Morphology of ion-sputtered surfaces
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.
Section snippets
Scaling theory
In the last decade we witnessed the development of an array of theoretical tools and techniques intended to describe and characterize the roughening of nonequilibrium surfaces and interfaces [37]. Initiated by advances in the statistical mechanics of various nonequilibrium systems, it has been observed that the roughness of many natural surfaces follows rather simple scaling laws, which can be quantified using scaling exponents. Since kinetic roughening is a common feature of ion-bombarded
Experimental results
The morphology of surfaces bombarded by energetic ions has long fascinated the experimental community. Lately, with the development of high resolution observation techniques such as atomic force and scanning tunneling microscopies, this problem is living a new life. The various experimental investigations can be classified into two main classes. First, early investigations, corroborated by numerous recent studies, have found that sputter eroded surfaces develop a ripple morphology with a rather
Theoretical approaches
The recent theoretical studies focusing on the characterization of various surface morphologies and their time evolution have revolutionized our understanding of growth and erosion phenomena (for reviews, see [37]). The physical understanding of the processes associated with interface roughening require the use of the modern concepts of fractal geometry, universality and scaling. In Subsection 4.1 we review the major theoretical contributions to this area, necessary to describe the morphology
Continuum equation for the surface height
Sigmund’s theory, while offering a detailed description of ion bombardment, is not able to provide direct information about the morphology of ion-sputtered surfaces. While Eq. (19) provides the erosion velocity, in the present form it cannot be used to make analytical predictions regarding the dynamical properties of surface evolution. To achieve such a predictive power, we have to eliminate the nonlocality contained in the integral (19) and derive a continuum equation describing the surface
Analysis of the growth equations
This section is devoted to the study of the morphological properties predicted by Eq. (40). This is not a simple task, due to large number of linear and nonlinear terms, each of which influence the surface morphology. The complexity of the problem is illustrated by some special cases of Eq. (40), for which the behavior is better understood. For example, when nonlinear terms and the noise are neglected (ξx=ξy=λx=λy=0, η=0), Eq. (40) reduces to a linear generalization of BH theory, which predicts
Comparison with experiments
In this section we compare the predictions of the theory, presented in this paper with experimental results on ripple formation and surface roughening. For a better presentation, we choose to structure the material around well known features of the morphological evolution, present the theoretical predictions and discuss to which extent are they supported by the available experimental data. We also discuss predictions that have not been tested in sufficient detail but could offer future tests of
Conclusions
In this paper we investigated the morphological properties of surfaces eroded by ion bombardment. Starting from the expression for the erosion velocity derived in the framework of Sigmund’s theory of sputtering of amorphous targets, we derived a stochastic partial differential equation for the surface height, which involves up to fourth order derivatives of the height, and incorporates surface diffusion and the fluctuations arising in the erosion process due to the inhomogeneities in the ion
Acknowledgements
We would like to acknowledge discussions with E. Chason, B. Kahng, H. Jeong, F. Ojeda and L. Vázquez. This research was supported by NSF-DMR CAREER and ONR-YI awards (A.-L.B. and M.M.) and DGES (Spain) grants PB96-0119 and BFM 2000-0006 (R.C.).
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