Description of x-ray reflection and diffraction from periodical multilayers and superlattices by the eigenwave method

Abstract

The analytical solution of recurrent equations for amplitudes of electromagnetic field is found for description of x-ray reflection and diffraction from periodical multilayered media. The method proposed uses the Bloch eigenwaves approach for periodical structure, which reduces considerably the computer time required for simulation of diffracted/reflected x-ray intensity and, therefore, accelerates the fitting trial-and-error procedure for sample model parameters. Numerical examples and fit results for experimental x-ray data are provided to demonstrate the effectiveness of method. A new parameter describing the fluctuation of superlattice period is introduced and its influence on experimental data interpretation is discussed.

X-ray dynamical diffraction from partly relaxed epitaxial structures

“…The present work is a further 14,15 extension of the matrix method for HRXRD. We developed x-ray dynamical diffraction theory from multilayered epitaxial structures, including partly relaxed samples.…”

“…thus the sum amplitude of the reflected waves coincides with the solution of the matrix equations for the pseudomorphic structure. 4,15 The following result should be especially emphasized, despite the fact that the recurrent equations in Eq. ͑23͒ express the amplitudes in partly relaxed structure via the amplitudes in pseudomorphic one, the wave-field distribution is considerably different in these cases.…”

“…10 However, as mentioned above, the kinematical theory is not applicable for the diffraction in thick layers and the Takagi-Taupin equations fail for the diffraction in grazing-incidence or grazing-exit geometry. In opposite, the matrix method of x-ray diffraction is applicable for arbitrary experimental geometry and number ͑thickness͒ of the layers in the sample 4,15 but it demands the continuity condition for the lateral component of the wave vector at interfaces, which is satisfied for the pseudomorphic layers only. As demonstrated above, the additional harmonics of the diffraction field caused by different mismatches in layers contribute negligibly to the integrated x-ray intensity.…”

“…For this sample, the matrix method can be used for calculation of the wave-field amplitudes D i,n for and polarizations for arbitrary number and thickness of the layers and arbitrary diffraction geometry. 4,15 Here the index i =1,2,3,4 determines four roots of the dispersion Eq. ͑9͒ for nth layer; n =0,1, ... ,N + 1, where n =0 corresponds to vacuum and n = N + 1 to substrate.…”

Reflection and transmission of weakly inhomogeneous anisotropic and bianisotropic layers calculated by perturbation method

“…There is a variety of exact and approximate techniques for analysis of electromagnetic wave propagation in stratified structures including the bianisotropic ones. In particular, those are Brillouin zone theory [29], approximation of geometrical optics [30][31][32], operator methods [33][34][35][36][37][38][39][40][41][42][43][44][45][46] based on Fedorov's works on SO(3) covariant crystal optics and crystal acoustics [3,[47][48][49], Green function techniques [50][51][52][53][54], vector circuit theory [55], wave splitting in the time-domain [52,56,57], transformation equation [58] and vector Fresnel equation [59] techniques; eigenwave technique for description of x-ray reflection and diffraction [60] and others. A number of new methods is used to reveal interdependence between symmetry of multilayered media including fractal ones and their spectral periodicity [61][62][63][64].…”