Swelling as a stabilizing mechanism in irradiated thin films: III. Effect on critical angle in a composite model

Abstract

Ion-beam irradiation of an amorphizable material such as Si or Ge may lead to spontaneous pattern formation, rather than flat surfaces, for irradiation beyond some critical angle against the surface normal. It is observed experimentally that this critical angle varies according to many factors, including beam energy, ion species and target material. However, many theoretical analyses predict a critical angle of about 45 degrees independent of energy, ion and target, disagreeing with experiment. Previous work on this topic has suggested that isotropic swelling due to ion-irradiation may act as a stabilization mechanism, potentially offering a theoretical explanation for the elevated critical angles in Ge compared to Si for the same projectiles. In the present work, we consider a composite model of stress-free strain and isotropic swelling with a generalized treatment of stress modification along idealized ion tracks. We obtain a highly-general linear stability result with a careful treatment of arbitrary spatial variation functions for each of the stress-free strain-rate tensor, a source of deviatoric stress modification, and isotropic swelling, a source of isotropic stress. Comparison with experimental stress measurements suggests that the presence of angle-independent isotropic stress may not be a strong influence on critical angle for the 250eV Ar+ on Si system. At the same time, plausible parameter values suggest that the swelling mechanism may, indeed, be important for irradiated Ge. As secondary results, we show the unexpected importance of the relationship between free and amorphous-crystalline interfaces in the thin film model for critical angle selection. We also show that under simple idealizations used elsewhere, spatial variation of stress may not contribute to critical angle selection. These findings prompt modeling refinements which will be the focus of future work.

Interfacial phase-change and geometry modify nanoscale pattern formation in irradiated thin films

“…We note that (a) Anisotropic Plastic Flow causes a linear shear flow in the x direction, (b) Ion-Induced Swelling induces a nontrivial z-dependence in most quantities, and (c) Boundary Amorphization multiplies all instance of the velocity V by ρc ρa > 1. In the "small swelling" limit discussed below and in our previous work [59][60][61], we find that this steady state also has an associated steady in-plane stress of…”

“…However, whereas Equation ( 13) predicts that the film thickness goes to zero at grazing incidence, experiments show that the film retains a comfortably non-zero thickness in this limit [21,42,43,68,71]. Even more worryingly, when this interface location is used, the predicted bifurcation angle drops from 45 • to only 30 • , far below the observed value [61]. These concerns motivate a more careful treatment than the simple geometric approaches described above, in hopes of obtaining a result somewhere between Equations ( 12) and ( 13).…”

“…with f again the ion flux, and A I a proportionality constant. Together, these equations describe the phenomenon of Ion Induced Swelling (IIS) [54][55][56][57][58][59][60][61]. Here ρ a is the initial density of the freshly-amorphized material, but over time, the accumulation of radiation-induced damage results in an increase in film volume by a (relative) amount ∆(x, z, t), which accumulates at rate f A I .…”

“…which we will refer to later. See also discussion in [61] where the above expression was compared with experimental results due to [41].…”

“…The resulting equations do not admit a simple closed-form solution, so for analytical tractability, we employ two simplifying limits that we have described more fully elsewhere [61]:…”