Crater Function Research Articles

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.

Nanohole morphologies on photoresist surface produced by low-energy Ar+ ion bombardment under normal and near-normal incidence

Random nanohole morphologies on a photoresist surface were produced spontaneously by ion bombardment, from normal to near-normal incidence. At an optimized ion energy, the critical parameters (diameter and depth) and uniformity of these nanoholes can be tailored with multi-ion parameters, including ion fluence (i.e. bombardment time), ion flux, and incidence angle. The temporal evolution of the nanohole structures, at normal incidence, showed growth and stabilization regimes. The mass redistribution effect and the distribution of energy deposition led to the transition from nanoholes to nanoripples, which were deduced from the crater shape evolution at low incidence angles. The incidence angle-dependent evolution of the crater shape (or rim) on the resist surface is consistent with the crater function simulation, and confirms the ion bombardment-induced surface atomic currents and mass redistribution. Finally, X-ray photoelectron spectroscopy characterization confirmed ion bombardment-induced decomposition, in particular, graphitization in the surface layer of the resist.

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.

Prediction of ion-induced nanopattern formation using Monte Carlo simulations and comparison to experiments

Ion induced nanopattern formation has been experimentally investigated for many different ion-target combinations and different ion irradiation conditions. Several theories and models have been developed throughout the past few years to explain the observed boundary conditions for pattern formation as well as features of the patterns like wavelengths, growth rates, shapes, and amplitudes. To compare specific experiments with the predictions of analytical theories, it is necessary to calculate the linear and non-linear coefficients of the respective equation of motion of a surface profile. Monte Carlo simulations of ion–solid interactions based on the binary collision approximation provide a very fast, rather universal, and accurate way to calculate these coefficients. The universality expresses the broad range of ion species, ion energies, and target compositions accessible by the simulations. The coefficients are obtained from the moments of calculated crater functions, describing ion erosion, mass redistribution, and ion implantation. In this contribution, we describe how most linear, non-linear, and higher order coefficients can be determined from crater function moments. We use the obtained data to compare the results of selected experimental studies with the predictions of theoretical models. We find good quantitative agreement, e.g., for irradiation of Si with Ar and Kr ions, Al2O3 with Ar and Xe ions, and amorphous carbon with Ne ions.

Ion-induced nanopatterning of silicon: Toward a predictive model

We review recent progress toward the development of predictive models of ion-induced pattern formation on room-temperature silicon, with a particular emphasis on efforts to eliminate fit parameters in the linear regime by means of experimental measurements or atomistic simulations. Analytical approaches considered include “mechanistic” models of the impact-induced collision cascade, the Crater Function Framework, and continuum treatments of ion-induced stress and viscous flow. Parameter evaluation methods include molecular dynamics and binary collision approximation simulations, as well as wafer curvature measurements and grazing incidence small-angle x-ray scattering. Mathematical detail is provided in the context of key results from pattern formation theory, which are also briefly summarized.

Pattern formation on ion-irradiated Si surface at energies where sputtering is negligible

The effect of low energy irradiation, where the sputtering is imperceptible, has not been deeply studied in the pattern formation. In this work, we want to address this question by analyzing the nanoscale topography formation on a Si surface, which is irradiated at room temperature by Ar+ ions near the displacement threshold energy, for incidence angles ranging from 0° to 85°. The transition from the smooth to ripple patterned surface, i.e., the stability/instability bifurcation angle is observed at 55°, whereas the ripples with their wave-vector is parallel to the ion beam projection in the angular window of 60°–70°, and with 90° rotation with respect to the ion beam projection at the grazing angles of incidence. A similar irradiation setup has been simulated by means of molecular dynamics, which made it possible, first, to quantify the effect of the irradiation in terms of erosion and redistribution using sequential irradiation and, second, to evaluate the ripple wavelength using the crater function formalism. The ripple formation results can be solely attributed to the mass redistribution based mechanism, as erosion due to ion sputtering near or above the threshold energy is practically negligible.

Neon ion beam induced pattern formation on amorphous carbon surfaces

We investigate the ripple pattern formation on amorphous carbon surfaces at room temperature during low energy Ne ion irradiation as a function of the ion incidence angle. Monte Carlo simulations of the curvature coefficients applied to the Bradley-Harper and Cater-Vishnyakov models, including the recent extensions by Harrison-Bradley and Hofsäss predict that pattern formation on amorphous carbon thin films should be possible for low energy Ne ions from 250 eV up to 1500 eV. Moreover, simulations are able to explain the absence of pattern formation in certain cases. Our experimental results are compared with prediction using current linear theoretical models and applying the crater function formalism, as well as Monte Carlo simulations to calculate curvature coefficients using the SDTrimSP program. Calculations indicate that no patterns should be generated up to 45° incidence angle if the dynamic behavior of the thickness of the ion irradiated layer introduced by Hofsäss is taken into account, while pattern formation most pronounced from 50° for ion energy between 250 eV and 1500 eV, which are in good agreement with our experimental data.

Self-organized surface ripple pattern formation by ion implantation

Ion induced ripple pattern formation on solid surfaces has been extensively studied in the past and the theories describing curvature dependent ion erosion as well as redistribution of recoil atoms have been very successful in explaining many features of the pattern formation. Since most experimental studies use noble gas ion irradiation, the incorporation of the ions into the films is usually neglected. In this work we show that the incorporation or implantation of non-volatile ions also leads to a curvature dependent term in the equation of motion of a surface height profile. The implantation of ions can be interpreted as a negative sputter yield; and therefore, the effect of ion implantation is opposite to the one of ion erosion. For angles up to about 50°, implantation of ions stabilizes the surface, whereas above 50°, ion implantation contributes to the destabilization of the surface. We present simulations of the curvature coefficients using the crater function formalism and we compare the simulation results to the experimental data on the ion induced pattern formation using non-volatile ions. We present several model cases, where the incorporation of ions is a crucial requirement for the pattern formation.

Argon ion beam induced surface pattern formation on Si

The development of self-organized surface patterns on Si due to noble gas ion irradiation has been studied extensively in the past. In particular, Ar ions are commonly used and the pattern formation was analyzed as function of ion incidence angle, ion fluence, and ion energies between 250 eV and 140 keV. Very few results exist for the energy regime between 1.5 keV and 10 keV and it appears that pattern formation is completely absent for these ion energies. In this work, we present experimental data on pattern formation for Ar ion irradiation between 1 keV and 10 keV and ion incidence angles between 50° and 75°. We confirm the absence of patterns at least for ion fluences up to 1018 ions/cm2. Using the crater function formalism and Monte Carlo simulations, we calculate curvature coefficients of linear continuum models of pattern formation, taking into account contribution due to ion erosion and recoil redistribution. The calculations consider the recently introduced curvature dependence of the erosion crater function as well as the dynamic behavior of the thickness of the ion irradiated layer. Only when taking into account these additional contributions to the linear theory, our simulations clearly show that that pattern formation is strongly suppressed between about 1.5 keV and 10 keV, most pronounced at 3 keV. Furthermore, our simulations are now able to predict whether or not parallel oriented ripple patterns are formed, and in case of ripple formation the corresponding critical angles for the whole experimentally studied energies range between 250 eV and 140 keV.

Tests of the modified Sigmund model of ion sputtering using Monte Carlo simulations

Monte Carlo simulations are used to evaluate the Modified Sigmund Model of Sputtering. Simulations were carried out for a range of ion incidence angles and surface curvatures for different ion species, ion energies, and target materials. Sputter yields, moments of erosive crater functions, and the fraction of backscattered energy were determined. In accordance with the Modified Sigmund Model of Sputtering, we find that for sufficiently large incidence angles θ the curvature dependence of the erosion crater function tends to destabilize the solid surface along the projected direction of the incident ions. For the perpendicular direction, however, the curvature dependence always leads to a stabilizing contribution. The simulation results also show that, for larger values of θ, a significant fraction of the ions is backscattered, carrying off a substantial amount of the incident ion energy. This provides support for the basic idea behind the Modified Sigmund Model of Sputtering: that the incidence angle θ should be replaced by a larger angle Ψ to account for the reduced energy that is deposited in the solid for larger values of θ.

Kinetic Monte Carlo simulation of self-organized pattern formation induced by ion beam sputtering using crater functions

The production of self-organized nanostructures by ion beam sputtering has been of keen interest to researchers for many decades. Despite numerous experimental and theoretical efforts to understand ion-induced nanostructures, there are still many basic questions open to discussion, such as the role of erosion or curvature-dependent sputtering. In this work, a hybrid MD/kMC (molecular dynamics/kinetic Monte Carlo) multiscale atomistic model is developed to investigate these knowledge gaps, and its predictive ability is validated across the experimental parameter space. This model uses crater functions, which were obtained from MD simulations, to model the prompt mass redistribution due to single-ion impacts. Defect migration, which is missing from previous models that use crater functions, is treated by a kMC Arrhenius method. Using this model, a systematic study was performed for silicon bombarded by ${\mathrm{Ar}}^{+}$ ions of various energies (100 eV, 250 eV, 500 eV, 700 eV, and 1000 eV) at incidence angles of ${0}^{\ensuremath{\circ}}$ to ${80}^{\ensuremath{\circ}}.$ The simulation results were compared with experimental findings, showing good agreement in many aspects of surface evolution, such as the phase diagram. The underestimation of the ripple wavelength by the simulations suggests that surface diffusion is not the main smoothening mechanism for ion-induced pattern formation. Furthermore, the simulated results were compared with moment-description continuum theory and found to give better results, as the simulation did not suffer from the same mathematical inconsistencies as the continuum model. The key finding was that redistributive effects are dominant in the formation of flat surfaces and parallel-mode ripples, but erosive effects are dominant at high angles when perpendicular-mode ripples are formed. Ion irradiation with simultaneous sample rotation was also simulated, resulting in arrays of square-ordered dots. The patterns obtained from sample rotation were strongly correlated to the rotation speed and to the pattern types formed without sample rotation, and a critical value of about 5 rpm was found between disordered ripples and square-ordered dots. Finally, simulations of dual-beam sputtering were performed, with the resulting patterns determined by the flux ratio of the two beams and the pattern types resulting from single-beam sputtering under the same conditions.

A modification to the Sigmund model of ion sputtering

We modify the Sigmund model of ion sputtering so that the distribution of deposited energy is in better accord with the results of molecular dynamics simulations. For a flat target surface, the model gives a sputter yield that displays a maximum as the angle of ion incidence is increased, as observed experimentally. The model also predicts that for sufficiently high angles of incidence, the curvature dependence of the crater function tends to destabilize the solid surface.

Crater function approach to ion-induced nanoscale pattern formation: Craters for flat surfaces are insufficient

In the crater function approach to the erosion of a solid surface by a broad ion beam, the average crater produced by the impact of an ion is used to compute the constant coefficients in the continuum equation of motion for the surface. We extend the crater function formalism so that it includes the dependence of the crater on the curvature of the surface at the point of impact. We then demonstrate that our formalism yields the correct coefficients for the Sigmund model of ion sputtering if terms up to second order in the spatial derivatives are retained. In contrast, if the curvature dependence of the crater is neglected, the coefficients can deviate substantially from their exact values. Our results show that accurately estimating the coefficients using craters obtained from molecular dynamics simulations will require significantly more computational power than was previously thought.

Model-independent test of the truncated crater function theory of surface morphology evolution during ion bombardment

A broad class of ``local response'' theories seeks to predict morphology evolution during energetic particle irradiation in terms of average surface height response to individual impacts---an approach that has been generalized by the crater function formalism of Norris et al. [J. Phys.: Condens. Matter 21, 224017 (2009); Nat. Commun. 2, 276 (2011)]. Keeping only the terms in the crater function formalism associated with the response of a flat surface has facilitated the use of molecular dynamics simulations of individual ion impacts to predict the stability or instability of a flat surface to ion bombardment. Here we report a sensitive experimental test of this truncated crater function theory that is independent of any a priori knowledge of the crater function itself. Existing measurements for 1 keV ${\mathrm{Ar}}^{+}/\mathrm{Si}$ and ${\mathrm{Kr}}^{+}/\mathrm{Ge}$ are inconsistent with the predictions of truncated crater function theory, for any conceivable crater function, at high bombardment angles. The failure of the theory suggests that the prediction of surface evolution from simulations of single-ion impacts will be more challenging than had been assumed.

Surface instability and pattern formation by ion-induced erosion and mass redistribution

The contribution of curvature dependent sputtering and mass redistribution to ion-induced self-organized formation of periodic surface nanopatterns is revisited. Ion incidence angle-dependent curvature coefficients and ripple wavelengths are calculated from 3-dimensional collision cascade data obtained from binary collision Monte Carlo simulations. Significant modifications concerning mass redistribution compared to the model of Carter and Vishnyakov and also models based on crater functions are introduced. Furthermore, I find that curvature-dependent erosion is the dominating contribution to pattern formation, except for very low-energy irradiation of a light matrix with heavy ions. The major modifications regarding mass redistribution and ion-induced viscous flow are related to the ion incidence angle-dependent thickness of the irradiated layer. A smaller modification concerns the relaxation of inward-directed mass redistribution. Ion-induced viscous flow in the surface layer also depends on the layer thickness and is thus strongly angle dependent. Simulation results are presented and compared to a variety of published experimental results. The simulations show that in most cases curvature-dependent erosion is the dominant contribution to surface instability and ripple pattern formation and also determines the pattern orientation transition. The simulations predict the occurrence of perpendicular ripple patterns at larger ion incidence angles, in agreement with experimental observations. Mass redistribution causes stabilization of the surface at near-normal ion incidence angles and dominates pattern formation only at very low ion energies.

Crater functions for compound materials: A route to parameter estimation in coupled-PDE models of ion bombardment

During the ion bombardment of targets containing multiple component species, highly-ordered arrays of nanostructures are sometimes observed. Models incorporating coupled partial differential equations, describing both morphological and chemical evolution, seem to offer the most promise of explaining these observations. However, these models contain many unknown parameters, which must satisfy specific conditions in order to explain observed behavior. The lack of knowledge of these parameters is therefore an important barrier to the comparison of theory with experiment. Here, by adapting the recent theory of “crater functions” to the case of binary materials, we develop a generic framework in which many of the parameters of such models can be estimated using the results of molecular dynamics simulations.As a demonstration, we apply our framework to the recent theory of Bradley and Shipman, for the case of Ar-irradiated GaSb, in which ordered patterns were first observed. In contrast to the requirements therein that sputtered atoms form the dominant component of the collision cascade, and that preferential redistribution play an important stabilizing role, we find instead that the redistributed atoms dominate the collision cascade, and that preferential redistribution appears negligible. Hence, the actual estimated parameters for this system do not seem to satisfy the requirements imposed by current theory, motivating the consideration of other potential pattern-forming mechanisms.

Influence of local curvature on sputtering

Sputtering depends on the local surface curvature; it is higher for convex and lower for concave surfaces than for a flat reference target. We analyze this dependence theoretically and show that the sputter yield primarily depends on a dimension-free curvature parameter, and second on the relative variance of the deposited-energy distribution in the direction along the ion beam. Using molecular dynamics, we study the effects in an exemplary way for 500 eV Ar impact on an a-Si target; here, we compare a wire and a valley structure (radii of curvature 2.5 nm) with a planar target. Our findings demonstrate that the sputter yield, the crater function, and the angular distribution of sputtered particles strongly depend on the curvature.

Atomistic simulation of ion beam patterning with crater functions

In this study, an atomistic model is developed to simulate ripple pattern formation when a surface is irradiated by incident low-energy energetic ions. The model treats individual ion impacts using crater functions, which represent the average change in the surface shape due to a single-ion impact. These functions incorporate the complete redistribution of mass along the surface due to an impact, and not just that due to sputtering. While most models only treat erosion, analysis of the craters reveals that the amount of mass redistributed across the surface is an order of magnitude greater than the mass removed by sputtering. Simulations in this study are conducted for 500eV Ar+ bombardments of Si at angles of 0° to 60° with 5° increment at temperature of 350K. Initial simulations with this model have shown agreement with prior observations of ripple pattern formation. However, some significant departures from other models based on the Bradley–Harper theory have emerged; the key difference is that the presence of crater rims plays a key role in ripple formation, which could explain phenomena such as maximum ripple amplitudes which most models do not account for. These results show that atomistic crater functions are a viable method for modeling ion beam patterning. They indicate that mass redistribution is a key mechanism for surface patterning.

Nanoscale topographic pattern formation on Kr+-bombarded germanium surfaces

The nanoscale pattern formation of Ge surfaces uniformly irradiated by Kr+ ions was studied in a low-contamination environment at ion energies of 250 and 500 eV and at angles of 0° through 80°. The authors present a phase diagram of domains of pattern formation occurring as these two control parameters are varied. The results are insensitive to ion energy over the range covered by the experiments. Flat surfaces are stable from normal incidence up to an incidence angle of θ = 55° from normal. At higher angles, the surface is linearly unstable to the formation of parallel-mode ripples, in which the wave vector is parallel to the projection of the ion beam on the surface. For θ ≥ 75° the authors observe perpendicular-mode ripples, in which the wave vector is perpendicular to the ion beam. This behavior is qualitatively similar to those of Madi et al. for Ar+-irradiated Si but is inconsistent with those of Ziberi et al. for Kr+-irradiated Ge. The existence of a window of stability is qualitatively inconsistent with a theory based on sputter erosion [R. M. Bradley and J. M. Harper, J. Vac. Sci. Technol. A 6, 2390 (1988)] and qualitatively consistent with a model of ion impact-induced mass redistribution [G. Carter and V. Vishnyakov, Phys. Rev. B 54, 17647 (1996)] as well as a crater function theory incorporating both effects [S. A. Norris et al., Nat. Commun. 2, 276 (2011)]. The critical transition angle between stable and rippled surfaces occurs 10°–15° above the value of 45° predicted by the mass redistribution model.