0^m({\theta _1})\}], and the right vertical face of the structure, namely [\{ S_n^{{M_x}}({\theta _1})\}], with additional phase factors:

[\eqalignno{S({\theta _1},{\theta _2}) &= \textstyle\sum\limits_{m = 0}^{{M_x}} {S_0^m({\theta _1})} \exp (i\varphi _{x,{\rm ex}}^m) &\cr&\quad+ \sum\limits_{n = 1}^{{N_z}} S_n^{M_x}({\theta _1}) \exp (i\varphi _{z,{\rm ex}}^n)\exp (i\varphi _{x,{\rm ex}}^{M_x}), &(8)\cr}]

where the additional phase factors [\varphi _{x,{\rm ex}}^m = - (2\pi /\lambda)m\Delta x\cos {\theta _2}] and [\varphi _{z,{\rm ex}}^n = (2\pi /\lambda)nd\sin {\theta _2}] depend on [{\theta _2} = {\theta _{\rm B}} + \Delta {\theta _2}]. The phase factor [\exp (i\varphi _{x,{\rm ex}}^{{M_x}})] takes into account the x coordinate of the right vertical face of the structure, namely [{L_x} = {M_x}\Delta x].

Thus, the amplitude [S({\theta _1},{\theta _2})] depends on both [{\theta _1}] and [{\theta _2}]. Therefore equation (8)[link] allows one to simulate RSMs. Also one can use equation (8)[link] to calculate the appropriate directional scans in reciprocal space. For instance, a qz-scan (CTR) simulation can be done if [{\theta _1} = {\theta _2}] (or [2\omega = \varepsilon]).

4. Numerical modelling

The numerical modelling of RSMs and directional scans in reciprocal space is performed using equations (6)[link] and (8)[link]. In our simulations we use Cu K[alpha]1 radiation (the wavelength is 0.154056 nm) for the (111) reflection of a Ge lateral crystalline structure. The thickness of this structure is Lz = 10000 d111 = 3.27 µm. We use several widths of the structure: Lx = 1.35, 5.39, 13.5, 53.9 µm, which correspond to the following number of nodes Mx = 1000, 4000, 10000, 40000, respectively. In an effort to carry out a proper normalization all simulated reflected wave intensities were normalized on the maximum intensity of the Darwin curve for a plane-parallel crystal of the same thickness.

Rocking curves shown in Fig. 5[link] were simulated using equation (8)[link] for the case of [\omega]-[2\theta] scans (i.e. [{\theta _1} = {\theta _2}]). Then the optical path difference [\Delta = {\Delta _1} + {\Delta _2} = 2z\sin \theta] ([\theta = {\theta _1} = {\theta _2}]) (see Fig. 4[link]) does not depend on the x coordinate, and the entire phase shift at the top surface of the structure is [\varphi _x^m = \varphi _{x,{\rm in}}^m + \varphi _{x,{\rm ex}}^m = 0].

[Figure 5]
Figure 5
[\theta]-[2\theta] scans (i.e. [\Delta {\theta _1} = \Delta {\theta _2}] or qx = 0) of X-ray diffraction on plane-parallel lateral crystalline structures (thick blue line) with thickness Lz = 3.27 µm for different widths: (a) Lx = 1.35 µm, (b) Lx = 5.39 µm, (c) Lx = 13.5 µm and (d) Lx = 53.9 µm. Compare with [\theta]-[2\theta] scans of X-ray diffraction on a plane-parallel crystal (thin red line) with thickness Lz = 3.27 µm and Lx = [infinity].

The extinction length (Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]) for the Ge(111) reflection for a semi-infinite crystal is 0.67 µm, the full width at half-maximum (FWHM) of the Darwin curve is 15.4 arcsec (Stepanov & Forrest, 2008[Stepanov, S. & Forrest, R. (2008). J. Appl. Cryst. 41, 958-962.]). In the case of small lateral width (Lx = 1.35 µm) the simulated rocking curve corresponds to the kinematical limit (see Fig. 5[link]a). As the width of the structure increases we observe a gradual transfer into dynamical diffraction (see Figs. 5[link]b, 5[link]c), where thickness oscillations are still observable even within the angular region of Darwin's `table'. For a large width of the structure (Lx = 53.9 µm) the simulated rocking curves for a lateral crystalline structure and a plane-parallel crystal are in close agreement (see Fig. 5[link]d).

Fig. 6[link] shows the RSM simulations for lateral crystalline structures of different width. If the lateral width is small, the shape of the RSM (see Fig. 6[link]a) is consistent with a typical kinematical diffraction case (Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]):

[I({q_x},{q_z}) = {\left| {S({q_x},{q_z})} \right|^2} \propto {\left| {{\rm{sinc}}({q_x}{L_x}/2)} \right|^2}{\left| {{\rm{sinc}}({q_z}{L_z}/2)} \right|^2}, ]

where sinc(x) = sin(x)/x. Both the width of the central peak and the period of lateral oscillations in the qx direction decrease as the lateral width of the structure increases. The period of lateral oscillations is inversely proportional to the lateral size of the structure as evident from Fig. 7[link], which shows qx-scans across the central peak of the RSM.

[Figure 6]
Figure 6
RSMs of X-ray diffraction on plane-parallel lateral crystalline structures with thickness Lz = 3.27 µm and different widths: (a) Lx = 1.35 µm, (b) Lx = 5.39 µm, (c) Lx = 13.5 µm and (d) Lx = 53.9 µm.
[Figure 7]
Figure 7
qx scans (across the central peak of the RSM) of X-ray diffraction on plane-parallel lateral crystalline structures with thickness Lz = 3.27 µm and different widths: (a) Lx = 1.35 µm, (b) Lx = 5.39 µm, (c) Lx = 13.5 µm and (d) Lx = 53.9 µm.

The obtained simulation results are in good agreement with the solution obtained by integration of Takagi's equations (Kolosov & Punegov, 2005[Kolosov, S. I. & Punegov, V. I. (2005). Crystallogr. Rep. 50, 357-362.]).

5. Conclusion

We demonstrated that Darwin's approach using algebraic recurrence relations can be extended to the case of lateral plane-parallel crystalline structures. This approach, being simple and transparent, is faster than the one based on Takagi's equations and allows simulations of RSMs. It is especially important for the solution of inverse problems using minimization of the discrepancy between experimental and simulated data (Pavlov et al., 1995[Pavlov, K. M., Punegov, V. I. & Faleev, N. N. (1995). J. Exp. Theor. Phys. 80, 1090-1097.]; Kirste et al., 2005[Kirste, L., Pavlov, K. M., Mudie, S. T., Punegov, V. I. & Herres, N. (2005). J. Appl. Cryst. 38, 183-192.]). Therefore this new approach can be widely used for non-destructive testing of lateral structures used in opto- and microelectronics devices, nonvolatile memory devices and X-ray optics. This approach can potentially be extended further to the three-dimensional case in both the Fourier space and real space.

Acknowledgements

This study was supported in part by the Russian Foundation for Basic Research (project No. 13-02-00272-a), the Presidium of the Russian Academy of Sciences (project No. 12-P-1-1014) and the Ural Branch of the Russian Academy of Sciences (basic research program project No. 12-U-1-1010). KMP acknowledges financial support from the University of New England.

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Acta Cryst (2014). A70, 64-71   [ doi:10.1107/S2053273313030416 ]