Periodic ripples generated from the off-normal-incidence ion-beam bombardment of solid surfaces have been observed to propagate with a dispersion in the velocity. We investigate this ripple behavior by means of a Monte Carlo model of the erosion process, in conjuction with one of two different surface-diffusion mechanisms, representative of two different classes of materials; one is a Arrhenius-type Monte Carlo method including a term ͑possibly zero͒ that accounts for the Schwoebel effect, while the other is a thermodynamic mechanism without the Schwoebel effect. We find that the behavior of the ripple velocity and wavelength depends on the sputtering time scale, which is qualitatively consistent with experiments. Futhermore, we observe a strong temperature dependance of the ripple velocity, calling for experiments at different temperatures. Also, we observe that the ripple velocity vanishes ahead of the periodic ripple pattern.

We study solid surface morphology created by off-normal ion-beam sputtering with an atomistic solid-onsolid model of sputter erosion. With respect to an earlier version of the model, we extend this model with the inclusion of lateral erosion. Using the two-dimensional structure factor, we found an upper bound Ӎ 2, in the lateral straggle , for clear ripple formation. Above this upper bound, for longitudinal straggle տ 1.7, we found the possibility of dot formation ͑without sample rotation͒. Moreover, a temporal crossover from a hole topography to ripple topography with the same value of collision cascade parameters was found. Finally, a scaling analysis of the roughness, using the consecutive gradient approach, yields the growth exponents ␤ = 0.33 and 0.67 for two different topographic regimes. The results are discussed in the context of the continuum theory.

We implement substrate rotation in a 2 + 1-dimensional solid-on-solid model of ion-beam sputtering of solid surfaces. With this extension of the model, we study the effect of concurrent rotation, as the surface is sputtered, on possible topographic regions of surface patterns. In particular, we perform a detailed numerical analysis of the time evolution of dots obtained from our Monte Carlo simulations at off-normal-incidence sputter erosion. We found the same power-law scaling exponents of the dot characteristics for two different sets of ion-material combinations, without and with substrate rotation.

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Non-equilibrum surfaces, at nanometer length scales, externally driven via bombardment with energetic particles are known to exhibit well ordered patterns with a variety of applications in nano-technology. These patterns emerge at time scales on the order of minutes. Continuum theory has been quite successful in giving a general picture of the processes that interplay to give the observed patterns, as well as how such competition might determine the properties of the nanostructures. However, continuum theoretical descriptions are ideal only in the asymptotic limit. The only other theoretical alternative, which happens to be more suitable for the characteristic lengthand time-scales of pattern formation, is Monte Carlo simulation.In this thesis, surface morphology is studied using discrete solid-on-solid Monte Carlo models of sputtering and surface diffusion. The simulations are perfomed in the context of the continuum theories and experiments. In agreement with the experiments, the ripples coarsen with time and the ripple velocity exhibits a power-law behaviour with the ripple wavelength, in addition, the exponent was found to depend on the simulation temperature, which suggests future experimental studies of flux dependence. Moreover, a detailed exploration of possible topographies, for different sputtering conditions, corresponding to different materials, was performed. And different surface topographies e.g. holes, ripples, and dots, were found at oblique incidence, without sample rotation. With sample rotation no new topography was found, its only role being to destroy any inherent anisotropy in the system.iii DedicationTo my wife Taiwo, and children: Seun, Tobi, and Tife. iv First and foremost I would like to thank my thesis advisor PD. Dr. Alexander Hartmann for invaluable advice not only as pertains to this work, but numerical modelling and computation in general. I would like to thank Prof.