Statistical Dynamical Theory Of Diffraction Research Articles

Modified statistical dynamical diffraction theory: analysis of model SiGe heterostructures

A modified version of the statistical dynamical diffraction theory (mSDDT) permits full-pattern fitting of high-resolution X-ray diffraction scans from thin-film systems across the entire range from fully dynamic to fully kinematic scattering. The mSDDT analysis has been applied to a set of model SiGe/Si thin-film samples in order to define the capabilities of this approach. For defect-free materials that diffract at the dynamic limit, mSDDT analyses return structural information that is consistent with commercial dynamical diffraction simulation software. As defect levels increase and the diffraction characteristics shift towards the kinematic limit, the mSDDT provides new insights into the structural characteristics of these materials.

Application of statistical dynamical diffraction theory to highly defective ion implanted SiGe heterostructures

AbstractThe statistical dynamical diffraction theory (SDDT) provides a method for performing high resolution X‐ray diffraction (HRXRD) analyses from materials that contain high levels of structural imperfection. SDDT is implemented by combining kinematical and dynamical diffraction formalisms into a single framework through the inclusion of two parameters (a static Debye–Waller factor and a correlation length) that can provide an adjustable coupling between the kinematic and dynamic extremes. Typically one of the prominent difficulties in implementing SDDT is the mathematical complexities that are characteristic of this theory. Recently we have demonstrated a simplified realization of SDDT [Shreeman and Matyi, J. Appl. Crystallogr. 43, 550 (2010)] that preserves the essential features of the theory while allowing it to be applied to a variety of structures. Here we show the viability of this approach by fitting various experimental HRXRD data from highly defective and partially relaxed Si0.70Ge0.30 ion implanted heterostructures. This study demonstrates the capabilities the SDDT theory provides for HRXRD analyses of highly defective semiconductor materials.

Precipitation in silicon wafers after high temperature preanneal studied by X-ray diffraction methods

We have investigated oxygen precipitation in Czochralski silicon wafers focusing on influence of nucleation temperature and high temperature pre-anneal during common three step treatment. Thick Si wafers were studied mainly by x-ray diffraction in Laue transmission geometry using Mo x-ray tube, but were also compared to reciprocal space maps obtained in Bragg reflection geometry. The analysis of measured diffraction scans in Laue geometry was performed by means of Takagi equations and statistical dynamical theory of diffraction. From the simulated Laue diffraction curves we find the size of the individual defect area and the fraction of strain area volume in the wafer. The results obtained from x-ray diffraction were compared to loss of interstitial oxygen according to infrared absorption spectroscopy and the size of SiO2 precipitate core was estimated. These techniques are in agreement with transmission electron microscopy images.

Oxygen Precipitation Studied by X-Ray Diffraction Techniques

We report on study of oxygen precipitates grown in Czochralski silicon wafers investigated by x-ray diffraction in Bragg reflection geometry and Laue transmission geometry. The analysis of diffraction curves in Laue geometry was done using Takagi equations and statistical dynamical theory of diffraction. These techniques allow us to determine as the radius of defect area as the defect concentrations from measurement in Laue geometry. These results obtained on silicon wafers exposed to two-step and three-step treatments were compared with other experimental techniques including transmission electron microscopy and infrared absorption spectroscopy, while only the largest precipitates are detected by other techniques. The results of all methods are in good agreement.

X-ray diffraction analysis of multilayer porous InP(001) structure

Multilayer structures composed of four porous bilayers have been studied by high-resolution X-ray diffraction using synchrotron radiation, and the photoluminescence of these structures has been investigated at 4 K. The porous structures were formed by anodic oxidation of InP(001) substrates in aqueous HCl solution. The structural parameters of the sublayers were varied by changing the electrochemical etching mode (potentiostatic/galvanostatic). The X-ray scattering intensity maps near the InP 004 reflection are obtained. A model for scattering from such systems is proposed based on the statistical dynamical diffraction theory. Theoretical scattering maps have been fitted to the experimental ones. It is shown that a mathematical analysis of the scattering intensity maps makes it possible to determine the structural parameters of sublayers. The reconstructed parameters (thickness, strain, and porosity of sublayers and the shape and arrangement of pores) are in satisfactory agreement with the scanning electron microscopy data.

X-ray scattering on multilayer porous structures

A model for description of the coherent and diffuse X-ray scattering from multilayer porous crystals has been developed based on a statistical dynamical theory of diffraction. Reciprocal space maps of the intensity of scattering from these structures are numerically simulated. It is shown that a significant role in the formation of equal-intensity contours is played by the statistical averaging with respect to pore sizes. Average parameters of InP porous structures, including the average pore sizes and lateral quasi-period, have been determined using a comparative analysis of the experimental data and results of numerical simulations.

Static Debye-Waller factor analysis of hydrogen precipitates in silicon

Abstract The hydrogen-induced defects in single crystals of silicon grown by the floating-zone method in a hydrogen atmosphere, have been measured by high-energy synchrotron radiation section topography. The static Debye-Waller factor (SDWF) produced by hydrogen precipitates, which were formed in silicon under isochronal annealing, has been calculated on the basis of a statistical dynamical diffraction theory. Within a H2 molecule cluster model which we proposed for a precipitate, the size and density of hydrogen precipitates have been determined using the SDWF method and the results of infrared absorption spectrum. The character of the early stage of hydrogen precipitation indicates that hydrogen in p-type silicon is mobile and less stable, compared with that in n-type silicon.

Models of spherically symmetric microdefects in the statistical dynamical theory of diffraction: I. Correlation function

Models of spherically symmetric microdefects in the statistical dynamical theory of diffraction: I. Correlation function

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.

Concerning order parameters in the statistical dynamical theory of diffraction

The development of the statistical dynamical theory of diffraction by real crystals involves two order parameters: a long-range and a short-range one. These parameters play a fundamental role in practical applications of the theory to real diffraction data. A reasonable probabilistic model is proposed to describe the phase correlation at two given scattering positions. This model allows for a tractable solution of the propagation equations for the beams.

On the statistical dynamical theory of diffraction: application to silicon

A detailed solution of Kato's equations describing the propagation of X-rays or neutrons in a crystal containing a statistical distribution of imperfections is presented: this solution makes use of propagation operators to describe multiple scattering events. Corrections to Kato's original solution are given which have a significant effect, even in the case of crystals with a high degree of long-range perfection. The present modified solution is applied to experimental measurement on parallel plates of silicon with different dislocation densities by Olekhnovich, Karpei, Olekhnovich & Puzenkova [Acta Cryst. (1983), A39, 116-122]. The theory reproduces observations quite well, in contrast to conclusions reached by Olekhnovich et al. on the basis of the original solution. It can be inferred that the basic ideas of Kato allow for a correct interpretation of diffraction reflectivities in highly perfect crystals, where a significant contribution from incoherent components of scattered intensities must be incorporated. However, the theory has to be modified for the case of lower long-range perfection: this involves the modification of the expressions for the effective correlation lengths that enter the theory.