For decades, it has been observed that broad-beam irradiation of semiconductor surfaces can lead to spontaneous self-organization into highly regular patterns, sometimes at length scales of only a few nanometers. Initial theory was largely based on erosion and redistribution of material occurring on fast time scales, which are able to produce good agreement with certain aspects of surface evolution. However, further experimental and theoretical work eventually led to the realization that numerous effects are active in the irradiated target, including stresses associated with ion-implantation and the accumulation of damage leading to the development of a disordered, amorphous layer atop the substrate. It was also shown that relaxation of this amorphous layer proceeds in a manner closer to viscous flow rather
than surface diffusion on a crystal lattice.

Observing the viscous character of the amorphous layer, it is natural to consider whether stress-based continuum models might help explain pattern formation under ion bombardment and the observations described above. Indeed, there are early indications from the experimental literature that this may be the case, and, at low energies (∼ 1keV), at least one experimental-theoretical study has shown that they may even dominate erosive and redistributive effects in their contribution to surface evolution.

In this thesis, we develop a continuum model based on viscous thin-film flow and ion-
induced stresses within the amorphous layer. This model is a composite of, and significant generalization of, a previously-studied “anisotropic plastic flow” (APF) mechanism and a previously-studied “ion-induced isotropic swelling” (IIS) mechanism. Previous work has shown that, with certain simplifying assumptions about the amorphous-crystalline interface and spatial homogeneity of anisotropic plastic flow, this mechanism produces an instability capable of predicting pattern formation beginning at 45◦ angle of incidence against the macroscopically-flat substrate, consistent with some experimental systems. Under similar simplifying assumptions, ion-induced swelling has been shown to be capable of suppressing pattern formation. Our generalizations allow the use of simulation data to inform both linear and nonlinear surface evolution due to the spatial localization of APF and IIS to certain regions of the bulk, improved treatment of the amorphous-crystalline geometry, and
boundary conditions suitable to the physical systems of interest. We are then able to provide insight into several phenomena that have previously been difficult to explain, but seem to emerge naturally from a more detailed treatment of the physical system.

Degree Date

Spring 2023

Document Type


Degree Name





Scott Norris

Second Advisor

Vladimir Ajaev

Third Advisor

Thomas Hagstrom

Fourth Advisor

Paul Krueger

Subject Area

Mathematics, Applied

Number of Pages




Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License