Enhancing the quality of self-organized nanoripples by Ar-ion bombardment of a bilayer system.

Ion bombardment (IB) is a promising nanofabrication technique for producing nanoripples. A critical issue that restricts the application of IB is the limited quality of IB-induced nanoripples. Photoresist (PR) and antireflection coating (ARC) are of technological relevance for lithographic exposure processes. Moreover, to improve the quality of IB-induced self-organized nanoripples, in this study, a PR/ARC bilayer was bombarded at an incidence angle of 50°. The surface normalized defect density and power spectral density, obtained via scanning atomic force microscopy, indicate the superiority of the PR/ARC bilayer nanoripples over those of single PR or ARC layers. The growth mechanism of the improved nanoripples, deciphered via the temporal evolution of the morphology, involves the following processes: (i) formation of a well-grown IB-induced nanoripple prepattern on the PR, (ii) transfer of nanoripples from the PR to the ARC, forming an initial ARC nanoripple morphology for subsequent IB, and (iii) conversion of the initial nonuniform ARC nanoripples into uniform nanoripples. In this unique method, the angle of ion-incidence should be chosen so that ripples form on both PR and ARC films. Overall, this method facilitates nanoripple improvement, including prepattern fabrication for guiding nanoripple growth and sustainable nanoripple development via a single IB. Thus, the unique method presented in this study can aid in advancing academic research and also has potential applications in the field of IB-induced nanoripples.

Nanoscale pattern formation produced by ion bombardment of a rotating target: The decisive role of the ion energy.

We study the nanoscale patterns that form on the surface of a rotating sample of an elemental material that is bombarded with a broad noble gas ion beam for angles of incidence θ just above the critical angle for pattern formation θ_{c}. The pattern formation depends crucially on the ion energy E. In simulations carried out in the low-energy regime in which sputtering is negligible, we find disordered arrays of nanoscale mounds (nanodots) that coarsen in time. Disordered arrays of nanodots also form in the high-energy regime in which there is substantial sputtering, but no coarsening occurs close to the threshold angle. Finally, for values of E just above the sputter yield threshold, nanodot arrays with an extraordinary degree of hexagonal order emerge for a range of parameter values, even though there is a broad band of linearly unstable wavelengths. This finding might prove to be useful in applications in which highly ordered nanoscale patterns are needed.

Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation.

Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE). The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results.

Topological measures of order for imperfect two-dimensional Bravais lattices.

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. The measures generalize previous measures of order using persistent homology that were applicable only to imperfect hexagonal lattices in two dimensions. We illustrate the sensitivities of these measures to the degree of perturbation of perfect hexagonal, square, and rhombic Bravais lattices. We also study imperfect hexagonal, square, and rhombic lattices produced by numerical simulations of pattern-forming partial differential equations. These numerical experiments serve to compare the measures of lattice order and reveal differences in the evolution of the patterns in various partial differential equations.

Theory of the nanoscale surface ripples produced by ion irradiation of a miscut (001) gallium arsenide surface.

We develop a theory for the surface ripples produced by near-normal-incidence ion bombardment of a (001) GaAs surface with a small miscut along the [110] direction. We restrict our attention to the case in which the energy of the incident ions is below the sputter yield threshold and the sample temperature is just above the recrystallization temperature. Highly ordered, faceted ripples with their wave vector aligned with the [110] direction form when the ion beam is normally incident and there is no miscut. Two additional terms appear in the equation of motion when the beam is obliquely incident and/or there is a miscut: a linearly dispersive term and a nonlinearly dispersive term. The coefficients of these terms can become large as the threshold temperature for pattern formation is approached from above. In the absence of strong nonlinear dispersion, strong linear dispersion leads to ripples with a dramatically increased degree of order. These ripples are nearly sinusoidal even though they are on the surface of a single crystal. The exceptionally high degree of order is disrupted by nonlinear dispersion if the coefficient of that term is sufficiently large. However, by choosing the angle of ion incidence appropriately, the coefficient of the nonlinearly dispersive term can be made small. Ion bombardment will then produce highly ordered ripples. For a different range of parameter values, nucleation and growth of facets and spinodal decomposition can occur.

Theory of the oscillatory instability of a rigid tip scraped over a polymer surface.

The patterns produced by dragging an atomic force microscope (AFM) tip over a polymer surface are studied using a mesoscopic model introduced by Gnecco and co-workers [E. Gnecco et al., New J. Phys. 17, 032001 (2015)1367-263010.1088/1367-2630/17/3/032001]. We show that the problem can be reduced to solving a closed integrodifferential equation for a single degree of freedom, the position of the AFM tip. We find the steady-state solution to this equation and then carry out a linear stability analysis of it. The steady state is only stable if the dimensionless indentation rate α is less than a critical value α_{c} which depends on the dimensionless velocity of the rigid support r. Conversely, for α>α_{c}, periodic stick-slip motion sets in after a transient. Simulations show that the amplitude of these oscillations is proportional to (α-α_{c})^{1/2} for α just above α_{c}. Our analysis also yields a closed equation that can be solved for the critical value α_{c}=α_{c}(r). If the steady-state motion is perturbed, as long as the deviation from the steady state is small, the deviation of the tip's position from the steady state can be written as a linear superposition of terms of the form exp(λ_{k}t), where the complex constants λ_{k} are solutions to an integral equation. Finally, we demonstrate that the results obtained for the two-dimensional model of Gnecco et al. carry over in a straightforward way to the generalization of the model to three dimensions.

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.

Spontaneous pattern formation induced by ion bombardment of binary compounds.

A theory is developed that explains the genesis of the strikingly regular hexagonal arrays of nanoscale mounds that can form when a flat surface of a binary compound is subjected to normal-incidence ion bombardment. We find that the species with the higher sputter yield is concentrated at the peaks of the nanodots and that hysteretic switching between the flat and the hexagonally ordered state can occur as the sample temperature is varied. Surface ripples are predicted to emerge for a certain range of the parameters.

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.

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.

Emergence and detailed structure of terraced surfaces produced by oblique-incidence ion sputtering.

We study the nanoscale terraced topographies that arise when a solid surface is bombarded with a broad ion beam that has a relatively high angle of incidence θ. We find that the surface is not completely flat between the regions in which the surface slope changes rapidly with position: Instead, small-amplitude ripples propagate along the surface. Our analytical work on these ripples yields their propagation velocity as well as the scaling behavior of their amplitude. Our simulations establish that the surfaces exhibit interrupted coarsening, i.e., the characteristic width and height of the surface disturbance grow for a time but ultimately asymptote to finite values as the fully terraced state develops. In addition, as θ is reduced, the surface can undergo a transition from a terraced morphology that changes little with time as it propagates over the surface to an unterraced state that appears to exhibit spatiotemporal chaos. For different ranges of the parameters, our equation of motion produces unterraced topographies that are remarkably similar to those seen in various experiments, including pyramidal structures that are elongated along the projected beam direction and isolated lenticular depressions.

Producing virtually defect-free nanoscale ripples by ion bombardment of rocked solid surfaces.

Bombardment of a solid surface with a broad, obliquely incident ion beam frequently produces nanoscale surface ripples. The primary obstacle that prevents the adoption of ion bombardment as a nanofabrication tool is the high density of defects in the patterns that are typically formed. Our simulations indicate that ion bombardment can produce nearly defect-free ripples on the surface of an elemental solid if the sample is concurrently and periodically rocked about an axis orthogonal to the surface normal and the incident beam direction. We also investigate the conditions necessary for rocking to produce highly ordered ripples and discuss how the results of our simulations can be reproduced experimentally. Finally, our simulations show that periodic temporal oscillations of coefficients in the Kuramoto-Sivashinsky equation can suppress spatiotemporal chaos and lead to patterns with a high degree of order.

Highly ordered square arrays of nanoscale pyramids produced by ion bombardment of a crystalline binary material.

A theory is developed for the nanoscale patterns formed when the (001) surface of a crystalline binary material with fourfold rotational symmetry is subjected to normal-incidence ion bombardment. The deterministic nonlinear continuum equations account for the Ehrlich-Schwoebel barrier, which produces uphill atomic currents on the crystal surface. We demonstrate that highly ordered square arrays of nanopyramids can form in a certain region of parameter space. An Ehrlich-Schwoebel barrier is required for patterns of this kind to develop. For another range of parameters, a disordered square array of nanodots forms and the pattern coarsens over time.

Nanoscale patterns formed by ion bombardment of rotating binary materials.

We explore the effects of sample rotation during ion sputtering of binary materials, as well as its effects during surfactant sputtering. We find that the rate with which the surface roughens or smooths depends on the period of rotation t(0) in the early time regime, in contrast to the behavior of elemental materials. In addition, the characteristic length scale l of the patterns that emerge can be tuned merely by changing the value of t(0). Finally, we demonstrate that l can even exhibit a jump discontinuity as t(0) is varied.

Theory of terraced topographies produced by oblique-incidence ion bombardment of solid surfaces.

When a solid surface is bombarded with a broad ion beam at a relatively large angle of incidence, the surface often develops a terraced form. We introduce a model that includes an improved approximation to the sputter yield and that produces a terraced surface morphology at long times for a wide range of parameter values. Numerical integrations of our equation of motion reveal that the terraces coarsen as time passes, just as observed experimentally. We also show that the terrace propagation direction can reverse as the amplitude of the surface disturbance grows. This highlights the important role higher order nonlinearities play in determining the propagation velocity at high fluences.

Producing nanodot arrays with improved hexagonal order by patterning surfaces before ion sputtering.

When the surface of a nominally flat binary material is bombarded with a broad, normally incident ion beam, disordered hexagonal arrays of nanodots can form. Shipman and Bradley have derived equations of motion that govern the coupled dynamics of the height and composition of such a surface [Shipman and Bradley, Phys. Rev. B 84, 085420 (2011)]. We investigate the influence of initial conditions on the hexagonal order yielded by integration of those equations of motion. The initial conditions studied are hexagonal and sinusoidal templates, straight scratches, and nominally flat surfaces. Our simulations indicate that both kinds of templates lead to marked improvements in the hexagonal order if the initial wavelength is approximately equal to or double the linearly selected wavelength. Scratches enhance the hexagonal order in their vicinity if their width is close to or less than the linearly selected wavelength. Our results suggest that prepatterning a binary material can dramatically increase the hexagonal order achieved at large ion fluences.

Electromigration-induced propagation of nonlinear surface waves.

Due to the effects of surface electromigration, waves can propagate over the free surface of a current-carrying metallic or semiconducting film of thickness h(0). In this paper, waves of finite amplitude, and slow modulations of these waves, are studied. Periodic wave trains of finite amplitude are found, as well as their dispersion relation. If the film material is isotropic, a wave train with wavelength lambda is unstable if lambda/h(0)<3.9027 ..., and is otherwise marginally stable. The equation of motion for slow modulations of a finite amplitude, periodic wave train is shown to be the nonlinear Schrödinger equation. As a result, envelope solitons can travel over the film's surface.

Diffusion in a two-dimensional channel with curved midline and varying width: reduction to an effective one-dimensional description.

Diffusion in a narrow two-dimensional channel with a midline that need not be straight and a width that may vary is reduced to an effective one-dimensional equation of motion. This equation takes the form of the Fick-Jacobs equation with a spatially varying effective diffusivity. The effective diffusivity includes a contribution that comes from the slope of the midline as well as the usual term stemming from variations in the channel width along the length of the channel. Our derivation of our equation of motion is completely rigorous and is based on an asymptotic expansion in a small dimensionless parameter that characterizes the channel width. For a channel that has a straight midline or wall, our equation of motion reduces to Zwanzig's equation [R. Zwanzig, J. Phys. Chem. 96, 3926 (1992)]. Our derivation therefore provides a rigorous proof of the validity of the latter equation. Finally, the equation of motion is solved analytically for channels with curved midline and constant width.