Generalized Laue dynamical theory for x-ray reflectivity at low and high incidence angles on strained multilayers

In this work, we present a generalized dynamical theory for strained multilayers within the Laue formalism valid for x-ray Bragg diffraction. In the generalized theory we take into account all the four solutions of the secular equation, the asymptotic sphericity of the dispersion surface, the difference between electric and displacement fields, and the boundary conditions of continuity at the heterointerfaces for the tangential components of the electric and magnetic fields. Only the following approximations are made: to consider a two-beam case, and to neglect quadratic terms of the dielectric susceptibility. Our general equations give the correct x-ray reflectivity in the whole angular range between 0 and pi/2, i.e., also very far from the Bragg peaks. Great differences between the predictions of the conventional and generalized theories are obtained in several cases, for important features of the rocking curves, like angular position, peak intensity, and full width at half maximum of superlattice satellite or epitaxial layer peaks. We discuss in detail the following cases for which only the generalized theory describes correctly the x-ray scattering: coplanar diffraction (i) very far from the Bragg condition, (ii) near to the Bra,og condition for strong asymmetric reflections, and (iii) for highly mismatched heterostructures.