One-dimensional island size distribution: From non-equilibrium to equilibrium

Highlights

• A critical view of the analysis of experimental one-dimensional island size distribution.

• Time-dependent evolution of one-dimensional island size distribution.

• The change in one-dimensional island size distribution during cooling or annealing.

Abstract

We present a critical view of the analysis of experimental one-dimensional island size distribution as a function of time. We study the processes of island growth using large-scale kinetic Monte Carlo simulations with diffusion barriers calculated within the framework of the density functional theory. We have shown that one-dimensional island size distribution depends significantly on the time of the experiment. Our model predicts that during annealing or cooling, the transition from one state of thermodynamical equilibrium to another occurs through a non-equilibrium state. This transition consists of Ostwald ripening and decay of one-dimensional islands. The results of our work demonstrate that considering experimental one-dimensional island size distribution as an equilibrium is a big misconception.

Introduction

One-dimensional nanostructures may be used to create a new generation of microelectronic devices because their properties usually differ considerably from those of the corresponding two-dimensional and three-dimensional nanostructures. Properties of one-dimensional nanostructures that are attractive for nanotechnology include quantized conductance [1], [2], [3], electronic ‘end’ states [4], charge/spin density waves [5], potential non-Fermi liquid behavior [6], [7], and giant Rashba split bands [8]. From the point of view of microelectronic devices, it would be desirable that one-dimensional nanostructures were of similar length and well-ordered. The self-assembly of well-ordered one-dimensional structures of various lengths is observed in some epitaxial systems [9], [10], [11], [12], [13], [14], [15]. Knowledge of the mechanisms which define sizes of one-dimensional nanostructures is necessary to develop techniques of their controlled growth. Therefore, the problem of characterization of island-size distribution in epitaxial growth has become one of the central problems in one-dimensional physics.

Gambardella et al. [14] were among the first who compared experimental data with analytical expression for the size distribution of monoatomic chains in the framework of a one-dimensional lattice gas model at thermodynamic equilibrium. This expression can be written as [16]

$$q_l=\frac{q^2}{n_1}{(1-\frac{q}{n_1})}^{l-1},$$

where q — number of chains, n₁ — number of occupied sites, l — length of a chain. However, satisfactory agreement between experiment and theory was found only for long chains (see Fig. 1 (a)). It was suggested that the disagreement between theory and experiment in the short-chain limit may be due to the presence of the epitaxial strain that was not accounted for in one-dimensional lattice gas model. Later it was shown that if additional interactions that the model does not account for introduce a sufficiently large positive curvature of the chain energy, the size distribution in thermodynamic equilibrium can be fitted to the experimental data with high accuracy [17]. Another work claims [18] that presence of defects may change the island size distribution from monomodal to monotonically decreasing in the case of the irreversible model.

Despite continuing efforts, most of theoretical descriptions describe only growth of one-dimensional islands which occurs during the deposition process [13], [14], [16], [17], [18], [19], [20], [21]. Of course, to create a realistic theoretical model, it is necessary to take into account all interactions and the presence of defects, as well as the phenomena of decay and coarsening. All these factors make the growth process extremely intricate, therefore theoretical description is a very important and nontrivial problem even in the one-dimensional case. In addition, it is incredible that none of the works took into account one important detail of the experiment. On the one hand, the distribution of lengths of one-dimensional structures is obtained by analyzing STM images at low temperatures. On the other hand, theoretical models use high temperature, at which one-dimensional structures were formed. Nobody before us paid attention to this simple and seemingly obvious fact.

Here we perform large-scale kinetic Monte Carlo simulations to determine equilibrium distribution of one-dimensional island sizes. We simulate the growth of one-dimensional islands and their further evolution. We focus on Ag chains on Pt(997) because of the availability of experimental data [14] that can be used for comparison. Strikingly, the average length of one-dimensional islands varies significantly with time. Our findings shed light on the understanding of the difference between non-equilibrium and equilibrium distribution of one-dimensional island sizes, and propose a new approach for analyzing experimental data.