Model-independent determination of the strain distribution for aSi0.9Ge0.1

Abstract

The strain distribution in a Si 0.9 Ge 0.1 /Si superlattice is determined from x-ray diffractometry data with a 25 Å depth resolution. A logarithmic dispersion relation is used to determine the phase of the structure factor with information available a priori on the sample structure. Phase information is obtained from the observed reflection intensity via a logarithmic Hilbert transform and the a priori information is used to select the zeros to be included in the solution. The reconstructed lattice strain profile clearly resolves SiGe and Si layers of 90-160 Å thickness alternately stacked on a silicon substrate. The SiGe layer is found to have a lattice spacing in the surface-normal direction significantly smaller than predicted by Vegard's law. The result gives simulated rocking-curve profiles in very good agreement with the observation. The apparent deviation from Vegard's law could be confirmed by chemical analysis.

Reply to a comment on `The interpretation of x-ray rocking cures by the Fourier transform method'

“…This example shows clearly that flipping the zeros into the lower half plane is indeed physically meaningful, contrary to the statement of Nikuin in [1]. The author of [1] cited a few of his own papers [7][8][9][10][11] to show that we should establish a firmer understanding of the inverse problem first. In those papers a procedure to retrieve the complex diffraction amplitude by analysing two measured intensities with two radiation energies was proposed.…”

A generalized Cahn–Hilliard equation incorporating geometrically linear elasticity

“…The resulting stress-strain relationship for the mixture is linear: this is Vegard's law. Vegard's law has been observed to fail in situations of great technological interest where the microstructure is known to be decisive for macroscopic properties, like, e.g., for Ga/As precipitates [56] and Ge/Si heterostructures [47]. An overview when Vegard's law fails is attempted in [28].…”

High-resolution X-ray diffraction imaging of non-Bragg diffracting materials using phase retrieval X-ray diffractometry (PRXRD) technique

“…Due to its one-dimensional nature, this method of X-ray phase retrieval is complicated by the difficulties presented by 'zero localization' [16], in which the 'true' zeros of the complex diffraction amplitude (CDA [14]), arising from interference suppression of the scattered waves, generally cannot be distinguished from the 'virtual' zeros arising from the numerical implementation of the mathematical formalism [18]. This uncertainty could result in non-unique solutions [19]. Two-radiation energy approach to the problem, however, has allowed the unique solution of the phase-retrieval procedure [14].…”