A theoretical study of forward and backward sputtering produced by the impact of single 20 keV Ar ions on freestanding amorphous Si membranes is carried out. We use three techniques: Monte Carlo (MC) and molecular dynamics (MD) simulations, as well as analytical theory based on the Sigmund model of sputtering. We find that the analytical model provides a fair description of the simulation results if the film thickness d exceeds about 10%–30% of the mean depth of energy deposition a. In this regime, backward sputtering is nearly independent of the membrane thickness and forward sputtering shows a maximum for thicknesses . The dependence of forward sputtering on the ion's incidence angle shows a qualitative change as a function of d: while for , the forward sputter yield has a maximum at oblique incidence angles, the maximum occurs at normal incidence for . As the membrane thickness is reduced below 0.1–, the theory's predictions increasingly deviate from the MC results. For example, the predicted forward sputter yield approaches a finite value but the MC result tends to zero. This behavior is interpreted in terms of energy deposition and sputtering efficiency. Near-perfect agreement is observed between the sputter yields calculated by MD and MC simulations even for the thinnest membranes studied (d = 5 Å).
I. INTRODUCTION
Sputtering of nanosystems has garnered considerable interest recently,1 and investigations of the sputtering of nanospheres2–10 and nanowires11–15 have been carried out. A membrane—defined here as a free-standing planar thin film whose thickness is comparable to the ion range—is another example of a spatially constrained system in which a dimension reaches the nanoscale. As for nanospheres and nanowires, forward sputter yields from membranes can be comparable to or exceed the backward yield.
Thin membranes are used in a number of applications. In biomolecular detection systems, for example, nanopores are formed in a membrane so that the passage of single molecules can later be monitored.16 Ion beams can be used to fabricate these nanopores17–19 as well as to controllably thin the membranes.20 While sputtering may not be the only physical mechanism determining the final target membrane geometry,17 an understanding of sputtering from membranes is essential if they are to be optimized for their chosen purpose.
In the present paper, we use a simple and well understood system—20 keV Ar ion impact on amorphous Si—as a case study to investigate the physics of sputtering from membranes. We use Monte Carlo (MC) simulations as our main tool, since they allow us to efficiently investigate how sputtering depends on the membrane's thickness and the ion's angle of incidence. Molecular dynamics (MD) simulations are used to corroborate the results for thin membranes, where the effects of the membrane surfaces are the most pronounced. Finally, the simulation results are compared to approximate analytical results obtained from Sigmund's theory of ion sputtering.21,22
II. ANALYTICAL MODEL
According to the Sigmund model of ion sputtering, the average number of atoms sputtered from an infinitesimal surface element dA centered on a point is
Here, denotes the nuclear energy deposition (NED) density at the point , i.e., the average energy per unit volume deposited in nuclear collisions at . The proportionality constant Λ is referred to as the sputtering efficiency. Sigmund argued that Λ only depends on the target material,21 but it has been subsequently found that it also depends on the ion species and energy.23
Sigmund also assumed that the NED density is given by a Gaussian,22 i.e.,
Here, is the average energy deposited in nuclear collisions per incident ion, and α and β are the longitudinal and transverse straggling lengths, respectively. The point of maximum NED is taken to lie a distance a along the ion incidence direction from the point of impact. and denote components of the vector leading from the point of maximum NED to the point . In particular, and are the components of that are parallel and perpendicular to the ion's incidence direction.
The membrane surface that the ion contacts first will be called the backward surface; the other surface will be referred to as the forward surface. The forward sputter yield is obtained by integrating Eq. (1) with the NED density ED given by Eq. (2) over the forward surface, and the backward sputter yield is defined in an analogous fashion. As is shown in the Appendix, the forward sputter yield Yf is a Gaussian function of the membrane thickness d with parameters that depend on Λ, , a, α, and β as well as the ion's incidence angle θ. The backward yield Yb is independent of d since it is obtained by setting the membrane thickness to zero, i.e., .
III. SIMULATION METHODS
The MC simulations were performed with the IMSIL code in its static mode24,25 in laterally infinite membranes of amorphous Si with given thickness d. Atoms interacted via the Ziegler-Biersack-Littmark (ZBL) potential.26 Electronic stopping was calculated using a mixed Lindhard/Oen-Robinson model with an equipartition rule.27–29 Parameters that may influence the results for sputtering and NED include the surface binding energy Es, the displacement energy Ed, and the cutoff-energy for trajectories Ef. A planar surface potential was assumed,21,30 i.e., recoil atoms needed to surmount a planar potential barrier of height Es before they were emitted from the target. We took Es to be 4.7 eV, the heat of sublimation of Si. The values of Ed and Ef are not critical for the purposes of our study as long as they are not larger than Es near the surface and not too large elsewhere. We therefore used , and so effectively eliminated all recoil motion at energies below Es. While this may have overestimated defect production—which is of no interest here—it faithfully reproduces sputtering and provides a simple model in which to discuss the NED. NED was calculated by summing up the energies of recoils that were not followed and the remaining energies of the ions and recoils when their trajectories were terminated. In all simulations, cascades of 100 000 impinging ions were simulated, except for some thick-membrane simulations in which up to 107 ions were used.
In addition, we performed MD simulations for thin a-Si membranes with thickness nm and impact at normal incidence. The membranes were cut out of an a-Si target that was prepared according to the recipe of Luedtke and Landman.31 The membranes had a lateral extension of 100 Å × 100 Å; their lateral boundaries were fixed. The thickest membrane (d = 7 nm) contained 35 152 atoms. The silicon atoms interacted via the Stillinger-Weber potential.32 For small interaction distances, the potential was fitted to the ZBL potential.26 Ar and Si atoms interacted via the ZBL potential. In each case, 250 impacts were simulated for a time of 3 ps. The impacts differed in that in each case a different impact point was chosen at random in the central part of the membrane, at a distance of at least 25 Å from the boundaries to prevent interaction with the boundaries. For thin membranes, the simulation time 3 ps is sufficient; for example, for a 2 nm membrane, no atoms are sputtered after 1 ps. For a 7 nm membrane (the thickest membrane we studied), we carried out an extrapolation of the time distribution of sputtering events that occurred before 3 ps. This showed that if our simulations had been continued until all sputtering had ceased, the resultant change in the sputter yield would have been around 5%. This is on the order of the statistical error in the MD simulation results.
IV. RESULTS AND DISCUSSION
Unless otherwise noted, all results presented here are for 20 keV Ar ion impact on an amorphous Si membrane of thickness d.
A. Sputter yields
Figure 1 summarizes the main results of this study for perpendicular ion impact. The forward sputter yield Yf increases with membrane thickness and has a maximum at around . Its shape follows the Gaussian form that the Sigmund model yields for Yf quite well. The parameters in the Gaussian NED distribution have been obtained from MC simulations. In IMSIL, it is possible to start the ions in the interior of the target, thus simulating energy deposition in an “infinite medium.” The moments of the MC data and the total amount of NED provide us with the parameters of the Gaussian function Eq. (2): a = 205.5 Å, Å, Å, and keV. The value of the sputtering efficiency Å/eV has been chosen so that the analytical prediction has the same maximum forward sputter yield as the MC simulations.
Forward and backward sputter yields as a function of membrane thickness d for perpendicular impact (). In panel (a), the MC results obtained with IMSIL (symbols connected with lines) are compared with SRIM results (crosses) and the theoretical estimate (solid lines). The height of the theoretical forward sputter yield has been adjusted to the MC data. Panel (b) provides a comparison of the MC and MD results for small thicknesses. For the purpose of comparison, the MC data shown in (b) have been obtained by switching off electronic stopping.
Forward and backward sputter yields as a function of membrane thickness d for perpendicular impact (). In panel (a), the MC results obtained with IMSIL (symbols connected with lines) are compared with SRIM results (crosses) and the theoretical estimate (solid lines). The height of the theoretical forward sputter yield has been adjusted to the MC data. Panel (b) provides a comparison of the MC and MD results for small thicknesses. For the purpose of comparison, the MC data shown in (b) have been obtained by switching off electronic stopping.
Note that the forward sputter yield is sizable and larger than the backward sputter yield for a wide range of membrane thicknesses around d = a. This is caused by the fact that NED is larger in the center of the cascade at depths around a from the surface rather than at the surface itself or deep within the target. The analytical model describes this feature well. In contrast, as the membrane thickness is reduced below , the model's predictions increasingly deviate from the MC results. In particular, the predicted forward sputter yield approaches a finite value, but the MC result tends to zero.
The backward yield has a monotonic dependence on d. It attains its saturated value at thicknesses . This is relevant information for the setup of the MD simulations in that it predicts the minimum thickness a target should have in order to model an “infinite” target satisfactorily. As mentioned in Sec. II, the analytical model predicts the backward yield to be independent of d. This obvious failure is caused by a reduction in surface NED and sputtering efficiency with decreasing membrane thickness; this is not taken into account by the model. In addition, if the Λ value obtained by fitting to the forward sputter yield is employed, the theoretical backward sputter yield does not match the MC data for thick membranes perfectly. These features will be discussed further in Secs. IV B and IV C.
Also shown in Fig. 1(a) are results obtained by the widely used SRIM code.33 They confirm the general trends discussed above, although the sputter yield values produced by SRIM are consistently higher than those obtained using IMSIL. Unphysical nonmonotonic behavior of the forward sputter yield computed using SRIM is observed for the smallest membrane thicknesses. It has been argued previously that SRIM cannot be relied upon to give accurate sputter yields.34
Near-perfect agreement is observed between the sputter yields calculated by MD and MC simulations, as shown in Fig. 1(b): The MC data are within the error bars of the MD results. In the MC simulations we carried out to make this comparison, electronic stopping was switched off in order to define a model consistent with the MD simulations. Differences between MD and MC results have been found previously in sputtering of a-Si nanospheres and nanocylinders,8,15 and have been ascribed to nonlinear collision-cascade (“spike”) effects which are beyond the MC collision models. Our findings indicate that for membranes, which are confined only in one spatial dimension, collision spikes do not play as important a role as they do for nano-objects confined in two or three spatial dimensions. The excellent agreement between the results of the two different simulation techniques, MC and MD, shows that the physics of collision cascades and of sputtering is well represented by both. This is a nontrivial observation, particularly in the case of thin membranes, where the structure of the surface—which is atomistically represented in the MD simulations but is only schematically described as a planar potential barrier in MC simulations—might be thought to influence the sputtering results the most. As Fig. 1(b) demonstrates, even for the thinnest membranes studied (d = 5 Å), the MD and MC results coincide nicely.
To test the range of validity of this finding, we performed a MD simulation in which the ions were incident at 80° on a 20 Å thick membrane. In this case, we find the MD sputter yields to be 15% ± 8% larger than the values determined by MC. We believe that this is due to the fact that the center of NED nears the surface when the angle of incidence is increased. This situation is similar to the bombardment of nano-spheres and -cylinders. In those cases, the surface can be closer to the center of NED if it is curved or if the ions impinge on the inclined parts of the surface in non-central impacts. However, the 15% discrepancy between the MD and MC sputter yields for oblique incidence impact on the membrane is much lower than those for cylinders15 and spheres8 in comparable impact conditions. For those nano-objects, the differences amounted to 50% or more, and increased with decreasing size of the nano-objects.
Figure 2 displays a few snapshots of membranes of various thicknesses after bombardment. These results are provided by the MD simulations. For each thickness, we show both a representative event, where the forward and the backward sputter yields are about average, and an event with abundant sputtering. For the thinnest membranes studied (those with d = 5 Å), even the emission of a few atoms gives rise to a hole through the film (Fig. 2(a)). In the exceptional cases of abundant sputtering, we see such hole-forming events even for 20-Å thick membranes. Note that in the cases of abundant sputtering shown, the sputter yields are an order of magnitude higher than the average yields. Such fluctuations, however, are typical for sputter yield statistics.35–37
MD results of single ion impacts on membranes with various thicknesses d. The snapshots of the sputtered membranes, viewed from the backside, were taken at 3 ps after ion impact. Atoms are colored according to the local temperature. The black dot marks the ion's impact point. The figure shows events with (a) representative and (b) abundant sputtering. The backward sputter yield Yb and the forward sputter yield Yf are given for each event.
MD results of single ion impacts on membranes with various thicknesses d. The snapshots of the sputtered membranes, viewed from the backside, were taken at 3 ps after ion impact. Atoms are colored according to the local temperature. The black dot marks the ion's impact point. The figure shows events with (a) representative and (b) abundant sputtering. The backward sputter yield Yb and the forward sputter yield Yf are given for each event.
The analytical model introduced in Sec. II and studied in detail in the Appendix predicts a qualitative change in the dependence of the forward sputter yield on the incidence angle as the membrane thickness is increased (Fig. 3). While for thin membranes () the forward sputter yield has a maximum at oblique or even near-glancing incidence angles, thick membranes () have their maximum sputter yield for normal incidence. The reason for this lies in the fact that the maximum sputter yield is expected when the center of the collision cascade coincides with the surface. For further discussion of this point, see the Appendix.
Dependence of the forward sputter yield on the incidence angle θ for various membrane thicknesses: (a) ; (b) . For the smaller membrane thicknesses in (b), both forward and backward sputter yields are shown. MC data (symbols connected with dashed or dotted lines) are compared to the theoretical predictions (solid lines). The same value of the sputtering efficiency Λ as we used in Fig. 1 was used to convert NED densities to sputter yields.
Dependence of the forward sputter yield on the incidence angle θ for various membrane thicknesses: (a) ; (b) . For the smaller membrane thicknesses in (b), both forward and backward sputter yields are shown. MC data (symbols connected with dashed or dotted lines) are compared to the theoretical predictions (solid lines). The same value of the sputtering efficiency Λ as we used in Fig. 1 was used to convert NED densities to sputter yields.
Figure 3(a) demonstrates that the predicted behavior agrees well with the MC results provided that the membranes are not too thin. For the thinnest membranes ( and , Fig. 3(b)) the forward sputter yield is significantly smaller than the prediction of the Sigmund model. Also shown in Fig. 3(b) are the backward sputter yields. Interestingly, with decreasing membrane thickness, the forward and backward sputter yields approach each other, in agreement with the analytical result.
In order to investigate the sensitivity of these results to the ion energy and mass, we repeated the simulations that yielded Figs. 1 and 3 with 2 keV Ar and 20 keV Xe ions. The results were qualitatively the same as for 20 keV Ar, but the thickness required for the backward sputter yield to saturate was slightly higher ( and for 2 keV Ar and 20 keV Xe ions, respectively). We also repeated the 20 keV Ar simulations that led to Figs. 1(a) and 3, but this time we neglected electronic stopping. The results (not shown) indicate that while electronic stopping has an influence on NED, it does not significantly affect the behavior of the forward and backward sputter yields if they are plotted versus d / a. We conclude that the choice of electronic stopping model did not have a crucial effect in the present study.
B. Energy deposition
To explore the subtler details of the physics of sputtering from membranes, we consider NED density profiles calculated by MC as a function of the depth as measured from the backward surface into the membrane (see Fig. 4). The maximum of the approximate Gaussian profile (the dashed line in Fig. 4) is shifted with respect to the MC data for the infinite medium (green histogram). This is due to the fact that the MC data have only a short tail towards negative depths, while the Gaussian's tail is considerable (not shown). The values of the two distributions on the backward surface are close to each other, but they are not identical (see Fig. 4(b)). It is therefore not surprising that the backward sputter yield predicted by the model does not match the MC data perfectly (Fig. 1).
NED in an infinite medium as calculated by MC (solid green histogram) and Gaussian approximation Eq. (2) (dashed blue line). In addition, the NED distributions in membranes of thicknesses (a) and (b) are shown. The vertical dotted lines indicate the positions of the forward surfaces.
NED in an infinite medium as calculated by MC (solid green histogram) and Gaussian approximation Eq. (2) (dashed blue line). In addition, the NED distributions in membranes of thicknesses (a) and (b) are shown. The vertical dotted lines indicate the positions of the forward surfaces.
Also shown in Fig. 4 are NED profiles in membranes of various thicknesses. According to Sigmund's theory, only the NED density values close to the surface affect sputtering. As can be seen, the NED density near the forward surface is smaller than it would be in an infinite medium. This is caused by the truncation of collision cascades by the surface:38 Recoils that leave the target as well as recoils that would be generated in the subcascades initiated by them do not reenter the target. This effect is weaker at the backward surface than at the forward surface. At the smallest membrane thicknesses, the NED is reduced across the whole membrane, because the collision cascades cannot fully develop.
The influence of the surface on NED can be investigated more fully by plotting the ratio of the surface NED density in a membrane to the infinite-medium NED density at the corresponding depth as a function of membrane thickness and incidence angle (Fig. 5). The surface density is taken to be the density in the histogram bin closest to the surface. Since the bin width of our MC simulations is , this corresponds to averaging the NED density over a sub-surface layer of thickness Å. The most obvious feature seen in Fig. 5 is the drastic reduction in NED for very thin membranes. It explains why the sputter yield tends to zero for thin membranes, see Fig. 1. In addition, we observe three trends in Fig. 5: (i) In most cases, and, in particular, for normal incidence (), the NED ratio is smaller at the forward surface than at the backward surface. In other words, the forward surface has a larger effect on the NED density than the backward surface. (ii) The NED ratios at the forward surface only have a weak dependence on the incidence angle and monotonically increase as a function of membrane thickness within the range of thicknesses and incidence angles studied. (iii) The NED ratio at the backward surface decreases with increasing incidence angle.
The ratio of the surface NED density to the NED density in an infinite medium as a function of membrane thickness for the incidence angles , and .
The ratio of the surface NED density to the NED density in an infinite medium as a function of membrane thickness for the incidence angles , and .
Feature (ii) may be explained by the size of the subcascades generated near the surfaces: In the infinite medium, there are subcascades that reach into the region that is vacuum in the simulations of sputtering from membranes. These subcascades send recoils into the region occupied by the membrane. Naturally, these recoils do not occur when the membrane rather than the infinite medium is bombarded. The number of “missing” recoils is larger the more energetic the subcascade is. According to SRIM,33 nuclear stopping of Si for Ar has its maximum at 20–25 keV. Therefore, the average energy of a primary recoil is largest when the 20-keV Ar ion enters the target and it decreases as the ion slows down. This explains why the forward NED ratio increases with membrane thickness: The thicker the membrane is, the greater distance the ion has to travel to reach the forward surface and the lower the energy it has when it arrives there. It does not explain, however, features (i) and (iii): If the size of the subcascades were the only relevant parameter, then we would expect the NED ratio at the backward surface to be lower than at the forward surface for all membrane thicknesses, and we would expect the backward NED ratio to be independent of the incidence angle.
To explain features (i) and (iii), we need to consider the direction of the subcascades. The more a subcascade is directed towards the surface, the larger the portion of its infinite-medium counterpart that is outside the real target. Basic collision kinetics dictates that recoils must have a positive momentum component in the direction of the projectile.39 Thus, when an ion has just entered the target at normal incidence and has not yet been scattered significantly, primary recoils are mainly directed away from the backward surface and towards the forward surface, which favors reduction in NED at the forward surface (feature (i)). As the incidence angle is increased, the component into the target of the mean primary recoil momentum is reduced, which allows the backward surface to exercise a larger effect on the NED distribution (feature (iii)).
C. Sputtering efficiencies
The sputter yield is the product of the sputtering efficiency Λ and the actual amount of NED at the surface in question. As we shall see in this subsection, the trends in NED discussed in Sec. IV B are in some cases amplified and in others partially compensated for by trends in the sputtering efficiency Λ.
In Fig. 6, the sputtering efficiency is plotted as a function of membrane thickness for incidence angles of , and . The forward sputtering efficiencies have an even weaker dependence on the incidence angle than the forward NED ratios do. In contrast to the NED ratio for the forward surface, the forward sputtering efficiency is almost independent of the membrane thickness for . As the membrane thickness is decreased below , the sputtering efficiencies decrease and so amplify the trend seen in the NED ratios. In contrast, sputtering efficiencies are lower for sputtering from the backward surface than for the forward surface, and the backward sputtering efficiency is an increasing function of incidence angle. This is opposite to the situation for the NED ratios. Therefore, the trends in the NED ratio and the sputtering efficiency partially compensate.
The sputtering efficiency as a function of membrane thickness for the incidence angles of , and .
The sputtering efficiency as a function of membrane thickness for the incidence angles of , and .
We recall that according to Sigmund's theory of sputtering, the sputtering efficiency should depend only on the target material and the ion species. The dependence of the sputtering efficiency on the membrane thickness and the difference between the efficiencies for the forward and backward surfaces shown in Fig. 6 may be explained by considering the energy and angular distributions of the sputtered atoms. In Sigmund's theory,21,38 it is assumed that the recoil energy distribution is proportional to and that the angular distribution is a cosine distribution. When traveling through the planar surface potential well, the recoils lose the surface binding energy Es, and they are refracted in a way such that the cosine angular distribution before the well remains a cosine distribution beyond the well.38 In Fig. 7, we therefore plot the MC results for the energy distribution of the sputtered atoms shifted by the surface binding energy Es (Fig. 7(a)), as well as the MC results for the angular distribution of the sputtered atoms (Fig. 7(b)), and compare them with Sigmund's predictions. The energy distributions show a behavior between and with a behavior closer to for backward sputtering. Accordingly, our finding that forward sputtering has a greater efficiency than backward sputtering makes perfect sense: More energetic recoils have a greater chance of surmounting the surface binding energy, but they do not necessarily deposit more energy near the surface. The slower decay of the energy spectrum for atoms sputtered from the forward surface therefore leads to a higher forward sputtering efficiency.
(a) Energy and (b) angular distribution of the sputtered atoms for perpendicular ion incidence. The energy scale in (a) is shifted by the surface binding energy Es to provide the energy spectrum of the (sputtered) recoils before sputtering. The dashed lines in (b) are cosine distributions. (c) The fraction of primary recoils among the sputtered atoms. The color coding in panels (b) and (c) is identical to panel (a).
(a) Energy and (b) angular distribution of the sputtered atoms for perpendicular ion incidence. The energy scale in (a) is shifted by the surface binding energy Es to provide the energy spectrum of the (sputtered) recoils before sputtering. The dashed lines in (b) are cosine distributions. (c) The fraction of primary recoils among the sputtered atoms. The color coding in panels (b) and (c) is identical to panel (a).
The angular distribution of sputtered atoms (Fig. 7(b)) is slightly over-cosine for thick membranes. This is somewhat more so for forward sputtering than for backward sputtering. For thin membranes, the angular distribution is under-cosine. This is because in thin membranes a larger fraction of the recoils are primary recoils and these are more likely to be emitted obliquely. This under-cosine angular distribution explains the decline in the sputtering efficiency for small membrane thicknesses: Recoils have a higher probability of being reflected by the surface potential well if they occur at high angles to the surface normal.
The fraction of primary recoils among the sputtered atoms is given in Fig. 7(c). Large values are observed for , and the values are still larger than normal at , while the fraction of primary recoils varies weakly for larger membrane thicknesses. This is consistent with the explanation of the non-cosine angular distributions given in the preceding paragraph. Note also that the fraction of primary recoils is larger for forward than for backward sputtering. This explains the larger deviation of the energy distribution from the form for forward sputtering (Fig. 7(a)): Primary recoils tend to have higher-than-average energies. Therefore, the high-energy tail of the energy distribution is more pronounced when there are more primary recoils.
With the exception of the energy spectrum at the highest energies, the forward energy and angular distributions do not change appreciably with membrane thickness for thicker membranes. Our finding that the sputtering efficiency does not depend on membrane thickness for thick membranes is therefore reasonable.
V. CONCLUSIONS
We used the concrete case of 20 keV Ar impact on a-Si membranes to investigate forward and backward sputtering from thin membranes. The main results of this study are:
For membrane thicknesses , our Monte Carlo simulations indicate that the backward sputter yield is nearly independent of the membrane thickness for normally incident ions. The forward sputter yield has a maximum around , on the other hand. The forward sputter yield also exhibits an intriguing dependence on the ion's angle of incidence. While for the forward sputter yield has a maximum at oblique angles of incidence, the maximum occurs at normal incidence for . The analytical model based on Sigmund's theory of sputtering describes these trends quite well.
For membrane thicknesses , both the forward and the backward sputter yields are increasing functions of the membrane thickness with the latter reaching a saturated value at . Both yields tend to zero as the membrane thickness vanishes. This may be explained by the cooperative effect of two mechanisms: First, collision cascades cannot fully develop in thin membranes, and therefore, energy deposition is reduced. Second, the sputtering efficiency is reduced because of the under-cosine angular distribution of recoils in the case of thin membranes.
Interestingly, there are opposing trends in energy deposition and the sputtering efficiency: Energy deposition is reduced to a greater extent at the forward surface than at the backward surface, and energy deposition at the backward surface decreases with the incidence angle. In contrast, the sputtering efficiency is higher for forward sputtering than for backward sputtering, and it increases as a function of incidence angle for backward sputtering.
Molecular-dynamics simulations performed for thin membranes corroborate the Monte Carlo results. The good agreement between our MD and MC results is remarkable in view of previous studies of nanospheres and nanowires8,15 in which significant differences between MD and MC results were found for the smallest nano-objects. Although the situation may be different for heavier ions and/or heavier target atoms, our results indicate that there is a trend toward reduced many-body effects in sputtering from membranes as compared to laterally confined nano-objects. Finally, the MD simulations also show that thin membranes with thickness nm can be perforated by a single ion impact in the statistically rare case of abundant sputtering.
These results show that the Sigmund model can be used to study the general trends in sputtering as long as the membranes are not too thin (– for the ion/target combination studied). Our comparison of the MD and MC results indicates that MC simulations are sufficient to investigate sputtering of thin membranes, at least under impact conditions similar to those investigated in this study. MC simulations could therefore be used to gain additional insight into the sputtering process—for example, the spatial distribution of the points on the membrane surfaces where sputtering occurs could be determined.
We note that plastic flow may occur during sputtering of membranes.17 The exact conditions for this to happen are currently not well characterized. However, if plastic flow does occur during ion bombardment, simulations can be carried out by combining a viscoelastic model with a MC simulation.40 The effect plastic flow has on membrane thinning will be investigated in future work.
ACKNOWLEDGMENTS
R.M.B. would like to thank the National Science Foundation for its support through Grant No. DMR-1305449.
APPENDIX: ANALYTICAL RESULTS
Consider a free-standing planar membrane of uniform thickness d. For the time being, we will take the membrane to be composed of an arbitrary elemental material, although in our simulations we will specialize to the case in which the material is amorphous silicon. Suppose an ion impacts the membrane with the angle of incidence θ. We place the origin at the point where the ion first makes contact with the membrane. The z-axis will be taken to point out of the film, normal to the backward surface. The x-axis will be oriented so that the direction of ion incidence is , where .
Recall that the vector leads from the point of maximum NED to the surface point , and that and are the components of that are parallel and perpendicular to the ion's incidence direction. These components appear in Eq. (2) and are given by
and
as shown in Ref. 41.
According to Eq. (1), the forward sputter yield is
Utilizing Eqs. (2), (A1), and (A2) in Eq. (A3) and evaluating the resultant Gaussian integrals, we obtain
Here, is the dimensionless membrane thickness, ,
and
Note that the argument of the exponential function in Eq. (A4) is quadratic in Δ, and so, Yf is a Gaussian function of the membrane thickness. For normal-incidence ion impact
The Gaussian is centered on the value in this case.
The backward sputter yield Yb is obtained simply by setting Δ to zero in Eq. (A4). This yields
a result that has been obtained previously.42 Notice that Yb is independent of the membrane thickness d in the Sigmund theory. As our simulations show this is approximately true only for sufficiently large values of .
It is illuminating to consider the special case in which . In this case, the distribution of deposited energy is spherically symmetric about its maximum. Equation (A4) reduces considerably because is independent of θ and C vanishes. A straightforward analysis shows that for d < a, the forward sputter yield has a maximum at . This is easily understood: for this value of θ, the maximum of the distribution of deposited energy lies on the forward surface. If d > a, on the other hand, the maximum value of is achieved for θ = 0. This makes perfect sense because the maximum in the deposited energy is closest to the forward surface for θ = 0.
Of course, α is in reality greater than β. Nevertheless, as shown in Fig. 3, the Sigmund theory's prediction for has its maximum at a nonzero value of θ for Δ in excess of a critical value Δc. Conversely, for , the maximum value of is achieved for θ = 0.