Possible origin for the experimental scarcity of KPZ scaling in non-conserved surface growth

Abstract

The Kardar-Parisi-Zhang (KPZ) equation is generically expected to describe the scaling properties of rough surfaces growing in the absence of conservation laws. However, very few experimental realizations are known of this universality class. Here we focus on the role of instabilities, whether of di usional origin or other, as physical mechanisms hindering the observation of KPZ scaling. Examples are drawn from various growth processes, such as electrochemical deposition (ECD), chemical vapor deposition (CVD), and erosion by ion-beam sputtering. We moreover consider an interface equation which, starting from the corresponding constitutive equations, can be derived to describe growth by either ECD or CVD depending on the interpretation of the quantities appearing. This approach makes contact with phenomenological parameters, and suggests that a more generic description of non-conserved growth may be provided by the KuramotoSivashinsky equation and generalizations thereof. In this case, the experimental scarcity of KPZ scaling would be due to exceedingly long transients determined by the instabilities that occur.

Dynamics of nanodots formation on irradiated materials

“…To generate a structure with a characteristic wavelength, which is related to the diameter of the nanodots, the balance between the free energy term (∂g à =∂f) and the interfacial energy term (22r 2 f) acting to smooth the boundary in equation 7was essential and decisive. [14][15][16] The morphology and length scale of our simulation results are comparable to many experimental observations, [7][8][9]15,32 as shown in figure 1(a) to (c) of Facsko et al 32 for amorphous GaSb thin films on different matrixes. It should be noted that the simulation results of nanodots formation in this study are considered in a condition of irradiated thin film, which is different from the bulk material.…”

“…Subsequently, the Bradley-Harper (BH) model was generalized to describe the stabilization and ordering of nanodot patterns in semiconductors. 15,16 Owing to the limitations of their partial differential equations, which relied on the variable of surface height, this theoretical model encountered difficulty in dealing with a complicated surface structure or interlaced morphology underneath the surface, such as the void/fiber structures in the subsurface [17][18][19] or embedded nanofibers with flat surfaces on the top. 20 Moreover, no radiation effects were involved in the model, e.g.…”

Kinetic roughening in a realistic model of non-conserved interface growth

“…These are those of the KPZ equation for slow attachment kinetics, while they correspond to a different universality class for k D → ∞. Thus, one would expect KPZ scaling for a wide range of parameters, which has indeed been found experimentally within ECD growth [31], but not with the generality that was expected [30,32]. One possibility is that the crossovers induced by the instabilities are so long lived as to prevent actual asymptotic scaling from being accessed by experiments.…”

Universal non-equilibrium phenomena at submicrometric surfaces and interfaces

“…A paradigmatic example is provided by kinetic roughening, wherein the Kardar-Parisi-Zhang (KPZ) equation (equivalent to the stochastic Burgers equation) is in principle expected to describe the interface fluctuations whenever growth takes place in the absence of conservation laws for the height field. In spite of this expectation, paradoxically very few experiments have been reported in which KPZ exponents have been consistently measured (see one example in [18], and overviews in [12,22]). Moreover, the problem with this type of "derivations" is that they do not enable a detailed connection with phenomenological parameters describing specific experimental systems.…”

“…While kinetic roughening is associated with the lack of typical time and length scales in the evolution of the surface morphology, it is of course possible (to some extent, almost unavoidably [22]) to encounter experimental situations in which surface features are characterized by well defined time and length scales, the morphology displaying a well defined pattern as a result of some morphological instability [23]. This is the case, for instance, in the nanostructuring of many materials by ion-beam sputtering (IBS).…”