Diffraction by a randomly distorted crystal. II. General theory

Becker, Petra; Haddad, Moussa

doi:10.1107/s0108767391009376

Cited by 43 publications

(8 citation statements)

References 4 publications

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“…The latter originates from an enlarged angular acceptance for the incident spherical wave generating diffuse intensity proportional to the sample thickness. These two effects are correlated within the statistical dynamical theory [23][24][25][26] through the static Debye-Waller factor E, which can be determined in our measurements with an accuracy of 10 À4 . Furthermore, the initial slope of the increase of the Bragg signal and the timescale of the process give access to the misfit strain per precipitate, the diffusion constant, and the precipitate density.…”

Section: Experimental Methodsmentioning

confidence: 99%

Diffusion-driven precipitate growth and ripening of oxygen precipitates in boron doped silicon by dynamical x-ray diffraction

Will

Gröschel

Bergmann

et al. 2014

Journal of Applied Physics

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X-ray Pendellösung fringes from three silicon single crystals measured at 900 °C are analyzed with respect to density and size of oxygen precipitates within a diffusion-driven growth model and compared with TEM investigations. It appears that boron doped (p+) material shows a higher precipitate density and a higher strain than moderately (p-) boron crystals. In-situ diffraction reveals a diffusion-driven precipitate growth followed by a second growth regime in both materials. An interpretation of the second growth regime in terms of Ostwald ripening yields surface energy values (around 70 erg/cm2) similar to published data. Further, an increased nucleation rate by a factor of ∼13 is found in the p+ sample as compared to a p- sample at a nucleation temperature of 450 °C.

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Section: Experimental Methodsmentioning

confidence: 99%

Diffusion-driven precipitate growth and ripening of oxygen precipitates in boron doped silicon by dynamical x-ray diffraction

Will

Gröschel

Bergmann

et al. 2014

Journal of Applied Physics

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“…These SiO x precipitates build up strain fields, which disturb the perfect silicon host lattice. Because of the high density of the agglomerates, this can be described in a statistical approach within the statistical dynamical theory [15][16][17][18] which predicts as the most prominent feature an incoherent increment of the average Bragg intensity originating from the enhanced phase space becoming available for Bragg reflection. In addition, the Pendell€ osung length becomes elongated and the amplitude of the oscillations appears damped.…”

Section: B Pendell € Osung Oscillationsmentioning

confidence: 99%

“…The measurements have been made with the characteristic K a1 -line of a high voltage tungsten X-ray tube at 59.31 keV, which provides the opportunity to illuminate through complex sample environments like high temperature scattering furnaces. , which is a consequence of the dissolution of the quartz crucible during crystal pulling and the high oxygen solubility of 2.5 Â 10 18 1/cm 3 at the melting point. Usual temperatures in the production line of integrated circuits are in the range of several hundred degrees and thus way below the melting temperature of silicon.…”

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confidence: 99%

Oxygen diffusivity in silicon derived from dynamical X-ray diffraction

Will

Gröschel

Kot

et al. 2013

Journal of Applied Physics

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Diffusion of the vacancy defect leading to the formation of multi-shell structures in the nanowire and nanobridge J. Appl. Phys. 112, 114301 (2012) Characterization of defect evolution in ultrathin SiO2 layers under applied electrical stress J. Appl. Phys. 112, 103513 (2012) Fluid like behavior of oxygen in cubic zirconia under extreme conditions Appl. Phys. Lett. 101, 181906 (2012) Out-diffusion of deep donors in nitrogen-doped silicon and the diffusivity of vacancies J. Appl. Phys. 112, 013519 (2012) Additional information on J. Appl. Phys. Thickness dependent Pendell€ osung oscillations are highly sensitive to strain fields from defects in a host crystal. Based on this, we present a novel technique to measure the precipitation kinetics of oxygen in silicon already at its early stage of clustering at high temperatures. At 900 C, precipitates with a radius smaller than 4 nm and with a density of 1 6 0:5 Â 10 13 1/cm 3 were observed. The technique was calibrated by complementary scanning transmission electron microscope and energy dispersive X-ray measurements in the range of normal diffusivity yielding a diffusion constant of 1:7 6 0:1 Â 10 À12 cm 2 =s, which is close to the literature value of 2:074 Â 10 À12 cm 2 =s. The measurements have been made with the characteristic K a1 -line of a high voltage tungsten X-ray tube at 59.31 keV, which provides the opportunity to illuminate through complex sample environments like high temperature scattering furnaces. V C 2013 American Institute of Physics. [http://dx

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“…The well known statistical dynamical treatment of Kato (1980a) developed on the basis of the equations of Takagi (1962) and Taupin (1964), and the modifications by Al-Haddad & Becker (1988) and Becker & Al-Haddad (1992), as well as the study by Guigay & Chukhovskii (1995), explain theoretically the diffraction process in the whole range of crystal imperfections from almost perfect to ideally imperfect crystals. Nonetheless, because of uncertainty regarding the correlation function of the phase factor, which characterizes the relation of crystal imperfections to scattering amplitude, no practical solution to this theory is available to this day.…”

Section: Introductionmentioning

confidence: 99%

X-ray extinction changes due to dislocations

Masimov

2007

J Appl Cryst

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A correct interpretation of the diffraction phenomenon in crystals with low dislocation content, for which the kinematical scattering theory does not yield any reasonable results, was carried out by X‐ray extinction treatments. Different approaches to investigating the dislocation‐induced X‐ray extinction changes are reviewed and the applicability of the statistical dynamical theory to analysis of the substructure characteristics of crystals is discussed. A relationship between the scattering amplitudes and the dislocation density in crystals is established by using pair correlation functions. A procedure for determination of the substructure characteristics by means of measurements of the integrated intensities is proposed. The tests illustrate satisfactory agreement of the investigated approaches with experimental data.

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Diffraction by a randomly distorted crystal. II. General theory

Abstract: A method is presented to deal with the propagation of X-rays or neutrons in a statistically distorted crystal in the general case where both short-range and longrange order are present: it is the generalization of a theory presented in a previous paper [Becker & A1

Cited by 43 publications

References 4 publications

Diffusion-driven precipitate growth and ripening of oxygen precipitates in boron doped silicon by dynamical x-ray diffraction

Diffusion-driven precipitate growth and ripening of oxygen precipitates in boron doped silicon by dynamical x-ray diffraction

Oxygen diffusivity in silicon derived from dynamical X-ray diffraction

X-ray extinction changes due to dislocations

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