Nanoscale pattern formation at surfaces under ion-beam sputtering: A perspective from continuum models

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Abstract

Although reports on surface nanostructuring of solid targets by low to medium energy ion irradiation date back to the 1960s, only with the advent of high resolution tools for surface/interface characterization has the high potential of this procedure been recognized as a method for efficient production of surface patterns. Such morphologies are made up of periodic arrangements of nanometric sized features, like ripples and dots, with interest for technological applications due to their electronic, magnetic, and optical properties. Thus, roughly for the last ten years large efforts have been directed towards harnessing this nanofabrication technique. However, and particularly in view of recent experimental developments, we can say that the basic mechanisms controlling these pattern formation processes remain poorly understood. The lack of nanostructuring at low angles of incidence on some pure monoelemental targets, the role of impurities in the surface dynamics and other recent observations are challenging the classic view on the phenomenon as the mere interplay between the curvature dependence of the sputtering yield and surface diffusion. We review the main attempts at a theoretical (continuum) description of these systems, with emphasis on recent developments. Strong hints already exist that the nature of the morphological instability has to be rethought as originating in the material flow that is induced by the ion beam.

Introduction

Observations of nano-scale patterns on the surfaces of solid targets that undergo irradiation by 100 eV to 10 keV ions, date back at least to the early 1960’s [1], [2]. They correspond to materials that either are amorphous like glass, or become amorphized by this type of bombardment like monocrystalline semiconductors such as Si [3]. These are the type of targets that we will be considering in the present work. Actually, quite similar patterns are found on other materials, such as metallic [4] or insulating targets, see e.g. [5], [6] for reviews, but additional physical effects are expected to come into play into their pattern forming properties.

Already in the first accounts of these shapes, and due to the naked eye similarities, natural analogies were drawn to more familiar patterns, like ripples that form in a self-organized fashion on the surface of sandy dunes, due to the action of wind or water. Examples illustrating these analogies can be found in Fig. 1. As it turns out, this analogy has played a fruitful role in the modeling of the nanostructures formed by ion-beam sputtering (IBS), and is expected to maintain as an inspiration for future developments, as seen below.

With the advent of high resolution surface characterization techniques, important geometrical features of these structures were revealed, such as the large degree of in-plane ordering that they can develop [7], which is of clear practical importance to applications ranging from optoelectronic to catalytic. Moreover, the universality of their production has been assessed [6], in the sense of the high degree of independence on the specific ion-target combination, as well as the dependence of their shape (e.g. rippled vs dot-like) only on rather general geometrical constraints, such as the symmetry of the experimental set-up (ripples for oblique angles of incidence θ  0 and dots for normal incidence θ = 0, or else for oblique incidence onto a rotating target), although some exceptions to these rules can be found, see e.g. [8]. From the point of view of understanding the physical processes that govern this pattern formation phenomenon, a crucial input concerns the availability of data on the time evolution of these systems. At this, one has to take into account the range of fluxes that is usually employed, typically 1 to 10 ion nm−2 s−1. Thus, the time scale associated with ion beam nanostructuring (≃1 s) is orders of magnitude slower than the typical times (≃1 ps) associated with the relaxation of collision cascades or with surface diffusion hopping attempts. This induces the occurrence of non-trivial changes in the morphological evolution at macroscopic time scales (from seconds to hours).

Coming back to the analogy with macroscopic patterns, it is instructive to compare typical magnitudes both in IBS nano-patterned systems and, e.g., ripple formation on aeolian sand dunes. Thus, ripple formation and coarsening has been observed for off-normal Ga+ bombardment of Si at 30 keV [9]. The ripple wavelength at instability onset was near 50 nm, ripples being transported in-plane with a velocity close to 0.3 nm/s. Quite similar figures were obtained more recently for off-normal irradiation of glass with similar beam characteristics [10]. Meanwhile, field and laboratory experiments on the formation of aeolian ripples indicate typical wavelengths around 6 cm and ripple transport speeds at 0.07 cm/s [11]. Hence, while there is a difference of 6 orders of magnitude in length scales, the typical times needed to travel the corresponding wave-length distances are both of the order of seconds. We conclude that IBS nanopatterning is quite a slow phenomenon (incidentally, 0.3 nm s−1  1 cm year−1 is a typical speed for tectonic plate motion!).

As time scales of the order of seconds remain beyond reach for microscopic or atomistic approaches like Molecular-Dynamics (MD) or even kinetic Monte Carlo [12], [13], continuum descriptions appear as a natural choice in order to account for the main properties of these pattern forming systems. The universality of the phenomena in the sense of their occurrence for a wide choice of ion-target combinations and incidence geometries adds credit to the applicability of this type of models. During the last twenty years or so, we have witnessed the application of continuum descriptions to IBS nanopatterns with a varying (and in general increasing) degree of success. However, a number of quite recent experimental observations are forcing us to rethink this modeling from a new perspective. It is the purpose of this work to overview the main steps taken in the process as well as to point out potential new avenues for this thriving activity.

Section snippets

Bradley–Harper-type models

Historically, the work that paved the way for further continuum modeling of IBS surface nanopatterns is due to Bradley and Harper (BH) [14]. The procedure amounts to assuming that collision cascades relax infinitely fast so that one can compute the local velocity of erosion at a surface point on the target as proportional to the total energy that is thus deposited in its surroundings. It had been already pointed out by Sigmund [15] that this assumption leads to what is known as a morphological

Two-field “hydrodynamic” models

Having reached this point in the continuum description of IBS induced nanopatterns, by 2005 there was a need for an improved continuum model that: (i) introduced an increased number and type of relaxation mechanisms in a natural (non ad-hoc) way that allows for an interaction among them and for the expected interplay between transport and morphology; (ii) improves upon consistency issues (cancellation modes, etc.); (iii) can in principle be adapted to improvements in the description of energy

Some caveats

Although the two-field model discussed in the previous section can account for a large number of experimental properties of IBS surface nanopatterns, a number of issues still seem to require a deeper reflection on the basics of this pattern formation process from a general point of view. We can mention some of these issues that have become increasingly relevant in recent years, such as (i) special properties of order and coarsening; (ii) the role of impurities and/or preferential sputtering,

Recent data on the morphological instability

In view of the relatively confusing situation2 in which different (sometimes conflicting) observations were reported under similar conditions, recently a number of experiments have been addressed at clarifying the situation. These works [50] focus on Si targets as a representative case of materials that become amorphous under irradiation, explicitly

Fully hydrodynamical model

In face of the new experimental data discussed in the previous Section, some of us [51] have very recently reconsidered the basic ingredients that are required in the continuum description of IBS surface nanopatterns in order to account within a single framework for, at least, the most salient features of the recent experimental picture of the process. In order to do this, and in view of the large predictive power that ensues when considering the dynamics of material transport, as seen in

Conclusions and outlook

As a general conclusion, we believe that continuum models still hold promise for describing surface nanopatterning under IBS, although, at present, many issues remain quite open. Already working within the scope of Section 6 there are many details, such as the energy dependence of the bulk force or an accurate representation of erosive contributions, that would indeed benefit from complementary approaches, such as more atomistic models, and of course from experimental assessment. At any rate,

Acknowledgments

This work has been partially supported by the Spanish Ministry of Science and Innovation (Grants Nos. FIS2009-12964-C05-01, FIS2009-12964-C05-03, FIS2009-12964-C05-04, and CSD2008-00023).

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