Interplay between kinetic roughening and phase ordering

– We studied interplay between kinetic roughening and phase ordering in 1+1 dimensional single-step solid-on-solid growth model with two kinds of particles and Ising-like interaction. Evolution of both geometrical and compositional properties was investigated by Monte Carlo simulations for various strengths of coupling. We found that the initial growth is strongly affected by interaction between species, scaling exponents are enhanced and the ordering on the surface is observed. However, after certain time, ordering along the surface stops and the scaling exponents cross over to exponents of the Kardar-Parisi-Zhang universality class. For sufficiently strong strength of coupling, ordering in vertical direction is present and leads to columnar structure persisting for a long time. Recently there has been considerable interest in growth induced surface roughening called kinetic roughening [1] and in phase ordering [2] independently. In kinetic roughening one is interested in the evolution of roughness during a nonequilibrium growth process. Phase ordering deals with the approach to equilibrium of a system quenched from a homogeneous high temperature phase into a two-component region. In both fields concept of scaling allowed to classify physical processes into universality classes. However, not very much is known so far about scaling exponents in systems where both processes, roughening as well as ordering, are relevant. Growth of many-component systems is in fact quite common situation. Many real materials are composed of two or more components but often the dynamics of one component is dominant and one can use a single-component growth model. For example, one can consider only kinetics of Ga atoms in studies of GaAs growth [3]. We are interested here in a more complex case when dynamics of Typeset using EURO-LaTEX 2 EUROPHYSICS LETTERS both components is important. It is a problem of practical interest, for example, microscopic understanding of growth of alloys is desirable [4, 5, 6]. One can study different aspects: kinetics, morphology, scaling behaviour etc. We are not going to model growth of any specific material but we shall concentrate on the scaling behaviour. The problem of growth in a system with two or more components is interesting from pure statistical-mechanical point of view, because growth process may belong to a new universality class [7, 8, 9]; such system might also exhibit a nonequilibrium phase transition between the low and high temperature region. Let us consider a surface in a d-dimensional space given by a single valued function h(r, t) of a d-dimensional (d=d+1) substrate coordinate r. The surface roughness is described by the surface width w(t, L) = 〈 √ h2 − h 2 〉, where t is the time, L is the linear system size and the bar denotes a spatial average, 〈...〉 a statistical average. The surface roughness often obeys dynamical scaling law w(t, L)∝Lf(t/L) where the scaling function f(x) has properties: f(x)= const., x≫1 and f(x)∝x , x≪1 (β=ζ/z). The exponents ζ and z (or ζ and β) characterize scaling behaviour of the surface width in a particular model and determine its universality class [1]. This universal behaviour has been observed in a wide variety of growth models and there has been considerable effort in finding different possible universality classes. Many of the growth models studied so far (for example ballistic deposition, Eden model, restricted solid-on-solid model, etc.) belong to the Kardar-Parisi-Zhang (KPZ) universality class [10]. There is a large menagerie of single-component growth models which can be potentially generalized to heterogeneous case. Moreover, there are different possible ways of generalization. Nevertheless, little is known so far about kinetic roughening in two-component growth models. This problem was probably first considered by Ausloos et al. [7]. They introduced a generalization of the Eden model coined as magnetic Eden model (MEM), which contains two types of particles with probabilities of growth given by Ising-like interaction. Ausloos et al. found a variety of morphologies, and they also measured the perimeter growth exponent [7]. They suggested that the model does not belong to the KPZ universality class. Recently Wang and Cerdeira [8] studied kinetic roughening in two 1+1 dimensional two-component growth models with varying probability of deposition of a given particle type. They did not found, however, a new universal behaviour. Although the phase ordering was apparently present it was not studied in these works. The characteristic length in phase ordering is a domain size D. It increases with time according to a power law, D ∝ t, ψ being one of the exponents specifying a universality class. Phase ordering is usually a bulk process, but growth induced ordering may be present just on the surface. Then evolution of the domain size on the surface is of interest. The scaling in this case has been recently studied by Saito and Müller-Krumbhaar [9] in another generalization of the Eden model (different from MEM). They M. KOTRLA et al.: INTERPLAY BETWEEN KINETIC ETC. 3 obtained ψ = 2/3 for growth of domain size, but they did not investigate kinetic roughening. In this Letter we study interplay between kinetic roughening and phase ordering. We introduce a new growth model with two types of particles suitable for the study of scaling and present results of simulations in 1+1 dimensions. Our model is based on the single-step solid-on-solid (SOS) model, a discrete model with constraint that the difference of heights between two neighbouring sites is restricted to +1 or −1 only. Advantage of this choice is that in contrast to cluster geometry, kinetic roughening can be more easily studied and that the single-step constraint allows simple interpretation of the deposition process in a binary system. We consider two types of particles, and denote the type of a particle by a variable σ which assumes values +1 or −1 (not to be confused with the step size). It is convenient to use an analogy with magnetic systems and to consider σ as a spin variable. We shall use this terminology in the following, but it should be warned that in the context of crystal growth this may be misleading because magnetic interactions of atoms are rather weak and do not usually control the growth process. One should rather think in terms of two types of particles with different bonding energies in this case. Therefore we shall call our model two-component single-step (TCSS) model. We describe our growth model for simplicity in 1+1 dimensions but it can be straightforwardly generalized to any dimension. Several realization of the single-step geometry differing by the number of nearest neighbours for a new particle are possible. Here we consider a variant with three nearest neighbours which can be represented as stacking of rectangular blocks with the height twice the width. During growth, particles are only added, there is neither diffusion nor evaporation. Once a position and a type of a particle are selected, they are fixed forever. Due to the single step constraint, particles can be added only at sites with a local heigh minimum, called growth sites. The probability of adding a particle with a spin σ to a growth site i depends only on its local neighbourhood in a way analogous to rules used in the magnetic Eden model [7]. It is proportional to exp(−∆E(i, σ)/kBT ), where kB is Boltzmann’s constant, T , and ∆E(i, σ) denote thermodynamic temperature and change of energy associated with deposition of a new particle, respectively. The energy ∆E(i, σ) is given by Ising-like interaction of a new particle with particles on the surface within nearest neighbours of a growth site (which are three in the selected realization left, bottom and right). When we denote the spin of a particle on the top of a column of spins at site i (surface spin) by σ(i) then ∆E(i, σ) = −Jσ [σ(i− 1) + σ(i) + σ(i+ 1)]−Hσ. Here J is a coupling strength and H is an external field. In the following we use for convenience dimensionless constants K = J/kBT and h = H/kBT . We studied both geometrical and compositional properties. Geometry is described by the surface width w(t, L) or by height-height correlation function 4 EUROPHYSICS LETTERS G (r, t) = 1 L ∑L i=1〈[h (i+ r, t) −h (i, t)] 〉. Evolution of geometry is affected by composition of the surface and vice versa. Therefore we introduce quantities characterizing composition on the surface. We concentrate only on ordering on the surface, we do not study bulk properties here. We consider the average domain size along the surface D(t), the spin correlation function of surface spins S(r, t) = 1 L ∑L i=1〈σ(i + r, t)σ(i, t) 〉, and magnetization M(t) = 1 L ∑L i=1〈σ(i, t)〉. We performed simulations for various coupling strengthsK in both ferromagnetic (K > 0) and antiferromagnetic (K < 0) regimes, mostly for zero external field h. System sizes ranged from L = 250 to L = 80000, time was up to 3×10 monolayers (ML). Presented results are averages over ten or more independent runs. We start from the flat surface as usual, but in two-component models the evolution strongly depends on initial composition of the substrate. We considered various possibilities (ferromagnetic, antiferromagnetic, random composition) and finally we have used growth on a neutral substrate, i.e. a substrate composed of particles with zero spin, and we let the system to evolve spontaneously in order to avoid initial transient effects. Fig. 1 shows examples of time evolution of morphologies obtained for selected couplings. We can see that with the increasing coupling the surface becomes more and more rough (facetted) and at the same time larger and larger domains of the same type of surface particles are formed. Notice also that for the larger coupling there is correlation between domain walls and changes of the local slope, and a columnar structure is observed. The time dependence of the surface width and the average domain size for L = 10000 are shown in Fig. 2 and Fig. 3, respectively. When the coupling is weak evolution of the roughness is almost the same as in the ordinary single-step model (β = β = 1/3). For a larger coupling it can be seen that the average surface width at a given time is an increasing function of the coupling and that ordering leads to an enhancement of the exponent βeff . However, after a time tcross, that increases with the coupling strength, the exponent βeff crosses over back to β (KPZ) = 1/3. To be sure that this unexpected crossover is not a finite size effect we performed for K = 1 the calculation for significantly larger system size L = 80000 and observed the same crossover (see inset in Fig. 2). When the coupling is even stronger the enhanced exponent βeff is observed during the whole simulation; for K = 2 we have βeff = 0.52± 0.02 for times up to 3× 10 5 ML. The crossover can be understood from behaviour of the average domain size. It increases initially according to a power law D ∝ teff , ψeff being an increasing function of K; ψeff = 0.33 for K = 1.1. However, after a certain time t (D) sat , D saturates to Dsat(K). We have checked that when the saturation is observed it is not a finite size effect (see inset in Fig. 3), rather an intrinsic property of the model. For large K the domain size increases during whole simulation, after some transient effect we observe ψeff = 0.44 for K = 2. The calculation of the spin-spin correlation function, and the correlation length ξ derived from M. KOTRLA et al.: INTERPLAY BETWEEN KINETIC ETC. 5 it, leads to similar results: during ordering the correlation length increases as a power law ξ(t,K) ∝ teff ; κeff ≈ 1/2 for K = 2. Then after a time increasing with the coupling ordering stops, and ξ saturates to ξ sat ∝ e γK ; γ = 3.25±0.08. Both effects, the crossover in time dependence of roughness and the saturation of the domain size (correlation length), are apparently related, tcross ≈ t (D) sat (see inset in Fig. 2). To complete the picture we need to know the second exponent for kinetic roughening, the roughening exponent ζ . We measured it from the spatial dependence of the height-height correlation function G (r, t) at t (w) sat (not shown here). The exponent has a value close to ζ (KPZ) = 1 2 for a weak coupling, but for a larger coupling we have found that there is a crossover behaviour provided the system is sufficiently large. We observed that the exponent ζeff is increasing with the coupling (ζeff = 0.7 for K = 0.7) on distances smaller than the correlation length ξ, and ζeff crosses over to ζ (KPZ) = 1 2 on a larger distances. If the coupling is strong we again do not see the crossover but only larger exponent, ζeff ≈ 1 for K = 2, because the simulated system sizes are not large enough to get into the regime r > ξ. Exponent ζ = 1 is the same as in some models with surface diffusion in which often anomalous scaling is observed [11]. However, this cannot be the case here since the step size is restricted. Our results show that for sufficiently strong coupling the TCSS model exhibits an intermediate growth regime with new scaling exponents which we estimate as β = 1/2, ζ = 1 (z = 2) and ψ = 1/2. However, these exponents are only effective and there is a crossover to the KPZ exponents for the medium coupling strengths. We estimated the dependence of t (D) sat (≈ tcross) on K. We have found that the saturated domain size as a function of K can be well fitted in the form Dsat(K) = 1 + 1 2 (e + e) (see inset in Fig. 3). Comparing Dsat ∝ (t (D) sat ) 1 2 with the fit we obtain that t (D) sat is a rapidly increasing function of K, t (D) sat (K) ∝ e 7K , e.g. t (D) sat (K = 2) ≈ 10 . Due to the progressively increasing time and system sizes needed for simulation we cannot exclude that there is a phase transition to a new true asymptotic phase at some value Kc. However, we expect that the crossover in the roughness is present for any value of K, but that it is hard to see it for strong coupling because tcross is larger than any possible simulation time. When K = ∞ then the model becomes trivial: type of a new particle added to a growth site is dictated by the majority of particle types in the neighbourhood and evolution of composition is fully determined by the initial configuration. So far all results were for zero external field. Non-zero field leads to nonzero surface (as well as bulk) magnetization, or in the context of alloy growth to changing stoichiometry. We can still define the exponent ψ for growth of the dominant domain size as well as the exponents for kinetic roughening. Effect of this symmetry breaking on the values of exponents remains to be studied. 6 EUROPHYSICS LETTERS One would like to know if the crossover to the KPZ exponents is generic behaviour or if facetted phase can be present also in the asymptotic regime. We have also tested a different variant of two-component growth, namely the singlestep model which corresponds to stacking of squares rotated by 45 degrees in which a new particle interacts with only two instead of three nearest neighbours. We observed the crossover as well, stronger coupling was needed to see enhanced exponents and for the same coupling the crossover time was smaller than in the variant with three nearest neighbours. Is is of interest to examine kinetic roughening and phase ordering in a different two-component growth model, in particular to reexamine the MEM model in the strip geometry since in the cluster geometry used by Ausloos et al. finite size effect are expected to be stronger. Situation in higher dimensions can be investigated by straightforward generalization. Finally, there is question of continuous description of the twocomponent growth. In conclusion we have generalized the single-step SOS model to two-component growth model. It can be considered as a model for growth of binary alloys from a fluid containing two types of particles, or for growth of a colony of two kinds of bacteria etc. The introduction of two kinds of interacting particles leads to new phenomena, like an increase of the roughness with the increasing coupling strength, larger effective scaling exponents and ordering on the surface with corresponding pattern formation during growth. However, after a certain time new behaviour stops and ordinary dynamic behaviour is restored. Hence, growth of a crystal can be divided into two stages. In the first, ordering is essential and influences the evolution of geometric properties. In the second stage, after saturation of domain size, the evolution of geometric properties is similar to that of the ordinary single-step model. Crossover time between two regimes is independent of system size and is a rapidly increasing function of coupling so that practically only the first regime may be observed. Patterns of the grown material in saturation regime are similar to lamellar structure observed in eutectic growth [4]. At the moment, it is not clear why ordering stops and the characteristic length (given by the saturated domain size) is selected.