Takagi Taupin Research Articles

X-ray diffraction investigation of diffusion in PbTe–PbSe superlattices

Diffusion intermixing in PbTe–PbSe epitaxial superlattice was investigated with help of the double-crystal X-ray diffractometry. Diffusion coefficients were defined for different annealing temperatures from the regular fading of satellites intensities on the X-ray diffraction patterns in the course of annealing. Theoretical grounds and computational treatment of experimental data are presented. Diffraction patterns have been simulated by numerical solution of the Takagi–Taupin equation and calculated curves were fitted to experimental ones accounting the superlattice profile smoothing due to diffusion. Diffusion coefficients determined by both procedures are in a good agreement.

Focusing and reflection by a bent crystal for high-energy synchrotron radiation.

The focusing properties and resolution of a doubly bent crystal in the Bragg case have been analytically studied from a geometrical viewpoint. Simulation using the Takagi-Taupin equations was also performed for singly bent crystal reflections to study the reflectivity. The critical radius of curvature for changing from dynamical to kinematical diffraction is calculated to be of the order of a few tens of metres for an Si 400 reflection of 110 keV X-rays.

Matrix method for the x-ray rocking curve simulation

A matrix representation is developed for the solution of the Takagi–Taupin equations of x-ray diffraction. By the virtue of its unimodular property, the solution matrix substantially reduces the calculation time for the superlattice (SL) structure with a large periodicity. Also, the simplified form of the solution makes it easier to understand and quantify the inherent properties of the x-ray diffraction such as the interference fringes and the SL peaks.

Fine interference effects in X-ray diffraction from multilayered structures

Attention is drawn to the importance of interference effects in the X-ray diffraction spectra for the precise characterization of multilayered structures. A novel simulation procedure, based on direct summation of waves scattered by atomic planes, permits to obtain analytical expressions for diffraction spectra and to expose interference contributions which are usually hidden within the Takagi–Taupin formalism. The simulation routine is suitable for the introduction of local variations of the structural and geometrical parameters, such as interface roughness or defect-induced local fluctuations of interplanar spacing. A new interference effect—a strong influence of lattice disorder on the periodicity of intensity fringes—was found as a result of the careful analysis of the diffraction spectrum from a triple-layered structure. Selected applications of this approach to the study of local lattice disorder in SiGe/Si heterostructures, to the characterization of ultrathin GaAs/GaInAs/GaAs quantum wells, amorphous SiO 2 layers buried in Si, and nearly matched InGaP/GaAs heterostructures, are given.

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

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.

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.

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

Simulation of X-ray reflection topographs from misfit dislocations

Abstract Images of misfit dislocations as obtained from X-ray reflection topography with a low incidence angle are simulated. New algorithms for integrating numerically the Takagi–Taupin equations are presented with decoupled, variable and adaptively chosen step sizes. This reduces drastically the computation time required. The adaptive integration is faster by a factor of ten or more over integration with fixed steps, is more accurate and requires a minimum of free parameters. Rocking topographs are introduced to show the contrast variation over the Bragg peak. It is shown that the contrast of dislocations parallel to the surfaces vanishes as the depth goes to zero, for any anisotropic medium and any Burgers vector.

X-ray reflection properties of elastically bent perfect crystals in Bragg geometry

Reflection curves of bent crystals were calculated using the Takagi–Taupin theory of dependence on bending radius, wavelength, crystal thickness and Bragg angle. The reflection properties of bent crystals were measured for bending radii down to 90 mm. Usually, for the measurement of rocking curves of crystals with large bending radii, the double-crystal diffractometer was used in parallel position (n, −n). A special achromatic diffractometer consisting of a plane and a bent crystal is proposed. It is used to measure rocking curves of bent crystals with small bending radii (R < 1 m). Experimental values show close agreement with the theory. The reflection properties are important for X-ray microscopy with two-dimensionally bent crystals and for X-ray spectroscopy with bent crystals.

The Images of Dislocations in Synchrotron Bragg-Case Section Topography of Diamond

Bragg-case synchrotron section topographs were studied in parallel slabs cut from a synthetic diamond of a good quality. The topographs revealed the Pendellosung fringes and images of dislocations and other defects containing several fringe systems. The experiment provided the opportunity for studying of the theoretical dislocation images obtained by numerical integration of the Takagi—Taupin equations. A program employing a variable step of integration in the Bragg-case has been presented. The influence of the finite slit width and of the limited beam divergence on the theoretical images is also discussed.

The RED extinction model. III. The case of pure primary extinction

Bragg diffraction in nearly perfect crystals is treated within the framework of the random elastic deformation (RED) model. Similar to the previous applications it is again assumed that each individual reflection event takes place within an elastically deformed domain and hence may be described by either an exact or a quasiclassical solution to the Takagi–Taupin equations. But because of the high degree of perfection and/or small thickness of the crystal the interaction is confined to one single domain so that the total reflected intensity is obtained just by summing up the contributions from the whole ensemble of domains characterized by different values of the deformation gradient and thickness. The present approach profoundly differs from Kato's statistical dynamical theory which starts with the averaging procedure within the Takagi–Taupin equations before solving them.

Interference effects in x-ray double-crystal rocking curves of Ga1−xAlxAs/GaAs laser structures and superlattices

Interference effects in the x-ray double-crystal rocking curves of Ga1−xAlxAs/GaAs laser structures and superlattices have been discussed in the framework of a reliable simulation model based on the Takagi–Taupin dynamical theory. It has been found that in the laser structures the presence of the active GaAs layer between the Ga1−xAlxAs confining layers causes a splitting of the confining layer Bragg peak in subpeaks of comparable intensity; in a similar way, the presence of GaAs layers inside a superlattice is able to split each satellite peak in subpeaks. Interference effects have been experimentally revealed in an AlAs/GaAs heterostructure made of three identical superlattices separated by thin GaAs layers.

A foundation for the statistical dynamical theory of diffraction

The statistical dynamical theory [Kato (1980). Acta Cryst. A36, 763–769] is reformulated on a sounder basis. The starting wave equation is free from the so-called Takagi–Taupin (T–T) approximation. Functional calculus, an operational technique and the concept of the Green function are used as mathematical tools. Integro-differential equations are derived for the coherent (averaged) wave field and the energy flow vector of the incoherent intensity field. The formulae are exact except for assuming a model in which the fluctuation of the lattice phase is a set of Gaussian random variables defined in three-dimensional space. The general framework of the previous theory is justified within the T–T approximation. In general, however, new terms must be added and some terms have to be revised by introducing a Green function matrix. The theory may be used as a starting point when any approximate theory is developed for practical purposes.