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

Abstract

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 /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 ͑i͒ very far from the Bragg condition, ͑ii͒ near to the Bragg condition for strong asymmetric reflections, and ͑iii͒ for highly mismatched heterostructures.

Atomic Structure Based Simulation of X-Ray Scattering from Strained Superlattices

“…Using GDT [16] (for the (004) reflection) it was found that the peak positions coincide very closely with those from Bragg's condition…”

“…Improved treatments of the deviation to higher order than in the original formulation of Takagi-Taupin [4] have been recently presented [14,15]. In [16] a generalized DT (GDT) has been presented in which the only approximations are the two-beam case and the neglection of higher powers of the dielectric susceptibility. It is claimed that this model allows to describe the interaction of the X-ray beam with strained heterostructures in the whole angular range 0 to pa2 [16].…”

Structural and photoluminescence characteristics of molecular beam epitaxy-grown vertically aligned In0.33Ga0.67As/GaAs quantum dots

“…The position for the zeroth-order satellite peak can be estimated from Eq. (1) by setting m ¼ 0 and from this, the average indium composition is calculated [10]. The thickness of the constituent layers in the period is now estimated using the growth rates of GaAs and InGaAs determined from RHEED oscillations.…”

“…4a) consists of a narrow substrate peak and several satellite peaks at angles smaller than the substrate Bragg angle. Using the kinematical approximation, the thickness of the superlattice period and the average alloy composition can be estimated by using the following relationship [10]:…”

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

“…The exact solution of dispersion equation and complete matrices ͑33͒ have to be used for the grazing-incidence ͑-exit͒ geometry, 14 and for the diffraction from superlattices with short period, when satellite peaks are located far from the exact Bragg condition. 15 The numerical problems arise for superlattices with large number of periods N, where multiple production of (4ϫ4) matrices has to be done. 14 These problems are caused by operations with matrices, the elements of which grow exponentially with the increasing value of N. In order to avoid these difficulties, new algorithm was suggested in Ref.…”