Coupling free-surface geometry and localized ion dose for continuum models of radiation-induced nanopatterning

TL;DR

This work addresses the problem of radiation-induced nanopatterning by linking the localized collision cascade to the evolving free surface within a continuum hydrodynamic framework. It develops asymptotic approximations that map ion dose to depth-dependent deposition and to the amorphous-crystalline interface, including closed-form expressions for $h_0(\theta)$ and $x_0(\theta;k)$ and a small-curvature expansion. It demonstrates that the detailed coupling between surface geometry and depth-dependent dose can dramatically alter predictions for $\theta_c$, $\lambda(\theta)$, and surface roughness, often overturning results obtained with simpler, vertically displaced-interface assumptions. It then applies these results to a linear and weakly nonlinear theory (IIS and APF) to derive a nonlinear PDE for the height field, revealing new nonlinear behaviors such as mound formation and enhanced roughness sensitivity. The findings underscore the importance of realistic collision-cascade geometry in interpreting nanopatterning phenomena and guide future integrated modeling that couples depth-dependent stresses to surface evolution.

Abstract

A first-principles understanding of the self-organization of highly regular, nanometer-scale structures atop irradiated semiconductor surfaces has been sought for decades. While numerous models exist which explain certain aspects of this phenomenon, a unified, physical model capable of explaining all details of pattern formation has remained elusive. However, it is increasingly apparent that such a model will require understanding the dual influence of the collision cascade initiated by ion implantation: first, as a source of material transport by sputtering and atomic displacements occurring over short time scales, and, second, as a source of defects permitting viscous flow within the thin, amorphous layer that results from sustained irradiation over longer time scales. To better understand the latter, we develop several asymptotic approximations for coupling the localized ion dose with an evolving free interface. We then show how theoretical predictions of quantities commonly used for comparison with experimental observations -- such as ripple wavelengths, critical irradiation angle for patterning onset, and surface roughening -- exhibit surprising sensitivity to the details of this coupling.

Figures (4)

• Figure 1: Example of level sets of localized dose (or fluence) of ions through a deformed free surface (black curve) at $\theta=45^{\circ}$ off-normal incidence according to Equation \ref{['deposition-integral']}. Red regions receive the highest local dose, and purple regions receive the lowest. We have supposed that the free interface is described by $h(X,t) = \frac{3}{10}\sin(\frac{3X}{10})$. We choose parameters associated with 250eV Ar$^+$ irradiation of Si: $a=1.8$ nm, $\alpha=0.7$ nm, and $\beta=0.8$ nm srim-2000.40. Where the intensity of ion implantation falls off, the amorphous-crystalline interface begins. The goal of the first part of this paper is to determine closed-form expressions for (i) the amorphous-crystalline interface and (ii) the local, depth-dependent dose as a function of the vertical ($z$) axis and wavenumber $k$.
• Figure 2: Examples of computed $h_0(\theta),x_0(\theta;k)$ and $r_0(\theta;k)$ based on Equations (\ref{['interfaces-arbitrary']}) for 1000keV Ar$^+$-irradiated Si. Left: comparison of theoretical angle-dependent film thickness with film thickness inferred from experiments norris-etal-SREP-2017. Center: the lateral shift separating free and amorphous-crystalline interfaces plotted for three wavenumbers. Right: the flattening factor plotted for three wavenumbers.
• Figure 3: Left: Comparison of wavelength $\lambda(\theta)$ predictions from norris-PRB-2012-linear-viscous (dash-dotted red curve), experimental data madi-etal-2008-PRL, and how they are changed by incorporating the present work. The solid blue curve uses the full-spectrum (arbitrary $k$) results (\ref{['interfaces-arbitrary']}) for the description of the interfaces, while the dashed green curve uses the long-wave $(k\approx 0)$ results (\ref{['interfaces-x0-longwave']}. Center, right: Value of the dimensionless coefficient appearing in the linear dispersion relation due to IIS when depth dependence is connected to the distribution of implanted ions for irradiated Si and Ge.
• Figure 4: Left: simulated time series showing the evolution of surface roughness for three systems where $fA_D=3\times10^{-3} \frac{1}{s}, \gamma=1.36\frac{\text{J}}{\text{m}^2}, \eta=100 \text{ GPa}\cdot \text{s}$ and $\theta=65^{\circ}$. The only differences are in the assignment of $a,\alpha,\beta$. Center, right: cross-section of the irradiated surfaces at the same time. For 500eV Ar$^+$-irradiated Si, surface roughness has already saturated around 1.9 nm, while the 500eV Xe$^+$-irradiated Ge surface is still experiencing exponential growth of ripples.