Crystal mosaicity determined by a novel layer deconvolution Williamson–Hall method

Abstract

The application of conventional Williamson–Hall (WH) plot analysis to crystals often results in broadening not proportional to the scattering length vector. Several reasons may influence the broadening such as composition or strain heterogeneities, wafer curvature, dislocation anisotropy and other defects. In this work, linearization of WH plots is achieved given the assumption that the total crystal size may be deconvoluted into a finite number of layers, each with a constant thickness, strain and mosaic spread. A novel linearization algorithm, the layer deconvolution WH (LdCWH) method, employs a finite number of pseudo-Voigt (PV) functions for each measurement. Afterwards, it searches for possible solutions by changing the PV coefficients until r² of the conventional WH representation is above 0.999. The searching procedure consists in a combination of a genetic algorithm (GA) to generate randomly the PV coefficients within a specified range and a Marquardt–Levenberg algorithm to fit simultaneously the measured reflections using the PV coefficients as inputs. The possible solutions further allow estimating the upper and lower bounds of the mosaicity. Conventional WH plots and the implementation of the LdCWH are applied to a commercial AlGaN thick layer and to bulk α-MoO₃ crystals and discussed. For the former, the lateral and perpendicular coherence lengths, tilt angle and heterogeneous strain derived are 616 ± 7 nm, 510 ± 10 nm, 0.069 ± 0.001° and 0.0345 ± 0.0002%, respectively, while for the latter, a vertical coherence length of 3883 ± 56 nm and heterogeneous strain of 0.0556 ± 0.0002% are found. The nature of peak broadening regarding integral breadth and full width at half-maximum is discussed.