Kinetic pattern formation at solid surfaces

Abstract

The emergence of regular spatial patterns from inanimate natural processes has been a source of fascination and wonder since the beginnings of scientific exploration. A quantitative understanding of the mechanisms underlying pattern formation is a rather recent achievement, which has been possible through a strong concerted research effort in nonlinear physics and mathematics during the past two or three decades. This, by now classic, body of work [1] has been mostly concerned with pattern formation phenomena in macroscopic systems, such as hydrodynamic instabilities and chemical oscillations. More recently, patterns in granular materials such as sand ripples have also been addressed from related points of view [2, 3]. The focus of the present contribution is on submicron-scale patterns like mounds, ripples and step bunches, which form on solid surfaces under nonequilibrium conditions. This subject differs from macroscopic pattern formation in that (i) the visualisation of the patterns of interest requires advanced microscopy techniques, notably scanning probe microscopes, and (ii) the formation of the patterns often relies on specific atomic processes, which must be analyzed in detail to reach quantitative agreement between theory and experiment [4]. Nevertheless a phenomenological continuum approach to pattern formation at solid surfaces has been developed, to some extent in analogy to the established macroscopic theories of pattern formation. Such an approach has proven to be useful, because it allows for a compact, unified description of a variety of mechanisms, as well as for the efficient analytical or numerical modeling of global aspects of the surface morphology. The purpose of this article is to provide an elementary introduction to the continuum approach. The article is based on lectures on pattern formation during epitaxial growth and erosion, which were delivered jointly with Thomas Michely. In addition to the topics covered in these lectures, phenomena related to surface electromigration and steering (the deflection of the trajectories of depositing atoms due to the attraction by the growing film surface) will be included here. The lectures also addressed atomistic aspects of growth and erosion, and the detailed relationship between the atomistic and continuum viewpoints. As extensive recent reviews on these issues are available elsewhere [4, 5, 6, 7], they will be treated only briefly here. We further emphasize that we are concerned only with kinetically (rather than energetically) driven morphological instabilities. This excludes the formation of hill-and-valley structures at thermodynamically unstable surfaces, as well as the broad class of patterns which form in heteroepitaxial growth due to the strain caused by the lattice mismatch between the substrate and the growing film [6, 7, 8]. The basic instability mechanisms are described in the next section. Section 3 discusses the emergence of the characteristic length scale of the patterns in the early time regime, while in Sect. 4 the nonlinear, late time evolution is discussed. Experimental examples are presented as appropriate to illustrate the theoretical concepts. © Springer-Verlag Berlin Heidelberg 2005.