PHYSICAL REVIEW E 97, 052802 (2018) Island size distribution with hindered aggregation Diego Luis González, 1 , * Manuel Camargo, 2 , † and Julián A. Sánchez 1 , ‡ 1 Departamento de Física, Universidad del Valle, A.A. 25360, Cali, Colombia 2 CICBA, Universidad Antonio Nariño–Campus Farallones, Km 18 vía Cali-Jamundí, Cali, Colombia (Received 9 February 2018; published 10 May 2018) We study the effect of hindered aggregation on the island formation processes for a one-dimensional model of epitaxial growth with arbitrary nucleus size i . In the proposed model, the attachment of monomers to islands is hindered by an aggregation barrier, ǫ a , which decreases the hopping rate of monomers to the islands. As ǫ a increases, the system exhibits a crossover between two different regimes; namely, from diffusion-limited aggregation to attachment-limited aggregation. The island size distribution, P (s ), is calculated for different values of ǫ a by a self-consistent approach involving the nucleation and aggregation capture kernels. The results given by the analytical model are compared with those from kinetic Monte Carlo simulations, finding a close agreement between both sets of data for all considered values of i and ǫ a . As the aggregation barrier increases, the spatial effect of fluctuations on the density of monomers can be neglected and P (s ) smoothly approximates to the limit distribution P (s ) = δ s,i +1 . In the crossover regime the system features a complex and rich behavior, which can be explained in terms of the characteristic timescales of different microscopic processes. DOI: 10.1103/PhysRevE.97.052802 I. INTRODUCTION Epitaxial growth (EG) has long been a subject of study due to both its academic and industrial importance. From an academic point of view, this out-of-equilibrium process is interesting as it displays a rich and complex behavior arising from the several timescales involved [1–14]. On the other hand, an understanding of the microscopic mechanisms affecting the growth process is a requirement to achieve an accurate description of material properties in industrial applications. A typical example of the latter is the use of atomic chains in nanoscale devices, life sciences, and fuel cells [15–17] which can be formed, for example, by using stepped surfaces [18,19] or by anisotropic diffusion on two-dimensional substrates [20–23]. In general terms, the microscopic mechanisms of EG in- volve three basic processes: nucleation, aggregation, and trans- port of basic growth units, usually referred to as monomers, which may be atoms, molecules, or colloidal particles. Dur- ing EG, monomers are deposited onto a flat substrate or a stepped surface at a constant deposition rate, F . The latter is well controlled in experimental setups and therefore can be considered as a known parameter in theoretical models. The time evolution of the deposition process is normally described in terms of the coverage θ , which is defined as the number of monomers per lattice site on the substrate at time t . If evaporation of monomers from substrate is negligible, then θ ≈ Ft . After its deposition, a monomer diffuses on the substrate with (lateral) diffusion constant D until they nucleate or aggregate. Nucleation occurs when a number of monomers * diego.luis.gonzalez@correounivalle.edu.co † manuel.camargo@uan.edu.co ‡ julian.a.sanchez@correounivalle.edu.co form an island, i.e., a stable cluster, and the aggregation process takes place when a monomer attaches to a previously nucleated island. A paramount concept in standard models of epitaxial growth is that of the critical nucleus size i , which is defined as the size of the largest unstable cluster, i.e., clusters with size larger than i are static and stable. Consequently clusters with size smaller than i + 1 are considered unstable and the monomers belonging to such clusters can diffuse away with diffusion constant D. Therefore, each monomer forming an unstable cluster behaves as a free monomer. In most EG models, nucleation and aggregation are instantaneous processes, i.e., monomers are incorporated to the clusters once they reach the interaction range; in such a case, the aggregation belongs to the diffusion-limited-aggregation (DLA) regime. Nevertheless, in more realistic situations nucleation and/or aggregation could be hindered by additional energy barriers which would increase the time required for each reaction. For instance, experiments on nucleation and growth of Ge islands on a Pb overlayer covering a Si(111) surface suggest that such a barrier could appear due to strain [24–27]. Also, nucleation hindered by attachment barriers has been observed in Fe deposition on graphene [28] and in metal (111) homoepitaxial systems [29,30]. Similarly, attachment barriers must be considered to properly explain the formation of graphene sheets on metal [31–34] and oxide [35] substrates. In the former case, individual graphene islands spread at a constant rate, suggesting that their growth is controlled by the attachment rate of carbon adatoms to the island edges. Motivated by previous theoretical [2,3,12,14,36–46] and experimental [21,22,47–51] studies, in this work we propose a one-dimensional model in which the aggregation of monomers is hindered by an additional attachment barrier ǫ a . As explained in the next sections, this barrier decreases the hopping rate of monomers to islands. Thus, for large barriers the monomers 2470-0045/2018/97(5)/052802(11) 052802-1 ©2018 American Physical Society