Sputter yield of curved surfaces

The mean sputter yield produced by the impact of a single ion depends on the radii of curvature of the target surface at the point of impact. Using the Sigmund model of ion sputtering, we develop analytical formulas for this dependence for the case in which the radii of curvature are large compared to the size of the ion-induced collision cascade; both locally perpendicular and oblique ion impact are considered. The sputter yield is increased for impact on convex surfaces. The influence of surface curvature along the incident-ion azimuth and perpendicular to it are discussed separately. Our analytical results are in good agreement with Monte Carlo simulations for the specific case of 20 keV Ar ion impact on a cylindrical nanowire consisting of amorphous silicon. We also extend the results for this case to small radii of curvature using both Monte Carlo and molecular dynamics simulations.

Figure 1

Impact of a single ion at the point $O$ on the solid surface. The $z$ axis is normal to the surface at $O$ and the ion's direction of incidence $-\widehat{\mathbf{e}}$ is in the x-z plane. The local angle of incidence is $\theta$.

Figure 2

Sketch of the definition of impact angles on a cylindrical surface. (a) Normal-to-axis impacts are characterized by the incidence angle $\vartheta$ and the impact parameter $b$. (b) Central impacts are characterized by the incidence angle $\varphi$.

Figure 3

Sputter-yield correction functions (a) $g_1\left(\vartheta \right)$ for normal-to-axis impacts and (b) $g_2\left(\varphi \right)$ for central impacts, as obtained by analytical theory for a cylinder surface, Eqs. (11) and (12).

Figure 4

Sputter yield of a cylinder for perpendicular impact $Y_R(\vartheta =0,\varphi =0)$, normalized to the sputter yield of a flat surface, $Y_{\infty }$. (a) MC data for cylinders with large radii compared to the theoretical result Eq. (19). (b) MC data for all cylinders, compared to the data for spheres. Data plotted vs $\sqrt{R/a}$ in order to compress the abscissa axis.

Figure 5

Emission-site distribution for impact on a cylinder of radius $R/a=0.19$ with impact parameter $b=0.6R$ and $\varphi =0$. Direction of ion incidence and impact point are indicated by the red line. The distribution has been obtained with 1000 incident projectiles.

Figure 6

(a) Ratio of the sputter yield of a cylinder of radius $R, Y_R$, to the sputter yield of a flat target, $Y_{\infty }$, as a function of the impact angle $\vartheta$ for normal-to-axis impacts, i.e., for $\varphi =0$. (a) MC data for a cylinder of radius $R=9.574a$ compared to the analytical result given by Eq. (17). (b) All MC data.

Figure 7

(a) Ratio of the sputter yield of a cylinder of radius $R, Y_R$, to the sputter yield of a flat target, $Y_{\infty }$, as a function of the impact angle $\varphi$ for central impacts, i.e., for $\vartheta =0$. (a) MC data for a cylinder of radius $R=9.574a$ compared to the analytical result given by Eq. (18). (b) All MC data.

Figure 8

Average sputter yield, $\langle Y_R\rangle$, of a cylinder of radius $R$ for central impact (i.e., for $\varphi =0$) as a function of the dimensionless inverse cylinder radius, $a/R$. Data have been normalized to the value for a flat target. Data for impacts on a sphere of radius $R$ (Ref. [12]) have been added for comparison.

Figure 9

Absolute sputter yields for central impact ($\vartheta =\varphi =0$) as a function of the dimensionless inverse cylinder radius, $a/R$. The MC data are compared to the results of MD simulations.

Figure 10

Perspective view of sputtered cylinders 3 ps after ion impact in MD simulations. In all cases, the ion was incident from the top $(\vartheta =\varphi =0)$ onto the middle of the cylinder segments displayed. Atoms are colored according to the local temperature. (a) Representative events in which impact leads to the average sputter yield. (b) Events with abundant sputtering. Note the change in color code between (a) and (b).