In this review we cover and describe the application of grazing incidence x-ray scattering techniques to study and characterize nanopattern formation on semiconductor surfaces by ion beam erosion under various conditions. It is demonstrated that x-rays under grazing incidence are especially well suited to characterize (sub) surface structures on the nanoscale with high spatial and statistical accuracy. The corresponding theory and data evaluation is described in the distorted wave Born approximation. Both ex situ and in situ studies are presented, performed with the use of a specially designed sputtering chamber which allows us to follow the temporal evolution of the nanostructure formation. Corresponding results show a general stabilization of the ordering wavelength and the extension of the ordering as a function of the ion energy and fluence as predicted by theory. The in situ measurements are especially suited to study the early stages of pattern formation, which in some cases reveal a transition from dot to ripple formation. For the case of medium energy ions crystalline ripples are formed buried under a semi-amorphous thick layer with a ripple structure at the surface being conformal with the crystalline/amorphous interface. Here, the x-ray techniques are especially advantageous since they are non-destructive and bulk-sensitive by their very nature. In addition, the GI x-ray techniques described in this review are a unique tool to study the evolving strain, a topic which remains to be explored both experimentally and theoretically.
