Stability and symmetry of ioninduced surface patterning
 Christopher S. R. Matthes^{1}Email authorView ORCID ID profile,
 Nasr M. Ghoniem^{1} and
 Daniel Walgraef^{2}
DOI: 10.1186/s4131301700051
© The Author(s) 2017
Received: 9 March 2017
Accepted: 9 May 2017
Published: 21 June 2017
Abstract
We present a continuum model of ioninduced surface patterning. The model incorporates the atomic processes of sputtering, redeposition and surface diffusion, and is shown to display the generic features of the damped KuramotoSivashinsky (KS) equation of nonlinear dynamics. Linear and nonlinear stability analyses of the evolution equation give estimates of the emerging pattern wavelength and spatial symmetry. The analytical theory is confirmed by numerical simulations of the evolution equation with the Fast Fourier Transform method, where we show the influence of the incident ion angle, flux, and substrate surface temperature. It is shown that large local geometry variations resulting in quadratic nonlinearities in the evolution equation dominate pattern selection and stability at long time scales.
Introduction
The erosion of surface material by ion sputtering is a fundamental process, which leads to the formation of surface roughness and patterns at the nanoscale. In some technological applications, sputtering erosion can be a significant factor in material degradation, while in others, nanopatterning by energetic particles can be a useful fabrication tool. The bombardment of solid surfaces with energetic ions initiates near surface collision cascades and the ejection of surface atoms. While a fraction of the ejected atoms may find their way back to be deposited on the surface, the majority travel farther away as the surface is eroded. The result of such atomistic events is a complex process of roughening, pattern formation, erosion and redeposition; all of which have the ingredients of producing patternforming instabilities (Makeev et al. 2002).
Experimental evidence shows that ion sputtering can result in the formation of periodic surface patterns (Costantini et al. 2001; Habenicht 2001; Navez et al. 1962; Rusponi et al. 1997; Valbusa et al. 2002). The nature of these patterns, including their wavelength, amplitude, and orientation depends on many factors (e.g. ion energy, flux, angle of incidence, substrate temperature, and material properties). Considerable research has been performed by a number of groups to examine the effects of these many parameters on the surface features that develop under ion bombardment. The many experimental variables involved in the determination of the pattern spatial symmetry and periodicity make it imperative to develop a theoretical understanding that can guide experimental research. Notable existing theories of ioninduced patterning have been developed by Sigmund (1973), Bradley and Harper (1988), and Makeev et al. (2002).
Among the key observations of ioninduced nanopatterning relate to the effects of the ion incidence angle, the ion fluence, and the surface temperature. It has been observed by Navez et al. (1962) that the bombardment of a clean glass surface with an ion beam produces surface morphology that is dependent on the ion beam incidence angle, θ. The observed morphology was the formation of ripple structures oriented perpendicular to the ion beam for incidence angles close to normal (θ=0), and parallel to the ion beam for incidence angles close to grazing. Habenicht studied the effect of ion fluence on surface dynamics by exposing a highly ordered pyrolytic graphite (HOPG) surface with (0001) orientation increasing levels of ion fluence (Habenicht 2001). At high ion fluence the surface roughness increased as the nanostructures grew in size. Rusponi et al. (1997) studied the influence of temperature on the surface morphology of ionbombarded silver. The results of their work showed that ion bombardment at low temperatures (160K) resulted in a rough surface, but as the temperature was raised to 290 K and above, a ripple structure began to appear aligned along the \(<1\bar {1}0>\) direction. The tendency of surface structures to orient along a particular direction is attributed to the enhanced surface diffusion that occurs at higher temperatures. On the surface of metals such as Ag(110) and Cu(110), the inchannel direction, \(<1\bar {1}0>\) is an easy pathway for the diffusion of adatoms and vacancies compared with the crosschannel, or <001> direction. Therefore, at higher temperatures where anisotropic surface diffusion is enhanced, the ripple structures have a quicker path to organize in that orientation (Costantini et al. 2001).
The early ideas of Sigmund provided the basis for Bradley and Harper to develop a continuum equation for surface evolution, from which the wavelength of the emerging pattern can be determined. More recently, Makeev et al. provided a comprehensive extension of the theory, where additional nonlinearities were incorporated as a result of a more rigorous analysis of the local geometry around the ion impact region. Nevertheless, the general framework remained consistent with earlier developments, with additional insights on casting the evolution equation into more familiar forms in the nonlinear dynamics literature. In the present work, we extend these theoretical efforts further in two regards. First, we examine the stability and symmetry of evolving surface patterns with an analytical procedure. Second, we develop numerical solutions for the evolution of surface patterns that are consistent with the developed analytical method.
The main objective of the present work is to develop analytical and computational methods to further the understanding of surface pattern evolution under ion bombardment. We build on previous efforts by Sigmund (1973), Bradley and Harper (1988), and Makeev et al. (2002). Specifically, we aim at determination of the pattern wavelength and spatial symmetry at the later stages of surface evolution beyond the linear regime. Our efforts are confined to the main aspects of ioninduced surface instabilities, where the effects of surface stress, subsurface point defect generation, surface composition, impurities, mass redistribution, hydrodynamic effects, stochastic noise, and other contributing processes are excluded. We then develop a numerical method to describe the evolution of surface patterns, where the competition between erosion, redeposition, and surface diffusion is considered. We begin by reviewing the framework used to explain the underlying erosion mechanisms controlling surface morphology, as well as theoretical attempts to model deposition in “Background theory” section. Stability analysis of the governing equations is then explored in “Stability analysis” section, providing insight into the effects of nonlinearities on surface evolution and expected pattern formation. The numerical method for simulating surface evolution using an FFT algorithm is then put forward, and the results of the numerical simulations are presented in “Numerical simulations of pattern evolution” section. The interest here is to compare analytical theory to numerical simulations in order to reveal the role of nonlinear phenomena on pattern selection, stability, and longterm evolution. Lastly, a discussion of the numerical results of surface stability is given in “Numerical simulation results” section, and a summary of conclusions of the study is provided in “Conclusions” section.
Background theory
The continuum theory of surface erosion and stability due to energetic particle sputtering is reasonably wellestablished, and dates back to the work of Sigmund (1973). Several authors have added more features to the theory, and applied it to the understanding of surface nanopatterning and roughening. We will briefly introduce the theory here for completeness, while references (Makeev et al. 2002) and (Bradley and Harper 1988) provide more detailed descriptions.
When an obliquelyincident ion bombards the surface, it initiates a collision cascade downstream, leading to the removal of surface atoms that are energized by the Primary Knockon Atom (PKA). Surface atoms that receive enough energy to break their bonds will be sputtered. If the surface location where the cascade initiates is concave (a local trough), more surface atoms will be closer to the PKA position than a convex surface, and thus more material will be removed. This fundamental idea was introduced by Bradley and Harper (1988), and it obviously leads to surface instabilities, since troughs will continue to be deeper as disproportionately more atoms are removed. Other phenomena can lead to modification of this behavior, for example, when surface diffusion is considered, it results in smoothing out of this geometrydictated instability, and a periodic structure is obtained. Additional factors can also be introduced to account for surface stresses and point defect generation (Lauzeral et al. 1997; Walgraef et al. 1997), as well as higherorder nonlinearities and noise (Makeev et al. 2002). Mass redistribution resulting from the momentum transfer by incident atoms has also been shown to be an influential factor in the dynamics of pattern formation (MuñozGarcía et al. 2014). The presented study, however, is constrained to the primary known mechanisms for modeling ion sputtering, including curvatureinduced erosion, temperatureinduced surface diffusion, and the effect of nonlinearities and linear damping.
where the coefficients Γ _{ X } and Γ _{ Y } are functions of the angle of incidence φ, as well as the collision cascade dimensions, a and β. Here a is the penetration depth of the ion along its trajectory, and β is the energy distribution width, assuming isotropy in the X and Y directions (i.e. a spherical distribution) to greatly simplify the terms, as utilized by Valbusa et al. (2002). Y _{0}(φ) represents the sputtering yield, and is defined by the ion energy, angle of incidence, and collision cascade dimensions. The atomic surface density is n, while the radii of curvature in the X and Y direction, R _{ X } and R _{ Y }, respectively, are equivalent to the second derivatives of the local surface height, Z, in each direction.
This linear BH equation is a useful model for predicting ripple formation and explaining experimentally observed processes, such as ripple orientation and the flux and ion energy dependence of the wavelength (Makeev et al. 2002). However, the equation is limited to small local changes in the surface curvature, where differences between local and global (lab) frames are small.
The damping term results in smoothing of all spatial frequencies, thereby inhibiting kinetic roughening (Keller and Facsko 2010). Its presence in the continuum equation for ionbombarded surfaces has been used to account for redeposition of sputtered material (Facsko et al. 2004; Keller and Facsko 2010), although this has been disputed (Bradley 2011).
where the coefficients are defined as: \(\hat \nu _{x}=2s^{2}c^{2}a_{\beta }^{2}s^{2}c^{2}\,\quad \hat \nu _{y}=c^{2},\quad \hat v_{x}=3s^{2}cc^{3}a_{\beta }^{2}s^{2}c^{3},\quad \hat v_{y}=c^{3}\), with c= cosφ and s= sinφ. In this expression, the following is defined as \( F=\frac {J\varepsilon {pa}_{\beta }^{2}e^{a_{\beta }^{2}/2}}{2(2\pi)^{1/2}}\), where J is the ion flux, ε is the ion energy, and a _{ β }=a/β characterizes the collision cascade size. Equation (6) is the time evolution equation of the surface in the laboratory frame. This expression has been simplified for normal ion incidence (θ=0), and by assuming an isotropic, or spherical energy distribution for the collision cascade (i.e. a radius of β).
The damping term depends directly on the surface position, h, rather than its derivatives, indicating that its effect goes beyond the scope of Sigmund’s theory of sputtering (Sigmund 1973). The influence of this term on the surface morphology may be explained as the selfdeposition of sputtered material resulting from ion bombardment (Facsko et al. 2004). When patterning is present, a significant amount of material may be deposited due to lineofsight interaction of the sputtered material with adjacent surface features. This effect is more significant in the surface troughs rather than the peaks, thereby producing a damping effect to the curvature instabilities described in the BradleyHarper model (Bradley and Harper 1988).
Stability analysis
Linear stability analysis
where \(\epsilon = \frac {\hat {\nu }_{x}^{2}}{4\bar {K}}\bar {\alpha }\). For positive values of ε, where \(\bar {\alpha }\) is less than \( \frac {\hat {\nu }_{x}^{2}}{4\bar {K}}\), spatial modes with \(\vec {q}=\pm \, q_{c}\vec {1}_{x}\) are unstable, that is \(\vec {q}=\pm \, \sqrt {\frac {\hat {\nu }_{x}}{2\bar {K}}}\vec {1}_{x}\).
On varying the beam incidence angle, the difference between \(\hat {\nu }_{x}\) and \( \hat {\nu }_{y}\) varies, and \( \hat {\nu }_{y}\) may become greater than \(\hat {\nu }_{x}\). In this case, the maximum linear growth rate corresponds to the modes with \(\vec {q}=\pm \,\sqrt {\frac {\hat {\nu }_{y}}{2\bar {K}}}\vec {1}_{y}\) and instability occurs at \(\bar {\alpha }= \frac {\hat {\nu }_{y}^{2}}{4\bar {K}}\).
The range of ripple sizes observed in plasmafacing materials has been well documented. As previously reviewed, nanoripples were observed on glass substrates early on by Navez et al. (1962). In addition, ripple formations have been seen to develop on the insulator rings of Hall thrusters at the mmscale (De Grys et al. 2010). The latter case likely involved the influence of stress effects due to plasma bombardment and thermal conditions. These experimental observations demonstrate the variety of size scales that may result from plasma exposure as surface features develop.
Weakly nonlinear stability analysis
As time proceeds, nonlinear terms grow and have to be taken into account. According to the distance from the instability threshold, these terms may saturate the linear growth of unstable modes and stabilize specific spatial patterns, or may induce spatiotemporal behavior that is irregular in space and time. To discuss the qualitative aspects of these regimes, let us consider the damping rate α as a varying parameter with other coefficients corresponding to anisotropies, diffusion, beam orientation, etc. as fixed.
where \(q^{2}_{c} = \frac {{\hat {\nu }}_{x}}{2\bar K}\), \(\epsilon = \frac {\hat \nu _{x}^{2}}{4\bar K}\bar \alpha \). Hence, on decreasing \(\bar \alpha \) below \(\frac {\hat \nu _{x}^{2}}{4\bar K}\), spatial modes with \(\vec q = \pm \, q_{c}\vec 1_{x}\) become first unstable, followed by modes different in wavenumber or in increasing orientation difference with \(\vec 1_{x}\) up to \(\bar \alpha = \frac {\hat \nu _{y}^{2}}{4\bar K}\), where modes with \(\vec q = \pm \, \frac {\vert \hat \nu _{x}\vert }{2\bar K}\vec 1_{y}\) become unstable. For \(\bar \alpha < \frac {\hat \nu _{y}^{2}}{4\bar {K}}\), modes with all orientations become thus unstable, but the maximum growth rate still corresponds to \(q_{c}\vec 1_{x}\). Finally, for \(\bar \alpha = 0\), the set of unstable wave vectors extends to \(\vec q = 0\), characteristic of the undamped KS equation.

For example, for critical ripples of uniform amplitude \(\left (\bar h = A_{1} e^{{iq}_{c}x} + A_{2} e^{i2q_{c}x} + \ldots + c.c \right)\) one obtains:$$\begin{array}{@{}rcl@{}} \partial_{t} A_{1} &=& \epsilon A_{1} + 4q_{c}^{2}v_{x} A_{2}A_{1}^{*} + \ldots \\ \partial_{t} A_{2} &=& \left(\epsilon 9{Kq}_{c}^{4}\right)A_{2}  q_{c}^{2}v_{x} A_{1}A_{1} + \ldots \end{array} $$(14)and the adiabatic elimination of harmonics leads to$$ \partial_{t} A_{1} = \epsilon A_{1}  u \vert A_{1}\vert^{2} A_{1} + \ldots $$(15)
where \(u=\frac {4q_{c}^{4}v2}{{Kq}_{c}^{4}  \epsilon }\simeq \frac {4v_{x}^{2}}{K}\). Similarly, amplitude equations for spatially varying amplitude may be obtained within the usual methods of pattern formation theory (Ghoniem and Walgraef 2008; Walgraef 1997), resulting in the formation of stable ripples with critical wave vector parallel the x axis and amplitudes \(\vert A_{1}\vert =\sqrt {\frac {\epsilon }{u}}\), \(\vert A_{2}\vert = \frac {}{} \epsilon \),...

The quadratic nonlinearity also couples triplets of modes satisfying the triangular relation \(\vec q_{1} + \vec q_{2} +\vec q_{3} =0\) which are able to destabilize onedimensional patterns and sustain stable hexagonal ones. For example, the wave vector of critical ripples, considered as \(\vec q_{1} = q_{c} \vec 1_{x}\), is coupled to \(\vec q_{2} = \frac {q_{c}}{2}\vec 1_{x} + \frac {\sqrt {3}}{2}q_{c}\vec 1_{y} \) and \(\vec q_{3} = \frac {q_{c}}{2}\vec 1_{x}  \frac {\sqrt {3}}{2}q_{c}\vec 1_{y} \). On writing the profile built on these wave vectors as \(\bar h = \sum _{n} (A_{n} e^{in\vec q_{1}\vec r} + B_{n} e^{in\vec q_{2}\vec r} +C_{n} e^{in\vec q_{3}\vec r} + c.c) \), the corresponding amplitude equations, for the dominant contributions, have the structure:$$\begin{array}{@{}rcl@{}} \dot A_{1} &=& \epsilon A_{1} + v_{A} B^{*}C^{*} u_{A} \vert A_{1}\vert^{2} A_{1}\\ \dot B_{1} &=& (\epsilon  \Delta)B_{1} + v_{B} A_{1}^{*}C_{1}^{*} u_{B} \vert B_{1}\vert^{2} B_{1}\\ \dot C_{1} &=& (\epsilon  \Delta) C_{1} + v_{C} B_{1}^{*}A_{1}^{*} u_{C} \vert C_{1}\vert^{2} C_{1} \end{array} $$(16)where \( \Delta =\frac 34 q^{2}_{c}(\vert \hat \nu _{x}\vert \vert \hat \nu _{y}\vert)\), \(v_{A} =\frac {q^{2}_{c}}{2}(3\hat v_{y}  \hat v_{x})\), \(v_{B}=v_{C}=q_{c}^{2}\hat v_{x}\), \(u_{A}=\frac {4\hat v_{x}^{2}}{9\bar K} \), \(u_{B}=u_{C}=\frac {4}{9\bar K+4\Delta }\left (\frac {(\hat v_{x}  3 \hat v_{y})^{2}}{4}\right)\). Critical ripples correspond to the steady state \({ A_{1} }_{s} = \sqrt {\frac {\epsilon }{u_{A}}}\), B _{1}=C _{1}=0. The linear evolution of perturbations of this state is given by:$$\begin{array}{@{}rcl@{}} \dot B_{1} &=& (\epsilon  \Delta)B_{1} + v_{B} A_{1s}^{*}C_{1}^{*}\\ \dot C_{1} &=& (\epsilon  \Delta) C_{1} + v_{C} B_{1}^{*}A_{1s}^{*} \end{array} $$(17)and the corresponding linear growth rate is positive for$$ \epsilon > \frac{v_{B}^{2}}{4u_{A}}\left[1+\sqrt{1+\frac{4\Delta u_{A}}{v_{B}^{2}}}\,\right]^{2}= \frac{9\bar K q_{c}^{4}}{16}\left[1+\sqrt{1+\frac{16\Delta }{9\bar K q_{c}^{4}}}\,\right]^{2} =\epsilon_{c} $$(18)
Hence, in large systems, for which the present discussion is valid, when ε<ε _{ c }, ripples should be selected, while for ε>ε _{ c }, steady state solutions of (16), corresponding to anisotropic hexagonal patterns (also viewed as “dotsonripples” patterns), should develop.
Note that in the twovariable model proposed by Motta et al. (2012,2014), the resulting set of unstable wave vectors consists of a finite domain around the critical wave vector, as in the DKS equation, plus a marginal mode with \(\vec q =0\). This is a classical problem of Turinglike instability coupled with a Goldstone mode (Cox and Matthews 2003; Dewel et al. 1995). The nonlinear couplings between the Goldstone mode and the finite wavelength unstable modes induce differences in the selection and stability ranges of patterns. In reference (Motta et al. 2012), a detailed amplitude equation analysis, including the effect of anisotropy, leads to a similar conclusion also supported by numerical analysis. The comparison between the outcome of the DKS and the twovariable model would of course be highly desirable in the analysis of specific experimental situations.
Beyond the weakly nonlinear regime
By decreasing the damping rate α (increasing ε), the range of unstable wave vectors increases, which allows more coupling between small and large scales. The weakly nonlinear approximation breaks down and the amplitude equation description is not valid anymore. Spatiotemporal patterns usually develop and numerical analysis is required to study them, as we will present in the Numerical simulations of pattern evolution Section. However, for vanishing damping, our evolution equation becomes a genuine anisotropic KS equation. While the onedimensional KS equation has been studied in detail in a huge number of papers (a few basic references are (Cvitanovíc et al. 2010; Hyman and Nicolaenko 1986; Hyman et al. 1986; Kevrekidis et al. 1990)), the 2D version, which in the undamped case has features of the KPZ equation, is much less investigated. In the damped case, the formation of steady hexagonal patterns, breathing hexagons or more complex spatiotemporal behavior has been reported (Gomez and Paris 2011; Paniconi and Elder 1997). In the 2D anisotropic case, coarsening ripples may appear in the undamped case (Rost and Krug 1995), while in the damped case, a numerical study shows competition between one and twodimensional patterns (Vitral 2015).

One dimension. For values of θ such that \(\vert \hat \nu _{x}\vert >\vert \hat \nu _{y}\vert \), the system is anisotropic and its behavior may be considered as essentially onedimensional, where the evolution equation is:$$ \partial_{\tau}{\bar{h}} = \vert\hat\nu_{x}\vert\bar h_{XX}  \left(\hat D_{xx}+ \bar{K}\right){\bar{h}}_{XXXX}+ {\hat{v}}_{x}({\bar{h}}_{X})^{2} $$(19)This equation is one of the simplest PDEs, which exhibits spatiotemporal chaotic behavior. When x∈[0,L], Eq. (19) is equivalent to an infinite set of ODEs:$$ \frac{d}{dt}\hat h_{k} = \left({\hat{\nu}}_{x} k^{2}  \left(\hat D_{xx}+ \bar K\right)k^{4}\right)\hat h_{k}  {\hat{v}}_{x}\Sigma_{k'}(kk')k'{\hat{h}}_{kk'}{\hat{h}}_{k'} $$(20)
with \({\bar {h}}(x,t)=i\Sigma _{k}{\hat {h}}_{k}(t)\exp (ikx)\), \(k=nq, q=\frac {2\pi }{L}\), n∈Z. The zero solution is unstable versus modes with \(\vert k\vert <\sqrt {\frac {{\hat {\nu }}_{x}}{{\hat {D}}_{xx}+ \bar K}}=\frac {2\pi }{L_{c}}\), or L>L _{ c } (the number of such modes increases with L). If L is taken as the bifurcation parameter and grows beyond L _{ c }, the solution passes through a complex hierarchy of bifurcations leading to cellular multimodal stationary, oscillatory and chaotic states. A typical behavior includes an irregular succession of windows with quasiperiodic and chaotic behavior.
In this analysis, it is the boundary condition that determines the wavenumbers which enter in the unstable domain. In our problem, on varying beam orientation, temperature (\(\bar K\)), or damping rate, one may increase or decrease the number of unstable modes in the system and expect similar results as in the previously mentioned finite domain case. Following a high T approximation, \({\bar {K}}\) dominates and D _{ xx } may be neglected (Bradley and Harper 1988; Makeev et al. 2002). In the absence of damping, the domain of unstable wave vectors extends to zero, which rules out an amplitude equation description, except, perhaps, for the first stages of the evolution, which is dominated by the fastest growing modes and the ones generated by nonlinear interactions. The evolution is described by Eq. (19) and the fastest growing mode corresponds to \(\vec q_{0}=\sqrt {\frac {{\hat {\nu }}_{x}}{2{\bar {K}}}}\vec 1_{x}\) with a growth rate \(q_{0}^{4}\bar K\). It is directly coupled, through the quadratic nonlinearity, with \(2\vec q_{0}\), \(\frac 12 \vec q_{0}\). The resulting evolution equations for these modes are:$$\begin{array}{@{}rcl@{}} \frac{d}{dt}{\bar{h}}(\vec q_{0},t)&=&{\bar{K}} q_{0}^{4}\bar h(\vec q_{0},t) +4q_{0}^{2}{\hat{v}}_{x} {\bar{h}}(2\vec q_{0},t)h(\vec q_{0},t)\frac{q_{0}^{2}}{2}\hat v_{x} {\bar{h}}\left(\frac{\vec q_{0}}{2},t\right)^{2} \\ \frac{d}{dt}{\bar{h}}(2\vec q_{0},t)&=&  8\bar K q_{0}^{4}{\bar{h}}(\vec 2q_{0},t)q_{0}^{2}{\hat{v}}_{x} {\bar{h}}(\vec q_{0},t)^{2}+ \ldots \\ \frac{d}{dt}{\bar{h}}\left(\frac{\pm\vec q_{0}}{2},t\right)&=&\frac{7}{16}{\bar{K}} q_{0}^{4}{\bar{h}}\left(\frac{\pm\vec q_{0}}{2},t\right)+q_{0}^{2}{\hat{v}}_{x} {\bar{h}}(\vec q_{0},t){\bar{h}}\left(\mp\frac{\vec q_{0}}{2},t\right) \\ \end{array} $$(21)\({\bar {h}}(2\vec q_{0},t)\) may be adiabatically eliminated and one obtains:$$\begin{array}{@{}rcl@{}} \frac{d}{dt}{\bar{h}}(\vec q_{0},t)&=&{\bar{K}} q_{0}^{4}{\bar{h}}(\vec q_{0},t) \frac{{\hat{v}}_{x}^{2}}{ 2{\bar{K}}}{\bar{h}}(\vec q_{0},t)^{2} h(\vec q_{0},t)\frac{q_{0}^{2}{\hat{v}}_{x}}{2} {\bar{h}}\left(\frac{\vec q_{0}}{2},t\right)^{2}\\ \frac{d}{dt}{\bar{h}}\left(\frac{\pm\vec q_{0}}{2},t\right)&=&\frac{7}{16}{\bar{K}}q_{0}^{4}{\bar{h}}\left(\frac{\pm\vec q_{0}}{2},t\right)q_{0}^{2}{\hat{v}}_{x} {\bar{h}}(\vec q_{0},t){\bar{h}}\left(\mp\frac{\vec q_{0}}{2},t\right) \end{array} $$On the fastest time scale, \(O\left (1/\bar K q_{0}^{4}\right)\), \({\bar {h}}\left (\vec q_{0},t\right)\) saturates to \( \frac {\sqrt {2}{\bar {K}}q_{0}^{2}}{\vert v_{x}\vert }\) with \(\bar h\left (\pm \frac {\vec q_{0}}{2}\right) =0\). However this solution is always unstable versus \(\bar h\left (\pm \frac {\vec q_{0}}{2},t\right)\). Effectively, the linear evolution of small perturbations \({\bar {h}}\left (\pm \frac {\vec q_{0}}{2},t\right)\) is given by:$$\begin{array}{@{}rcl@{}} \frac{d}{dt}{\bar{h}}\left(\frac{\vec q_{0}}{2},t\right)&=&\frac{7}{16}\bar K q_{0}^{4}{\bar{h}}\left(\frac{\vec q_{0}}{2},t\right) + \sqrt{2}{\bar{K}}q_{0}^{4}{\bar{h}}\left(\frac{\vec q_{0}}{2},t\right)\\ \frac{d}{dt}{\bar{h}}\left(\frac{\vec q_{0}}{2},t\right)&=&\frac{7}{16}{\bar{K}} q_{0}^{4}{\bar{h}}\left(\frac{\vec q_{0}}{2},t\right) + \sqrt{2}{\bar{K}}q_{0}^{4}{\bar{h}}\left(\frac{\vec q_{0}}{2},t\right) \end{array} $$(22)Hence, the quadratic coupling between \({\bar {h}}(\vec q_{0},t)\) and \(\bar h\left (\pm \frac {\vec q_{0}}{2},t\right)\) enhances the latter’s growth rate to about \( 1.85{\bar {K}}q_{0}^{4} \). A cellular state with wavelength λ=λ _{0}=2π/q _{0} grows thus first and, after some time, is replaced by another cellular state with λ=2λ _{0}. This situation has been studied in more detail by Misbah et al. in (Misbah and Valance 1994). Similar arguments may be performed for successive subharmonics and provide a qualitative picture for the first stages of the surface evolution.

Two dimensions. In this case, the fastest growing mode remains the mode with \(\vec q = \vec q_{0}= q_{0} \vec 1_{x}\). Besides the quadratic coupling with \(2\vec q_{0}\) and \(\frac 12 \vec q_{0}\), \(\bar h(\vec q_{0},t)\) may also be coupled with \(\bar h(\vec q_{1},t)\) and \({\bar {h}}(\vec q_{2},t)\) where \(\vec q_{0}= q_{0}\vec 1_{x}\), \(\vec q_{1}=\frac {1}{2}q_{0}\vec 1_{x} +\frac {\sqrt 3}{2}q_{0}\vec 1_{y}\), \(\vec q_{2}=\frac {1}{2}q_{0}\vec 1_{x} \frac {\sqrt 3}{2}q_{0}\vec 1_{y}\). The coupled evolution of these modes is then:$$\begin{array}{@{}rcl@{}} \frac{d}{dt}{\bar{h}}\left(\vec q_{1},t\right)&=& \left({\bar{K}} q_{0}^{4}\Delta\right) {\bar{h}}\left(\vec q_{1},t\right)+q_{0}^{2}{\hat{v}}_{x}{\bar{h}}\left(\vec q_{0},t\right){\bar{h}}\left(\vec q_{2},t\right) + \ldots\\ \frac{d}{dt}{\bar{h}}\left(\vec q_{2},t\right)&=& \left({\bar{K}} q_{0}^{4}\Delta\right){\bar{h}}\left(\vec q_{2},t\right)+q_{0}^{2}{\hat{v}}_{x}{\bar{h}}\left(\vec q_{0},t\right){\bar{h}}\left(\vec q_{1},t\right)+ \ldots \end{array} $$(23)
where \(\Delta = \frac 34 q_{0}^{2}(\vert \hat \nu _{x}\vert  \vert \hat \nu _{y}\vert)\). On the saturation scale of \({\bar {h}}(\vec q_{0},t)\), the maximum growth rate of \(\bar h(\vec q_{1,2},t)\) is \(((\sqrt 2  0.5){\hat {\nu }}_{x}\vert + 1.5\vert \hat \nu _{y}\vert)\frac {q_{0}^{2}}{2}\).
This growth rate is greater than the effective growth rate of subharmonics \(\left (1.85\bar {Kq}_{0}^{4}=1.85 \vert \hat \nu _{x}\vert \frac {q_{0}^{2}}{2}\right)\) for \( \vert \hat \nu _{x}\vert < 1.6 \vert \hat \nu _{y}\vert \). Hence, there is a range of incidence angles around θ=0 where twodimensional effects should dominate over subharmonic ones during the first stages of the evolution. Further analysis requires of course numerical simulations, as presented in the next section.
Let us recall that we considered, in this section, systems with large spatial extension, where boundary effects are irrelevant, and random initial conditions. Boundary effects in small systems and preexisting initial patterns may force spatial modes into the system able to compete with the dynamically selected ones, and affect pattern selection and evolution. We consider these effects in a separate publication D Walgraef, CSR Matthes, NM Ghoniem: The influence of surface architecture on ioninduced nanopatterning, in preparation.
Numerical simulations of pattern evolution
Method development
The Fourier transform of derivatives can easily be found subsequently from \(\hat {u}_{k}\): \(\textrm {FFT}\left (\frac {\partial ^{\nu } u_{j}}{\partial x^{\nu }}\right)\equiv (ik)^{\nu }\hat {u}_{k}\). To model the directional dependence of the expression, it is necessary to introduce the indices k _{ x } and k _{ y }, as the derivatives are applied across both dimensions.
This result may be calculated for all time steps, and converted back to real space. The operation is performed over many time steps to simulate the evolution of the surface profile over a specified span. In the case of linear evolution, the nonlinear term in the numerator, expressed by the convolution, may be removed.
Convergence and accuracy tests
Parameters used to define the surface evolution
Parameter  Description  Quantity  Units 

J  Ion flux  5×10^{21}  [m] ^{−2}[s] ^{−1} 
ε  Ion energy  300  [eV] 
a  Ion penetration depth  2×10^{−9}  [m] 
β  Collision cascade dimension  1×10^{−9}  [m] 
D _{0}  Surface diffusivity  2.4025×10^{−7}  [m] ^{−2} 
γ  Surface free energy density  2.9  [J]/[m] ^{−2} 
Ω  Atomic volume  1.5825×10^{−29}  [m] ^{3} 
ρ _{ s }  Atomic surface density  7.0811×10^{18}  [m] ^{−2} 
T  Temperature  500  [K] 
The wavelength spectrum can by averaged at each time step in the following way: \(\lambda _{ave} = \frac {\Sigma \left (\lambda _{i}\cdot p_{i}\right)}{\Sigma \lambda _{i}}\), and is shown over a range of time steps on the left in Figs. 3 and 4 in the x and ydirection, respectively. It can be observed that the average wavelength begins at a value corresponding to the initial surface, and grows to a steadystate value as the surface morphology changes, representing the final “evolved” wavelength spectrum of the surface. The behavior of the average wavelength demonstrates the emergence of surface ripples of a particular wavelength, which cause the average spectral wavelength to grow toward a converged result as the most unstable wavelength takes over the morphology.
The “dominant” wavelength refers to the single wavelength with the highest spectral density at any given time. This value is shown on the right side in Figs. 3 and 4, and is observed to begin at a particular value determined by the initial configuration. In this case the dominant λ is shown to switch after 9 h of simulated exposure to a value close to the analytical result, indicated by the horizontal black line. The steadystate value corresponding to the wavelength expected by linear stability analysis further indicates an accurate converged value.
Numerical simulation results
Numerical simulations using the FFT method were performed to provide insight into the longterm evolution of the surface height, and extend the scope of analytical predictions described above. The simulations were performed without the presence of damping (i.e. \(\bar \alpha =0\)), such that the bifurcation parameter ε, and thus the range of unstable wave vectors is maximized. Under these conditions, surface symmetries according to the linear evolution equation may not match directly with theory, and pattern formation according to weakly nonlinear analysis becomes less predictable. Periodic boundary conditions have been applied. Spectral analysis was performed on the surface profiles to allow for a determination of dominant frequencies associated with the ripple wavelengths present in the developed symmetries. The parameter values used in all of the present numerical calculations are given in Table 1.
Effects of nonlinearities in the transient regime
The quadratic nonlinearity introduced in the surface evolution equations is physically a result of large local geometric variations, in violation of the small slope approximation of the original BH approach. Therefore, one expects that as time proceeds, the influence of higher order nonlinearities will become more significant. Our previous analytical results indicated that couplings between the dominant surface mode and other modes that are twice and half of the dominant wavelength will appear. Moreover, if the initial surface is intentionally structured, as previous experimental efforts have done using Chemical Vapor Deposition (Li et al. 2017; Matthes et al. 2017), it would be interesting to see the role of nonlinearities in pattern selection. A full exploration of the relationship between the intial surface architecture and dynamically selected pattern is presented in reference D Walgraef, CSR Matthes, NM Ghoniem: The influence of surface architecture on ioninduced nanopatterning, in preparation. Here, we consider a case of an initial structured surface and numerically follow its linear and nonlinear evolution.
System behavior in 2 dimensions
It can be observed that when using the linear equation, the evolved surface (Fig. 8 b) demonstrates the emergence of “dotsonripples” type patterning, where organized ripples mixed with hexagonal features have been produced. The ripple formation corresponds to previous experimental work on graphite (Habenicht 2001) and glass (Navez et al. 1962) surfaces, as well as certain metals (Costantini et al. 2001; Valbusa et al. 2002). The ripples are oriented at an angle relative to the incident ions traveling in the \(\vec 1_{x}\) direction. Because the angle of incidence is set to the surface normal (θ=0), the curvature coefficients \(\hat \nu _{x}\) and \(\hat \nu _{y}\) are equal, such that orientational degeneracy is expected. The simulations displayed this degeneracy in the direction of the emergent ripples (i.e. the x and y components changed between positive and negative with successive simulations), but both components were shown to be equal in magnitude, as expected analytically. The wave vectors in the displayed case can be defined as \(\vec q_{1}=q_{c}\vec 1_{x}\), \(\vec q_{2} = \frac {q_{c}}{2}\vec 1_{x}  \frac {q_{c}}{2}\vec 1_{y} \), and \(\vec q_{3} = \frac {q_{c}}{2}\vec 1_{x} + \frac {q_{c}}{2}\vec 1_{y} \), satisfying the triangular relation \(\vec q_{1}+\vec q_{2}+\vec q_{3}=0\). The linear result demonstrates the presence of all three wave vectors, with more dominant critical ripples corresponding to \(\vec q_{2}\), having components in both the x and ydirections. In these simulations, we are operating in a regime where the damping rate α is set to zero, making ε a maximum value. Therefore, the evolution of the surface is determined according to a KPZ equation form. The result shows the emergence of a number of coupled unstable wave vectors, resulting in the “dotsonripples” pattern that has emerged. Successive simulations showed the orientation of the dominant ripples to vary between \(\vec q_{2}\) and \(\vec q_{3}\), as the randomized initial surface allowed for the competition between wave vectors to vary.
The evolution results according to the nonlinear equation (Fig. 8 c) show the formation of striations similar to that of the linear evolution. However, the wavelength scale differs. It can be seen that the nonlinear evolution produced wavelengths that are noticeably smaller in size than the linear evolution, replicating the nonlinear behavior demonstrated in Fig. 6, where the wavelength is half the magnitude as the linear result. The orientation of the critical ripples, like the linear result, corresponded to the \(\vec q_{2}\) mode, indicating the presence of other modes to be less consequential in determining the resulting symmetries for the shown simulation. In other simulations with a randomized initial profile, it was observed that the critical mode varied between \(\vec q_{2}\) and \(\vec q_{3}\), indicating a degenerate solution. The scale of the ripple wavelength appears to be half the magnitude of the wavelength in the linear case, indicating the emergence of \(2\vec q_{2}\) as the dominant wave vector, which was consistent for all simulations.
Conclusions
This study highlights the patternforming characteristics of ionbombarded surfaces according to a theoretical understanding of the physical processes. The outcomes have been achieved through both an analytical examination of stability, as well as a computational investigation to provide insight where analytical theory is questionable. The comprehensive stability analysis provides a background for understanding the types of patterns that occur under various conditions relating to the evolution equation. Linear stability analysis provides the expected dominant wavelength magnitude of ripple formations, determined by the ion and material parameters. This furnishes a baseline to which the size of ripples may be compared, in order to track the presence of coupled modes resulting from the quadratic nonlinearity. The orientation of ripples is shown to be dependent on the most unstable wave vector. In addition, weakly nonlinear analysis demonstrated that the coupling of other modes can destabilize onedimensional stripes in order to sustain hexagonal patterns. When the damping coefficient is set to zero (\({\bar {\alpha }}=0\)), the evolution operates outside the weakly nonlinear regime, such that the range of unstable wave vectors is maximized. Under such conditions, the evolution proceeds following the undamped KS equation form, presenting an area that has not been well investigated. In the undamped regime, the ripple orientation is not easily predicted and depends on the competition between modes. The selected mode for a particular initial surface configuration is controlled largely by the angle of ion incidence θ, which controls the magnitude of the coefficients \({\hat {\nu }}_{x}\) and \({\hat {\nu }}_{y}\).
Computational simulations offer insight into the evolution behavior outside the weakly nonlinear regime, where the competition between wave vectors is not easily predicted. Therefore, the resulting patterns are dependent on the initial surface architecture, which influences the emergence of certain modes. For the 2dimensional simulations, a randomized surface height profile has been selected in order to observe overall trends associated with the evolutionary mechanisms, rather than relying on the consistent influence of a particular surface configuration.
Simulations performed using the linear evolution equation have produced numerical results that demonstrated the formation of “dotsonripples” type patterning. The orientation of these ripples depends on the relationship of the curvature coefficients \(\hat \nu _{x}\) and \(\hat \nu _{y}\), where the components of the wave vectors are equal at angles where their magnitudes match (i.e. θ=0° and 45°). Performing spectral analysis on the profiles show that the most dominant wavelength closely matches the analytical calculations for stable ripple formation. By introducing the nonlinear term to the evolution equation, it has been found that the effect of the nonlinearities is significant over long time spans, producing sharpened or enhanced symmetries. Spectral analysis as well as qualitative results have shown that at short times, the nonlinear term has little effect on the morphology of the surface profile. After a sufficient number of time steps, however, the nonlinear term grows rapidly and dominates the evolution equation. The nonlinear influence manifests itself by producing surface ripples with a dominant wavelength that is half the magnitude of what is expected according to linear stability. This result demonstrates the coupling of \(2\vec q_{c}\) as the fastest growing growing mode, following the effect of the quadratic nonlinearity. It is worth mentioning here that the nonlinearities in the evolution equation actually reflect the large local variations in the surface structure; an aspect that has not been considered in the original BH theory.
In the nonlinear case, the angle of ion incidence is found to be influential in the selected pattern formation. While normal incidence only resulted in dotsonripples patterning, θ=45° resulted in the ripples becoming a clear dot pattern. That is, at θ=0°, the nonlinear mode oriented along \(\vec q_{2}\) dominates the coupled modes of other orientations, while θ=45° permits strong influence of multiple differently oriented modes. The nonlinear dotsonripples again reveal \(2\vec q_{c}\) to be the fastest growing mode, which contrasts with the linear result. The presence of nonlinearities has also been shown to require an extended speed of convergence compared with linear evolution, as the nonlinear term is initially not present in the symmetry, but eventually dominates the evolution.
Declarations
Acknowledgements
This material is based upon work supported by the US Air Force Office of Scientific Research (AFOSR), under award number FA95501610444.
Authors’ contributions
CSRM and NMG prepared the manuscript and conceived the study. DW lead the nonlinear stability analysis and CSRM formulated the numerical simulations. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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