In the previous chapter we have studied reflection from a multilayer with flat interfaces. However, the sample growth is a complicated process with a certain amount of randomness, and therefore it introduces structural imperfections. Since the X-ray reflectivity feels the profile of the index of refraction, we will consider inhomogeneities of two kinds: interface roughness and interdiffusion.
An interface between two layers is a (mathematical) surface between the two materials constituting both layers. Deposited atoms impact randomly on the surface of the layer already grown. Since their mobility is finite and many other conditions influence the layer growth, the layer does not grow constantly in all places of the layer surface. Therefore perfectly flat interfaces cannot be achieved and the modulation of the actual interface between the layers with respect to the ideal interface is referred to as roughness.
For these reasons it is indispensable to incorporate the roughness, i.e. the random character of the interfaces, into the reflectivity calculation.
The influence of the interface roughness and of the interdiffusion is indistinguishable in the specular reflectivity. Both can be characterized by the same root mean square roughness and a mean interdiffusion width. However, a laterally homogeneous interdiffusion does not scatter the radiation into other than the specular direction. On the other hand, statistically random interface roughness produces diffuse scattering.
The preliminary task before the reflectivity calculation of samples with rough interfaces is the description of the randomness. Therefore we start this chapter by studying the statistical properties of single rough interfaces and of the correlation between different interfaces of a multilayer. We show how the rough interfaces change locally the layer thickness. We use both the dynamical and kinematical theories for calculating the specular intensity from a rough interface and from a rough multilayer. Our calculation leads to the well-known exponential diminution factors that decrease the value of the Fresnel coefficients. We use this model to fit experimental reflectivity curves of sandwich and periodic multilayers.
Laterally inhomogeneous random roughness produces diffuse (incoherent) scattering. We use a distorted-wave Born approximation to calculate quantitatively the scattered intensity. We demonstrate all the features of the diffuse scattering on the measured map of a periodic multilayer.