Reflection from rough surfaces and interfaces

The previous chapters treated surfaces and interfaces with perfectly flat surface. Real surfaces are rarely perfect. Induced by the surface structure of the substrate, thermal treatment, growth mode during deposition, diffusion into the substrate, etc. a variety of surface morphologies can be produced. Usually the surface has a rough structure with randomly distributed heights. The roughness can be described by a Gaussian distribution with the width $\sigma$ as sketched in the inset of figure . The fluctuations in the height diminish the reflectivity of the sample exponentially. The reflectivity of a rough surface or interface can be calculated from the known Fresnel reflectivity $r$ by

$$r_{\sigma}(Q)=r(Q)e^{-Q^{2}\sigma^{2}}.$$ (32)

A similar formula can be obtained in the case of scattering from graded (but flat) interfaces when $\sigma$ is the width of the graded region [17]. Unfortunately this means, that reflectivity experiments cannot unambiguously reveal the true structure of an interface.

Figure: Fresnel reflectivity of a 10 nm Fe$_{3}$Si thick film on a MgO substrate calculated with three different surface roughnesses: 0 Å(solid line), 2 Å(dashed line), 4 Å(pointed line). The faster decay of the reflectivity with increasing the roughness is clearly visible.