Introduction

Ion beam irradiation (IBI) with an ion energy between 10 and 10 5 eV can produce self-assembled patterns of dots or ripples on most solid surfaces. This observation has raised questions regarding the physical mechanisms that lead to the self * Author to whom any correspondence should be addressed.

assembly. It has also led to a great deal of interest in the possible applications of IBI that exploit its scalable, single-step, low-cost nano-patterning capabilities [1][2][3][4].

Silicon targets have taken center stage in the study of nanopatterning by IBI. That is in part because of the ease of analysis, because at room temperature an initially crystalline Si target becomes amorphous near the surface during IBI, obviating the need to consider the anisotropic kinetic and energetic processes that act on crystalline surfaces, and in part because of silicon's importance in modern industrial applications [5,6].

A consensus seems to have been reached on the pattern evolution on Si as the angle of incidence Θ is increased [7][8][9][10][11]. (Θ is the polar angle of the ion beam measured with respect to the surface normal.) Below the threshold angle for pattern formation Θ th , the surface is stable and it remains almost completely flat. (The value of Θ th depends on the choice of ion beam. It is approximately equal to 55 • in our experiments.) For Θ th < Θ < Θ r ∼ 80 • , a parallel-mode ripple pattern develops with its wave vector parallel to the surface projection of ion beam's direction. Moreover, at least according to the consensus, for Θ > Θ r , a perpendicular-mode ripple pattern prevails with its wave vector perpendicular to the surface-projected ion beam direction. At Θ r , the instability is biaxial, and mounds form on the surface. This apparent reorientation of the ripple pattern is rather insensitive to the incident ion energy E if E 2 keV, and is also observed for Ge [12][13][14] and mica [15] targets. A switch from parallel-mode to perpendicular-mode ripples as Θ is increased is predicted by the Bradley-Harper theory, the first theory that was able to explain the genesis of rippled topographies on surfaces subjected to IBI [7]. The equation of motion (EOM) that describes the surface dynamics is commonly taken to be the anisotropic Kuramoto-Sivashinsky (AKS) equation [5,16].

Difficulties with this conventional viewpoint have started to become increasingly evident. The nature of the so-called perpendicular-mode ripple pattern is controversial because the patterns show little order as judged by the absence of the expected peaks in their power spectral densities [17]. The kinds of patterns that are observed in experiments are also more diverse than those mentioned above. Raised and depressed triangular regions that are traversed by parallel-mode ripples are frequently observed [5,11,12,[18][19][20][21][22][23][24][25][26][27], for example, although typically little attention has been paid to them. Terraced surfaces and 'fins' (tilted ridges that are aligned with the projected ion beam direction) form in other experiments [12,13,18,22,[28][29][30][31][32][33][34][35][36][37][38]. The AKS equation does not produce triangular nanostructures, terraces or fins. Loew and Bradley [39] recently found that once the effect of linear dispersion has been added to the AKS equation, triangular nanostructures develop. On the other hand, the addition of cubic nonlinearities to the AKS equation results in the Harrison-Pearson-Bradley (HPB) equation, which yields terraces and fins similar to the ones seen in many experiments [40,41].

In this paper, we study the nanoscale patterns that develop on the (100) surface of silicon when it is irradiated with a 2 keV Kr ion beam. Our work complements and extends earlier experimental studies with this ion-target combination [9,37]. We find that the patterns observed at high ion fluences depend sensitively on the angle of incidence. In particular, for angles of incidence between 74 • and 85 • , we observe five distinctly different kinds of topographies-triangular nanostructures traversed by ripples, complex terraced structures, fins decorated by parallel-mode ripples, an intriguing mesh-like morphology, and a so-called perpendicular-mode ripple pattern. In contrast, only parallel-mode ripples are present for low ion fluences except for Θ = 85 • . In the latter case, the solid surface is apparently stable, and roughens only as a result of shot noise in the ion beam. We also place our experimental results in the context of theories developed since the earlier experimental work on the bombardment of silicon with a 2 keV Kr ion beam was carried out. Triangular nanostructures that closely resemble those in our experiments emerge if a conserved Kuramoto-Sivashinsky nonlinearity is appended to the EOM studied by Loew and Bradley. Fins traversed by parallel-mode ripples, on the other hand, are found in simulations of the HPB equation for a range of parameter values. 'Perpendicular-mode ripples' akin to those we observe experimentally are also reproduced in our simulations. Complex terraced surfaces and mesh-like morphologies did not develop in simulations of any of the equations of motion we have examined, however. As such, they represent a challenge to future theoretical work.

This paper is organized as follows. We discuss the details of our experimental set-up in section 2. Our experimental results for a range of ion fluences and incidence angles are presented in section 3. Our observations are compared with existing theoretical models in section 4 and some improvements to these models are advanced. In section 5, we compare our experimental results with past studies of IBI of silicon with a 2 keV Kr ion beam. Finally, in section 6, we summarize the results of our investigations, in part by giving a 'phase diagram' of patterns as a function of incidence angle and fluence that represents a more refined and complete version of previous diagrams.

Experimental details

IBI was performed in a vacuum chamber with a base pressure in the low 10 -9 Torr range. B-doped, 10 mm × 10 mm Si(100) chips were irradiated by a 2 keV Kr ion beam employing a Kauffman-type ion gun (Physical Electronics, 04-161 Sputter Ion Gun). The (global) beam flux ( f ) estimated from the total beam current was f ∼ 10 μA cm -2 at Θ = 0 • . Since secondary electrons also contribute to the beam current, f only sets an upper limit on the real ion flux. Θ was adjusted by an a.c. servo motor controlled by a LabView program we built.

The patterns formed on the Si substrate were imaged ex situ by an atomic force microscope (AFM; Park Systems XE-100) in the non-contact mode. The images were then analyzed using both the SPIP package (Image Metrology) and home-built programs.

The ion gun was operated in the fully focused mode so that the beam diameter was less than 10 mm. A photograph that gives an indication of the beam cross-section at the surface is shown in figure 1(a) for a case in which the beam was obliquely incident on the sample. Figure 1(b) shows a contour map of the surface width W measured by AFM. The beam center, which was defined to be the surface point with the greatest value of f and W, was used as the reference position for the purpose of specifying a value of Θ. We place the origin of the coordinate system at the beam center, the x axis in the same direction as the surface projection of the ion beam direction, and the z axis normal to the surface. The height of the surface above the x-y plane will be denoted by h. 5Although the beam was focused, it remained slightly divergent. We exploited this fact in two different ways. First, because the beam was focused, moving away from the beam center in the y direction produces a reduction in the flux f and hence in the fluence F. By moving far enough away, the fluence could be reduced almost to zero. Secondly, because the beam was divergent, we were able to make detailed surveys of the patterns formed at a sequences of closely spaced angles of incidence by imaging at different locations along the x axis. We used this ability to identify the angles of incidence Θ at which the five different kinds of pattern are most clearly defined at the beam center. The range of Θ that could be accessed within a single sample exceeded 2.7 • at normal incidence. This range decreases as Θ increases due to the limited lateral sample size.

It has been known for some time that unintended codeposition of metal impurities can completely alter the patterns produced by ion bombardment of a silicon surface [9,[42][43][44][45][46][47][48][49][50][51]. We therefore took great care to ensure that impurities were not co-deposited during IBI. In particular, by targeting a point close to the center of the silicon sample with a focused beam, we made sure that virtually all of the incident ions struck the sample rather than the stainless steel sample holder or chamber walls. In addition, we placed a sacrificial portion of a silicon wafer over the part of the sample holder that could possibly have been exposed to the ion beam to prevent the sputter-deposition of the metallic species that make up the holder.

Experimental results

Figure 2 shows images of the patterns on the sample surface at the beam center following irradiation with the 2 keV Kr ion beam for selected values of Θ. We arranged for the fluence F to have the value 1950 ions nm -2 in each case by adjusting the irradiation time to compensate for the geometric factor cos Θ.

Parallel-mode ripples form for Θ = 74 • , as seen in figure 2(a). The topography is much richer than this simple, traditional description would suggest, however, since raised and depressed triangular regions are evident, and these are traversed by the ripples. The obliquely-oriented edges of the triangular nanostructures are marked in red and blue in figure 2(a ). The PSD of the surface height appears in the inset of figure 2(a). It displays an X-shaped structure centered on the origin in addition to the two peaks that arise from the parallelmode ripple pattern. This 'X' in the PSD is an important signature of the triangular nanostructures. The angle subtended by the two legs of the X-shaped structure, φ ∼ = 46 • , matches the mean angle subtended by the beam-facing apex of the triangular structures well. (See the guidelines in figure 2(a) and its inset.) The length of the legs of the X-shaped structure in the PSD is an indication of the spread in the distances between the similarly oriented edges of neighboring triangular structures that are shown in red and blue in figure 2(a ).

Figure 3(a) shows the pattern at the same angle of incidence as figure 2(a), but this image was taken far from the origin on the y-axis and hence for a much reduced value of the fluence F. Only parallel-mode ripples are observed-the triangular nanostructures do not appear in the real space image of the surface, and the X-shaped structure is not present in the corresponding PSD. This observation shows that nonlinear effects contribute to the formation of the triangular structures in figure 2(a). The work of Loew and Bradley strongly suggests that linear dispersion is also an important factor in the formation of these nanostructures [39].

For Θ = 78 • , neither parallel-nor perpendicular-mode ripples are immediately apparent in figure 2(b), an AFM scan for the fluence F = 1950 ions nm -2 . Traditionally, this angle of incidence would be thought to be close to the angle of incidence Θ r where the instability is biaxial and mounds form on the surface [7]. However, the real-space image for a much lower ion fluence and the same value of Θ that is shown in figure 3(b) establishes that the instability is in fact not biaxial, since parallel-mode ripples form at early times. This conclusion is supported by the corresponding PSD that appears in the inset of figure 3(b).

The surface topography for Θ = 78 • and F = 1950 ions nm -2 is complex and has structure on a broad range of length scales, as seen in figures 2(b) and (b ). A histogram of the surface slope angle δ ≡ -arctan(∂h/∂x) displays two peaks (see figure 4), which indicates that the surface does not consist of mounds but instead is terraced. The plot of h x ≡ ∂h/∂x versus x and y shown in figure 5(a) confirms this, but also shows that the surface steps are not straight and so the terraces are irregular. In fact, a significant portion of the terrace edges are nearly parallel to two preferred diagonal directions. (See the guidelines in figure 2(b) and the edge-enhanced image in figure 5(a).) Some of the edges even form wedges with their apexes facing the ion beam. The mean angle subtended by the two edges of a wedge is ∼56 • . The PSD shown in the inset of figure 2(b) also displays an X-shaped structure, albeit faintly. The value of φ determined from the 'X' in the PSD agrees well with the angle subtended by the two preferred directions in the real-space image. It is therefore likely that dispersion once again makes a significant contribution to the pattern formation. At the same time, the terraces present on the surface strongly suggest that the cubic nonlinearity h 3

x also plays an important role at this angle of incidence [41,52]. . Parallel-mode ripples on terraces were previously reported for both silicon [53] and germanium [12] targets. For Θ = 80.5 • and the fluence F = 1950 ions nm -2 , the surface is no longer terraced. Instead, it is made up of truncated ridges or 'fins' running along the x direction, as shown in figure 2(c). The three-dimensional surface image in figure 2(c ) reveals that the fins have a face that faces the beam and a gradually descending ridge with sloped flanks. The ridges are narrow and the slope h y ≡ ∂h/∂y changes quite rapidly across them. Similar structures were observed on ion-irradiated silicon surfaces by Macko et al [9].

The fin-shaped structures are not regularly spaced, and accordingly there is no discernible peak along the k y axis in the PSD of the surface height-see the inset of figure 2(c). The pattern for Θ = 80.5 • should therefore not be described as a perpendicular-mode ripple, although that has been done elsewhere [8][9][10]. In fact, parallel-mode ripples decorate the whole surface, as seen in figure 2(c ) and its inset as well as in figure 5(b). In addition, only parallel-mode ripples are present on the surface for low fluences, as shown by figure 3

(c).

A faint 'X' is present in the PSD shown in figure 2(c). This suggests that dispersion may also play a role in the pattern formation for Θ = 80.5 • . Correspondingly, the ends of the fins are shaped and arranged like vees, as highlighted by the guidelines in the real-space image in figure 2(c). This is most apparent in the edge enhanced image in figure 5(b). When Θ is increased to 82 • , the fins are replaced by a parallel-mode ripples, as shown in figure 2(d). Interestingly, nanoholes are present between the ripple crests, giving the surface an appearance that is somewhat reminiscent of a mesh. (Note that in this case the image size was reduced to 2 μm × 2 μm so that details of the pattern could be seen clearly.) An analogous pattern has been observed on a Ge surface bombarded at a comparable high angle of incidence [12]. The PSD in the inset of figure 2(d) shows an X-shaped structure centered on the origin as well as two peaks coming from the parallel-mode ripple pattern. Once again, only parallel-mode ripples can be discerned for low fluences (see figure 3(c)).

To investigate the structure in the mesh pattern shown in figure 6(a), we smoothed the surface height to reduce the shortwavelength roughness produced by shot noise in the ion beam. A nanohole was defined to be a connected set of points that have positive surface curvatures in both the x-and y-directions. The nanoholes in figure 6 Our next step was to explore the order in the arrangement of nanoholes. We took the position of each nanohole to be its centroid position. These locations were then used to construct the corresponding two-dimensional Delaunay triangulation. This places a bond between each nearest-neighbor pair of nanoholes, as shown in figure 6(c). A histogram of the angles that the bonds in the Delaunay triangulation make with the x axis is shown in figure 6(d). There are peaks near ±19 • and ±90 • . These peaks demonstrate that there is orientational order in the arrangement of nanoholes. The bond angles near ±90 • come from neighboring pairs of nanoholes with the bond that joins them approximately parallel to the y direction. These peaks in the bond angle distribution occur because the nanoholes have a tendency to be located in the troughs of the parallel-mode ripples. The two peaks of the bond angle distribution closer to the origin in figure 6(d) were fitted with Gaussians. The fits show that these peaks occur at the angles ±19 • (±2 • ). These angles coincide with the angles that the X in the PSD in figure 6(a) makes with the k y axis. Thus, the X that is observed in the PSD of the mesh pattern is due to the orientational order of the nanoholes. The peaks in the histogram at ±19 • show that the nanoholes tend to be arranged in chains that are oblique to the projected ion direction.

For Θ = 85 • , the mesh-like pattern has disappeared. Instead, a so-called perpendicular-mode ripple pattern develops, as shown in figure 2(e). Patterns of this kind have been reported by numerous groups for both silicon [9][10][11] and germanium [13] targets. The PSD in the inset of figure 2(e), however, does not exhibit any peaks, as previously reported [17]. In addition, no peaks are present in the PSDs for lower fluences shown in figure 7. As a consequence, the surface morphology should not be described as a perpendicular-mode ripple pattern.

Our results show that when the angle of incidence is changed by just a few degrees, the topography of the surface can change dramatically. These changes are attended by a wide variation in the surface width W (the root-mean square deviation of the surface height from its average value), as seen in figure 8. As Θ is increased, W passes through a maximum value of 7.3 nm in the vicinity of 78 • and declines to less than 4.1% of its maximum value for Θ = 85 • .

Comparison with theoretical models

In section 3, we discussed the different kinds of patterns that emerge as Θ is increased from 74 • to 85 • . In our comparison of these experimental results with theoretical models, however, it will be convenient to consider the five kinds of patterns formed in a different order. We will begin with the morphologies found for Θ = 74 • and 80.5 • since these patterns are best modeled by equations of motion that have already been studied, or by modifications of those models. The patterns that develop for Θ = 78 • will be discussed next because they seem to be a complex hybrid of the patterns found for Θ = 74 • and 80.5 • . We will then advance a model for the noisy, streaked surfaces that form for Θ = 85 • . Finally, we will close this section with a brief discussion of the mesh-like pattern that is found for Θ = 82 • . In this case, no plausible model is known at the present time.

The most widely employed model of pattern formation on the surface of solid that is bombarded with an obliquelyincident ion beam is the AKS equation. However, for angles of incidence Θ in excess of the threshold angle for pattern formation Θ th but close to it, linear dispersion has been shown to have an important effect on the pattern formation [54]. Such a term does not appear in the AKS equation. Simulations carried out by Loew and Bradley have shown that once a linearly dispersive term has been appended to the AKS equation, raised and depressed triangular regions traversed by parallel-mode ripples can emerge as time passes [39]. These triangular nanostructures are similar to the ones present in figure 2(a), but their edges are less distinct. Moreover, the PSDs of patterns generated by the EOM studied by Loew and Bradley have a much weaker X-shaped structure than the one that appears in the experimental PSD shown in the inset of figure 2(a).

The conserved Kuramoto-Sivashinsky (CKS) nonlinearity produces ripple coarsening [55,56], a phenomenon that is frequently observed in experiments [5,57]. This term is also needed if the pattern formation that occurs on a silicon target that is rocked during ion bombardment is to be reproduced [58]. Although the effect of the CKS term is negligible near the threshold angle for pattern formation [54], the angle of incidence Θ in the case of figure 2(a) was 74 • and so was significantly larger than Θ th ∼ 55 • . The CKS term may therefore have played an important role in the dynamics. Adding such a term to Loew and Bradley's EOM yields

Here h = h(x, y, t) is the height of the solid surface above the point with coordinates x and y in the x-y plane at time t, as measured in a suitably chosen moving frame of reference. The subscripts x, y and t on h denote partial derivatives with respect to these variables. A 1 , A 2 , B, λ 1 , λ 2 , α, β, r, μ 1 and μ 2 are constants that depend on Θ. For μ 1 = μ 2 = 0, no CKS term is present and (1) reduces to the EOM studied by Loew and Bradley. When, in addition, α and β are both zero, linear dispersion is absent and (1) becomes the AKS equation. Finally, we note that the most general form of ( 1) is a special case of the EOM for an ion-irradiated solid surface that results from a 'hydrodynamic' model in which a layer near the surface of the solid is mobilized by the impinging ions [55,56]. We carried out numerical integrations of ( 1) with low amplitude spatial white noise initial conditions for selected parameter values. Periodic boundary conditions were employed. A nonzero value of μ 1 was chosen so that a nontrivial CKS term appears in the EOM. The results are shown in figure 9. Triangular nanostructures are evident for α equal to 1, 5 and 10. In addition, the corresponding PSDs display a prominent X-shaped structure centered on the origin, just as in the experimental PSD shown in the inset in figure 2(a). The apex opening angle of the triangular nanostructures φ is a decreasing function of α. For α = 50, the opening angle is small enough that the triangular nanostructures evident for smaller α values have been reduced to streaks.

The triangular nanostructures appear briefly for a time after the linear regime in simulations of Loew and Bradley's EOM but soon disappear: see figures 10 The experimental surface pattern shown in figure 2(c) has many features in common with surfaces produced by simulations of the HPB equation

(2) The terms proportional to h 2

x and h 2 y in this equation result from expanding the slope dependence of the sputter yield to second order in the surface gradient [16,59]. Similarly, the terms proportional to h 3

x and h x h 2 y appear if terms up to third order in h x and h y are retained in this expansion [40,41]. For a = b = 0, the HPB equation ( 2) reduces to the AKS equation. Just as the terms proportional to h xxx and h xyy in (1) do, the cubic nonlinearities in the HPB equation ( 2) break the symmetry under the transformation x → -x that the AKS equation possesses even though the experimental setup does not.

The results of a simulation of the HPB equation with a low amplitude spatial white initial condition are shown in figure 11. For the chosen parameter values, parallel-mode ripples appear at early times, as shown in figure 11(a). Fins develop at longer times and grow larger as time passes, as seen in figures 11(b) and (c). Each fin has a sloped facet facing the beam as well as sloped facets that flank its gradually descending central ridge. The enlargement of a portion of the fins in figure 11(c) that is shown in figure 11(c ) clearly shows that ripples are present on the fins. The simulated surfaces therefore resemble the AFM images we obtained for Θ = 80.5 • in a number of key respects. The experimental surfaces do not have well-defined facets, however.

As we saw in section 3, at high fluences, the distribution of surface slope angles is bimodal and the surface is terraced for Θ = 78 • . In addition, parallel-mode ripples are evident at early times and also on the terraces at longer times. The HPB model does yield surfaces with these characteristics for a range of parameter values, as illustrated by figure 12. The model fails to reproduce the very complicated surface structure seen in the experiments in the high fluence regime, however. This may be because the HPB equation includes neither linear dispersive terms nor the CKS nonlinearity, and, as we have seen, these terms play an important role in the dynamics for the somewhat smaller angle of incidence Θ = 74 • .

As we have discussed, the pattern obtained for Θ = 85 • shown in figure 2(e) should not be described as perpendicularmode ripples. The corresponding PSD has a central maximum but does not have peaks on either the k x or k y axes. This is also true for lower ion fluences, as seen in figure 7. These results strongly suggest that the surface is stable for Θ = 85 • . The surface, however, does not remain flat when ion bombardment begins because there is shot noise in the incident ion flux. As a first approximation, the effects of shot noise are negligible for the smaller angles of incidence we studied. For Θ = 85 • , however, the root-mean square surface width is quite small (see figure 8) and so the effects of noise are more important in this case. Including the effect of shot noise, the AKS equation is

where ν 1 and ν 2 are both positive if the surface is stable and η = η(x, y, t) is uncorrelated Gaussian white noise with zero mean and a variance proportional to the ion flux [16,59]. The term -B∇ 2 ∇ 2 h can be omitted from (3) because long wavelength disturbances are strongly suppressed by the smoothing effect of the terms ν 1 h xx and ν 2 h yy . If, in addition, the effect of noise is relatively weak, the amplitude of the surface disturbance will remain small and the nonlinear terms may be omitted from (3), yielding

Equation ( 4) is an anisotropic version of the Edwards-Wilkinson equation [60]. Atomistic simulations combined with the crater function formalism show that for 1 keV Ar ion bombardment of a silicon target, the coefficient of h xx in the EOM can change sign from negative to positive and then increase rapidly as Θ nears 90 • [61]. We therefore took ν 1 to be substantially larger than ν 2 : for the sake of illustration, we adopted the values ν 1 = 1, ν 2 = 0.01 and took the noise amplitude to be 0.1. The results of integrating (4) numerically with these parameter values are shown in figure 13. The surface is quite noisy, but there are 'streaks' that are elongated in the x direction, much as in the AFM image shown in figure 2(e). These streaks arise because ν 1 was chosen to be significantly larger than ν 2 . The relatively small value of ν 2 means that smoothing in the y direction is weak, and consequently shot noise must be taken into account. There are no peaks away from the origin in the PSD shown in figure 13(b) and the central peak is elongated in the k y direction, just as in the inset of figure 2(e). The model therefore produces results consistent with our experiments. A detailed comparison between the theory and the experiments would require precise values for the parameters in (4), however.

The genesis of the pattern in figure 2(d) is a puzzle and an interesting topic for future study. The X-shaped structure that passes through the origin in the PSD suggests that dispersion and the CKS nonlinearity play important roles in the pattern formation, just as they do in the case of figure 2

(a).

There is no evidence of terracing or fin formation, which indicates that if cubic nonlinearities are present in the EOM, their coefficients are small. We might therefore conclude that the EOM ( 1) is an appropriate model of the surface dynamics. However, at least for the choices of parameter values we have investigated, (1) does not produce mesh-like topographies that resemble figure 2(d). It is possible that if additional terms were included in (1), the resulting EOM would generate a meshlike pattern for a range of parameter values. It is unclear at the present time just what those additional terms should be, or whether the models considered to date omit some essential physics.

Comparison with previous experimental work

As we mentioned in the introduction, the nanoscale patterns that develop on the (100) surface of silicon when it is irradiated with a 2 keV Kr ion beam have been studied in the past by Macko et al [9] and Engler et al [37]. Our work complements and extends those studies.

There are of course numerous studies of the patterns produced by bombardment of silicon targets with ions of various species and energies. (See references [5,37] for reviews.) In this section, we will confine ourselves to comparing the results The most detailed study of the 2 keV Kr + -Si ion-target combination that appeared prior to our work is that of Engler et al [37]. Engler et al concluded that patterns form on the silicon surface for angles of incidence Θ between 58 • and 79 • , and that for Θ outside this interval, the surface remains flat. We, however, found that a mesh pattern is formed for Θ = 82 • . Experiments for this value of Θ were not carried out by Engler et al, and, as a result, they did not observe this fascinating type of pattern formation. In addition, we found that for Θ = 80.5 • , the surface does not remain flat, contrary to the assertion of Engler et al. Instead, fins develop. This type of topography was not reported in Engler et al's paper, although a somewhat similar morphology appears in Macko et al [9] for Θ = 79 • .

Engler et al found that parallel-mode ripples form for 58 • Θ 63 • and that terraced surfaces develop for 67 • Θ 79 • and high ion fluences. For Θ = 78 • , we found that a complex terraced surface forms, and that this surface morphology closely resembles the one observed by Engler et al for Θ = 75 • . In contrast, for Θ = 74 • , we observed the formation of triangular nanostructures traversed by parallel-mode ripples rather than terraces. It is possible that terraces would have emerged had we continued bombarding the surface to the

Conclusions

In this paper, we studied the nanoscale patterns that develop on the (100) surface of silicon when it is irradiated with a broad 2 keV krypton ion beam. The results of our investigations are summarized by the 'phase diagram' given in figure 14, which shows the patterns we observed for selected incidence angles Θ and fluences F. In the span of only 11 • , a total of five markedly different morphologies occur for high ion fluences. In contrast, for low fluences, only parallel-mode ripples are present for Θ equal to 74 • , 78 • , 80.5 • and 82 • . For Θ = 85 • , the solid surface appears to be stable, and to roughen only as a result of shot noise in the ion beam.

We have been able to reproduce some of the experimentally-observed patterns in simulations. Triangular nanostructures that closely resemble those found in our experiments for Θ = 74 • form if a CKS nonlinearity is appended to the EOM proposed by Loew and Bradley [39]. This nonlinearity is known to produce ripple coarsening, as observed in many experiments [55,56]. Our results once again highlight the importance of this term. Fins traversed by parallel-mode ripples are observed for Θ = 80.5 • . They also emerge in our simulations of the HPB equation [41] for a range of parameter values, which shows that the cubic nonlinearities that result from expanding the sputter yield to third order in the surface slope must be included in the EOM in this case. Finally, simulations of an anisotropic version of the Edwards-Wilkinson equation yield disordered surfaces with 'streaks' aligned with the projected ion direction, much like the surfaces in our experiments with Θ = 85 • .

For Θ = 78 • and high ion fluences, the surface topography is complex and has structure on a broad range of length scales. At long length scales, the surface is terraced, but the parts of the terrace edges are wedge-shaped. The terraces are also decorated by short-wavelength, low amplitude parallel-mode ripples. The mesh-like pattern observed for Θ = 82 • can be thought of as a parallel-mode ripple pattern with nanoholes in the troughs. The arrangement of nanoholes displays a degree of orientational order: in particular, the nanoholes tend to be arranged in chains along two directions that are oblique to the projected ion direction. So far as we have been able to determine, these two kinds of patterns are not reproduced by simulations of any of the equations of motion discussed in this paper. Explaining them remains a challenge for future theoretical work.