Nonuniversality due to inhomogeneous stress in semiconductor surface nanopatterning by low-energy ion-beam irradiation

A lack of universality with respect to ion species has been recently established in nanostructuring of semiconductor surfaces by low-energy ion-beam bombardment. This variability affects basic properties of the pattern formation process, like the critical incidence angle for pattern formation, and has remained unaccounted for. Here, we show that nonuniform generation of stress across the damaged amorphous layer induced by the irradiation is a key factor behind the range of experimental observations, as the form of the stress field is controlled by the ion/target combination. This effect acts in synergy with the nontrivial evolution of the amorphous-crystalline interface. We reach these conclusions by contrasting a multiscale theoretical approach, which combines molecular dynamics and a continuum viscous flow model, with experiments using ${\text{Xe}}^{+}$ and ${\text{Ar}}^{+}$ ions on a Si(100) target. Our general approach can apply to a variety of semiconductor systems and conditions.

Figure 1

Schematic summary of the methods and approaches employed in this work.

Figure 2

(a) AFM top view of experimental ripples obtained by ${\text{Xe}}^{+}$ IBS of Si for $E=500$ eV and $\theta ={65}^{\circ }$. (b) HRTEM cross-sectional high-resolution image along the $\langle 110\rangle$ direction. The free interface is marked by dots. (c) MD simulation after one ${\text{Xe}}^{+}$ impact under the same conditions as in (a). Different colors stand for different values of the temperature, which decreases with distance from the ion entry.

Figure 3

(Small, upper image) Cross-sectional multibeam image along the $\langle 110\rangle$ direction, corresponding to Fig. 2. The bar corresponds to 20 nm. (Large, lower image) Corresponding HRTEM image of the area marked with a square in the center of the upper image. The HRTEM image allows to appreciate the characteristic $\left\{111\right\}$ Si-lattice fringes of the crystalline phase. Thickness variations in the amorphous-Si layer are also visible, together with the Pt layer. The bar corresponds to 5 nm.

Figure 4

MD simulations of ${\text{Xe}}^{+}\to$ Si. (a) Diagonal element of the excess (ion-induced) stress tensor in the normal direction, ${\tau }_{zz}$, for $E=300$ eV. Note the nonuniform distribution of stress across the amorphous layer. (b) Dependence of the total stress $\overline{\tau }$ with ion energy (circles are simulation data). In this range of energies, the numerical results are compatible with different models for defect relaxation [48], as detailed in the legend (nonsolid lines). Simple Davis' scaling [41] (solid line) seems to capture the simulation results better.

Figure 5

Laboratory coordinate system and geometrical quantities defined in the main text. We illustrate the geometrical relations between the distance along the ion beam trajectory, $d_{z^{\prime }}$, and the thickness of the amorphous layer, $R_0$; and between the laboratory and local ion incidence angles, $\theta$ and ${\gamma }_0$, respectively, and the local inclination angle, $\gamma$. The amorphous layer is represented as the grey area, being bounded by the free and the amorphous-crystalline interfaces, which are denoted by $h$ and $h_{\mathrm{ac}}$, respectively.

Figure 6

Dependence of the critical angle ${\Theta }_c$ with the stress ratio $g$, Eq. (25). The horizontal gray bar indicates the experimental critical angle range for ${\text{Xe}}^{+}\to \text{Si}$ $\left({\theta }_c=58\pm 2^{\circ }\right)$. The symbols show values from MD simulations, see legend. The inset is a zoom to show the weak dependence of ${\theta }_c$ with $E$. The orange dashed horizontal line is a guide to the eye with asymptote $\theta =61.8^{\circ }$.

Figure 7

Ripple propagation velocity vs incidence angle under a non-BH scenario for ripple formation with $g=-0.5$ [the value obtained in Fig. 4], $\Sigma =0.046,a/\sigma =2.04$ and $\mu /\sigma =0.66$ (see Ref. [59]), for $B=1$ (blue solid line) and $B=0.1$ (red circles). The red dotted line is a guide to the eye. The shaded area is the region where the surface is morphologically stable (there are no ripples), delimited by the critical angle for pattern formation (vertical black dashed line) at $\theta ={\theta }_c$. The values of the transition angles mentioned in the text, ${\widehat{\theta }}_c$ and ${\tilde{\theta }}_c$, are indicated.

Figure 8

(a) Experimental ripple wavelength vs $\theta$ for Si at 500 eV, using ${\text{Ar}}^{+}$ (black circles, from Ref. [11]) and ${\text{Xe}}^{+}$ (red squares) ions. The solid black and the dashed red lines are fits to Eq. (27) for ${\theta }_c={46}^{\circ }$ and ${58}^{\circ }$, respectively. (b) $\lambda$ vs $E$ for ${\text{Xe}}^{+}$ according to Eq. (26) using MD data (red squares) and experiments (black circles). Straight lines are linear fits.