X-ray diffraction of multilayers and superlattices

Abstract

Recursion formulae for calculating the reflected amplitude ratio of multilayers and superlattices have been derived from the Takagi-Taupin differential equations, which describe the dynamical diffraction of X-rays in deformed crystals. Calculated rocking curves of complicated layered structures, such as nonideal superlattices on perfect crystals, are shown to be in good agreement with observed diffraction profiles. The kinematical theory can save computing time only in the case of an ideal superlattice, for which a geometric series can be used, but the reflectivity must be below 10% so that multiple reflections can be neglected. For a perfect crystal of arbitrary thickness the absorption at the center of the dynamical reflection is found to be proportional to the square root of the reflectivity. Sputter-deposited periodic multilayers of tungsten and carbon can be considered as an artificial crystal, for which dynamical X-ray diffraction calculations give results very similar to those of a macroscopic optical description in terms of the complex index of refraction and Fresnel reflection coefficients.

Growth and structural characterization of PbTe/PbEuTe double barrier

“…To obtain the structural parameters of the DB samples, the spectra were calculated within the framework of the dynamical theory of x-ray diffraction using Takagi-Taupin equations [3,11] and compared to the measured ω/2θ scans.…”

Structural investigations of the α12Si–Ge superstructure

“…The integrated intensity curves taken from the 004 RSMs were fitted with fully dynamical Takagi-Taupin simulations (Bartels et al, 1986). With this approach it is possible to determine the thickness, composition and strain of the individual layers.…”

Dynamical X-ray Diffraction from ZnS y Se1−y /GaAs (001) Multilayers and Superlattices with Dislocations

“…Calculation of the rocking curve for an arbitrary semiconductor heterostructure is based on the solution of the Takagi-Taupin equation [4][5][6][7] for dynamical diffraction,…”

“…Dynamical [1][2][3][4][5][6] and kinematical [7][8][9][10][11] simulations have been used in conjunction with a curve-fitting procedure to extract the profiles of strain and composition, but are based on perfect, dislocation-free laminar crystals, and this renders the analysis inapplicable to mismatched structures with dislocation densities greater than 10 6 cm À2 . Krivoglaz and Ryaboshapka 12 and Levine and Thomson 13 have analyzed the line profiles of Bragg peaks from crystals containing straight, parallel screw dislocations with precisely known atomic displacements.…”