Dynamic scaling in conserved systems with coupled fields: Application to surfactant-mediated growth

– We present an analytical study of the interaction of two nonequilibrium conservative fields. Due to the conservative character of the relaxation mechanism, the scaling exponents can be obtained exactly using dynamic renormalization group. We apply our results to surfactant-mediated growth of semiconductors. We find that the coupling between the surfactant thickness and the interface height cannot account for the experimentally observed layered growth, implying that reduced diffusion of the embedded atoms is a key mechanism in surfactant-mediated growth. Presently there exists considerable interest in understanding the dynamical properties of growing nonequilibrium interfaces [1], [2]. While originally this research was motivated by studies of interface roughening, the formalism and the knowledge accumulated can be applied to a wider range of nonequilibrium phenomena [3]. In studies of interface roughening it is customary to consider only one relevant field, the height of a d-dimensional interface h(x, t) [4]. However, recent work indicates that in many physically relevant cases a single field is not sufficient for the complete characterization of the system [5]. In this paper we consider a coupled-field approach to discuss a technologically important problem, the growth of semiconductors by molecular beam epitaxy (MBE) in the presence of surfactants. The surfactant is a thin layer of atoms, deposited on the surface, which segregate on top of the growing interface [6]. For a complete characterization of the surfactant-interface system, one has to consider the coupling between the height of the interface and the thickness of the surfactant layer. On the assumption that the surfactant severely curtails the diffusion length of the embedded atoms, it was shown [7] that coupled continuum equations predict the existence of a phase in which both the interface and the surfactant layer are smooth. This phase is reminiscent of the layer-by-layer growth mode observed experimentally. An unresolved issue of critical importance is whether the coupled dynamics of the interface and the surfactant c © Les Editions de Physique 130 EUROPHYSICS LETTERS are able by themselves to produce the smooth growth, without requiring that the embedded atoms are immobile. In the present work we show that in surfactant-mediated growth the coupling of two conserved fields is not adequate to induce the experimentally observed smoothing effect, implying that reduced diffusion of embedded atoms is a key element in the process. We arrive at this conclusion by performing a dynamical renormalization group (DRG) study of the interaction between two fields, for which the main relaxation mechanism is conservative. Due to the conservative character of the relaxation mechanism, all the exponents can be obtained exactly for the fully coupled system. In addition to surfactant-mediated growth, this study allows us to discuss the possible universality classes arising from the coupling of two nonequilibrium processes, a broader question of current interest in statistical mechanics. We study the following set of equations ∂th = −Kh∇h+ λ∇(∇h) + β∇(∇v) + ηh , (1a) ∂tv = −Kv∇v + γ∇[(∇h)(∇v)] + ηv (1b) as a generic model for the dynamics of two coupled nonequilibrium fields, h(x, t) and v(x, t), for which the relaxation mechanism is strictly conservative. As we discuss below, h can be interpreted as the height of a surface on which a surfactant layer of thickness v resides. We first motivate the choice of coupled equations by symmetry arguments. To simplify the notation we assume that we have a two-component field ψ = (h, v). The conservative requirement of the relaxation mechanism means that any local variation in the magnitude of ψ is the result of a transport mechanism described by the current jψ = (jh, jv), such that ψ obeys the continuity equation ∂tψ = −∇jψ + ηψ. In many physically relevant systems the local variations in the current are governed by the spatial variations in the local chemical potential, μ, giving jψ ∼ −∇μψ. If a diffusion bias at step edges is present (an effect known as “Schwoebel” barrier), or gravity plays a role in the relaxation process, the chemical potential is simply proportional to the field μψ ∼ ψ. The possible effects of terms generated by a Schwoebel barrier will be discussed below. The next relevant functional form for the chemical potential is μ ∼ ∇ψ, i.e. the potential depends on the local curvature of the field (the explicit ∇ψ dependence can be excluded since it results in an unphysical broken x→ −x symmetry). Combining the above relations, we obtain ∂tψ = −Kψ∇ψ + ηψ , (2) which gives the linear terms in (1) [8]. In order to account for nonequilibrium effects, we write the current as jψ = j eq ψ + j ne ψ , where j eq ψ is the equilibrium part discussed previously, while j ne ψ contains the nonequilibrium terms that cannot be obtained by variations of a Hamiltonian. λ∇(∇h), β∇(∇v) and γ∇[(∇h)(∇v)] represent such terms, which can be associated with the local chemical potentials μ λ ' (∇h), μ β ' (∇v) and μ γ ' ∇h∇v. The terms β∇(∇v) and γ∇[(∇h)(∇v)] are the lowest-order relevant coupling terms, while the nonlinear term λ∇(∇h) is generated by β and γ. In principle one can consider further nonlinear terms, but they do not add to the relevant physics [9]. If the system is strictly conservative, the noise ηψ = (ηh, ηv) has to obey this conservation property. It is, however, possible to have conservative relaxation processes but nonconservative noise. Accordingly, we include both conservative and nonconservative noise terms in (1). The noise is assumed to have zero average and correlations 〈ηψ(x, t)ηψ(x, t′)〉 = Dψδ(x− x′)δ(t− t′) , (3) where Dψ = D ψ − D ψ∇ + D′ ψ∇. The D′ ψ term is generated by D ψ and D ψ, as will be shown below. A.-L. BARABÁSI et al.: DYNAMIC SCALING IN CONSERVED SYSTEMS WITH ETC. 131 The coexistence of the conservative and nonconservative noise terms introduces a length scale Lψ ∼ (D ψ/D ψ ) which delimits two different scaling regimes: one dominated by the conservative noise (with the system size L¿ Lψ), and one dominated by the nonconservative noise (L À Lψ). Moreover, D h can be different from D v (and similarly, D h 6= D v), which introduces two different length scales Lψ = (Lh, Lv), resulting in a rather complicated crossover behavior. We analyzed the coupled system (1) using a DRG scheme, limiting the calculations to the one-loop approximation. Our main results are expected to be valid for higher loops as well, presuming that a coupled phase exists. After integrating out the fast modes in the momentum shell eΛ0 ≤ |k| ≤ Λ0, we have performed the rescaling x → ex, t → et, h → ehh, and v → evv, where χh and χv are the roughness exponents of the fields h and v, respectively, and z is the dynamic exponent of the system. Since we are interested in the properties of the coupled phase, we assume that there is only one dynamic exponent controlling the time evolution of both fields [5]. The first important result is related to the nonrenormalization of the nonlinear term λ, ∂λ ∂l = λ[χh + z − 4], a consequence of the invariance of eq. (1) under a generalized Galilean transformation [10]. It is interesting that the coupling to an additional field v does not destroy the validity of this transformation. A consequence of this invariance is that the flow equation for λ is free from higher-loop corrections, and the resulting scaling relation