X-Ray Topography Studies of Oxygen Precipitates in MCZ Silicon
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X-RAY TOPOGRAPHY STUDIES OF OXYGEN PRECIPITATES IN MCZ SILICON ANTHONY J. HOLI4AND, G. STEPHAN GREEN, BRIAN K. TANNER AND MAI ZHENHONG Physics Department, University of Durham, South Road, Durham. DH1 3LE, UK *Instituteof Physics, Chinese Academy of Sciences, Beijing, P.R. China. ABSTRACT X-ray section topography has been used to study the distribution and size of precipitates resulting from heat treatment of MCZ silicon. A low density of precipitates was found, enabling individual precipitate images to be studied. Images have been simulated by numerical solution of Takagi's equations and the magnitude of the strain field deduced by comparison with experiment. Excellent agreement has been found in the details of simulated and experimental images. The effective defect volume increased monotonically with annealing temperature. The effect of surface relaxation and long range curvature on the accuracy of determining the microscopic strain field by matching simulated and experimental images has been investigated. INTRODUCTION X-ray topography [1,2] has, for many years, been used for the characterization of the defect content of crystals. By examining the distribution of diffracted intensity with position a two-dimensional map of the X-ray scattering is recorded, usually photographically. The strain field due to the defect gives rise to different diffraction conditions from that of the perfect crystal and hence this leads to contrast in the topograph. Although electron microscopists have for many years used simulation to deduce microscopic strain fields from the diffracted intensity, only recently has simulation become widespread in X-ray topography. This is principally due to the difficulties encountered due to the large Bragg angle and non-parallel wave in the X-ray case. The stationary section topograph, where the incident ribbon beam is narrow compared with the width of the emerging fan of X-rays from the crystal, is the easiest to simulate and the literature is now quite extensive [3,4]. By adjustment of model parameters to obtain matching between simulation and experimental images, it is possible to determine directly the real physical parameters associated with the defect. COMPUTATIONAL TECHNIQUE The simulation program uses Epelboin's algorithm [3,4] to solve numerically Takagi's equations [5] which describe X-ray propagation in a deformed crystal. In it we model a precipitate by a spherically symmetric strain field with the displacement ur given by ur = Cr/ r3
(1)
where the "deformation parameter", C, defines the magnitude and sense of the deformation [6]. It is proportional to the volume of a coherent spherical precipitate. Isotropic elasticity theory is assumed and this has previously been shown to be satisfactory for simulation of section topograph images of hydrogen precipitates in silicon [6,7]. The effect of surface relaxation was incorporated by using the method of images to determine the strain field. Then the total strain field becomes the sum of that due to the precipitate itself and that 6f an image
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