Origin of nonlinear sputtering during nanocluster bombardment of metals

Recent experiments [S. Bouneau et al., Phys. Rev. B 65, 144106 (2002)] show that the sputtering of atoms from surfaces by cluster impacts behaves differently from what is expected in the classical stopping theory. The most significant unresolved questions are that the sputtering yield divided by cluster nuclearity squared $(Y∕N^2)$ is independent of $N$ in the size range $N=2–13$ and that the energy maximum is not at the position expected from the nuclear stopping power. We use classical molecular dynamics simulations to examine the energy deposition from $0.1–18.5\mathrm{MeV}$ ${\mathrm{Au}}_5$ and ${\mathrm{Au}}_{13}$ clusters to the Au(111) surface to investigate this question. The simulations show that only a portion of the energy deposited from the cluster atoms to the crystal contributes to the formation of the displacement cascade because either the cluster channels through the surface layers as one entity or, if strong collisions occur, the energy deposited in these collisions is mostly carried away from the collision region by fast knock-on atoms. Based on the observations in the simulations, we develop an analytical model that explains the $N^2$ effect and an energy maximum that differs from the nuclear stopping.

Figure 1

(Color online) Experimental (Ref. 6) and simulated total sputtering yields as function of cluster energy (keV/atom). The experimental ${\mathrm{Au}}_5$ yields are shown in the left frame, and all three experimental ${\mathrm{Au}}_{13}$ yield points available in Ref. 6 are plotted in the right frame (squares). The ${\mathrm{Au}}_5$ yields in the middle frames are simulated with 0° and 6° angles of incidence and the ${\mathrm{Au}}_{13}$ yield in the right frame with 0°. The simulated yields are averages of three runs and the errors are standard deviations. The initial yields just after the sputtering has stopped are marked with circles, and the final yields after the fragmentation of sputtered clusters are marked with crosses. The blue lines show fittings of Eq. (4) to the data with parameters given in Table .

Figure 2

(Color online) Visualizations of velocities induced by different events. Top row: $1500\mathrm{keV}$ ${\mathrm{Au}}_{13}$, 0°; $1000\mathrm{keV}$ ${\mathrm{Au}}_5$, 0°; $140\mathrm{keV}$ ${\mathrm{Au}}_5$, 0°; and $60\mathrm{keV}$ ${\mathrm{Au}}_5$, 0°. Bottom row: $77\mathrm{keV}$ ${\mathrm{Au}}_{13}$, 0°; $1000\mathrm{keV}$ ${\mathrm{Au}}_5$, 6°; $140\mathrm{keV}$ ${\mathrm{Au}}_5$, 6°; and $60\mathrm{keV}$ ${\mathrm{Au}}_5$, 6°. Only atoms moving faster than $100\mathrm{km}∕\mathrm{s}$ are shown. The arrows are relative velocities on the plane. It follows that the atoms with small arrows may have considerable velocity perpendicular to the plane. Cluster atoms are marked with diamond (red in the online version). Snapshots are taken $15–40\mathrm{fs}$ after the impact depending on energy. ${\mathrm{Au}}_{13}$ results are from Ref. 13.

Figure 3

(Color online) A $1500\mathrm{keV}$/atom ${\mathrm{Au}}_{13}$ cluster penetrating into to the Au(111) crystal. The rightmost frame shows disorder in the same region after $55\mathrm{fs}$.

Figure 4

(Color online) Cluster energy loss in a $0–4.5\mathrm{nm}$ deep surface layer as function of the cluster’s angle of incidence. Each point is the average of 32 ${\mathrm{Au}}_5$ impacts. The lines are intended as guides for the eye.

Figure 5

(Color online) Energy deposited at $0–4.5\mathrm{nm}$ depth from the surface in ${\mathrm{Au}}_5$ impacts at various incidence angles. The results are averages of 32 simulations.

Figure 6

(Color online) Sums of horizontal and vertical velocities (total momenta) of the crystal atoms located in a rectangular volume $(4\times 4\times 14{\mathrm{nm}}^3)$ surrounding the cluster track. Notice the different velocity scale in the ${\mathrm{Au}}_{13}$ results. The horizontal velocity is defined as the sum of absolute values of atomic velocities parallel to the surface. The vertical velocity is the component perpendicular to the surface. The velocities of the cluster atoms are not included in the summation.

Figure 7

(Color online) Correlation of energies and horizontal velocities transferred in ${\mathrm{Au}}_5$ collision events. Each frame shows results from 32 runs. The negative energy values indicate that in some collisions, the cluster atoms gain energy from other cluster atoms either directly or transmitted by the crystal atoms.

Figure 8

(Color online) Cluster energy loss in the $8\mathrm{nm}$ deep surface layer divided in components coming from collisions of different strengths. The energy $E_c$ gained by the knock-on atoms in the collision is used to classify the collisions in the strength categories. The upper frames show the original energy losses and the lower frames the losses after filtering the collision events. The functional forms fitted in the yield distributions (Fig. 1) are shown in the lower frames (continuous lines) for comparison to the total energy loss curve.

Figure 9

(Color online) Visualization of cascade development after an impact of $280\mathrm{keV}$ ${\mathrm{Au}}_5$. Times of snapshot from left to right and from top to bottom are $100\mathrm{fs}$, $400\mathrm{fs}$, $1\mathrm{ps}$, $4\mathrm{ps}$, and $47\mathrm{ps}$. The frames show $4\times 22\times 2{\mathrm{nm}}^3$ slices; thus, the bottom in each frame is located at $20\mathrm{nm}$ from the surface. The stripes shown around the crater in the $47\mathrm{ps}$ frame are coherent displacements induced by the lateral pressure (Ref. 29).

Figure 10

(Color online) Left: Examples of simulated collision cascades of ${\mathrm{Au}}_5$ clusters. Each figure shows a $0.5\mathrm{nm}$ thick and $16\mathrm{nm}$ wide slice of the crystal $1.5–3.0\mathrm{ps}$ after the collision. At this stage, the cluster has broken apart and its energy is deposited to the secondary atoms, which has moved away from the initial collision region. Right: Schematic figure of the droplet model. Sputtering yield depends on the intersection area of the surface and the droplet shape, which describes the average shape of collision cascades. The portions of the imaginary droplets inside the solid have variable shapes depending on energy, and the cross section area varies. At low energies (a), the shape is almost perfect hemisphere, and at high energies (c), the shape is almost a cylinder. Between these extremes, a droplet shape appears (b).