Self-consistent approach to x-ray reflection from rough surfaces

A self-consistent analytical approach for specular x-ray reflection from interfaces with transition layers [I. D. Feranchuk et al., Phys. Rev. B 67, 235417 (2003)] based on the distorted-wave Born approximation (DWBA) is used for the description of coherent and incoherent x-ray scattering from rough surfaces and interfaces. This approach takes into account the transformation of the modeling transition layer profile at the interface, which is caused by roughness correlations. The reflection coefficients for each DWBA order are directly calculated without phenomenological assumptions on their exponential decay at large scattering angles. Various regions of scattering angles are discussed, which show qualitatively different dependence of the reflection coefficient on the scattering angle. The experimental data are analyzed using the method developed.

Figure 1

Scattering of x rays from a rough surface. The reflection $R\left(\alpha \right)$ and transmission $T\left(\alpha \right)$ coefficients depend on the scattering angle $\alpha$ and are expressed through the scattering amplitude $T(\vec{k},\vec{p})$ for different wave vectors of the final state (2).

Figure 2

The simulated x-ray reflectivity curves from a rough ${\mathrm{Si}}_{0.65}{\mathrm{Ge}}_{0.35}$ layer on a Si substrate by different methods: numerical solution of Parratt’s equation for the erf profile (solid line), renormalized by the Nevot-Croce factor the reflection coefficient (dots), and SCA in the first DWBA order (dashed line)

Figure 3

The fitted by SCA experimental reflectivity from a ${\mathrm{Si}}_{0.65}{\mathrm{Ge}}_{0.35}$ sample with large roughness (Ref. 6). For the fitting of the entire area of scattering angle, formula (45) has been used with the following parameters: $\sigma =45\mathrm{\AA }$, $h=2$, and $\nu =k^2{\alpha }_c{\delta }_x{\delta }_y{L}_c^2=0.012$. For comparison the dashed line shows the reflected intensity renormalized by conventional Nevot-Croce factor. The vertical lines distinguish the angle areas with different behavior of x-ray reflectivity.