Morphology of ion-sputtered surfaces

Abstract

We derive a stochastic nonlinear continuum equation to describe the morphological evolution of amorphous surfaces eroded by ion bombardment. Starting from Sigmund’s theory of sputter erosion, we calculate the coefficients appearing in the continuum equation in terms of the physical parameters characterizing the sputtering process. We analyze the morphological features predicted by the continuum theory, comparing them with the experimentally reported morphologies. We show that for short time scales, where the effect of nonlinear terms is negligible, the continuum theory predicts ripple formation. We demonstrate that in addition to relaxation by thermal surface diffusion, the sputtering process can also contribute to the smoothing mechanisms shaping the surface morphology. We explicitly calculate an effective surface diffusion constant characterizing this smoothing effect and show that it is responsible for the low temperature ripple formation observed in various experiments. At long time scales the nonlinear terms dominate the evolution of the surface morphology. The nonlinear terms lead to the stabilization of the ripple wavelength and we show that, depending on the experimental parameters, such as angle of incidence and ion energy, different morphologies can be observed: asymptotically, sputter eroded surfaces could undergo kinetic roughening, or can display novel ordered structures with rotated ripples. Finally, we discuss in detail the existing experimental support for the proposed theory and uncover novel features of the surface morphology and evolution, that could be directly tested experimentally.

Introduction

Sputtering is the removal of material from the surface of solids through the impact of energetic particles [1], [2], [3]. It is a widespread experimental technique, used in a large number of applications with a remarkable level of sophistication. It is a basic tool in surface analysis, depth profiling, sputter cleaning, micromachining and sputter deposition. Perhaps the largest community of users is in the thin film and semiconductor fabrication areas, sputter erosion being routinely used for etching patterns important to the production of integrated circuits and device packaging.

To have a better control over this important tool, we need to understand the effect of the sputtering process on the surface morphology. In many cases sputtering is routinely used to smooth out surface features. On the other hand, some investigations indicate that sputtering can also roughen the surface. Consequently, sputter erosion may have different effects on the surface, depending on many factors, such as incident ion energy, mass, angle of incidence, sputtered substrate temperature and material composition. The experimental results on the effect of sputter erosion on the surface morphology can be classified in two main classes. There exists ample experimental evidence that ion sputtering can lead to the development of periodic ripples on the surface [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Depending on the sputtered substrate and the sputtering conditions these ripples can be surprisingly straight and ordered. However, a number of recent investigations [28], [29], [30], [31], [32], [33], [34], [35], [36] have provided rather detailed and convincing experimental evidence that under certain experimental conditions ion eroded surfaces become rough and the roughness follows the predictions of various scaling theories [37]. Moreover, these investigations did not find any evidence of ripple formation on the surface. Up to recently these two morphological features were treated separately and no unified theoretical framework describing these morphologies was available.

The first widely accepted theoretical approach describing the process of ripple formation on amorphous substrates was developed by Bradley and Harper (BH) [38]. This theory is rather successful in predicting the ripple wavelength and orientation in agreement with numerous experimental observations. However, a number of experimental results have systematically eluded this theory. For example, the BH theory predicts an unlimited exponential increase in ripple amplitude in contrast with the observed saturation of the surface width. Similarly, it cannot account for surface roughening, or for ripple orientations different from those defined by the incoming ion direction or perpendicular to it. Finally, recent experiments [12], [13] have observed ripples whose wavelength is independent of the substrate temperature, and linear in the ion energy, in contrast with the BH prediction of a ripple wavelength which depends exponentially on temperature and decreases with ion energy.

In the light of the accumulated experimental results, it is clear that a theory going beyond the BH approach is required, motivating the results described in this paper. Thus here we investigate the morphology of ion-sputtered amorphous surfaces aiming to describe in an unified framework the dynamic and scaling behavior of the experimentally observed surface morphologies. For this we derive a nonlinear theory that describes the time evolution of the surface morphology. At short time scales the nonlinear theory predicts the development of a periodic ripple structure, while at large time scales the surface morphology may be either rough or dominated by new ripples, that are different from those existing at short time scales. We find that transitions may take place between various surface morphologies as the experimental parameters (e.g. angle of incidence, energy deposition depth) are varied. Usually stochastic equations describing growth and erosion models are constructed using symmetry arguments and conservation laws. In contrast, here we show that for sputter eroded surfaces the growth equation can be derived directly from a microscopic model of the elementary processes taking place in the system. A particular case of our theory was presented in [39].

In addition, we show that the presented theory can be extended to describe low temperature ripple formation as well. We demonstrate that, under certain conditions, ion-sputtering can lead to preferential erosion that appears as a surface diffusion term in the equation of motion, even though no mass transport along the surface takes place in the system. To distinguish it from ordinary surface diffusion, in the following we refer to this phenomenon as effective smoothing (ES). We calculate analytically an effective surface diffusion constant accounting for the ES effect, and study its dependence on the ion energy, flux, angle of incidence, and energy deposition depth. The effect of ES on the morphology of ion-sputtered surfaces is summarized in a morphological phase diagram, allowing for direct experimental verification of our predictions. A restricted study along these lines appeared in [40].

The paper is organized as follows. In Section 2 we review the recent advances in the scaling theory of rough (self-affine) interfaces. Section 3 is dedicated to a brief overview of the experimental results on surface morphology development under ion sputtering. A short summary of the theoretical approaches developed to describe the morphology of ion sputtered surfaces is presented in Section 4. This section also contains a short description of Sigmund’s theory of sputtering, that is the basis for our calculations. In Section 5 we derive the nonlinear stochastic equation describing sputter erosion. Analysis of this equation is presented in Section 6, discussing separately both the high and low temperature ripple formation. We compare the predictions of our theory with experimental results on surface roughening and ripple formation in Section 7, followed by Section 8, that summarizes our findings.