Generalized statistical dynamical theory of x-ray diffraction by imperfect multilayer crystal structures with defects

The generalized statistical dynamical theory of x-ray scattering by imperfect single crystals with randomly distributed Coulomb-type defects has been extended to characterize structure imperfections in the real multilayers of arbitrary thickness in Bragg diffraction geometry. The recurrence relations for the coherent amplitude reflection and transmission coefficients of such multilayers, which consist of any number of layers with constant strain and randomly distributed defects in each one, have been derived within the concept of the dynamical wave field, i.e., the so-called Ewald-Bethe-Laue approach, with rigorous accounting for boundary conditions at layer interfaces. The analytical expression for the differential dynamical diffuse component of the reflection coefficient of an imperfect multilayer system has been obtained as well. This expression establishes direct connection between the distribution of the diffuse scattering intensity in a momentum space and statistical characteristics of defects in each layer. In addition, the integrals from differential diffuse scattering intensity over the Ewald sphere and over vertical divergence have been found, which correspond to diffuse components in measurements of rocking curves and reciprocal space maps, respectively. The developed theory provides the dynamical description for the one- and two-dimensional angular distributions of the mutually consistent coherent and diffuse components of x-ray scattering intensities, which are measured by the high-resolution double- and triple-crystal diffractometers, respectively, from imperfect films, multilayer structures, superlattices, etc. Examples of simulated rocking curves for the imperfect superlattice with defects of several types and reciprocal space map for the ion-implanted sample of yttrium iron garnet film with defects are given and discussed.

Figure 1

Schematic diagram of relations between the wave vectors of incident, transmitted, and diffracted coherent plane waves in the single-crystal plate (a), arbitrary single layer of a multilayer crystal system (b), and epitaxial film on the substrate (c).

Figure 2

Relations between the wave vectors of coherent (${\mathbf{K}}_0^{\delta }$ and ${\mathbf{K}}_{\mathbf{H}}^{\delta }$) and diffusely scattered plane waves (${\mathbf{K}}_0^{\delta }+{\mathbf{q}}_{\delta \tau }$ and ${\mathbf{K}}_{\mathbf{H}}^{\delta }+{\mathbf{q}}_{\delta \tau }$) with momentum transfers ${\mathbf{q}}_{\delta \tau }$ on dispersion branches for the asymmetric Bragg diffraction geometry of the absorbing crystal (Ge 333, $\mathrm{Cu}\mathrm{K}{\alpha }_1$, asymmetry angle $\psi =5^{\circ }$). Angular deviations $\mathrm{\Delta }\theta$ and $\mathrm{\Delta }\theta {}^{\prime }$ of the wave vectors of incident ($\mathbf{K}$) and diffusely scattered ($\mathbf{K}{}^{\prime }$) plane waves in vacuum, respectively, from their exact Bragg reflection directions (shown by dashed lines) lead to the deviation $\mathbf{k}$ of the $\mathbf{K}{}^{\prime }$ vector from the reciprocal lattice point $H$. Two of four possible cases of the angular deviation combinations are shown: (a) $\mathrm{\Delta }\theta >0$ and $\mathrm{\Delta }\theta {}^{\prime }>0$, and (b) $\mathrm{\Delta }\theta >0$ and $\mathrm{\Delta }\theta {}^{\prime }<0$.

Figure 3

Schematic plot of the integration path of the diffuse scattering intensity in a momentum space, where the dash-dotted line represents the intersection of the coherent scattering plane $(\mathbf{K},\mathbf{H})$ with the integration plane tangent to the Ewald sphere, and ${\mathbf{k}}_{0j}$ is the deviation of the Ewald sphere from the reciprocal lattice point ${\mathbf{H}}_j$. At $|{\mathbf{k}}_{0j}|\le k_{\mathrm{m}j}$ the integration area includes both Huang ($|{\mathbf{k}}_{0j}+{\mathbf{k}}^{\prime }|\le k_{\mathrm{m}j}$) and Stokes-Wilson ($|{\mathbf{k}}_{0j}+{\mathbf{k}}^{\prime }|\ge k_{\mathrm{m}j}$) scattering regions.

Figure 4

Simulated rocking curves ($R$) and their coherent ($R_{\mathrm{coh}}$) and diffuse ($R_{\mathrm{diff}}$) components for the (004) reflection of $\mathrm{Cu}\mathrm{K}{\alpha }_1$ radiation from the imperfect InGaAs/GaAs superlattice containing structural defects of various types: thermal atom vibrations (a) and dislocation loops in GaAs substrate (b).

Figure 5

Simulated rocking curve ($R$) and its coherent ($R_{\mathrm{coh}}$) and diffuse ($R_{\mathrm{diff}}$) components for the imperfect $\mathrm{I}{\mathrm{n}}_{0.2}\mathrm{G}{\mathrm{a}}_{0.8}\mathrm{As}$ superlattice containing disk-shaped InAs particles (see Table 1); symmetric GaAs (004) reflection, and $\mathrm{Cu}\mathrm{K}{\alpha }_1$ radiation. The peak of the diffuse component of multilayers is located at the Bragg angle, which corresponds to the structure of $\mathrm{I}{\mathrm{n}}_{0.2}\mathrm{G}{\mathrm{a}}_{0.8}\mathrm{As}$ layers, and disposed between two satellites on the coherent component; mainly the right decreasing tail of the diffuse scattering intensity distribution can be seen in the considered angular range.

Figure 6

Strain profile in the YIG film implanted with $90-\mathrm{keV}{\mathrm{F}}^{+}$ ions at the dose $D=6\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-2}$.

Figure 7

Calculated reciprocal space map (in center), its coherent component (left), and longitudinal cross section (right) for 444 reflection of $\mathrm{Cu}\mathrm{K}{\alpha }_1$ radiation from YIG/GGG film system implanted with $90-\mathrm{keV}{\mathrm{F}}^{+}$ ions at the dose $D=6\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-2}$. Coherent peaks created from YIG film, GGG substrate, and ion-implanted layer are denoted by letters $F, S$, and $L$, respectively. Contours of equal intensity around these peaks are formed due to diffuse scattering from growth defects in the substrate and film, as well as owing to secondary radiation defects in the ion-implanted layer.