Takagi-Taupin Equations Research Articles

Primary Extinction and Absorption: a Theoretical Approach Based on the Takagi-Taupin Equations. Application to Spherical Crystals

The primary-extinction problem for X-ray diffraction by perfect crystals is treated using the Becker–Coppens iterative procedure within the Takagi–Taupin equations. An analytical approximation for the primary-extinction factor yp describing both the effects of the X-ray multiple scattering and the absorption processes within the perfect crystal of an arbitrary shape is derived. The solution differs from the known expressions given by Zachariasen and Becker & Coppens on the basis of the Hamilton–Darwin intensity transfer equations and in the limiting case of a non-absorbing crystal it concurs with the Kato–Becker formula found in the Laue approximation of the dynamical theory. The theoretical results are consistent with experimental data of a number of reflections of Ge and Si single-crystal spheres measured at X-ray wavelengths λ = 0.56, 0.71 and 1.54 Å with a laboratory CAD-4 and a Huber four-circle diffractometer at HASYLAB, DESY, Hamburg, Germany. Two novel features are discussed. First, it is shown that by neglecting the X-ray absorption effect the calculated extinction factor yp is close to the value given by the Becker–Coppens formula. Second, it was found that for absorbing spherical crystals with μR ≥ 1 absorption effects cannot be treated separately from the primary-extinction phenomenon because of imaginary dispersion corrections to the atomic form factors. The experimental data are fitted to the Becker–Coppens and present theoretical models. The best fits are found to relate to the present model and produce relatively low R factors of 3 to 6% for the Bragg intensities measured in the cases of Si and Ge spherical crystals.

Three-Beam Diffraction in Finite Perfect Crystals. II. Influence of Absorption, Resonant Scattering and Polarization

By using the boundary-value Green-function technique, effects of absorption, resonant scattering and polarization are incorporated in the solution of the Takagi–Taupin equations for three-beam diffraction in finite perfect crystals. It is shown how such features may influence the dynamical interactions. The potential of three-beam interference acting as a phase-sensitive probe for resonant scattering is demonstrated. Depending on the applied model used for the calculations of the corrections to the atomic scattering factor, profile asymmetry reversal is theoretically predicted to occur near the photoelectric threshold in a finite germanium crystal. Significant distortions of the three-beam profiles are found for polarized incident radiation, especially for reflection triplets having an invariant phase sum near 90°.

Three-Beam Diffraction in Finite Perfect Crystals. I. Exploring Laue–Laue and Bragg–Laue Scattering Using a Series-Expansion Approach for the Solution of the Takagi–Taupin Equations

Using the boundary-value Green-function technique, analytical expressions in the form of finite series expansions are obtained for the relative change in the integrated power of the primary reflection due to the gradual excitation of a secondary reciprocal-lattice point on the Ewald sphere. Solutions are found for both a Laue–Laue and a Bragg–Laue case in finite shaped crystals confined by the scattering vectors. When the crystal sizes do not exceed the Pendellösung length of the involved reflections, the \psi profiles exhibit the same qualitative features in the two cases. The solutions do however indicate a strong dependence on the outer crystal dimensions – which add a geometrical aspect to the interpretation of the Aufhellung and Umweganregung concepts.

Bragg-Case Synchrotron Section Topography of Silicon Implanted with High-Energy Protons and α Particles

Back-reflection section topography using white-beam synchrotron radiation has been applied for the investigation of silicon implanted with 1 and 1.6 MeV protons and 4.8 MeV α particles. The beam width was limited to 5 μm, and a series of spots in the vicinity of a centrally adjusted reflection were indexed and analysed. The back-reflection section pattern of implanted crystals usually exhibits fringes corresponding to the reflection from the surface and a series of fringes corresponding to the rear region of the shot-through layer, the destroyed layer and the bulk. The patterns were used for direct evaluation of ion ranges and thicknesses of the shot-through layer. The overall characteristics of the obtained patterns were successfully reproduced in simulations based on numerical integration of the Takagi–Taupin equations. The agreement between the simulation and experiment proves that the lattice-parameter depth-distribution profiles can be assumed to be proportional to interstitial-vacancy distributions obtained using the Monte Carlo method from the Biersack–Ziegler theory. The simulation also reproduced interference tails observed in some section patterns. It was found that these tails are caused by the ion-dose change along the beam and they were probably formed due to the interference between the radiation reflected from the bulk and those rays reflected by the rear region of the shot-through layer.

Interference Fringes in Synchrotron Section Topography of Implanted Silicon with a Very Large Ion Range

Silicon crystals implanted with 9 MeV protons to the dose of 5x 10 17 cm-2 were studied with X-ray topographic methods using both conventional and synchrotron radiation sources. After the implantation the crystals were thermally and electron annealed. The implantation produced large 600 pm thick shot-through layer while the total thickness of the samples was 1.6 mm. It was confirmed by means of double crystal topography that the whole crystal was elastically bent. The transmission section patterns revealed both parts of the implanted crystal separated by strong contrasts coming from the most damaged layer and distinct interference fringes which appeared on one side of the topograph only. The location of the fringes changed when the beam entered the other side of the sample. The mechanism of fringe formation was studied with numerical integration of the Takagi—Taupin equations, especially studying the intensity distribution in the diffraction plane. The simulations reproduced the location of the fringes in different geometries and indicate that they can be caused both by variable crystal curvature and variable ion dose. PACS numbers: 61.10.—i, 61.80.—x

Synchrotron White Beam Topographic Studies of Gallium Arsenide Crystals

A series of samples cut out from different types of gallium arsenide crystals with low dislocation density were studied by means of white beam synchrotron topography. The investigation was performed with transmission and back-reflection projection methods and transmission section method. Some of the topography in transmission geometry provided a very high sensitivity suitable for revealing small precipitates. The transmission section images significantly differed depending on the wavelength and absorption. In some cases a distinct Pendellosung fringes and fine details of dislocation and precipitates images were observed. It was possible to reproduce the character of these images by means of numerical simulation based on integration of Takagi-Taupin equations. Due to more convenient choice of radiation, synchrotron back-reflection projection topography provided much better visibility of dislocations than analogous methods realized with conventional X-ray sources.

Standing Waves Investigation of InAs/InP Superlattices

X-ray standing waves measurements have been performed on a set of six lnAsP/InP superlattices exhibiting a wide range of lattice strain values and on both the (002) and (004) reflections. The objective was to assess the XSW technique as a structural probe in superlattices. For samples with perpendicular lattice misfit less than 1.5% we fmd clear XSW modulations in the As fluorescence; this signal is well reproduced by simulations performed by solving the Takagi-Taupin equations. For samples with higher misfit a negligible XSW signal is found, indicating considerable lattice disorder.

Interference effects in Bragg-case synchrotron section topography of elastically bent silicon implanted crystals

White-beam synchrotron section topography was applied to a silicon crystal implanted with 4.8 MeV α-particles elastically bent with the radius of curvature close to 100 m. A number of section patterns corresponding to different reflections was analysed. It was found that the section pattern in the bent sample was drastically different from those observed in a flat sample. The difference consists in the occurrence of the sets of additional interference fringes covering a long distance up to several millimetres behind the main diffraction maximum. It was possible to reproduce some characteristic features of the fringes both in the implanted and non-implanted region by numerical integration of the Takagi-Taupin equation.

The images of misfit dislocations in Bragg-case synchrotron section topography

Silicon epitaxial layers deposited on substrates doped with boron and containing certain amount of misfit dislocations were studied by means of back-reflection synchrotron section topography. Samples with different curvature determined by substrate thickness and status of their back sides were examined. When the curvature of the sample was negligible the section pattern consisted of two stripes corresponding, respectively, to the layer and to the substrate. Misfit dislocations were revealed in the direct contrast located mostly in the vicinity of the substrate stripe. In the samples with radius of curvature smaller than 100m additional interference fringes were observed in a wide area behind the main two stripes. The sequence of these interference fringes was dependent on the curvature of the sample and differed from that of bent substrate wafers of the similar curvature. As a consequence the images of misfit dislocations became much more extended and contained many characteristic details. The character of experimental misfit dislocation images both in the case of flat and bent crystals was reproduced using numerical integration of the Takagi-Taupin equations.

X-Ray Diffraction Imaging Using Perfect Crystals

The imaging properties of perfect crystals, used for controlling and directing x-ray beams in imaging systems, are analyzed using optical transfer functions. The optical transfer functions are related to the point-spread functions for the crystal imaging system and are derived from a one-dimensional Fourier transform of the Takagi–Taupin equations. Images obtained using diffracting crystals as optical elements are simulated for the Laue and Bragg geometries using a Fourier transform method and the imaging characteristics of each of these crystal configurations are analyzed. It is demonstrated that the perfect crystals act as spatial filters of the object wave.

Simulation of Synchrotron White-Beam Topographs. An Algorithm for Parallel Processing: Application to the Study of Piezoelectric Devices

This paper presents a new algorithm for the integration of Takagi–Taupin equations taking into account the fact that X-ray diffraction is a parallel phenomenon. The diffraction equations show that the propagation of the waves is independent in each incidence plane. It is thus possible to compute in parallel the propagation of the waves in different planes. Two algorithms are presented: the first one for multiprocessor machines where the processors share a common memory, the second one for massively parallel computers. The program is written to achieve a high vectorization ratio and to make it as efficient as possible with modem superscalar and array processors. The simulation of the image of a defect has been divided into two independent parts. In the first one, one computes the derivatives of the deformation inside the crystal; in the second one, these results are used to simulate the image. This allows one rapidly to change the model for a defect, something that was not feasible in all previously written simulation programs since the computation of the deformation was part of the simulation. The study of stroboscopic images of the propagation of acoustic waves in piezoelectric devices is given as an example of the possibilities of this new program.

X-Ray Diffraction Imaging Using Perfect Crystals

The imaging properties of perfect crystals, used for controlling and directing x-ray beams in imaging systems, are analyzed using optical transfer functions. The optical transfer functions are related to the point-spread functions for the crystal imaging system and are derived from a one-dimensional Fourier transform of the Takagi-Taupin equations. Images obtained using diffracting crystals as optical elements are simulated for the Laue and Bragg geometries using a Fourier transform method and the imaging characteristics of each of these crystal configurations are analyzed. It is demonstrated that the perfect crystals act as spatial filters of the object wave.

Remarks on the geometric parameters in describing diffraction optics of asymmetric Bragg crystals

This communication gives the asymmetric factor b and deviation parameter η describing the diffraction geometry in a more general way including very asymmetric scattering arrangements. Darwin's parameter Y defined on the dispersion surface and the associated deviation parameter ηY given by the Takagi–Taupin equations are proposed to characterize the angular deviation from the exact resonance of Bragg diffraction. With respect to ηY of a high accuracy, the validity of conventional forms of the deviation parameter is further discussed.

X-ray propagation and diffraction in distorted, anisotropic, dielectric and magnetic crystals

The interaction of X-rays in distorted anisotropic crystals is derived using a semi-classical approach based on Maxwell's equations. The X-ray wave is represented by a vector potential that is coupled to the charge and the magnetic moments of the electrons in the crystal by scattering factors obtained from non-relativistic quantum theory. Distortions are introduced through a slowly varying displacement field that shifts the positions of each unit cell in the crystal from its perfect-crystal position. The result is a generalization of the Takagi–Taupin equations in the form of a matrix equation for the electric field vector of the X-rays that includes magnetic scattering and the mixing of X-ray polarization states. The solutions of the equations in one-beam and two-beam cases are discussed.

Investigation of mosaicity of epitaxic multilayers by the statistical theory of X-ray dynamical diffraction

Based on the statistical theory of X-ray dynamical diffraction for thin films, the mosaicity of three types of semiconductor epitaxic layers has been investigated by analyzing their rocking curves by the X-ray double-crystal diffraction method. It is shown that the statistical theory can provide quantitative information on the mosaicity of the epitaxic layers such as the mean size and the mean disorientation of mosaic blocks in the layers. Some misunderstandings in interpreting experimental data are cleared up by taking into account the effect of diffuse scattering. It is emphasized that attempts to obtain structural parameters of specimens from their rocking curves by means of the Takagi-Taupin equations for coherent fields only are not strictly correct since diffuse scattering causes additional changes in the tails of the rocking curves.

Numerical simulation of Bragg-case topographic images of dislocations and precipitates in GaAs epitaxial layers

The numerical integration of Takagi-Taupin equations was used for the simulation of double-crystal topographic images of threading dislocations, misfit dislocation crossings and point-like precipitates in silicon and gallium arsenide. The simulations took account of lattice parameter profiles and the finite divergence of the beam. It was found that the images of dislocations, especially those outcropping on the surface, are dominated by a dilation-orientation contrast. Stronger interference effects were present in the images of point-like defects. A similarity was observed between the images of threading dislocations computed for equivalent positions in the layer and substrate maxima. A reasonable correspondence between simulated and experimental images was confirmed in several cases.

The transmission diffraction patterns of silicon implanted with high-energy alpha -particles

2 mm thick silicon wafers, implanted with 4.8 MeV alpha -particles are studied by means of transmission section topography and additionally by Lang and double-crystal methods. It was found that all three methods produced a negligible contrast in the symmetric transmission reflection apart from some fragments of the implanted area's boundaries. The interference fringes were observed in the case of asymmetric reflections. The asymmetric section topographs revealed distinct interference fringes, which cannot be explained in terms of simple bicrystal models. In particular, the curvature of these fringes may be interpreted as being due to the change in the implanted ion dose along the beam intersecting the crystal. Some features of the fringe pattern were reproduced by numerical integration of Takagi-Taupin equations.

Characterisation of semiconductor structures by high resolution X-ray diffraction

AbstractA simulation program based on the Takagi–Taupin equations from the dynamical theory of X-ray diffraction has been developed. The novelty of the program lies mainly in that it includes the elastic distortion of coherent epitaxial layers by employing a model based on the elasticity theory of materials. The simulation model is fairly flexible, allowing the calculation of any symmetric or asymmetric reflection, any substrate orientation, graded and relaxed layers, and instrumental effects. Some simulation results for different materials and multilayered structures are presented. A preliminary study of the influence of beam dispersion and detection noise has also been carried out.MST/3251

Dislocation contrast in X-ray reflection topography of strained heterostructures

Abstract X-ray reflection topography for imaging misfit dislocations in highly strained systems with a small critical thickness is assessed. The samples often contain complex heterostructures such as capping layers, quantum wells or superlattices. For the first time the strain within the heterostructure is taken into account explicitly into simulation of X-ray reflection topographs. A new algorithm to simulate such heterostructures is proposed on the basis of a local kinematical approximation within one step of the numerical integration of the Takagi-Taupin equations. Dramatic effects of the heterostructure configuration on the contrast of dislocation images are predicted by simulation, based on three effects: firstly the depth of the dislocation (referred to the free surface) determines the effect of surface relaxation on the distortion field; secondly the composition around the dislocation determines which part of the dislocation field takes part in image formation; thirdly interference between differen...

Observation of a triclinic lattice distortion of InxGa1− xAs (100)-oriented epitaxial layers by high-resolution double-crystal X-ray diffraction

In this work the lattice distortion of In x Ga 11− x As epitaxial layers with different InAs mole fractions (0.08 < x < 0.28), grown on (1 0 0)-oriented GaAs substrates by molecular beam epitaxy, is studied in detail by high-resolution X-ray diffractometry. The strain field and the chemical composition are determined by recording rocking curves in different diffraction geometries in the vicinity of the (4 0 0), (4 2 2) and (5 3 1) GaAs reflections. In order to analyze the experimental data we used a 2nd-order approximation of the strain function in the Takagi-Taupin equation. Our results reveal that a triclinic lattice distortion occurs in all of the investigated heterostructures despite the misorientation of the (1 0 0) substrate surface was less than 0.2°.