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

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 angleθcof 45^∘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 value ofθcin 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 onθcfor the 250 eV Ar+→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 forθcof the relationship between free and amorphous-crystalline interfaces in the thin film model. We also show that under simple idealizations used elsewhere, spatial variation of stress may not contribute toθcselection. These findings prompt modeling refinements which will be the focus of future work.