This thesis addresses the evolution of nanoscale ripple patterns on solid surfaces during low-energy ion sputtering. Particular attention is paid to the long-time regime in which the surface evolution is dominated by nonlinear processes. This is explored in simulation and experiment. In numerical simulations, the influence of anisotropy on the evolution of the surface patterns in the anisotropic stochastic Kuramoto-Sivashinsky (KS) equation with and without damping is studied. For a strong nonlinear anisotropy, a 90 rotation of the initial ripple pattern is observed and explained by anisotropic renormalization properties of the anisotropic KS equation. This explanation is supported by comparison with analytical predictions. In contrast to the isotropic stochastic KS equation, interrupted ripple coarsening is found in the presence of low damping. This coarsening seems to be a nonlinear anisotropy effect that occurs only in a narrow range of the nonlinear anisotropy parameter. Ex-situ atomic force microscopy (AFM) investigations of Si(100) surfaces sputtered with sub-keV Ar ions under oblique ion incidence show the formation of a periodic ripple pattern. This pattern is oriented normal to the direction of the ion beam and has a periodicity well below 100 nm. With increasing ion fluence, the ripple pattern is superposed by larger corrugations that form another quasi-periodic pattern at high fluences. This ripple-like pattern is oriented parallel to the direction of the ion beam and has a periodicity of around one micrometer. Interrupted wavelength coarsening is observed for both patterns. A dynamic scaling analysis of the AFM images shows the appearance of anisotropic scaling at large lateral scales and high fluences. Based on comparison with the predictions of different nonlinear continuum models, the recent hydrodynamic model of ion erosion, a generalization of the anisotropic KS equation, is considered as a potentially powerful continuum description of this experiment. In further in-situ experiments, the dependence of the dynamic scaling behavior of the sputtered Si surface on small variations of the angle of incidence is investigated by grazing incidence small angle X-ray scattering (GISAXS). A transition from strongly anisotropic to isotropic scaling is observed. This indicates the presence of at least two fixed points in the system, an anisotropic and an isotropic one. The dynamic scaling exponents of the isotropic fixed point are in reasonable agreement with those of the Kardar-Parisi-Zhang (KPZ) equation. It remains to be seen whether the hydrodynamic model is able to show such a transition from anisotropic to isotropic KPZ-like scaling.

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This thesis addresses the evolution of nanoscale ripple patterns on solid surfaces during low-energy ion sputtering. Particular attention is paid to the long-time regime in which the surface evolution is dominated by nonlinear processes. This is explored in simulation and experiment. In numerical simulations, the influence of anisotropy on the evolution of the surface patterns in the anisotropic stochastic Kuramoto-Sivashinsky (KS) equation with and without damping is studied. For a strong nonlinear anisotropy, a 90 rotation of the initial ripple pattern is observed and explained by anisotropic renormalization properties of the anisotropic KS equation. This explanation is supported by comparison with analytical predictions. In contrast to the isotropic stochastic KS equation, interrupted ripple coarsening is found in the presence of low damping. This coarsening seems to be a nonlinear anisotropy effect that occurs only in a narrow range of the nonlinear anisotropy parameter. Ex-situ atomic force microscopy (AFM) investigations of Si(100) surfaces sputtered with sub-keV Ar ions under oblique ion incidence show the formation of a periodic ripple pattern. This pattern is oriented normal to the direction of the ion beam and has a periodicity well below 100 nm. With increasing ion fluence, the ripple pattern is superposed by larger corrugations that form another quasi-periodic pattern at high fluences. This ripple-like pattern is oriented parallel to the direction of the ion beam and has a periodicity of around one micrometer. Interrupted wavelength coarsening is observed for both patterns. A dynamic scaling analysis of the AFM images shows the appearance of anisotropic scaling at large lateral scales and high fluences. Based on comparison with the predictions of different nonlinear continuum models, the recent hydrodynamic model of ion erosion, a generalization of the anisotropic KS equation, is considered as a potentially powerful continuum description of this experiment. In further in-situ experiments, the dependence of the dynamic scaling behavior of the sputtered Si surface on small variations of the angle of incidence is investigated by grazing incidence small angle X-ray scattering (GISAXS). A transition from strongly anisotropic to isotropic scaling is observed. This indicates the presence of at least two fixed points in the system, an anisotropic and an isotropic one. The dynamic scaling exponents of the isotropic fixed point are in reasonable agreement with those of the Kardar-Parisi-Zhang (KPZ) equation. It remains to be seen whether the hydrodynamic model is able to show such a transition from anisotropic to isotropic KPZ-like scaling.