Covariant description of X-ray diffraction from anisotropically relaxed epitaxial structures

Abstract

A general theoretical approach to the description of epitaxial layers with essentially different cell parameters and in-plane relaxation anisotropy has been developed. A covariant description of relaxation in such structures has been introduced. An iteration method for evaluation of these parameters on the basis of the diffraction data set has been worked out together with error analysis and reliability checking. The validity of the presented theoretical approaches has been proved with a-ZnO on r-sapphire samples grown in the temperature range from 573 K up to 1073 K. A covariant description of relaxation anisotropy for these samples has been estimated with data measured for different directions of the diffraction plane relative to the sample surface.

Fisher information for optimal planning of X-ray diffraction experiments

“…which is known to characterize the accuracy of parameter estimation by fitting (Prince, 2012;Zhylik et al, 2013). Alternatively, one can adhere to the Bayesian approach (Koch, 2007) and consider the posterior probability p(Z|Y) of the parameter vector Z, conditioned by the measured data set Y.…”

“…The bounds ( 16) and ( 17) are not necessarily satisfied if the estimation becomes biased, which is the case encountered when certain a priori constraints are taken into account and effectively reduce possible fluctuations in the resulting values [see e.g. Cutler & Flanagan (1994), Vallisneri (2008), Rodriguez et al (2013), Zhylik et al (2013) and Mikhalychev et al (2019)]. For specific tasks, when the estimate is biased, a priori information about the sample parameters is to be accounted for or a more accurate description of the fluctuations is requested [higher-order moments or a detailed probability distribution instead of variances and covariances, provided by equations ( 16) and ( 17)], one may need to go beyond the Fisher information formalism.…”

“…An infinite CRB-based lower bound for the parameter errors does not imply the complete impossibility of their determination. The CRB is valid for unbiased estimates only, while the physical constraints 0 R i 1 inevitably introduce bias to the estimates and reduce the actual variations in the reconstructed parameters [some details on the influence of constraints on the parameter estimation errors can be found in the work of Zhylik et al (2013)]. Here, we do not consider managing such a degenerate case in detail, because the errors ÁR i ' 1 achievable for the 004 reflection are impractically large relative to those obtained for all the other considered reflections.…”