Date Published: September 01, 2018
Publisher: International Union of Crystallography
Author(s): K. V. Nikolaev, I. A. Makhotkin, S. N. Yakunin, R. W. E. van de Kruijs, M. A. Chuev, F. Bijkerk.
A theoretical description is given of the novel X-ray diffraction effect in single-crystal structures with a distorted crystal subsurface based on the dynamical theory of diffraction.
The motivation for the characterization of crystal surfaces can be found in, for example, the development of topological insulators (Ngabonziza et al., 2015 ▸) and spin-injection structures (Aronzon et al., 2008 ▸) in which their properties depend on the crystal subsurface structure. However, the characterization of this crystal subsurface is challenging because it typically requires high-brilliance synchrotron radiation (Robinson, 1986 ▸) that is generally not as readily available as radiation from compact laboratory-scale X-ray source setups.
In this section, we review the matrix formalism (Stepanov et al., 1998 ▸; Caticha, 1993 ▸) of the dynamical diffraction theory (Pinsker, 1978 ▸) that we implement for the numerical simulations in §4. The problem of X-ray diffraction is approximated by the scalar wave equation (Pietsch et al., 2013 ▸): where E is the scalar amplitude of the polarized electric wave and is the wavenumber of the wave in vacuum with wavelength λ. This equation was derived by assuming that diffraction is an elastic scattering process and the magnetic permittivity is equal to unity . The crystal structure is represented with a dielectric susceptibility as a function of coordinate .
In general, the measurement of GID rocking curves is carried out through an azimuthal rotation of the sample (rotation around the normal to the surface). We denote the angle between vector and the xy plane as ψ. Fig. 1 ▸ shows a typical measurement geometry for GID, where for clarity of the drawing the lattice planes are chosen to be perfectly perpendicular to the crystal surface , although the equations in §2 allow any orientation of the lattice planes. For the reciprocal-lattice vector lies in the surface plane. The coordinate system is chosen such that the tangential projection of the incident wavevector lies in the plane yz and the xy plane is parallel to the surface. Thereby, the orientation of the crystal planes is described by the position of the vector , i.e. by azimuthal angle φ.
In this section, we discuss predictions of the dynamical theory for the diffraction of a GID wave on a distorted crystal subsurface. The model sample is a silicon single crystal incorporating a thin ‘distorted’ single-crystalline Si layer with a lattice mismatch on top. The layer and substrate consist of the same material, and the lateral lattice mismatch of 0.1% is not sufficient to change the optical density; hence there is no optical contrast between the subsurface layer and the bulk of the single crystal. In this example, the thickness of the layer is d = 9 nm.
Implementation of the matrix formalism of the dynamical X-ray diffraction theory allowed us to describe theoretically a novel scattering process for single crystals that have defects in the crystal structure of the subsurface layers, that were introduced by the interaction with the atmosphere. GID waves induced in the subsurface interface and on the surface yield strongly asymmetric azimuthal curves. That asymmetry allowed us to estimate the difference between lattice constants of the subsurface and crystal substrate, the thickness of the distorted subsurface structure, the difference in miscut between the subsurface and substrate, and the optical contrast. Based on the obtained approximate solutions of the dispersion equation we conclude that these parameters are uncorrelated. It was also shown by means of simulations that GID modulates the specular reflection which potentially allows one to take measurements using laboratory-based instruments, making the technique widely accessible to researchers.