Reciprocal Space Analysis of the Initial Stages of Strain Relaxation in SiGe Epilayers
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scattering angle of diffracted x-rays emerging from the sample over a wide range of 20 angles. Parallel analysis of multiple scattering angles allows one to map reciprocal space more than 100 times faster than can be done using a traditional, serially-scanned, triple-axis diffractometer. This efficiency offers unique views of reciprocal space -- especially so for weak, diffuse diffraction. Even though the detection scheme used here is very efficient, scan times of -12 hours were required in order to thoroughly examine the diffuse x-ray scattering for the earliest, very low-dislocation-density stages of relaxation. Thus, the present results are almost inaccessible to a conventional triple-axis system using a sealed-tube or rotating anode x-ray source because of the prohibitively long time required to serially map the same region of reciprocal space. There are tradeoffs when using a PSD-based mapping scheme: these are decreased resolution and dynamic range relative to that of a triple-axis system [3]. Following x-ray analysis, the defectivity of the samples was revealed using a Schimmel etch: 4(0.3 M CrO3 ):5(HF). The etchant was diluted to 2(Schimmel):3(H 2 0) to reduce the etch rate. Samples were dipped and examined in intervals to obtain optimum contrast for Nomarskidifferential-interference-contrast optical microscopy of the dislocation lines. DISPLACEMENT FIELDS AND THE DISLOCATION ARRAY GEOMETRY Both the evolution of reciprocal space maps with dislocation density, and the quantitative relationship between lineal-misfit density and integrated-diffuse-scattering intensity, are best
discussed within the framework of a rather simple geometric model of the orthogonal-dislocation array. A schematic of this model is shown in Fig. 1. It is assumed that the dislocation density is approximately equal along both line directions, and that the array can be approximated by a uniform grid of dislocation lines separated by an average spacing, d. A final assumption is that the displacement field extends out laterally from the dislocation line a finite distance, w/2, along the heterointerface. Within this lateral distance, the displacement field extends vertically throughout the epilayer thickness, t. This artificially partitions the epilayer into two domains -distorted domains within -w/2 of the dislocation line, and domains outside of this boundary which are considered to be nearly perfect crystalline material. Given this geometric framework it is easy to derive an equation for the total volume fraction of epitaxial material that lies within distorted domains, Vt, in terms of the dislocation density, l/d. This equation appears in Fig. 1. Also defined in Fig. 1 is Vi, the volume fraction of epitaxial material within overlapping distorted domains formed at intersections in the array. These volume fractions are plotted in terms of the dimensionless parameter, w/d, in Fig. 2. The relative fraction of displaced material within intersections, given by the ratio Vi/Vt, is also shown in Fig. 2. Note that ViNt
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