Unambiguous determination of crystal-lattice strains in epitaxially grown SiGe/Si multilayers

Abstract

A new method for unambiguous reconstruction of crystal-lattice strains in epitaxially grown layers from high-resolution x-ray diffraction data is proposed. The technique uses x-ray diffracted intensity profiles collected for two different radiation wavelengths. We enhance the theory for the previously developed algorithm for model-independent determination of crystal-lattice strain profiles in single crystals with epitaxially grown top-surface layers. The method relies on the retrieval of the scattered x-ray wave phase from its intensity profile via a logarithmic Hilbert transform. This phase-retrieval technique is always associated with the problem of complex polynomial root finding. A practical procedure for the mapping of complex polynomial roots is proposed to distinguish true and virtual zeros. This allows the phase of the diffracted x-ray wave to be retrieved unambiguously. The method was applied to determine physical dimensions and concentration composition of a Si/Si1−xGex/Si alloy multilayer structure typical for SiGe heterobipolar transistor device.

Application of the Phase-Retrieval X-Ray Diffractometry to an Ultra-High Spatial Resolution Mapping of SiGe Films near the Absorption Edge of Ge

“…The thickness and Ge concentration values studied in the present experiment are consistent with parameters pertaining in the manufacture of new-generation ultra-high frequency transistors (e.g. hetero-bipolar transistor with up to 160 GHz clock frequency [5]). The ability to readily acquire information, with atomic spatial resolution, on the fine structure of the near-surface and interface regions can be extremely useful for the precise control of the structural properties of artificially grown top-surface layered structures.…”

“…Thus, we can unambiguously determine the complete phase profile, jðQðEÞÞ, using the analytical formalism [4]. In fact, once the locations of the physical zeros of the CDA are determined, we can calculate the structure-factor profile from the data obtained using either one of the radiation energies [5]. The fundamental limit for the spatial resolution, Dz min , that can be achieved in the present technique, is Dz min !…”

“…The assumption that the true CDA zero locations will be invariant for any radiation energy [3,5,6] works perfectly in a numerical simulation. In a real experiment however, the significant difference in the radiation energies used for the diffraction data collection (0.8-1.5 A [5] and 0.7-1.5 A [6]) causes not only a difference in the linear attenuation coefficients to be sufficient for the numerical data analysis, but also introduces a considerable difference in all the parasitic scattering phenomena such as thermal diffuse scattering, inelastic scattering, surface and interface roughness scattering etc. All these parasitic effects "blur" the location of the true zeros of the CDA so that the phase-retrieval analysis technique is "sensitive" to the experimental set-up [7].…”

“…The two-radiation energy approach to the inversion problem has been recently experimentally developed and implemented [1,5,6]. The assumption that the true CDA zero locations will be invariant for any radiation energy [3,5,6] works perfectly in a numerical simulation.…”

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

“…The technique relies on the retrieval of the phase of a scattered X-ray wave from the experimentally measured one-dimensional intensity profile using a logarithmic dispersion relation [16,17]. 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].…”