X-ray diffraction tomography is an innovative method that is widely used to obtain 2D-phase-contrast diffraction images and their subsequent 3D-reconstruction of structural defects in crystals. The most frequent objects of research are linear and helical dislocations in a crystal, for which plane wave diffraction images are the most informative, since they do not contain additional interference artifacts unrelated to the images of the defects themselves. In this work the results of modeling and analysis of 2D plane wave diffraction images of a nano-dimensional Coulomb-type defect in a Si(111) thin crystal are presented based on the construction of numerical solutions of the dynamic Takagi-Taupin equations. An adapted physical expression for the elastic displacement field of the point defect, which excludes singularity at the defect location in the crystal, is used. A criterion for evaluating the accuracy of numerical solutions of the Takagi-Taupin equations is proposed and used in calculations. It is shown that in the case of the Coulomb-type defect elastic displacement field, out of the two difference algorithms for solving the Takagi-Taupin equations used in their numerical solution, only the algorithm for solving the Takagi-Taupin equations where the displacement field function enters in exponential form is acceptable in terms of the required accuracy-duration of the calculations.

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].

In the case of the point defect in a crystal, the inverse Radon’s problem in X-ray diffraction microtomography has been solved. As is known, the crystal-lattice defect displacement field function f(r) = h·u(r) determines phases − (±h)-structure factors incorporated into the Takagi–Taupin equations and provides the 2D image patterns by diffracted and transmitted waves propagating through a crystal (h is the diffraction vector and u(r) is the displacement field crystal-lattice-defects vector). Beyond the semi-kinematical approach for obtaining the analytical problem solution, the difference-equations-scheme of the Takagi–Taupin equations that, in turn, yield numerically controlled-accuracy problem solutions has been first applied and tested. Addressing the inverse Radon’s problem solution, the χ2-target function optimization method using the Nelder–Mead algorithm has been employed and tested in an example of recovering the Coulomb-type point defect structure in a crystal Si(111). As has been shown in the cases of the 2D noise-free fractional and integrated image patterns, based on the Takagi–Taupin solutions in the semi-kinematical and difference-scheme approaches, both procedures provide the χ2-target function global minimum, even if the starting-values of the point-defect vector P1 is chosen rather far away from the reference up to 40% in relative units. In the cases of the 2D Poisson-noise image patterns with noise levels up to 5%, the figures-of-merit values of the optimization procedures by the Nelder–Mead algorithm turn out to be high enough; the lucky trials number is 85%; and in contrast, for the statistically denoised 2D image patterns, they reach 0.1%.

Digital processing of 2D X-ray projection images of a Coulomb-type point defect in a Si(111) crystal detected against the statistical Gaussian noise background has been carried out using a guided filter and a wavelet filter with the 4th-order Daubechies function. The efficiency of 2D image filtering has been determined by calculating the relative square deviations of the intensities of the filtered and reference (noise-free) 2D images averaged over all points. A comparison of the calculated relative root-mean-square deviations of the intensities has shown that the investigated methods work quite well and can be effectively used in noise processing of X-ray diffraction images for 3D reconstruction of nanoscale defects in crystal structures.

A central point of validity of computer X-ray diffraction micro tomography is to improve the digital contrast and spatial resolution of the 3D-recovered nano-scaled objects in crystals. In this respect, the denoising issue of the 2D image patterns data involved in the 3D high-resolution recovery processing has been treated. The Poisson-noise simulation of 2D image patterns data was performed; afterwards, it was employed for recovering nano-scaled crystal structures. By using the statistical average and geometric means methods of the acquired 2D image frames, we showed that the statistical average hypothesis works well, at least in the case of 2D Poisson-noise image data related to the Coulomb-type point defect in a crystal Si(111). The validation of results related to the de-noised 2D IPs data obtained was carried out by both the 3D recovery processing of the Coulomb-type point defect in a crystal Si(111) and using the peak signal-to-noise ratio (PSNR) criterion.

The theoretical framework and a joint quasi-Newton–Levenberg–Marquardt–simulated annealing (qNLMSA) algorithm are established to treat an inverse X-ray diffraction tomography (XRDT) problem for recovering the 3D displacement field function f Ctpd ( r − r 0 ) = h  ·  u ( r − r 0 ) due to a Coulomb-type point defect (Ctpd) located at a point r 0 within a crystal [ h is the diffraction vector and u ( r − r 0 ) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function in a χ 2 sense in order to recover the function f Ctpd ( r − r 0 ) [ is the parameter vector that characterizes the 3D function f Ctpd ( r − r 0 ) in the algorithm search]. A theoretical framework based on the analytical solution of the Takagi–Taupin equations in the semi-kinematical approach is elaborated. In the case of true 2D imaging patterns (2D-IPs) with low counting statistics (noise-free), the joint qNLMSA algorithm enforces the target function to tend towards the global minimum even if the vector in the search is initially chosen rather a long way from the true one.

According to the diffraction tomography data, the efficiency of minimization algorithms used to reconstruct the displacement field of a defect in a crystal depends on the presence of a noise component and implies preliminary use of noise filtering algorithms. The quality of projection image filtering has been estimated using the root-mean-square deviations of the intensity of a denoised image from the intensity of the initial noiseless image in a fixed rectangular neighborhood of the defect under study and beyond it. A comparison of these quantities, calculated after application of different noise filtering algorithms, has shown that their minimum values are obtained simultaneously using a guided image filter. The 3D reconstruction based on the projection images denoised in this way has significantly improved the quality of reconstructing the defect displacement field as compared with the results based on noisy unfiltered images.

X-ray diffractometry, X-ray profilometry, atomic-force microscopy, and sclerometer tests are employed in the study of R -cut plates of sapphire single crystals grown with the aid of the Kyropoulos technique and used as substrates for the silicon-on-sapphire structures. Regions with different degrees of perfection can be found even on a single plate. Mutually consistent results of the four experimental methods based on different physical principles can be used to reveal regions of structural imperfection on the surface of a plate. It is expedient to employ a complex procedure for multipoint monitoring of the parameters of plates to reduce the number of defect samples that serve as substrates in the production of electronic devices.

The paper considers the data of measurement of electrophysical parameters of silicon pin-photodiodes after implantation of defect-forming ions and subsequent heat treatment, which open a new way to reduce the dark current and increase the output of suitable devices. The data of electrophysical measurements are compared with the results of structural studies. The efficiency of proton irradiation of the periphery of n+-p transitions to protect the surface of pin-photodiodes based on high resistance silicon was experimentally established. Optimal conditions - modes of proton irradiation and subsequent thermal annealing (E = 100+200+300 Kev, D = 21016 cm-2, T = 300С, t = 2 h), at which the formation of a surface layer with optimal characteristics for achieving minimum dark currents of photosensitive sites and the guard ring occurs. The application of these modes to serial pin-photodiodes with a depth of n+ - p-transitions  3 m allowed to reduce the dark current by an order of magnitude and increase the output of suitable devices.

The measured electrical parameters of silicon pin photodiodes subjected to the implantation of defect-forming ions and subsequent heat treatment are analyzed, which reveal a new way of reducing the dark current and enhancing the device yield. The data of the electrical measurements are compared with the results of the structural study. The experimental efficiency of proton irradiation of the n+–p junction periphery for protecting the surface of high-resistance silicon-based pin photodiodes is experimentally established. The optimal proton irradiation and subsequent annealing regimes (an energy of E = 100 + 200 + 300 keV, a dose of D = 2 × 1016 cm–2, a temperature of T = 300°C, and a time of t = 2 h) are determined, which ensure the formation of a surface layer with characteristics optimal for obtaining the minimum dark currents of photosensitive areas and a guard ring. The use of these regimes in commercial pin photodiodes with an n+–p junction depth of ~3 μm has allowed a reduction in the dark current by an order of magnitude and an increase in the device yield.

An R-cut sapphire wafers grown by Kyropoulos technique for fabrication of silicon-on-sapphire structures has been examined with the use of X- ray diffractometry, reflectometry, atomic-force microscopy and sclerometry methods. It is found that areas of different degree of crystal structure perfection are present even within the same wafer. It is shown that the results obtained by four different methods, based on different physical principles, are in good agreement and allow to recognize the areas of imperfect structure on the wafer surface. Implementation of comprehensive control of the quality parameters at several points of wafer surface is justified in order to reduce the production defects in the usage of sapphire wafers for fabrication of semiconductor devices.

To describe the 1D and 2D patterns of the grazing-incidence small-angle X-ray scattering (GISAXS) from a rough fractal surface, the novel integral equations for the amplitudes of reflected and transmitted waves are derived. To be specific, the analytical expression for the 2D total intensity distribution dR_tot( θ ,ϕ ;θ_0)/dΩ is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular dR_spec( θ ;θ_0)/dΩ and diffuse dR_dif( θ ,ϕ ;θ_0)/dΩ patterns, respectively. Hereafter, θ is the scattering meridian angle, ϕ is the scattering azimuth angle; θ_0 is the angle of incidence. By using the above analytical expressions, the 1D and 2D GISAXS patterns are numerically calculated. Some new experimental measurements of the specular reflectivity curves R spec ( θ_0 ) related to the fused quartz and crystal Si(111) samples have been carried out. Based on the theoretical approach developed, a direct least-squared procedure in a χ 2 -fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.

The structural features of the formation of radiation defects in proton-implanted layers of silicon wafers during their heat treatment are studied. New data on the nature, characteristics and concentration of microdefects in Si crystals irradiated with protons with energies of 100+200+300 Kev, with a total dose of 2·1016 ion/cm2, and the evolution of the defective structure during heat treatment in a wide temperature range from 200 to 1100°C were obtained from the analysis of the results of studies by high-resolution three-crystal X-ray diffractometry and transmission electron microscopy. This work was supported by the Ministry of Science and Higher Education within the State assignment FSRC «Crystallography and Photonics» RAS.

The structural features of the formation of radiation defects in proton-implanted layers of silicon plates during their heat treatment have been studied. New data on the nature, the characteristics, and the microdefect concentration in Si crystals irradiated with protons with energies 100 + 200 + 300 keV with the total dose 2 × 10 16 cm –2 and the evolution of the defect structure during heat treatment have been obtained by high-resolution three-crystal X-ray diffraction and transmission electron microscopy over wide temperature range from 200 to 1100°C.

This paper is a continuation of previous studies on the development of X-ray topo-tomography using laboratory equipment. The results on the spatial location of a single polygonal dislocation half-loop in a silicon single crystal were obtained as a result of testing the sensitivity of the X-ray topo-tomo diffractometer. A comparison was made with high-resolution experimental data obtained at the European synchrotron radiation facility (ESRF). The experimental procedure, software, and hardware for 3D reconstruction of the investigated single defect — a polygonal dislocation half-loop — are described.

AbstractIn this paper, we present the results of a three-crystal X-ray diffractometry (XRD) study of the state of a disturbed layer formed in silicon crystals by implantation of hydrogen ions with energies of 100 + 200 + 300 keV and a total dose of 2 × 10^16 cm^–2 during the subsequent heat exposure in the temperature range from 200 to 1100°C. Here, X-ray studies were performed in the three-crystal XRD scheme when the sample under consideration operates as a second stationary crystal with various fixed angular detuning α from the Bragg position while the third (perfect) crystal-analyzer sweeps the angular distribution of radiation diffracted by the second crystal. Based on a comparison of the shape of diffraction and diffuse maxima for the samples under study, a qualitative conclusion about a significant transformation of radiation defects at post-implantation annealing was derived.

The theoretical framework and a joint quasi-Newton–Levenberg–Marquardt–simulated annealing (qNLMSA) algorithm are established to treat an inverse X-ray diffraction tomography (XRDT) problem for recovering the 3D displacement field function f Ctpd ( r − r 0 ) = h · u ( r − r 0 ) due to a Coulomb-type point defect (Ctpd) located at a point r 0 within a crystal [ h is the diffraction vector and u ( r − r 0 ) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function {\cal F}\{ {\cal P} \} in a χ 2 sense in order to recover the function f Ctpd ( r − r 0 ) [{\cal P} is the parameter vector that characterizes the 3D function f Ctpd ( r − r 0 ) in the algorithm search]. A theoretical framework based on the analytical solution of the Takagi–Taupin equations in the semi-kinematical approach is elaborated. In the case of true 2D imaging patterns (2D-IPs) with low counting statistics (noise-free), the joint qNLMSA algorithm enforces the target function {\cal F} \{ {\cal P} \} to tend towards the global minimum even if the vector {\cal P} in the search is initially chosen rather a long way from the true one.

Features of formation and transformation of radiation defects in near-surface layers of silicon plates that are implanted with hydrogen ions are studied. Using the method of high-resolution double-crystal X-ray diffractometry, values of the main parameters, such as mean effective thickness Leff and mean relative deformation Δa/a of a doped layer, are determined depending on the implantation dose and substrate temperature.