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 diffusional 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 Kuramoto–Sivashinsky 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.

Introduction

The kinetic roughening of surfaces [1] is a subject of wide interest, due both to its implications for processes of technological relevance [2], [3], [4], and to the interesting instances that it offers of extended systems evolving in the presence of fluctuations [5]. A very successful theoretical framework for the study of rough interfaces has been the use of stochastic growth equations for the interface height. Among these, the one proposed by Kardar, Parisi and Zhang (KPZ) [6] has played a prominent role, since in particular it has allowed to make connections to other physical problems, like directed polymers in disordered media or randomly stirred fluids [1]. Denoting by $\text{h(}\text{r}\text{,t)}$ the surface height at time t above position $\text{r}$ on a reference substrate plane, the KPZ equation is the simplest nonlinear coarse-grained description of a surface growing (or eroding) at a constant average rate v along the local normal direction, and reads

$$\text{∂h}\text{∂t}\text{=v+ν}\text{∇}{}^2\text{h+}\text{v}\text{2}\text{(}\text{∇}\text{h)}{}^2\text{+η(}\text{r}\text{,t)}\text{,}$$

where ν is a positive constant, and $\text{η(}\text{r}\text{,t)}$ is an uncorrelated Gaussian noise representing fluctuations, e.g., in a flux of depositing particles. Thus, the KPZ equation was postulated to describe generically the dynamics of surfaces growing in the absence of additional conservation laws, and is thus expected to be relevant to such diverse physical growth systems as erosion by low energy ion-beam sputtering (IBS) [2], electrochemical deposition (ECD) [4] or chemical vapor deposition (CVD) [3]. The generality of KPZ scaling would be a consequence of the phenomenon of universality observed for the scaling properties of rough surfaces.

However, despite some attempts at measuring KPZ scaling in, e.g., IBS [7] or ECD [8], to date very few experiments have been reported which are unambiguously described by the KPZ equation [9], [10], [11]. The identification of the physical mechanisms responsible for this paradoxical (termed “anomalous” in some early studies of kinetic roughening) non-KPZ scaling behavior has been complicated by two main reasons: on the one hand, while detailed derivations of the KPZ equation have actually been achieved, they apply to discrete or continuous theoretical models [12] which are indirectly related with experiments, or else the derivations themselves need resort to approximations which are not free from ambiguities.¹ On the other hand, symmetry arguments such as those leading to the KPZ equation do not enable a detailed connection with phenomenological parameters describing specific experimental systems. These facts have led to invoking additional effects on a coarse-grained level, such as specific noise statistics, non-local effects, etc. [1], in order to account for the difference between the observed and the predicted scaling behaviors of rough surfaces. However, a wide range of scaling exponents ensued, there being no theoretical argument that could identify the correct exponents for a specific growth experiment.

Physically, many of the experimental growth systems expected to be in the KPZ universality class—and which seem to fail such expectations—feature dynamical instabilities leading to development of large slopes on the surface, and even to the production of characteristic surface features whose presence breaks scale invariance. A prototype example is IBS, where standard experimental conditions lead to the production of nanometric ripples [14] or dots [15] on the surface at short and intermediate times. Actually, it has been possible to derive [16] the relevant interface equation for these experiments, which for ions bombarding the target at perpendicular angles reads

$$\text{∂h}\text{∂t}\text{=−|v}{}_{\mathrm{<mtext>IBS</mtext>}}\text{|−|ν}{}_{\mathrm{<mtext>IBS</mtext>}}\text{|}\text{∇}{}^2\text{h−K}{}_{\mathrm{<mtext>IBS</mtext>}}\text{∇}{}^4\text{h+}\text{λ}{}_{\mathrm{<mtext>IBS</mtext>}}\text{2}\text{(}\text{∇}\text{h)}{}^2\text{+η(}\text{r}\text{,t)}\text{,}$$

the constants $\text{v}{}_{\mathrm{<mtext>IBS</mtext>}}\text{,}\text{ν}{}_{\mathrm{<mtext>IBS</mtext>}}$, K_IBS and λ_IBS being in this case functions of phenomenological quantities, such as the angle of incidence, ion energy, ion penetration depth, etc. Eq. (2) is the noisy generalization of the Kuramoto–Sivashinsky (KS) equation [17]. The KS equation has been seen to describe other dynamic processes such as flame front propagation or chemical turbulence [17], or terrace growth in epitaxial systems [18]. Eq. (2) displays a linear instability at short times, there being a most unstable Fourier mode which induces the observed [14], [15] periodic array of ripples or cells. At very long times, scale invariance is restored, the values of the scaling exponents being—at least for a one-dimensional substrate—those of the KPZ universality class [19]. Hence, the specific example of IBS suggests transients due to instabilities as a physical reason for the difficulty in observing the asymptotic KPZ scaling, even if it should occur, since probing asymptotic times may well lie beyond experimental capabilities. For instance, in the CVD experiment in Ref. [11], deposition runs up to 2 days long had to be carried out in order to confirm KPZ behavior, which is both unusual and almost prohibitive. The IBS example illustrates another important fact: trying to assess the origin of non-KPZ behavior implies addressing pre-asymptotic features of the growth process under study. Hence, symmetry arguments do not suffice to derive the interface equation of motion, since contributions which are very important in the early dynamics can be overlooked. For instance, in the IBS system symmetry suggests a simple KPZ description [20], which misses the correct negative sign of the linear Laplacian term in (2) accounting for the physical ripple instability.