Dynamical x-ray diffraction of multilayers and superlattices: Recursion matrix extension to grazing angles

A generalized dynamical theory has been developed that extends previous models of x-ray diffraction from crystals and multilayers with vertical strains to the cases of grazing incidence and/or exit below the critical angle for total specular reflection. This provides a common description for extremely asymmetric diffraction, surface (“grazing-incidence”), and grazing Bragg-Laue diffraction, thus providing opportunities for the applications of grazing geometries to the studies of thin multilayers. The solution, obtained in the form of recursion formulas for (2×2) scattering matrices for each individual layer, eliminates possible divergences of the (4×4) transfer-matrix algorithm developed previously. For nongrazing x-ray diffraction in the Bragg geometry and for grazing-incidence x-ray specular reflection out of the Bragg diffraction conditions, the matrices are reduced to scalars and the recursion formulas become equivalent to the earlier recursion formulas by Bartels et al. [Acta Cryst. A 42, 539 (1986)] and Parratt [Phys. Rev. 95, 359 (1954)], respectively. The theory has been confirmed by an extremely asymmetric x-ray-diffraction experiment with a strained AlAs/GaAs superlattice carried out at HASYLAB. A solution to the difficulties due to dispersion encountered in extremely asymmetric diffraction measurements has been demonstrated. Finally, the validity of Ewald’s expansion for thin layers and the relation of the matrix method to the Darwin theory, as well as the structure of x-ray standing waves in multilayers are discussed