Pattern formation on Ge by low energy ion beam erosion

Modification of nanoscale surface topography is inherent to low-energy ion beam erosion processes and is one of the most important fields of nanotechnology. In this report a comprehensive study of surface smoothing and self-organized pattern formation on Ge(100) by using different noble gases ion beam erosion is presented. The investigations focus on low ion energies (⩽ 2000 eV) and include the entire range of ion incidence angles. It is found that for ions (Ne, Ar) with masses lower than the mass of the Ge target atoms, no pattern formation occurs and surface smoothing is observed for all angles of ion incidence. In contrast, for erosion with higher mass ions (Kr, Xe), ripple formation starts at incidence angles of about 65° depending on ion energy. At smaller incident angles surface smoothing occurs again. Investigations of the surface dynamics for specific ion incidence angles by changing the ion fluence over two orders of magnitude gives a clear evidence for coarsening and faceting of the surface pattern. Both observations indicate that gradient-dependent sputtering and reflection of primary ions play crucial role in the pattern evolution, just at the lowest accessible fluences. The results are discussed in relation to recently proposed redistributive or stress-induced models for pattern formation. In addition, it is argued that a large angular variation of the sputter yield and reflected primary ions can significantly contribute to pattern formation and evolution as nonlinear and non-local processes as supported by simulation of sputtering and ion reflection.

in relation to recently proposed redistributive or stress-induced models for pattern formation. In addition, it is argued that a large angular variation of the sputter yield and reflected primary ions can significantly contribute to pattern formation and evolution as nonlinear and non-local processes as supported by simulation of sputtering and ion reflection.

Introduction
For particular sputtering conditions with low-and medium-energy ions self-organized nanostructures can evolve on solid surfaces. In the last decades, a large variety of evolving surface structures are observed, such as periodic ripple patterns and arrays of dots or holes. These pattern formations are also found on many different materials, for instance metals [1][2][3][4][5][6][7], semiconductors [8][9][10][11][12][13][14][15] as well as various compounds [16][17][18][19][20]. Special attention is focused on Si as an important material for technical applications. Additionally, it is a simple onecomponent system and readily amorphizes during ion irradiation that makes it a feasible model system for revealing the processes behind patterning and smoothing. Many investigations on the pattern formation process have been recently done for Si [21][22][23][24][25]. It has been found that, at low incident angles the surface remains flat up to a critical angle when ripple formation starts. This angle slightly depends on ion energy as well as on the ion species. Additionally, observations suggested that the metal incorporation during ion sputtering also plays an important role in the pattern formation. Dots and ripple formation takes place under near-normal incidence with simultaneous metal deposition whereas the surface remains flat without metal incorporation [25][26][27][28][29][30][31][32]. The first model that describes the ripple formation process is the erosionbased theory of Bradley and Harper (BH) [33], which combines surface-curvature-dependent sputtering for surface destabilization with a temperature-dependent surface diffusion which tends to smooth the surface. However, this model cannot explain a number of experimental observations, such as smoothing of the surface for small angles of incidence and a transition from smooth to rippled surfaces with increasing ion incidence angle as observed for Si or Ge. of experimental parameters. An overview of the evolving structures is given for the full angular range between 0 • and 85 • . The effect of ion species is studied for different ion energies ( 2 keV) as well as different time scales. All experiments are carried out in an experimental setup, which ensures negligible surface contamination with metallic or other impurities. The results are discussed in context to the recently proposed redistributive or stress-induced models for pattern formation. The experimental findings supported with TRIM.SP [48] simulations imply a strong correlation between the angular variation of sputter yield and formation of the surface pattern. The importance of gradient-dependent sputtering and the reflection of primary ions on pattern formation are shown. The angle-dependent sputter yield and the reflection coefficient are calculated for different ion species used in this work.

Experimental setup
The samples utilized in this work were commercially available: epi-polished Ge(100) substrate pieces (n-type) with a root-mean-square (rms) roughness of about 0.5 nm. These samples were mounted on a water-cooled substrate holder in a high vacuum chamber with a base pressure of 10 −6 mbar. The substrate holder can be tilted from 0 • (corresponding to normal ion incidence) up to 90 • with respect to the axis of the ion beam source. Furthermore, the sample holder is equipped with a Si shielding in order to prevent metallic contaminations that affect the evolving structures [25,31]. For that reason, the grid aperture of the ion source has been reduced compared to former experiments [45]. For the experiments a homebuilt Kaufman-type two grids ion source with a beam diameter of 100 mm is used. No iron or other metallic contaminations could be detected with Rutherford backscattering spectrometry as well as with x-ray photoelectron spectroscopy. This non-co-deposition setup [49] also prevents secondary collisions of scattered gas ions and re-deposition of sputtered silicon atoms. The current density was kept constant at 300 µA cm −2 during the experiments which results in an ion flux of J = 1.87 × 10 15 cm −2 s −1 in a plane perpendicular to the ion beam. The surface topography was analyzed by scanning force microscopy (atomic force microscopy (AFM)) using a Dimension FastScan system from Bruker operating in TappingMode™ or ScanAsyst mode. The measurements were conducted in air using silicon nitride cantilever with a nominal tip radius smaller than 5 nm (TappingMode™) or 2 nm (ScanAsyst). The analyzed area of each sample was 2 × 2 µm 2 or 4 × 4 and 10 × 10 µm 2 with a resolution of at least 1024 × 1024 pixels.

Topography evolution under low-energy ion beam erosion on Ge
In the following part, the experimental results are shown which demonstrate the role of different control parameters on the topography evolution of Ge under low-energy ion beam erosion. In particular, the influence of ion incidence angle, ion species, ion energy and total ion fluence is examined in detail. Figure 1 shows the general influence of the incidence angle on the evolution of the surface topography. The irradiation, for example, with Kr + ions is shown with an ion energy of E ion = 1200 eV and fluence of 6.7 × 10 18 cm −2 . Figure 1. AFM images of the initial Ge surface (a) and Kr + ion beam eroded Ge surfaces for different ion incidence angles:

Overview of topography evolution versus ion incidence angle
The white arrow indicates the projection of the ion beam direction. The image size is 4 × 4 and 1 × 1 µm 2 for magnification insets. The different height scales of the images are specified in each image. E ion = 1200 eV, j ion = 300 µA cm −2 and ion fluence of 6.7 × 10 18 cm −2 .
The first AFM image (figure 1(a)) shows the surface of the Ge substrate before erosion. The rms roughness R q is approx. 0.5 nm. After normal incidence irradiation (α = 0 • , figure 1(b)) the surface is flattened and a very-low surface roughness of R q = 0.11 nm is calculated. Up to an incidence angle of 60 • (figure 1(d)) the roughness slightly increases to R q = 0.20 nm, but it is still lower than the initial roughness of the Ge substrate. At 65 • (figure 1(e)), the first signs of correlated surface structures emerge. These weakly pronounced structures have a preferential orientation perpendicular to the ion beam, but there is no regular pattern visible yet in both real Root mean square roughness (rms) R q as a function of ion incidence angle for Kr + and Xe + irradiated samples. E ion = 1200 eV, j ion = 300 µA cm −2 and ion fluence of 6.7 × 10 18 cm −2 . The dotted line indicates the initial roughness (R q (t = 0)) of an untreated Ge wafer. and Fourier space. The ripple pattern that evolves at 70 • (figure 1(f)) is more pronounced. From the power spectral density (PSD) curve of the pattern, a wavelength of about 98 nm is obtained. From this, it can be concluded that in the case of 1200 eV Kr + ion beam erosion ripple formation starts at a critical angle between 65 • and 70 • . At 75 • , (figure 1(g)), a saw tooth profile with an rms roughness of 4.0 nm is observed. The profile exhibits facets, forming a small angle of around 5 • with respect to the global surface plane. This faceting is a clear indication that gradientdependent sputtering dominates the surface evolution at this angle and this ion fluence. At an incidence angle of 80 • (figure 1(h)), a mixture of a ripple-like and hole pattern is observed with a preferential orientation perpendicular to the ion beam direction and period of around 34 nm. This pattern is very similar to the fine structure found on top of the facets (downstream side) for 75 • incidence angle. This behavior is caused by the increased local angle of incidence on top of the facets which is almost 80 • . This pattern can be better ascertained at Xe-irradiated samples where larger facets evolve at 75 • . At grazing incidence of 85 • (figure 1(i)), only small structures are visible with a preferential direction parallel to the ion beam direction. Furthermore, the surface roughness decreases down to 0.24 nm. Figure 2 summarizes the evolution of the rms roughness as a function of ion incidence angle for Kr + and Xe + ion beam erosion. A maximum is clearly pronounced at 75 • and an enhancement of the surface roughness in comparison to the initial surface is visible in the range of 70-80 • . For all other ion incidence angles the rms surface roughness is smaller compared to the initial one, indicating a regime of surface smoothing.

Influence of ion species on pattern formation on Ge
Similar to the investigations presented in section 3.1, the surface evolution was examined over the whole range of incidence angles with respect to four different noble gas ion species (Ne, Ar, Kr and Xe). Apparently, the surface evolution is comparable for the case of Kr and Xe (see figure 2). Contrary to Kr (and Xe), where the surface destabilization is very pronounced in a narrow range of incidence angles around 75 • (see figure 2), the surface evolution behaves totally different for Ne and Ar irradiation. The topography evolution is compared for different ion species at 75 • in order to illustrate the effect of ion species. Figure 3 shows the surfaces irradiated at an ion energy of E ion = 1200 eV with a total fluence of 6.7 × 10 18 cm −2 for Ne, Ar, Kr and Xe ions.
In the case of Ne and Ar, no structures are observed (figure 3, top row). In fact, a smoothing effect of the surface occurs for both. Their surface roughness is smaller than the surface roughness of the initial Ge substrate (R q ≈ 0.5 nm). As indicated in figure 3, the rms roughness decreases to 0.18 and 0.16 nm, respectively. A lower surface roughness for Ar-ion-irradiated surfaces compared to Ne can also be observed for other incidence angles as well as for irradiation with larger ion energies (e.g. E ion = 2000 eV). This means that the surface generally becomes smoother after argon bombardment than for Ne irradiation. In contrast, there is a faceted topography evolving for Kr and Xe irradiation at 75 • associated with a higher surface roughness. The single facets are larger in the case of Xe. As described in section 3.1, these facets form a small angle of around 5 • with the global surface plane which results in a local ion incidence angle on top of the facets of around 80 • . This induces a ripple-like fine structure on top of the facets with a mean period which is comparable with this evolving at an ion incidence angle of 80 • after Kr or Xe irradiation. However, for Kr bombardment as shown in figures 1 and 3, it is hard to recognize due to the smaller facet size. Comparing Kr and Xe, the evolving structures and the general trend with the incidence angle are similar but the amplitude of the structures is higher in the case of Xe (see also figure 2).

Role of the ion energy on the topography evolution
This section describes the role of the ion energy E ion on the pattern formation. The ion energy is varied between 400 and 2000 eV, which is limited by the used ion beam source and power supply. The samples were eroded with Xe + ions at incidence angles of 65 • as well as 75 • with a total fluence of 6.7 × 10 18 cm −2 . The corresponding AFM images are presented in figures 4 and 6.
In particular, the angle of 65 • (figure 4) was chosen to have a closer look at the transition regime between smoothing and roughening which takes place around this value. Furthermore, the largest structures evolve around 75 • (figure 6) where the sputter yield has a maximum. A highly regular ripple pattern evolves for the lowest energy (400 eV) at 65 • (figure 4(a)). The ripples are oriented perpendicular to the ion beam direction (i.e. ripple wave vector is parallel to ion beam projection) with a wavelength slightly above 30 nm. This regular pattern is superimposed by long wavelength triangular and faceted hillocks and depressions. Along these structures the ripples are curved and the wavelength of the ripples slightly changes from the upstream to the downstream side depending on the local surface angle. This is illustrated in the magnified 400 eV image (figure 4(e)). For an ion energy of 800 eV (figure 4(b)), the ripple wavelength increases to approx. 52 nm and the amplitude of the ripples decreases compared to 400 eV that correlates with a smaller rms value, indicated in the top left corner of the image. In addition, the pattern for 800 eV becomes more irregular. If the ion beam energy increases further, the ripple pattern vanishes and rudiments of a pattern are visible only. The reduction of surface amplitude fluctuation is also seen from the lower rms roughness values, where the smallest roughness is observed for 2000 eV (figure 4(d)). Furthermore, no signs of a pattern are visible in the AFM images as well as in the Fourier analysis. The overall behavior is also demonstrated in figure 5 where the PSD curves are plotted for the AFM images from figure 4. It is clearly seen that the wavelength shifts to larger values (smaller spatial frequencies, indicated by the arrows) and the ripple regularity decreases with increasing ion energy, which can be observed from broader peaks and suppression of higher-order peaks in figure 5. It is obvious that the roughness decreases over the whole spatial frequency range encompassed by the AFM measurements.
The second series of experiments has been done at an ion incidence angle of 75 • , whereas other ion beam parameters were the same as for the series described before. The experimental results are summarized in figure 6. At this angle of incidence saw tooth profiles with rms roughness values between 6 and 8 nm evolve for all used ion energies, similar to the pattern seen for Kr + ion erosion of Si [25]. The topographies show extended facets at the downstream side and the angle of the local surface normal toward the global surface normal is determined to be 5 • . For increasing ion energy the size of the facets shrink and the topography is more redolent of a faceted ripple pattern. As described in section 3.2 the downstream facets are decorated by a sub-pattern which can be understood again by the enlarged local incidence angle on the facets. From the magnified amplitude error images (figures 6(d)-(f)) it can be seen that the period and amplitude become more pronounced with increasing ion energy.

Temporal evolution of surface pattern
The last set of experiments focused on the time evolution of the surface pattern. Among the different parameter regimes that have been investigated two time series will be shown. In the first case, E ion = 600 eV and an incidence angle of 65 • were chosen where ripple formation occurs (see figure 4). The second set of experiments was conducted using 1200 eV Xe ions and an incidence angle of 75 • , a regime where significant faceting appears (see figure 6). The samples were irradiated for durations from 1 up to 120 min that corresponds to a fluence range From these images, several important points can be noted. It is clearly seen that ripple pattern changes with fluence. The amplitude and regularity of the ripple pattern increase with time. Furthermore, larger wavelength structures, i.e. larger triangular and faceted hillocks and depressions become more and more evident. This is also observed from the PSD curves in figure 8 calculated from 4µm × 4 µm AFM images.
In the PSD curves, two distinct frequency ranges can be identified. In the low spatial frequency region ( f 10 µm −1 , corresponding to spatial wavelengths 100 nm) the roughness grows rapidly with fluence as seen as hillocks and depressions in the AFM images. Moreover, a high-frequency peak which correlates with the ripple pattern is evident. Beginning with the smallest adjustable fluence the associated peak shifts to lower frequencies (larger ripple wavelengths) with increasing erosion time. The arrows in figure 8 mark the ripple wavelength and the dotted lines indicate the peak positions (wavelength) at smallest and largest ion fluences, respectively. It is further evident that the width of the peaks decreases with the ion fluence correlating with a higher regularity of the pattern. These experimental findings are summarized in figure 9, which shows the temporal evolution of the surface roughness and ripple wavelength for more than two orders of magnitude varying ion fluence (or erosion time).
As is already expected from the PSD curves, R q and λ grow non-exponentially with time. It should be noted that for ion fluences less than or equal to 1.12 × 10 18 cm −2 , the presented data are averaged over two independent experimental runs, where the individual runs show a variation in surface roughness and ripple wavelength 5% related to the averaged values.
Finally, the evolution of faceted pattern at 75 • with ion fluence will be addressed. Characteristic AFM images taken after erosion with different fluences are presented in figure 10. It is noticed that the pattern after the shortest irradiation time ( figure 10(a)) shows a ripplelike appearance with wavelength of approx. 42 nm and the regularity is less than that for the corresponding case at 65 • . With increasing fluence the structures are subjected to a strong coarsening (i.e. smaller ones disappear while larger ones grow). Furthermore, the pattern becomes more and more faceted and again the downstream sides of the faceted structures form an angle of 5 • (± 0.5 • ) toward the global surface plane (as already addressed in Section 3.1). If the fluences are high enough ( 3.37 × 10 18 cm −2 ), a sub-pattern emerges at the downstream side of the facets with a ripple wavelength of approx. 45 nm, similar to the pattern formed at 80 • incidence angle at plane surfaces (figure 1), which is clearly visible in the AFM image for a fluence of 1.35 × 10 19 cm −2 ( figure 10(f)). The quantitative evolution of the surface roughness and ripple wavelength/facet size is summarized in figure 11. Both quantities exhibit a non-exponential growth, as is already observed in the case of 65 • ion incidence angle ( figure 9). Considering the time evolution of the pattern at 65 • and In addition, the pattern coarsening is drastically reduced at 65 • ion incidence from λ ≈ 31 to 42 nm. In the case of 75 • the mean size of the ripples/faceted structures increases more than one order of magnitude from 42 to approx. 700 nm.

Comparison of results to former reports on pattern formation
Compared to previous investigations [45,50] it is apparent that in the present study no ripple and dot pattern have been encountered at low incidence angles. In the modified experimental setup used here (see section 2), no simultaneous co-deposition of iron takes place. As is recently  shown for the ion erosion of Si [25,26,31,32] the insertion of silicide-forming metals is essential for the pattern formation at low incidence angles. In the same way, it is evident that iron incorporation is likewise the key factor for the formation of highly regular pattern on Ge at normal or low ion incidence conditions. If the formation of iron germanides is not possible, as in the present experiments, surface smoothing occurs. In the following part the presented results, selected from a comprehensive study varying the control parameters ion species, ion energy, ion incidence angle and erosion time will be set within the context of recently published investigations of Kr + ion beam erosion of Ge under partially comparable experimental conditions [46,47]. Both studies focused on the linear regime of pattern formation at rather low ion fluences.
In the first study by Anzenberg et al [46], the kinetics of surface evolution of pre-smoothed and pre-rippled Ge under 1 keV Kr ion irradiation for selected incidence angles was examined. The pre-patterning with 500 eV Kr using a broad beam ion source at an incidence angle of 60 • results in a ripple wavelength of approx. 45 nm as unfortunately no ion fluence was specified. This is consistent with a ripple wavelength between 32 and 42 nm observed for 65 • ion erosion with 600 eV Xe (figure 9) even considering the growth of wavelength near the critical angle. Furthermore, it can be concluded that the critical angle in this study is below 60 • . For 1 keV Kr ions erosion of pre-smoothed surfaces at 75 • with another ion source a wavelength of approx. 60 nm can be deduced from grazing-incidence small-angle x-ray scattering (GISAXS) measurements presented in [46], which is again comparable to the value obtained in the presented experiments for the smallest ion fluences (figure 11). From a fit to the experimental dispersion GISAXS data Anzenberg et al derived the curvature-dependent coefficients as a function of ion incidence angle with stability to instability transition at 62 • . The authors were not sure whether the difference of the critical angle determined for 500 eV results from the different ion energy or from the different experimental setup. Our results illustrated in figures 4 and 5 clearly show that the transition angle (i.e. where ripples starts to form) shifts to lower angles with decreasing ion energies.
In a related study of Perkinson et al [47], using the same ion beam facilities as for pre-patterning in the former case, pattern formation at 250 and 500 eV Kr ion erosion was investigated for an ion incidence angle interval ranging from 0 • to 80 • . From the associated phase diagram, the authors deduced a critical angle of 57.5 • , separating surface smoothing from pattern formation. This validates again the observed trend that the critical angle decreases as ion energy becomes smaller, as it can be seen in figures 4 and 5. At an incidence angle of 75 • , a hole pattern is reported by Perkinson et al, similar to the sub-pattern shown in figure 1 for E ion = 1200 eV and 80 • incidence angle. The sub-pattern can be observed also on the downstream side of the evolving facets at α = 75 • (figures 3, 6 and 10), where the local incidence angle is approx. 80 • as discussed in sections 3.2 and 3.3. This gives evidence that not only the critical angle shifts to larger values with increasing ion energy, but the whole regime of pattern moves to large incidence angles (also seen in figure 6). In addition, the reported wavelengths in [47] are comparable to the wavelength at 400 or 600 eV in this work (e.g. figures 5 and 9). Slightly different from our investigations a ripple amplitude saturation for fluences 1.8 × 10 17 cm −2 (at least up to fluences of 1.8 × 10 18 cm −2 ) was observed. Also, the larger ripple wavelength at 250 eV compared with 500 eV for an incidence angle of 60 • (see table 1 in [47]) cannot be seen. Finally, it should be mentioned that our figures 9 and 11 substantiate the observation of Perkinson et al that the growth rates of surface amplitudes/roughness increase as the incidence angle is shifted to a larger value with respect to the critical angle.

Discussion of pattern formation with recent models
In the discussion of experimental results, both groups [46,47] conclude that in the case of Kr ion erosion of Ge, the CV-model [34] as well as the modified by Madi et al [35] give the proper qualitative behavior; however, they fail to predict qualitatively the observed critical angle which significantly differs from the expected value of 45 • . In the presented experiments, a wider range of parameters has been included compared to [46,47]. In order to see if there are further potential conflicts with the simplified mass redistribution model, we follow Madi's approach [35] and calculate the curvature-dependent coefficients S x for all ion/Ge-target combinations as well as for different ion energies E ion relevant to the experiments. However, deviating from the calculation method proposed by Madi et al, some modifications have been made in the calculation of the erosive (S eros x ) and redistributive parts (S distr x ) of the curvature-dependent coefficients (S x = S eros x + S distr x ). For the calculation of the mean depth a and the longitudinal σ and lateral µ straggling parameters of the deposited energy distribution, required for determination of the Bradley-Harper coefficients x and y , the approach of Bolse [51] was used. In this approach the distribution of displaced atoms (vacancies, recoils) is used to deduce the distribution of the deposited energy. The vacancy/recoil distribution is extracted from the collision.txt output file calculated by SRIM2012 [52] and analyzed assuming a Gaussian distribution. Recently, an alternative method for the estimation of these parameters is proposed by Bobes et al [53]. The calculated parameters a, σ and µ are given in table 1 for 1200 eV Ne, Ar, Kr, Xe and for various energies with Xe. Usually, the determined values are smaller than the values calculated from ion range distributions [35]. In addition, the sputter yield curves are obtained from TRIM.SP [48] because there are some discrepancies in the calculation of sputter yields with former versions of SRIM [54]. For the TRIM.SP simulations 250 000 ion impacts are simulated to ensure a sufficiently good statistic. Furthermore, the potentials and free parameters that have been suggested by Eckstein [55] are used. The ion energy-dependent mean displacement distance of target atoms δ, required for the calculation of the redistributive part, is determined from the expressions in [56]. The calculated curvature-dependent coefficients are summarized in Independently of ion species and energy, both curves in figure 12 show qualitatively the expected shape, i.e. a transition from stability (smoothing) to instability (parallel mode ripple) at an incidence angle of approx. 45 • which is inherent to the simplified approach for the redistributive part. Small deviations from 45 • are caused by the contribution from the erosive part. Therefore, ripple pattern should occur irrespective of ion species, which contradicts the lack of ripple formation for Ne and Ar ions. In addition, the onset angle of ripple formation (Kr, Xe) significantly differs from 45 • (figures 1 and 4). The shift of critical angle with ion energy (figures 4 and 5) is also not reproduced from figure 12(b). Furthermore, the expected ripple growth rate should increase with S x (and energy according to figure 12(b)), which is again not seen (figures 4 and 5). One further conclusion from the mass redistribution models is the diverging ripple wavelength in the vicinity of the angular region that separates instable (pattern forming) from stable (smoothing) surface evolution. From the presented investigations and from a set of similar investigations for Si, fused silica and sapphire [57] no evidence of a diverging wavelength in the vicinity of the critical angle is observed. Solely a moderate increase of ripple wavelength can be seen accompanied with a decline of ripple amplitude and pattern regularity.
At this point it is not clear if the crater function approach by Norris et al [38] gives a more realistic picture compared to the simplified CV/Madi [35] model because of lack of supporting MD simulations. Thus, it is still an open question whether the absence of pattern formation for Ne and Ar ions can be attributed to redistributive processes. However, based on the similarity of the basic mechanisms of ion-target interaction a completely different behavior is not expected for Ne, Ar from one side and Kr, Xe from the other side. Therefore, from the presented results it appears that mass redistribution alone cannot describe the observed scenarios for pattern formation on Ge.
In this context, it should also be mentioned that recently proposed models based on ioninduced solid flow [39,41] fail to explain the experimental observation. Both approaches predict a critical angle of 45 • , in coincidence with experimental results for Ar ion erosion of Si [40,58], and a diverging wavelength at this angle. In both models this is the direct consequence of the implemented angular dependence of the stress tensor [41] or the proportionality of the so-called body force to the local ion flux at the surface [39]. As noted by Norris [41] this proportionality could be no longer valid for higher incidence angles. At high incidence angles, the amount of reflected ions increases, which could change the angular dependence of the stress tensor. But it is again not clear if a modified angular dependence of the stress tensor could explain the completely different surface evolution under inert gas ion erosion of Ge.

Role of surface-gradient-dependent sputtering and reflection of primary ions
From the experiments can be concluded that an evidence for coarsening (e.g. figures 7-11) and faceting (e.g. figures 1-3 and 6) of surface pattern starting already at low fluences exists. Both observations indicate that gradient-dependent sputtering and reflection of primary ions play crucial role, just at the earliest accessible stage of surface evolution. As was mentioned above, sputtering and the reflection of ions for all possible ion/target combinations were analyzed with TRIM.SP simulation tool. From these calculations it is found that the amount of reflected ions naturally increases with ion incidence angle. Otherwise the amount of reflected ions for a given incidence angle increases for lower energies and also for ions with a lower mass. For example, approx. 1/3 of all Xe + ions at 65 • incidence angle and E ion = 600 eV are reflected. This number increases to 3/5 for E ion = 1200 eV and an incidence angle of 75 • . In conjunction with the Hauffe mechanism [59], this accounts for the reduced ripple coarsening seen in figure 9 (Xe + , 65 • , 600 eV) compared to figure 11 (Xe + , 75 • , 1200 eV). In the Hauffe model coarsening of structures is attributed to reflected ions impinging adjoining structures. The local erosion rate is enhanced in the vicinity of larger structures as more particles are reflected from larger structures than from smaller ones. Consequently, smaller features close to the larger ones vanish and the surface coarsens. This effect strongly enhances at larger incidence angles, as already seen on SiO 2 surfaces [60].
Additionally, the angular dependence of the normalized sputter yields (normalized to their values at normal incidence) and the normalized deposited energy for all four ions with 1200 eV were extracted from TRIM.SP simulations and plotted in figures 13(a) and (b). Likewise the deposited energy, which is calculated from total energy of all incoming ions subtracting the total energy of all reflected and sputtered ions/atoms, was also normalized to the value at normal incidence. From figure 13(a) it is obvious that the angular variation of the sputter yield is more pronounced for Kr or Xe ions compared with Ne or Ar. Consequently, the difference in the amplification rate of two adjacent surface elements with different local surface slopes is higher for Kr and Xe irradiation as for Ar or Ne. In contrast, nearly the same quantitative curve shape for the deposited energy is obtained. This is due to the fact that for heavier ion masses less ions (and energy) are reflected but more atoms are sputtered. For lighter ions the reverse scenario is found. It should be underlined that these findings correlate with recent MD simulations of the deposited energy distribution and sputtering of Si and Ge by Hossain et al [61] The authors could validate the assumption of an ellipsoidal energy distribution as in Sigmund theory of sputtering [62]; however, the linear relationship between local sputter yield and local deposited energy as supposed by the BH model [33] is not valid. Nevertheless the overall deposited energy decreases with incidence angle ( figure 13(b)), the local surfacedeposited energy grows with the incident angle, up to the angle, where the sputter yield has maximum, is reached [61]. At oblique ion incidence the ellipsoid of the deposited energy is inclined to the surface and the center of the distribution is closer to the surface. Nevertheless, the angular increase of surface-deposited energy is significantly lower than the angular variation of sputter yield [61]. In figures 14(a) and (b) the angular curves of sputter yield and energy distribution were calculated for the 'limiting cases' of the experiments: lowest (400 eV) and highest (2000 eV) ion energy and lightest (Ne) and heaviest (Xe) ion. Typically, the normalized total deposited energy ( figure 14(b)) is smaller at lower ion energies due to the higher reflection coefficient at these energies. For larger incidence angles and higher ion energies the difference becomes smaller. Thus, comparable angular variations of the surface-deposited energy are expected independent of E ion and ion mass as can be seen in table 1.
However, large variations in the sputter yield curves in figure 14(a) are seen again. In general, the angular variations for Xe ions are much larger compared to Ne. For Xe the maximum in sputter yield shifts to lower angles with decreasing energy. The same tendency is observed for Ne, but here the angular variation Y (α)/Y (0 • ) grows at higher energies in contrast to Xe. Figure 14(a), together with figures 3 and 4 implies that pattern formation is strongly influenced by the angular variation in sputter yield. It seems that if the variation is too small no ripples or surface structures are formed. The model calculations of sputtering and reflection in conjunction with the experimental results (showing structure coarsening and faceting) indicate that gradient-dependent sputtering together with reflection of ions significantly contribute to the surface evolution at least for the studied experimental conditions (ion/target combinations, incidence angles, ion energies and ion fluences). Already for angles near the onset of pattern formation (approx. 65 • , dependent on ion energy) and earliest erosion time or fluences (approx. 1 min or 10 17 cm −2 ) these mechanisms can affect the pattern formation. One should be aware that both processes give a nonlinear and a non-local contribution to the patternforming mechanism. In this context it is interesting to note that no clear evidence for diverging wavelengths is seen in experiments, as only a moderate growth of wavelength approaching the transition angle accompanied by a decrease of ripple amplitudes is observed. This contradiction to present models of pattern formation, either of redistributive or stress-induced solid flow origin, might be solved by introducing an additional non-local process to the existing models as pointed out by Davidovitch et al [63] Originally, a non-local damping term for pattern formation was proposed by Facsko et al [64], which mimics the preferred re-deposition of sputtered material into valleys compared to hillocks. Simulations for the re-deposition mechanism in a first simple approximation predict a transition from instability to stability/smoothing under large damping conditions [64]. A similar effect, namely the preferred erosion of adjacent protrusions, can be caused by reflected ions. At this point it can only be speculated if a surface stabilization or smoothing can arise from a large portion of reflected particles. An analogous situation is assumed to be responsible for surface polishing under glancing incidence conditions where most of the ions incident on a nearly perfect plane surface will be specularly reflected after collisions with surface atoms. However, if the ion impinges surface irregularities (e.g. polishing scratches or bumps), where the probability of sputtering off atoms from the surface is significantly higher [65], a smoothing effect is observed. For somewhat smaller ion incidence angles, most ions are still reflected in the forward direction and result in a distinct forward sputtering of initial topographic defects, especially at the upstream side of concave surface regions (depressions). The defect size grows in the direction of ion beam, probably accompanied by a guiding of ions along the defects, similar to the formation of perpendicular mode ripples on metallic surfaces at 83 • via a network of coalesced elongated vacancy islands [66,67].
In this case the initial structure wavelength is set by mean defect separation while ripple coarsening can be again explained as a result of the ion reflection in ion beam direction along the sidewalls according to the Hauffe mechanism. This would coincide with the results for ion erosion of polycrystalline metal films at grazing incidence where the initial surface morphology affects the formation of nanostructures and drives the formation of ripples with a mean wavelength given by the original grain size [68]. Such a kind of pattern can be also interpreted as perpendicular mode ripple pattern, however, induced by ion reflection and sputtering under grazing incidence conditions, alternative to the change of ripple orientation within the BH model.
At this point, it becomes clear that more investigations are necessary for the full explanation of the influence of reflected ions on the surface evolution at large incidence angles. From an experimental point of view this is currently addressed with further investigations for different materials, especially for Si as direct counterpart for the Ge study, presented here, and two oxidic materials, i.e. SiO 2 and Al 2 O 3 .

Summary
In conclusion, surface evolution of Ge by low-energy ion beam erosion with different noble gas ions (Ne, Ar, Kr and Xe) was examined for a wide range of experimental parameters. The dependence on ion incidence angle (0-85 • ) and ion energy (400-2000 eV) for ion fluences from 1.1 × 10 17 up to 1.3 × 10 19 cm −2 were investigated. In detail it is found that no pattern formation occurs in the case of Ne and Ar ions and ripple formation with Kr and Xe starts at larger incidence angles (approx. 65 • ). Until this angle surface stabilization or smoothing is found. This supports that former results are affected by Fe co-deposition, allowing the formation of highly regular pattern at small ion incidence angles similar to Si. In the absence of co-deposition only a small angular range for patter formation exists. The incidence angle for the onset of ripple formation depends on ion mass and ion beam energy. With decreasing ion energy, the angle shifts to lower values. The time evolution at specific ion incidence angle reveals a coarsening and faceting of the pattern starting with the earliest erosion times or ion fluence accessible within the experiments. Both observations indicate that gradient-dependent sputtering and reflection of primary ions play crucial role for Ge surface evolution. The results are discussed in detail with respect to recently proposed redistributive or stress-induced models for pattern formation. Based on experimental findings and supported by TRIM.SP simulations, it is speculated that a large angular variation of the sputter yield seems to be a potential requirement for surface instabilities during ion erosion of Ge. Finally, it was assumed that reflected primary ions could contribute to surface stabilization and surface smoothing. This could give rise to a non-local contribution to surface evolution which could also explain the non-diverging ripple wavelength at the ion -incidence angle-dependent stability-instability transition observed in the experiments for Ge.