Local density profiles in thin films and multilayers from diffuse x-ray and neutron scattering

We develop a technique to determine local density profiles in conformally rough thin films and multilayers for which conventional reflectometry does not work. The main idea is to integrate the total scattered intensity for a given vertical momentum transfer over the parallel momentum transfer. Probing Fourier space globally results in a local probe in real space and the integrated intensity is proportional to the local reflectivity of the surface. We also discuss the influence of a finite range of integration as well as sample inhomogeneities, such as nonconformity of the roughness. This technique is limited to situations where the kinematic Born approximation is sufficient to describe the scattering process. However, in certain cases, the technique can be used in the vicinity of the critical angle of total external reflection as well.

Figure 1

Sketch of a conformal film on a rough substrate. The local density profile ${\rho }_0$ (shown on the right for the sample position $x_0$) is the same for all $x$ but shifted by the roughness $h\left(x\right)$. The density $\overline{\rho }$ averaged laterally over all $x$ (shown on the left) does not show the layer structure.

Figure 2

Three generic intensity distributions in the kinematic approximation observed from a multilayer on a substrate as a function of ${\alpha }_i$ and ${\alpha }_f$. (a) For smooth layers, one finds only a specular intensity contribution with peaks given by the Bragg position corresponding to the stacking (thick black diagonal). (b) Roughness leads to a decrease of the specular reflectivity and to diffuse scattering (sheets near the Bragg peaks). For dynamical effects, see Sec. 6. (c) The specular reflection vanishes for very rough samples (the expected position is indicated by the dotted diagonal line). Diagonals from the upper left to the lower right are lines of constant $q_z$.

Figure 3

X-ray or neutron scattering in reflection geometry. The incident beam (with momentum ${\mathbf{k}}_i$) lies in the $xy$ plane and impinges on the surface at an incidence angle ${\alpha }_i$. Scattered photons with final momentum ${\mathbf{k}}_f$ are detected at an exit angle ${\alpha }_f$ and out-of-plane angle $\theta$.

Figure 4

Density profile as a function of $x$ (or $y$) of a flat sample. The density depends only on the normal coordinate $z$.

Figure 5

Conformal density profile as a function of $x$ (or $y$) of a rough sample. The density profile is the same as in Fig. 4 for every lateral position $(x,y)$, but shifted along $z$ by $h(x,y)$.

Figure 6

Three one-dimensional sample surfaces generated with the same sequence of random numbers and with the same roughness and correlation length (indicated by the width of the dashed box) but for different roughness exponents $H=0.25$, 0.5, and 1.0. Note that the height variation within the box is smaller for larger $H$.

Figure 7

Illustration of a nonconformal layer on a one-dimensional rough substrate. The density as a function of $x$ and $z$ is shown in (d), while in (c), the density is shown as a function of $x$ and the distance from the substrate surface $\zeta$. (a) and (b) are density profiles as a function of $\zeta$ at $x_a$ and $x_b$, respectively.

Figure 8

Schematic of the scattered intensity (right) from a moderately rough layered system (left) as a function of ${\alpha }_i$ and ${\alpha }_f$. For ${\alpha }_{i∕f}⪢{\alpha }_c$, Born approximation is valid and the theory presented in this paper applies. For smaller angles, the specular intensity will dominate the integral of the intensity (checked area). The contribution to the integration from other areas are negligible, except for the Yoneda wings at ${\alpha }_{i∕f}\approx {\alpha }_c$ (hatched area). The dashed arrow indicates the path of an in-plane scan as presented in Fig. 9.

Figure 9

Simulated scattered intensity in a rocking scan (along the dashed arrow in Fig. 8) for a self-similar rough surface($2\pi ∕k_c=126{\xi }_{\Vert }$, $H=0.5$, and $q_z=50k_c$) as a function of $q_x$ (full line) in DWBA. In the center part, Born approximation is valid (dashed line) and Yoneda wings are present near the end of the rocking range. The product of transmission functions which gives rise to the Yoneda wings is shown as a dotted line.

Figure 10

(a) Illustration of the orientation of the ${\mathrm{H}}_2\mathrm{O}\mathrm{Si}{\mathrm{O}}_2$ assembly with respect to the scattering geometry. The solid line in (b) is the density profile resulting of a fit to the integrated intensity $I^{*}$ in Fig. 12. Fitting the partially integrated intensity (with respect to $q_y$ only, peak intensity $I_p$ in Fig. 11) results in a blurred profile (dashed line).

Figure 11

$q_y$-integrated in-plane scans for $q_z=0.25{\mathrm{\AA }}^{-1}$ (circles)and $q_z=0.16{\mathrm{\AA }}^{-1}$ (boxes). The sharp peak at $q_x=0$ depicts the resolution function, i.e., the expected width of the specular contribution. The inset shows the increase of the in-plane width $\Delta q_x$ as a function of perpendicular momentum transfer $q_z$.

Figure 12

Integrated intensity $I^{*}$ (circles) and peak intensity $I_p$ in the rocking scan (triangles, see Fig. 11) versus momentum transfer $q_z$. As illustrated in the inset, $I_p$ is the integral of the scattering cross section at $q_x=0$ with respect to $q_y$ for a given value of $q_z$ (fat line) while $I^{*}$ is the integral over the whole $q_x{q}_y$ plane. The full line and the dashed line are fits to $I^{*}$ and $I_p$, respectively, using a standard reflectivity analysis. The corresponding density profiles are shown in Fig. 10b.