X-ray scattering from real surfaces: Discrete and continuous components of roughness

Incoherent surface scattering yields a statistical description of the surface, due to the ensemble averaging over many independently sampled volumes. Depending on the state of the surface and direction of the scattering vector relative to the surface normal, the height distribution is discrete, continuous, or a combination of the two. We present a treatment for the influence of multimodal surface height distributions on crystal truncation rod scattering. The effects of a multimodal height distribution are especially evident during in situ monitoring of layer-by-layer thin-film growth via pulsed laser deposition. We model the total height distribution as a convolution of discrete and continuous components, resulting in a broadly applicable parametrization of surface roughness which can be applied to other scattering probes, such as electrons and neutrons. Convolution of such distributions could potentially be applied to interface or chemical scattering. Here we find that this analysis describes accurately our experimental studies of < 001 > SrTiO3 annealing and homoepitaxial growth.