de Gennes Narrowing and Relationship between Structure and Dynamics in Self-Organized Ion-Beam Nanopatterning

Investigating the relationship between structure and dynamical processes is a central goal in condensed matter physics. Perhaps the most noted relationship between the two is the phenomenon of de Gennes narrowing, in which relaxation times in liquids are proportional to the scattering structure factor. Here, a similar relationship is discovered during the self-organized ion-beam nanopatterning of silicon using coherent x-ray scattering. However, in contrast to the exponential relaxation of fluctuations in classic de Gennes narrowing, the dynamic surface exhibits a wide range of behaviors as a function of the length scale, with a compressed exponential relaxation at lengths corresponding to the dominant structural motif—self-organized nanoscale ripples. These behaviors are reproduced in simulations of a nonlinear model describing the surface evolution. We suggest that the compressed exponential behavior observed here is due to the morphological persistence of the self-organized surface ripple patterns which form and evolve during ion-beam nanopatterning.

Figure 1

Top: A schematic diagram of the GISAXS experiment. The ion source is placed at the polar angle $\theta$, which causes self-organized rippling on the silicon surface. The sample is positioned so that the x-ray incident angle ${\alpha }_i$ is slightly above the critical angle of total external reflection. The scattering is recorded as a function of the exit angles ${\alpha }_f$ and $\psi$ using a 2D detector. Bottom: A detector image during nanopatterning. The Yoneda wing, spread across $q_z=0.36{\mathrm{nm}}^{-1}$ or $q_z^{\prime }=0.16{\mathrm{nm}}^{-1}$, is the surface-sensitive scattering exiting the sample at the critical angle ${\alpha }_c$. For ${\mathrm{Ar}}^{+}$ patterning, correlation peaks at $q_{\parallel }=q_0\simeq \pm 0.18{\mathrm{nm}}^{-1}$ are due to the correlated nanoripples on the surface.

Figure 2

KWW fits of $g_2^T(q_{\parallel },\mathrm{\Delta }t)$ for select wave numbers at aging time $T=775\mathrm{s}$. For ${\mathrm{Kr}}^{+}$ patterning, correlation peaks are at $q_{\parallel }=q_0\simeq \pm 0.21{\mathrm{nm}}^{-1}$.

Figure 3

Correlation times $\tau (q_{\parallel })$ and exponents $n(q_{\parallel })$ extracted from $g_2^T$ fits at different aging times $T$ for ${\mathrm{Kr}}^{+}$ patterning. Normalized $I(q_{\parallel })$ plots at corresponding aging times $T$ are also shown.

Figure 4

Correlation times $\tau (q_{\parallel })$ and exponents $n(q_{\parallel })$ extracted from $g_2^T$ fits in ${\mathrm{Ar}}^{+}$ patterning and its simulation.