A Package for Fast Finite Element Method based Simulation of X-ray Diffraction From Nano-structures
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1228-KK01-08
A Package for Fast Finite Element Method based Simulation of X-ray Diffraction From Nano-structures
Eugen Wintersberger1 and Jay Oswald2 1
Department of semiconductor physics, University of Linz, A-4040 Linz, Austria
2
Theoretical and Applied Mechanics, Northwestern University, Evanston, IL 60208-3111, U.S.A.
ABSTRACT In this work a novel package for the calculation of the diffracted intensity from nanostructures based on finite element simulations is presented. Besides a short introduction into the algorithm which we have developed two examples namely the diffraction from Si/SiGe systems with ripples and quantum dots with dislocations are shown. INTRODUCTION X-ray diffraction is a powerful tool for the characterization of nano-structures. Properties like the chemical composition and strain state are accessible with x-ray diffraction techniques. This information can be drawn from 2D reciprocal space maps (RSMs). With novel 2D CCD detectors, now even 3D RSMs can be recorded within reasonable time. However, the evaluation of x-ray data usually requires computer simulations of the scattered intensity pattern. While 2D RSMs can be already pretty expensive to calculate, the 3D case can become virtually impossible with classical integration methods. To reduce computational effort, many programs are specialized to a particular system. This requires the implementation of new code anytime a new system is investigated. Finally, many of the available codes scale pretty bad on many processors and do not exploit the power of todays multicore-CPUs. THEORY An exact derivation of the expressions for the calculation of the scattered x-ray intensities from nano-structures used here can be found in [1]. In this work we will restrict ourself to
Figure 2:
Averaging is done by adding up the comFigure 1: h denotes the reciprocal lattice point of the reference lat- puted scattered intensities tice. Q = ks − k0 is the momentum transfer between the incidenting from many ensembles incoand the diffracted wave. herently.
the case of coplanar diffraction. Only single scattering processes are considered (kinematic approximation). Furthermore, we assume that all scattering processes are elastic. The scattered intensity can be written as a function of the reciprocal space coordinate q with I(q) ∝ |E(q)|2 . (1) Usually, the intensity would be expressed as a function of the momentum transfer Q = ks −k0 between the incidenting and the diffracted wave. However, since we are only interested in a small region around a particular reciprocal space point h using q is more convenient (see Fig. 1). h denotes a particular point in reciprocal space which belongs to a reference lattice (usually the substrate lattice of the sample). q is the position in reciprocal space relative to h. The averaging in eq. (1) is due to the fact that the incidenting x-ray beam is usually much larger then the structures under investigation. Therefore, the scattered intensity represents an average over many of such objects. The scattered wavefield E(q) in eq. (1) can
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