Influence of collision cascade statistics on pattern formation of ion-sputtered surfaces

Theoretical continuum models that describe the formation of patterns on surfaces of targets undergoing ion-beam sputtering are based on Sigmund’s formula, which describes the spatial distribution of the energy deposited by the ion. For small angles of incidence and amorphous or polycrystalline materials, this description seems to be suitable, and leads to the classic Bradley and Harper (BH) morphological theory [R. M. Bradley and J. M. E. Harper, J. Vac. Sci. Technol. A 6, 2390 (1988)]. Here we study the sputtering of Cu crystals by means of numerical simulations under the binary-collision approximation. We observe significant deviations from Sigmund’s energy distribution. In particular, the distribution that best fits our simulations has a minimum near the position where the ion penetrates the surface, and the decay of energy deposition with distance to ion trajectory is exponential rather than Gaussian. We provide a modified continuum theory which takes these effects into account and explores the implications of the modified energy distribution for the surface morphology. In marked contrast with BH’s theory, the dependence of the sputtering yield with the angle of incidence is nonmonotonous, with a maximum for nongrazing incidence angles.

Figure 1

Sample cascade originating from an impact of a $5\mathrm{keV}$ Cu ion on a Cu crystal. The angle of incidence is 60°. The cube shown acts just as scale and has size $2.65{\mathrm{nm}}^3$, while the full lattice simulated has size ${\left(10.6\mathrm{nm}\right)}^2\times 18\mathrm{nm}$.

Figure 2

Spatial distribution of ejected Cu atoms emerging from 6000 independent trials of hitting the $(x,y)$ crystal surface [oriented in (1,0,0) direction] with a single $5\mathrm{keV}$ Cu ion at normal incidence. Distances are measured in units of $a=3.61\mathrm{\AA }$.

Figure 3

Spatial distribution of ejected Cu atoms emerging from 6000 independent trials of hitting the $(x,y)$ crystal surface [oriented in (1,0,0) direction] with a single $5\mathrm{keV}$ Cu ion at 30° off-normal incidence. The arrow indicates the direction of the projection of the ion beam onto the surface. Distances are measured in units of $a=3.61\mathrm{\AA }$.

Figure 4

Distribution of ejected particles with different directions of velocity. Here, $\kappa$ denotes the angle between the projection of particle velocity onto the surface and the vector between point of ion impact and point of particle ejection.

Figure 5

Spatial distribution of ejected Cu atoms emerging from 6000 independent trials of hitting the $(x,y)$ crystal surface [oriented in $(58,32,\overline{39})$ direction] with a single $5\mathrm{keV}$ Cu ion at normal incidence. Distances are measured in units of $a=3.61\mathrm{\AA }$.

Figure 6

Probability of ejected particles vs distance $\rho$ from point of ion incidence (measured in units of $a=3.61\mathrm{\AA }$), as determined from the data of Fig. 5.

Figure 7

Surface density of mean energy of sputtered Cu atoms vs distance $\rho$ (measured in units of $a=3.61\mathrm{\AA }$) from point of ion incidence for $5\mathrm{keV}$ Cu ions on semilog scale. The solid line is the best fit of the data to an exponential with a polynomial prefactor, namely, $0.297({\rho }^2-0.392\rho )\exp (-1.27\rho )$. The dotted line, which corresponds to a fit to a Gaussian, is obviously inadequate.

Figure 8

Probability density of energy, which is transported to the surface by a single collision cascade emerging from a $5\mathrm{keV}$ Cu ion, on a log-log scale. The solid line corresponds to the function $5.259{(5.035+ϵ)}^{-1.874}$, which is the best fit to a simple power law $q∕{(b+ϵ)}^{\alpha }$, while the straight line represents a simple ${ϵ}^{-2}$ power law.

Figure 9

Conditional probability density $p(ϵ\mid \rho )$ to find energy $ϵ$ for ejected particles keeping the distance $\rho$ fixed. Different symbols correspond to different values $1.8⩽\rho ⩽10.0$.

Figure 10

Normalized sputtering yield $Y\left({\gamma }_0\right)∕Y\left(0\right)$ as a function of incidence angle ${\gamma }_0$, for the various one-dimensional energy distributions. Thick solid line: Bradley–Harper for $\overline{a}=3.8\mathrm{nm}$, $\alpha =2.2\mathrm{nm}$, $\beta =1.5\mathrm{nm}$. Thin solid line: modified Gaussian, Eq. (21), for $\overline{a}=3.8\mathrm{nm}$, ${\sigma }_z=2.2\mathrm{nm}$, ${\sigma }_x=1.5\mathrm{nm}$. Dashed line: exponential, Eq. (23), for $\overline{a}=3.8\mathrm{nm}$, ${\sigma }_z=2.2\mathrm{nm}$, ${\sigma }_x=0.28\mathrm{nm}$, $c=-0.14\mathrm{nm}$. Dotted line: truncated exponential, Eq. (23) with $c=0\mathrm{nm}$, and $\overline{a}=3.8\mathrm{nm}$, ${\sigma }_z=2.2\mathrm{nm}$, ${\sigma }_x=0.28\mathrm{nm}$.

Figure 11

Normalized sputtering yield $Y\left({\gamma }_0\right)∕Y\left(0\right)$ as a function of incidence angle ${\gamma }_0$, for the various two-dimensional energy distributions. Thick solid line: Bradley–Harper for $\overline{a}=3.8\mathrm{nm}$, $\alpha =2.2\mathrm{nm}$, $\beta =1.5\mathrm{nm}$. Thin solid line: modified Gaussian, Eq. (20), for $\overline{a}=3.8\mathrm{nm}$, ${\sigma }_z=2.2\mathrm{nm}$, ${\sigma }_{xy}=1.5\mathrm{nm}$. Dashed line: exponential, Eq. (19), for $c=-0.14\mathrm{nm}$, $\overline{a}=3.8\mathrm{nm}$, ${\sigma }_z=2.2\mathrm{nm}$, ${\sigma }_{xy}=0.28\mathrm{nm}$. Dotted line: truncated exponential, Eq. (19) with $c=0\mathrm{nm}$, and $\overline{a}=3.8\mathrm{nm}$, ${\sigma }_z=2.2\mathrm{nm}$, ${\sigma }_{xy}=0.28\mathrm{nm}$.

Figure 12

Normalized values of ${\Gamma }_x$ and ${\Gamma }_y$ for the distribution (1) using the same parameter values as in Fig. 11.

Figure 13

Normalized values of ${\Gamma }_x^g$ and ${\Gamma }_y^g$ for the distribution (20) using the same parameter values as in Fig. 11.

Figure 14

Normalized values of ${\Gamma }_x^e$ and ${\Gamma }_y^e$ for the distribution (19) using the same parameter values as in Fig. 11. The inset corresponds to $c=-0.14$, while the main panel corresponds to $c=0$.