Recurrence relations in the dynamical theory of X-ray diffraction in a lateral crystal

Darwin’s dynamical theory of X-ray diffraction is considered to the cases of plane-parallel and lateral (i.e., having a finite length in the lateral direction) crystals. Therefore this new approach can be widely used for non-destructive testing of lateral structures used in opto- and microelectronic devices, nonvolatile memory devices and X-ray optics. We provided a short review of Darwin’s approach in the case of plane-parallel crystals and analyzed the exact solution of this problem. The main consideration is devoted to the dynamical X-ray diffraction in the lateral crystal. We restrict ourselves to the case of a symmetrical coplanar Bragg diffraction. Recurrence relations describing the dynamical X-ray diffraction in the lateral crystal is obtained. The angular distribution of X-ray scattering intensity in the reciprocal space is investigated. Phase changes of X-ray waves in the crystal are analyzed in detail. It is shown that this approach allows one to calculate rocking curves as well as Reciprocal Space Maps (RSMs) for lateral crystals having rectangular cross-section. An algorithm for computing of rocking curves and RSMs from the lateral crystal is developed. In accordance with the model, we used a rectangular lattice having a fixed distance between nodes to describe the dynamical diffraction process. The simulation procedure based on recurrence relations consists of the external and internal cycles. The external cycle starts with the first column and goes up to the last column. The internal cycle (from top to bottom) starts with the first reflecting plane and goes up to the last one. Then substituting the amplitudes calculated for the first column we can calculate the amplitudes for the second column and so on. At the end of the external cycle we obtain arrays of the reflected and transmitted amplitudes. We perform numerical modelling for crystals with different lateral sizes. The obtained simulation results are in good agreement with the solution obtained by integration of Takagi’s equations. It is demonstrated that the kinematical approximation is valid for thick crystals having small length in the lateral direction. It is shown that this approach, being simple and transparent, is faster than the one based on Takagi’s equations and allows simulations of RSMs. The obtained results can potentially be extended further to the three-dimensional case in both the Fourier space and real space.