Microscopic Origin of the Apparent Activation Energy in Diffusion-Mediated Monolayer Growth of Two-Dimensional Materials

Abstract

Current trends indicate that mass production of high-quality, two-dimensional materials will likely be based on the use of on-surface synthesis (OSS) and/or chemical vapor deposition (CVD) on selected substrates. However, the success of these techniques heavily relies on a deeper understanding of terrace and perimeter diffusion of the adspecies across and around many self-generated, mobile clusters, and/or motionless islands with a variety of shapes (dendritic, compact, irregular, polygonal, ...). We show that, for typical monolayer growth conditions at constant deposition rate, the total square distance traveled by all adsorbed particles departs from the total number of diffusion hops due to the onset of correlations between subsequent hops along the perimeters of a growing density of obstacles. As a result, we propose a new expression to determine the tracer diffusivity of the adparticles, directly providing a simple understanding of the temperature and coverage dependence of this observable. Most importantly, we describe the microscopic origin of the apparent diffusion barrier extracted from a typical Arrhenius plot at any given coverage.

On the validity of the Arrhenius picture in two-dimensional submonolayer growth

“…The natural quantity describing submonolayer growth under a constant flux of adparticles is the tracer diffusivity [27]:…”

“…where the hat symbol ( ˆ) denotes the value of a timedependent variable at time t, δ is the dimensionality (= 2 for diffusion on a surface), na is the number of adsorbed particles, R2 = Σ na i=1 |x i −x * i | 2 is the total squared distance travelled by the adparticles, with x * i denoting the position of adparticle i when it was adsorbed, and X is the ensemble average of X over K samples in the limit of large K. In paticular, the diffusivity can be re-written as [27]:…”

“…is the time average of X, and f T is the correlation factor, which accounts for memory effects between consecutive hops at finite coverages, e.g. hopping from site i to site j leaves site i empty and, thus, at finite coverage the adparticle has a higher chance of returning to i [27].…”

“…Indeed, the event probabilities of equation 7 are very useful, providing a complete picture of the undergoing competition between the different diffusion events, directly indicating which events dominate and which are essentially irrelevant. Typically, the contribution from the correlation factor is small (E f ≈ 0) [27]. Thus, equation 5 shows that the temperature dependence of the diffusivity is essentially given by that of the total diffusion rate:…”

“…The quality of the 2D material is affected markedly by the density and structure of the formed grain boundaries, directly depending on the actual size and shape of the islands (dendritic, compact, polygonal, ...), which ultimately depends on the relative occurrence of the adsorption and diffusion events. In this context, the natural quantity describing the behaviour of the system is the tracer diffusivity [27].…”