The dynamics of island nucleation and growth —beyond mean-field theory

– Fully deterministic calculations of the dynamics of island nucleation and growth during thin-film deposition are presented, using a capture zone model that transcends the classical mean-field approximation. Nucleation rates for various critical island sizes i = 0, 1, 2 are accurately calculated from the time-dependent monomer density within the capture zones. The deterministic evolution of the Joint Probability Distribution (JPD) of island and capture zone size is then calculated. The JPDs are found to converge rapidly to approximately scale-invariant forms that are in excellent agreement with Monte Carlo simulation data. Island nucleation and growth during thin-film deposition is often encountered, and is of widespread technological and scientific interest [1]. Islands can be used as the active elements in applications such as catalysis or electronics, and they are the building blocks of thin films and nanostructures. Scientific interest focuses on the statistical properties of the island arrays: How does the island density depend on coverage, deposition rate and temperature? What is the island size distribution and why does it display (to a very good approximation) scale invariance with coverage and deposition rate? For decades the main theoretical approach to these questions has used mean-field rate equations [2–6], where islands of the same size are assumed to exist in the same environment, and success has been achieved in addressing the first of the questions concerning island density [5]. However, it has long been realised that this approach does not generally provide good predictions of the island size distribution, and cannot explain its scaling properties [6]. More recently, it has been shown that the island sizes are intimately related to their capture zones, which are the regions of substrate closest to the islands [7–9]. Material deposited into an island’s capture zone is more likely to diffuse to that island that to any other, thereby dictating the island’s growth rate. Therefore, a theory for the island size distribution must necessarily also consider the distribution of capture zones and how this varies as new islands nucleate, and so must look beyond the traditional mean-field approach. A suitable theoretical approach has recently been suggested where the evolution of the Joint Probability Distribution (JPD) of island and capture zone sizes is considered [10, 11]. The JPD contains information on the spatial arrangement of the islands as well as their sizes, and so is a quantity of fundamental interest to both theory and experiment. The scale invariance of the JPD can be explained by assuming that the dynamics of the island density follow the mean-field results, and the detailed form of the JPD can then be successfully calculated for critical island size i = 0, 1, where islands of size greater than i are stable [10]. However, the reliance on the mean-field description of island nucleation rates is clearly not justified given the failings described above. (Other recent work using the capture zone picture to calculate the i = 1 island size distribution also relies on the mean-field dynamics [12]). In this paper we therefore examine the dynamics of island nucleation within the capture zone model, and show how nucleation rates can accurately be calculated for i = 0, 1, 2. In addition, these rates are used in entirely deterministic calculations [13] of the evolution of the JPDs. We find excellent agreement with simulation results for island density and the JPDs, and in particular we observe how the JPDs rapidly evolve to scale-invariant forms as previously predicted. Therefore this work provides a versatile and successful method of calculation that completely transcends the mean-field approach for the first time, and so significantly adds to our understanding of the statistical properties of island nucleation and growth. We start by investigating the island nucleation rate within a single capture zone using Monte Carlo simulation (similar studies for island nucleation rates in one-dimension [14, 15] and on top of monolayer islands [16–18] have recently been carried out). The simulations are performed on a square lattice, with monomers being randomly deposited at a rate of F monolayers per unit time and diffusing with constant D. The behaviour of the simulation is dictated by the ratio R = D/F , and typically large values of R ≥ 10 represent experimental conditions. In these simulations there is an absorbing circular island of radius ris at the lattice origin, and the capture zone is a circle of radius rcz > ris. Thus, if a monomer diffuses within a distance ris of the origin, it is completely removed from the simulation, whereas if it diffuses to a distance greater than rcz, it is removed but re-inserted at random at a distance rcz from the origin. The latter process represents the fact that at a capture zone boundary the net flux of monomers is zero. Simulations start with no monomers in the capture zone, but the number of monomers rises as the simulation proceeds and will saturate if the simulation continues for a sufficient length of time. However, each simulation terminates if the conditions for the nucleation of a new island are satisfied. The condition for critical island size i > 0 is that i+1 monomers must coincide at a site at the same time. For spontaneous nucleation (i = 0) each monomer has a finite, low probability of causing a nucleation each time it takes a diffusive hop. The simulation is repeated 1000 times for a range of rcz and ris so that the average time to nucleation can be estimated in each case to about 3% accuracy. From this we estimate the nucleation rate of each capture zone and island size pair in terms of a probability per diffusive hop. Typical results are shown in fig. 1a for i = 0, 1, 2. As expected, the nucleation rate decreases monotonically as the island radius increases, tending to zero as it approaches the capture zone radius. In order to understand these nucleation rates, we calculate the monomer density profile n1(r, θ) using the driven diffusion equation: ∂n1(r, θ) ∂θ = 1 +R∇n1(r, θ), where θ = Ft is the fractional substrate coverage at time t after monomer deposition starts. This approach has been used before for i = 1 but assuming that the monomer density has saturated [19]; here we start with zero density. We implicitly use the circular symmetry of the island and capture zone system. From the monomer density we estimate the average time to nucleation 〈θ〉, and hence nucleation rate (4R〈θ〉), by making a local-density approximation 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 (b) Island Radius M o n o m e r N u m b e r i=0 Simulation