Crystal truncation rods from miscut surfaces

Crystal truncation rods are used to study surface and interface structure. Since real surfaces are always somewhat miscut from a low index plane, it is important to study the effect of miscut on crystal truncation rods. We develop a model that describes the truncation rod scattering from miscut surfaces that have steps and terraces. We show that non-uniform terrace widths and jagged step edges are both forms of roughness that decrease the intensity of the rods. Non-uniform terrace widths also result in a broad peak that overlaps the rods. We use our model to characterize the terrace width distribution and step edge jaggedness on three SrTiO$_3$ (001) samples, showing excellent agreement between the model and the data, confirmed by atomic force micrographs of the surface morphology. We expect our description of terrace roughness will apply to many surfaces, even those without obvious terracing.