High-Resolution Mapping of Two-Dimensional Lattice Distortions in Ion-Implanted Crystals from X-ray Diffractometry Data

Abstract

The triple-crystal synchrotron X-ray diffractometry data described by Nikulin, Stevenson, Hashizume, Wilkins, Cookson, Foran & Garrett [j Appl. Cryst. (1995), 28, 57-60] has been analyzed to map out two-dimensional (2D) lattice distortions in silicon (111) crystals implanted with B + ions of 100 keV energy through a periodic SiO2 strip pattern. The lateral periodic structure produced a series of satellite reflections associated with the 111 Bragg peak. The 2D reconstruction incorporates the use of the Petrashen-Chukhovskii method, which retrieves the phases of the Bragg waves for these satellite reflections, together with that for the fundamental. The finite Fourier series is then synthesized with the relative phases determined. Localized distortions perpendicular to the surface arising from deposited B + ions in nearsurface layers of the crystal are clearly displayed with spatial resolutions of 0.016 and 0.265 lam in the depth and lateral directions, respectively. For a sample with the oxide layer removed from the surface, two equally plausible strain maps have been obtained by assigning relative phases to 11 satellites using a sequential trial method and a minimum-energy method. Failed map reconstructions for the oxide-covered sample are discussed in terms of the non-unique solutions of the

<title>Phase-retrieval x-ray diffractometry in the case of high- or low-flux radiation source</title>

“…(9) In practice, the strain profile, Ad I d(z) , can be obtained either via equation (8) or by extracting the phase from the integrated inverse Fourier transform ofthe normalized complex diffraction amplitude. (9) In practice, the strain profile, Ad I d(z) , can be obtained either via equation (8) or by extracting the phase from the integrated inverse Fourier transform ofthe normalized complex diffraction amplitude.…”

“…The method has been applied successfully to determine the structure of one-and two-dimensional crystal-lattice strain distributions in SiGe/Si heterostructures 'and in silicon crystals, that have been implanted with high-energy ions [8][9][10] We have shown previously that a reliable structure-factor profile can be obtained by collecting data in a preferably symmetric range of the scattering vector length . In order to determine a crystal structure-factor without any a priori model of the sample under analysis, a new approach to a model-independent characterization of crystalline materials has been recently developed 16 The method is physically sound and self-consistent, and allows the unambiguous determination of the complex crystal-structure factor with a very high, 1-2 nm, depth resolution 46• The technique uses a logarithmic Hilbert transform (logarithmic dispersion relation) 1-6 to retrieve the phase of the x-ray wave diffracted by a single crystal under Bragg conditions.…”

Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals

“…The pioneering works concerning the direct imaging of the deformations in periodical crystalline structures from the diffraction patterns were published many years ago. 2,3 The theoretical studies of the advantages and limitations of the proposed method were published later. 4 The present work concentrates on the investigation of applicability of another modern technique called coherent diffractive imaging ͑CDI͒.…”

A comment on `The interpretation of x-ray rocking curves by the Fourier transform method'

“…The statement that 'The zeros cannot be obtained from the diffraction data alone' [1] (p 3296) is not substantiated. In fact, zeros of the complex diffraction amplitude can be 0022-3727/98/223323+02$19.50 c 1998 IOP Publishing Ltd obtained from the diffraction data alone [3,4,[7][8][9][10][11][12][13][14][15][16]. In [7], which was published in 1996, the inverse problem in x-ray Bragg diffraction was solved unambiguously using the physically substantiated formalism of the zero treatment.…”

“…The physical meaning of the zeros of the complex diffraction amplitude and their role in the inversion are elucidated in a recent paper [12]. The entire technique has been proven experimentally and applied for characterization of one- [4,[7][8][9][10][11][12] and two-dimensional [13][14][15][16] crystal lattice strains.…”