X-ray multiple diffraction from crystalline multilayers: Application to a 90° Bragg reflection

A generalized solution to the problem of multiple-beam dynamical x-ray diffraction from single and layered crystals with vertical strain is presented. The new formalism embraces all possible diffraction geometries, including extreme cases of grazing and normal incidence and emergence. The expressions for the diffracted wave amplitudes were obtained as recursion formulas. Numerical simulations were implemented and shown to be computationally stable. The formalism was successfully tested using experimental data obtained from a SiC/Si single-layered structure in a 90° diffraction geometry.

Figure 1

Diffraction geometry and the Cartesian coordinate system in reciprocal space.

Figure 2

Two-wave rocking curves for SiC/Si structure (incident energy $E=9.135\mathrm{keV}$, layer deformation $\Delta a_z∕a=-2.68\times {10}^{-4}$, layer thickness $t=1.4\mathrm{\mu }\mathrm{m}$) which are simulated using different theoretical approaches: the formalism presented in the paper (solid line), the extended dynamical theory (Ref. 38) (dashed line), and solution of the Takagi-Taupin equations (Ref. 27) (dash-dotted line). The angular deviation on the abscissa axis is with respect to the exact 90° position.

Figure 3

Diffraction geometry.

Figure 4

Experimental setup.

Figure 5

A two-dimensional map of the experimentally measured (008) reflected intensity for SiC/Si structure as a function of the incident radiation energy and the sample angular deviation from the exact 90° position.

Figure 6

Simulated two-dimensional map of the (008) reflected intensity for a SiC/Si structure as a function of the incident radiation energy and the sample angular deviation from the exact 90° position. The simulation takes into account the energy-angular resolution function of the experimental setup shown in Fig. 3 and multiple-beam diffraction condition in the vicinity of the 90° incident angle.

Figure 7

Comparison of the experimental (solid line) and simulated (dashed line) angular profiles of the diffracted intensity at the same incident radiation energy. The angular deviation on the abscissa axis is with respect to the exact 90° position.

Figure 8

Simulated two-dimensional map of the transmitted intensity for a perfect Si crystal under the multiple-beam diffraction condition as a function of the incident radiation energy and the sample angular deviation in the vicinity of the 90° incident angle (Ref. 22).

Figure 9

Simulated two-dimensional map of the (008) reflected intensity for a SiC/Si structure under the multiple-beam diffraction condition as a function of the incident radiation energy and the sample angular deviation in the vicinity of the 90° incident angle.

Figure 10

Enlargement of the multiple diffraction areas shown in Fig. 8: (a) Si substrate energy-angular-diffracted intensity plot in the vicinity of the 90° incident angle, (b) SiC film energy-angular-diffracted intensity plot in the vicinity of the 90° incident angle.