Crater function moments: Role of implanted noble gas atoms

Spontaneous pattern formation by energetic ion beams is usually explained in terms of surface-curvature dependent sputtering and atom redistribution in the target. Recently, the effect of ion implantation on surface stability has been studied for nonvolatile ion species, but for the case of noble gas ion beams it has always been assumed that the implanted atoms can be neglected. In this work, we show by molecular dynamics (MD) and Monte Carlo (MC) simulations that this assumption is not valid in a wide range of implant conditions. Sequential-impact MD simulations are performed for 1-keV Ar, 2-keV Kr, and 2-keV Xe bombardments of Si, starting with a pure single-crystalline Si target and running impacts until sputtering equilibrium has been reached. The simulations demonstrate the importance of the implanted ions for crater-function estimates. The atomic volumes of Ar, Kr, and Xe in Si are found to be a factor of two larger than in the solid state. To extend the study to a wider range of energies, MC simulations are performed. We find that the role of the implanted ions increases with the ion energy although the increase is attenuated for the heavier ions. The analysis uses the crater function formalism specialized to the case of sputtering equilibrium.

Figure 1

Retained NG fluence as a function of implanted fluence for 1-keV Ar, 2-keV Kr, and 2-keV Xe bombardment of Si at an incidence angle of ${60}^{\circ }$ as obtained with MD (solid lines) and MC simulations (dashed lines). Note the significantly higher steady-state retained fluence in the MC simulations than in MD.

Figure 2

Steady-state Ar concentration profiles after 1-keV Ar bombardment of Si as obtained with MD (solid line), MC (dashed line), and MEIS [54] (dashed line with symbols).

Figure 3

Number of Si atoms vs number of NG atoms in a constant volume $V$ (a) after each impact for 1-keV Ar, 2-keV Kr, and 2-keV Xe bombardment of Si at an incidence angle of ${60}^{\circ }$, and (b) after each of the last 750 impacts of 2-keV Kr at various incidence angles. The dotted lines are fits of the NG atomic volume to the last 750 impacts at ${60}^{\circ }$ assuming the atomic volume of Si to equal the experimental value of ${\mathrm{\Omega }}_{\mathrm{Si}}=20.4{\AA }^3$ [56].

Figure 4

Atomic volumes of Ar, Kr, and Xe as obtained by our MD simulations (squares with error bars) and experimental values in the solid state (green circles, [55]). In the MD simulations the NG-NG and NG-Si interatomic potentials have been chosen according to ZBL [38] (blue symbols and line) and Süle et al. [40] (red symbol). The lines are drawn to guide the eye.

Figure 5

MD results for the zeroth moment $M^{\left(0\right)}$ of the crater function of Si bombarded with 2 keV Kr vs incidence angle. (a) Contributions by implantation and erosion comparing low fluence and steady state; (b) sum of contributions in steady state. The difference between the blue solid line and the green dashed line in panel (b) should vanish in sputtering equilibrium.

Figure 6

MD results for the first moment $M^{\left(1\right)}$ of the crater function [(a) and (b)] and for the curvature coefficient $S_{11}$ as defined in Sec. 2 (c) for Si bombarded with 2 keV Kr as a function of incidence angle. (a) Contributions by implantation, erosion, and redistribution; [(b) and (c)] sum of all contributions. For the meaning of the line styles see Fig. 5. In addition, results neglecting only the Kr erosive contribution [Eq. (24)] are shown by the magenta lines with triangles. The differences between the solid blue line and the dashed green line in (b) and (c) indicate the role of the implanted Kr ions.

Figure 7

Analogous to Figs. 5 and 6, but as a function of ion species for an incidence angle of ${60}^{\circ }$. For the meaning of the line styles see Fig. 6.

Figure 8

MD results for the curvature coefficients (a) $C_{11}$ and (b) $C_{22}$ for 2-keV Kr bombardment of Si considering only flat targets and Si atoms $[S_{ii}$ (Si), green dotted lines], considering in addition the effect of surface curvature $[S_{ii}+T_{ii}$ (Si), red dashed lines], and considering all contributions including those of the NG atoms as proposed in this work $[S_{ii}+T_{ii}$ (Si+Kr), blue solid lines]. Note that the first zero of $C_{11}$ would occur at $\theta <{40}^{\circ }$ if data points at smaller incidence angles were added.

Figure 9

Contributions of the NG (blue circles) and Si (green squares) to the first moment $M^{\left(1\right)}$ as obtained by MD (solid lines) and MC simulations (dashed lines). In addition, the implant plus erosive contributions of Kr are shown by the magenta triangles. (a) 2-keV Kr as a function of incidence angle; (b) for an incidence angle of ${60}^{\circ }$ as a function of ion species (1-keV Ar, 2-keV Kr, and 2-keV Xe).

Figure 10

(a) Distribution of the projected distance from the impact point $(=x$ coordinate) for 2-keV Kr bombardment of Si at an incidence angle of ${60}^{\circ }$; (b) mean projected distance from the impact point as a function of incidence angle for 2 keV Kr bombardment of Si. Red lines: implanted ions; green lines: sputtered Kr atoms. Solid lines: MD in steady state; dashed lines: MC; dotted lines: MD at low fluence. In (a), the MD results have been averaged over the last 1500 ion impacts.

Figure 11

MC results for the mean projected distances from the impact point for the implanted ions $({\overline{x}}_{\mathrm{NG},\mathrm{impl}}$, red lines and symbols) and for the sputtered atoms $({\overline{x}}_{\mathrm{NG},\mathrm{eros}}$, green lines and symbols) as a function of impact energy. Data for the three ion species Ar, Kr, and Xe at an incidence angle of ${60}^{\circ }$ are shown.

Figure 12

MC results for the curvature coefficients (a) $C_{11}$ and (b) $C_{22}$, comparing simulations with and without consideration of the effect of the implanted NG ions (solid and dashed lines, respectively). Data are shown for Si bombarded with Ar, Kr, and Xe at energies of 0.2, 2, 20, and 200 keV as a function of incidence angle. The results have been scaled with the square root of the impact energy to better represent them in the plots.