Received 11 October 2013 | ## Darwin's approach to X-ray diffraction on lateral crystalline structures
Darwin's dynamical theory of X-ray diffraction is extended to the case of lateral ( |

There are several approaches to X-ray dynamical diffraction by crystals (see, *e.g.*, Authier, 2001; Pietsch *et al.*, 2004 and references therein). Among them Darwin's diffraction theory (Darwin, 1914*a*,*b*), 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); therefore it was not widely used. Borie (1966) first used Darwin's recurrence relations to describe the Borrmann effect. Using Borie's approach, Bezirganian & Navasardian (1969) 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; Kuznetsov & Fofanov, 1970), (ii) multiwave diffraction (Kuznetsov & Fofanov, 1972; Ignatovich, 1992) and (iii) properties of the dispersion surface (Borie, 1967). The recurrence relations for the amplitudes of transmitted and reflected waves were also employed to investigate neutron diffraction by polyatomic crystals (Ignatovich, 1990) and scattering of visible light in liquid crystals (Chandrasekhar & Rao, 1968). Darwin's recurrence relations were also used in multilayer materials optics (Dub & Litzman, 1999).

The crystal truncation rod (CTR) method (Robinson, 1986) 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), Nakatani & Takahashi (1994), Takahashi & Nakatani (1995), Durbin & Follis (1995) and Takahashi *et al.* (2000).

Yashiro & Takahashi (2000) analysed the reflection and transmission coefficients of a single atomic plane for an arbitrary two-dimensional Bravais lattice. Their results (Yashiro & Takahashi, 2000) are similar to those obtained by Borie (1967); however, they differ from ones obtained by Durbin (1995). Later, on the basis of Yashiro & Takahashi (2000) a variant of Darwin's theory for grazing-incidence geometry was developed (Yashiro *et al.*, 2001).

Prins (1930) 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) with the use of the Chebyshev polynomials (Abramowitz & Stegun, 1972). Chen & Bhattacharya (1993) showed that the Darwin recurrence relations are identical to Takagi's differential equations (Takagi, 1962) 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; Punegov, 1990, 1991, 1993; Pavlov & Punegov, 1998, 2000) using Takagi's equations (Takagi, 1962) is one possible way to describe X-ray dynamical diffraction in such structures. However, Darwin's theory (Darwin, 1914*a*,*b*), which was originally developed for an ideal crystal, can also be modified for use with non-ideal crystals. For instance, Chung & Durbin (1999) considered the thermal vibration effect. Li *et al.* (1997) 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). Their numerical results were confirmed later by analytical solutions (Kolpakov & Punegov, 1985; Punegov & Vishnjakov, 1995). Prudnikov obtained analytical solutions for the cases of crystals distorted by surface acoustic waves (Prudnikov, 1998) and non-ideal heterostructures (Prudnikov, 2000).

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; Bartels *et al.*, 1986). This approach was further extended to multiwave diffraction (Ladanov & Punegov, 1989; Punegov, 1993), highly asymmetrical diffraction geometry (Punegov & Ladanov, 1989*b*, 1990) and glancing geometry (Punegov & Ladanov, 1989*a*).

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; Minkevich *et al.*, 2011; Lee *et al.*, 2006 and references therein). Different dynamical diffraction approaches (Olekhnovich & Olekhnovich, 1978; Thorkildsen & Larsen, 1999*b*; Kolosov & Punegov, 2005) 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 & Kolosov, 2007) or an arbitrary cross-sectional shape (Punegov *et al.*, 2007).

However, using Takagi's equations to simulate dynamical X-ray diffraction on lateral crystalline structures (Becker, 1977; Becker & Dunstetter, 1984; Olekhnovich & Olekhnovich, 1980; Saldin, 1982; Chukhovskii *et al.*, 1998; Thorkildsen & Larsen, 1999*a*) 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.

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), the Darwin model of crystals assumes that all electron density is placed on those atomic planes. The amplitude reflection coefficients, *q* and , and transmission coefficient, 1 - *i**q*_{0}, of an atomic plane can be calculated using Fresnel diffraction (Borie, 1967; Yashiro & Takahashi, 2000). On the other hand, these transmission and reflection coefficients are expressible (Chen & Bhattacharya, 1993) in terms of the Fourier coefficients of dielectric susceptibility (polarizability) , where *F*_{g} is the structure factor (), is the wavelength, *V*_{c} is the volume of the elementary cell, *r*_{0} = *e*^{2}/(*m**c*^{2}) 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* = and , where is the Bragg angle. In this paper we consider a symmetrical coplanar Bragg diffraction case for -polarization. The equations can be extended for -polarization by incorporating into .

Let us consider a plane-parallel crystal having a finite thickness of *L*_{z} = *d**N*, where *N* is the number of reflecting atomic planes. The angle between the wavevector of an incident plane wave and the crystal surface is , where is a small deviation from the Bragg angle.

Then the transmitted, *T _{n}*, and reflected,

Here, *T*_{n - 1}, *S*_{n + 1} are the wave amplitudes for the (*n* - 1)th and (*n* + 1)th atomic planes, respectively. The additional phase shift, , 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, 1914*b*; Authier, 2001) because we use the following definition for plane waves, , instead of used by other authors. As the angular deviation is small, the phase shift can be written in the following form: . Taking into account the boundary conditions *T*_{0} = 1 and *S*_{N} = 0, the analytical solutions of equation (1) can be presented as follows (Punegov, 1992):

where ,

and *B* = - *i**q*. From equation (1) we can get the amplitude reflection, *S*_{0}, and transmission coefficient, *T*_{N}, of the entire crystal:

Taking into account that *u*_{1}^{n}*u*_{2}^{n} = 1, equation (3) can be further transformed into

This result is identical to equations (12) and (13) in Perkins & Knight (1984).

Equation (4) can also be rewritten in a more compact form (*cf*. Vardanyan *et al.*, 1985):

where

and .

Now we use Darwin's methodology of recurrence relations to describe X-ray diffraction on a lateral crystalline structure having width of *L _{x}* and thickness of

| 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 before being reflected by the next atomic plane. The projection of this distance (*i.e.*, ) on the lateral direction is . We can use as a step size along the *x* axis to indicate the positions (*m* is an integer) where this beam will be partially transmitted to the next atomic plane or partially reflected.

Let *T*_{n}^{m} and *S*_{n}^{m} 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). The total number of nodes, *M*_{x} + 1 and *N*_{z} + 1, along the *x* and *z* axes, respectively, is determined by the structure width, , and thickness, *L*_{z} = *N*_{z}*d*.

| Figure 2 The transmitted, T_{i}^{j}, and reflected, S_{i}^{j}, 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:

where , and *b*_{2} = . It should be noted that in equation (6) we use the expressions for the constants *q*_{0}, *q* and 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 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 (see Fig. 3). The reflected wave is registered in the direction of the wavevector . The angle between and the *x* axis is . Both the wavevectors, **k** and , lie in the diffraction plane and . We consider the case when the angular deviations and are small. The deviation of the scattering vector from the reciprocal-lattice vector **h** is defined by the vector **q** (see Fig. 3). The appropriate projections of the vector **q** are

We can also rewrite these relations as

| Figure 3 The wavevectors k, of the incident and reflected waves, respectively. h is the vector of the reciprocal lattice. The deviation of the scattering vector, , from the reciprocal-lattice vector h is defined by the vector q. The angular deviations of k and from the Bragg angle position are described by and , respectively. |

Traditionally, the angular deviations of the sample and analyser crystal in the so-called triple-crystal scheme (Iida & Kohra, 1979) are defined by and , respectively. Then using and we can rewrite equation (7) in the following form, which explicitly connects the position in reciprocal space and the experimentally measured angular parameters and :

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 and 4) 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 (see Fig. 4). The phase shift at the node positions *z*_{n}^{} = *n**d* along the *z* direction is . Thus the boundary conditions at the left face of the structure (*x* = 0) are and *S*_{n}^{0} = 0, where .

| Figure 4 Optical path differences and 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 . Therefore, the phase shift of the incident wave in both the lateral and vertical directions depends on . The appropriate boundary condition at the top surface of the structure can be written as , where and .

Considering that the exiting X-ray wave emerges from the top surface and the right vertical face of the structure (see Fig. 1), the boundary conditions for the reflected wave *S* at the bottom surface and the left vertical face of the structure are *S*_{Nz}^{m} = 0 and *S*_{n}^{0} = 0, respectively.

In accordance with the model (see Fig. 2), we use a rectangular *M*_{x} ×*N*_{z} lattice having a fixed distance between nodes to describe the dynamical diffraction process.

The simulation procedure based on equation (6) 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* = *M*_{x}). 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* = *N*_{z}). 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 and , respectively. Then substituting the amplitudes calculated for the first column in equation (6) we can calculate the amplitudes for the second column, namely { *S*_{n}^{2}} and { *T*_{n}^{2}}, and so on. At the end of the external cycle we obtain arrays of the reflected and transmitted amplitudes, namely { *S*_{n}^{m}} and { *T*_{n}^{m}}, at each node of the two-dimensional lattice. Note that according to the boundary conditions we apply for the amplitudes the following phase factors: , where , when we do calculations for the *m*th column, and , where , when we do calculations for the *n*th plane.

Hence, using equation (6) and the boundary conditions for the entering wavefield we are able to calculate the reflection amplitudes, , 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 or the angle .

The amplitude reflection coefficient, , of the lateral plane-parallel crystalline structure is a sum of the amplitudes of waves exiting the top surface of the structure, namely , and the right vertical face of the structure, namely , with additional phase factors:

where the additional phase factors and depend on . The phase factor takes into account the *x* coordinate of the right vertical face of the structure, namely .

Thus, the amplitude depends on both and . Therefore equation (8) allows one to simulate RSMs. Also one can use equation (8) to calculate the appropriate directional scans in reciprocal space. For instance, a *q*_{z}-scan (CTR) simulation can be done if (or ).

The numerical modelling of RSMs and directional scans in reciprocal space is performed using equations (6) and (8). In our simulations we use Cu *K*_{1} radiation (the wavelength is 0.154056 nm) for the (111) reflection of a Ge lateral crystalline structure. The thickness of this structure is *L*_{z} = 10000 *d*_{111} = 3.27 µm. We use several widths of the structure: *L*_{x} = 1.35, 5.39, 13.5, 53.9 µm, which correspond to the following number of nodes *M _{x}* = 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 were simulated using equation (8) for the case of - scans (*i.e.* ). Then the optical path difference () (see Fig. 4) does not depend on the *x* coordinate, and the entire phase shift at the top surface of the structure is .

| Figure 5 - scans ( i.e. or q_{x} = 0) of X-ray diffraction on plane-parallel lateral crystalline structures (thick blue line) with thickness L = 3.27 µm for different widths: (_{z}a) L = 1.35 µm, (_{x}b) L = 5.39 µm, (_{x}c) L = 13.5 µm and (_{x}d) L = 53.9 µm. Compare with - scans of X-ray diffraction on a plane-parallel crystal (thin red line) with thickness _{x}L = 3.27 µm and _{z}L = ._{x} |

The extinction length (Authier, 2001) 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). In the case of small lateral width (*L _{x} =* 1.35 µm) the simulated rocking curve corresponds to the kinematical limit (see Fig. 5

Fig. 6 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*a*) is consistent with a typical kinematical diffraction case (Authier, 2001):

where sinc(*x*) = sin(*x*)/*x*. Both the width of the central peak and the period of lateral oscillations in the *q _{x}* 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, which shows

| Figure 6 RSMs of X-ray diffraction on plane-parallel lateral crystalline structures with thickness L = 3.27 µm and different widths: (_{z}a) L = 1.35 µm, (_{x}b) L = 5.39 µm, (_{x}c) L = 13.5 µm and (_{x}d) L = 53.9 µm._{x} |

| Figure 7 q scans (across the central peak of the RSM) of X-ray diffraction on plane-parallel lateral crystalline structures with thickness _{x}L = 3.27 µm and different widths: (_{z}a) L = 1.35 µm, (_{x}b) L = 5.39 µm, (_{x}c) L = 13.5 µm and (_{x}d) L = 53.9 µm._{x} |

The obtained simulation results are in good agreement with the solution obtained by integration of Takagi's equations (Kolosov & Punegov, 2005).

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; Kirste *et al.*, 2005). 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.

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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