From crater functions to partial differential equations: a new approach to ion bombardment induced nonequilibrium pattern formation

Swelling as a stabilizing mechanism in irradiated thin films: II. Effect of swelling rate

It has long been observed experimentally that energetic ion-beam irradiation of semiconductor surfaces may lead to spontaneous nanopattern formation. For most ion/target/energy combinations, the patterns appear when the angle of incidence exceeds a critical angle, and the models commonly employed to understand this phenomenon exhibit the same behavioral transition. However, under certain conditions, patterns do not appear for any angle of incidence, suggesting an important mismatch between experiment and theory. Previous work by our group (Swenson and Norris 2018J. Phys.: Condens. Matter30304003) proposed a model incorporating radiation-induced swelling, which is known to occur experimentally, and found that in the analytically-tractable limit of small swelling rates, this effect is stabilizing at all angles of incidence, which may explain the observed suppression of ripples. However, at that time, it was not clear how the proposed model would scale with increased swelling rate. In the present work, we generalize that analysis to the case of arbitrary swelling rates. Using a numerical approach, we find that the stabilization effect persists for arbitrarily large swelling rates, and maintains a stability profile largely similar to that of the small swelling case. Our findings strongly support the inclusion of a swelling mechanism in models of pattern formation under ion beam irradiation, and suggest that the simpler small-swelling limit is an adequate approximation for the full mechanism. They also highlight the need for more-and more detailed-experimental measurements of material stresses during pattern formation.

Nanoscale pattern formation on solid surfaces bombarded by two broad ion beams in the regime in which sputtering is negligible

We study nanoscale pattern formation on the surface of a solid that is bombarded with two diametrically opposed, broad ion beams for ion energies low enough that sputtering can be neglected. We focus on the case in which the angle of ion incidence is just above the threshold angle for pattern formation. The equation of motion at sufficiently long times is derived using a generalized crater function formalism. This formalism also yields expressions for the coefficients in the equation of motion in terms of crater function moments. We find that virtually defect-free ripples with a sawtooth profile can emerge at sufficiently long times. The ripples also coarsen as time passes, in contrast to the near-threshold behavior of ripples in the higher energy regime in which sputtering is significant.

Parameter estimation for pattern formation induced by ion bombardment of solid surfaces using deep learning

The nanostructures produced by oblique-incidence broad beam ion bombardment of a solid surface are usually modelled by the anisotropic Kuramoto-Sivashinsky equation. This equation has five parameters, each of which depend on the target material and the ion species, energy, and angle of incidence. We have developed a deep learning model that uses a single image of the surface to estimate all five parameters in the equation of motion with root-mean-square errors that are under 3% of the parameter ranges used for training. This provides a tool that will allow experimentalists to quickly ascertain the parameters for a given sputtering experiment. It could also provide an independent check on other methods of estimating parameters such as atomistic simulations combined with the crater function formalism.

Theory of nanoscale ripple topographies produced by ion bombardment near the threshold for pattern formation

Nanoscale pattern formation on the surface of a solid that is bombarded with a broad ion beam is studied for angles of ion incidence, θ, just above the threshold angle for ripple formation, θ_{c}. We carry out a systematic expansion in powers of the small parameter ε≡(θ-θ_{c})^{1/2} and retain all terms up to a given order in ε. In the case of two diametrically opposed, obliquely incident beams, the equation of motion close to threshold and at sufficiently long times is rigorously shown to be a particular version of the anisotropic Kuramoto-Sivashinsky equation. We also determine the long-time, near-threshold scaling behavior of the rippled surface's wavelength, amplitude, and transverse correlation length for this case. When the surface is bombarded with a single obliquely incident beam, linear dispersion plays a crucial role close to threshold and dramatically alters the behavior: highly ordered ripples can emerge at sufficiently long times and solitons can propagate over the solid surface. A generalized crater function formalism that rests on a firm mathematical footing is developed and is used in our derivations of the equations of motion for the single and dual beam cases.

Intuitive Model of Surface Modification Induced by Cluster Ion Beams

Topography development is one of the main factors limiting the quality of depth profiles during depth profiling experiments. One possible source of topography development is the formation of self-organized patterns due to cluster ion beam irradiation. In this work, we propose a simple model that can intuitively explain this phenomenon in terms of impact-induced mass transfer. By coupling our model with molecular dynamics simulations, we can predict the critical incidence angle, which separates the smoothening and roughening regimes. The results are in quantitative agreement with experiments. It is observed that the problems arising from topography development during depth profiling with cluster projectiles can be mitigated by reducing the beam incidence angle with respect to the surface normal or increasing its kinetic energy.

Effect of dispersion on the nanoscale patterns produced by ion sputtering

Our simulations show that dispersion can have a crucial effect on the patterns produced by oblique-incidence ion sputtering. It can lead to the formation of raised and depressed triangular regions traversed by parallel-mode ripples, and these bear a strong resemblance to nanostructures that are commonly observed in experiments. In addition, if dispersion and transverse smoothing are sufficiently strong, highly ordered ripples form. Finally, dispersion can cause the formation of protrusions and depressions that are elongated along the projected beam direction even when there is no transverse instability. This may explain why topographies of this kind form for high angles of ion incidence in cases in which ion-induced mass redistribution is believed to dominate curvature-dependent sputtering.

Modeling of high-fluence irradiation of amorphous Si and crystalline Al by linearly focused Ar ions

Long time ion irradiation of surfaces under tilted incidence causes formation of regular nanostructures known as surface ripples. The nature of mechanisms leading to ripples is still not clear, this is why computational methods can shed the light on such a complex phenomenon and help to understand which surface processes are mainly responsible for it. In this work, we analyse the surface response of two materials, a semiconductor (silicon) and a metal (aluminium) under irradiation with the 250 eV and 1000 eV Ar ions focused at 70° from the normal to the surface. We simulate consecutive ion impacts by the means of molecular dynamics to investigate the effect on ripple formation. We find that the redistribution mechanism seems to be the main creator of ripples in amorphous materials, while the erosion mechanism is the leading origin for the pattern formation in crystalline metals.

Swelling as a stabilizing mechanism in irradiated thin films

Irradiation of semiconductor surfaces often leads to the spontaneous formation of rippled structures at certain irradiation angles. However, at high enough energies, in the keV range, these structures are sometimes observed to vanish for all angles, despite the absence of any identified, universally-stabilizing physical mechanisms in operation. Here, we examine the effect on pattern formation of radiation-induced swelling, which has been excluded from prior treatments of stress in irradiated films. After developing a suitable continuum model, we perform a linear stability analysis to determine its effect on stability. Under appropriate simplifying assumptions, we find swelling indeed to be stabilizing at all angles for wavenumbers typical of experimental observations. Therefore, this mechanism may account for the suppression of ripple formation observed at energies over 1 keV.

Self-Assembled Gold Nano-Ripple Formation by Gas Cluster Ion Beam Bombardment

In this study, we used a 30 keV argon cluster ion beam bombardment to investigate the dynamic processes during nano-ripple formation on gold surfaces. Atomic force microscope analysis shows that the gold surface has maximum roughness at an incident angle of 60° from the surface normal; moreover, at this angle, and for an applied fluence of 3 × 10¹⁶ clusters/cm², the aspect ratio of the nano-ripple pattern is in the range of ~50%. Rutherford backscattering spectrometry analysis reveals a formation of a surface gradient due to prolonged gas cluster ion bombardment, although the surface roughness remains consistent throughout the bombarded surface area. As a result, significant mass redistribution is triggered by gas cluster ion beam bombardment at room temperature. Where mass redistribution is responsible for nano-ripple formation, the surface erosion process refines the formed nano-ripple structures.

Distinguishing physical mechanisms using GISAXS experiments and linear theory: the importance of high wavenumbers

In this work we analyze GISAXS measurements of the structure factor of Si surfaces evolving during 1 keV Ar+ ion bombardment. Using newly-developed methods sensitive to the full range of experimentally-available wavenumbers q, we extract the linear amplification rate R(q) governing surface stability over a range of wavenumbers 4–5 times larger than has previously been obtained. Comparing with theoretical models also retaining full wavenumber-dependence, we find an excellent fit of the experimental data over the full range of irradiation angles and wavenumbers. Moreover, the fitted parameter values represent experimental evaluation of the magnitudes of most physical mechanisms currently believed to be important to the pattern-formation process. In all cases, the extracted values agree well with direct observations or atomistic simulations of the same quantities, suggesting that GISAXS analysis may allow more powerful comparison between experiment and theory than had previously been thought.

Ion beam nanopatterning of III-V semiconductors: consistency of experimental and simulation trends within a chemistry-driven theory

Several proposed mechanisms and theoretical models exist concerning nanostructure evolution on III-V semiconductors (particularly GaSb) via ion beam irradiation. However, making quantitative contact between experiment on the one hand and model-parameter dependent predictions from different theories on the other is usually difficult. In this study, we take a different approach and provide an experimental investigation with a range of targets (GaSb, GaAs, GaP) and ion species (Ne, Ar, Kr, Xe) to determine new parametric trends regarding nanostructure evolution. Concurrently, atomistic simulations using binary collision approximation over the same ion/target combinations were performed to determine parametric trends on several quantities related to existing model. A comparison of experimental and numerical trends reveals that the two are broadly consistent under the assumption that instabilities are driven by chemical instability based on phase separation. Furthermore, the atomistic simulations and a survey of material thermodynamic properties suggest that a plausible microscopic mechanism for this process is an ion-enhanced mobility associated with energy deposition by collision cascades.

Ripple coarsening on ion beam-eroded surfaces

The temporal evolution of ripple pattern on Ge, Si, Al 2 O 3, and SiO 2 by low-energy ion beam erosion with Xe (+) ions is studied. The experiments focus on the ripple dynamics in a fluence range from 1.1 × 10(17) cm(-2) to 1.3 × 10(19) cm(-2) at ion incidence angles of 65° and 75° and ion energies of 600 and 1,200 eV. At low fluences a short-wavelength ripple structure emerges on the surface that is superimposed and later on dominated by long wavelength structures for increasing fluences. The coarsening of short wavelength ripples depends on the material system and angle of incidence. These observations are associated with the influence of reflected primary ions and gradient-dependent sputtering. The investigations reveal that coarsening of the pattern is a universal behavior for all investigated materials, just at the earliest accessible stage of surface evolution.