Modeling the Decay of Nanopatterns:  A Comparative Study between a Continuum Description and a Discrete Monte Carlo Approach

Abstract

The surface-diffusion-driven decay of 2-dimensional periodic nanopatterns is studied by means of a kinetic Monte Carlo model. Activation energy barriers are computed by using a harmonic approach. Accounting for the proper rates of all processes involved allows us to implement a real time algorithm. Thus, the relationship between real time and Monte Carlo time is discussed. By using this discrete approach, we recover the most relevant results expected from the linear theory of surface diffusion for initial surfaces within the small-slope approximation. We also focus on the decay kinetics of periodic nanopatterns (sinusoidal and rectangular) starting from nonequilibrium states far from the small-slope approximation. In this case, we found a nonexponential decay as well as the emergence of several interesting nanostructures (overhangs, nanoislands, nanovoids, etc.), which depend on the geometrical properties (e.g., the aspect ratio) of the initial pattern. We compare results obtained by using the proposed discrete model with the expectations of the standard continuous theory of surface diffusion, and we show that many qualitative aspects are common to both approaches, in spite of the important differences between both descriptions.

Generalized Fourier analysis for nanopatterns with overhangs

“…As was shown in previous works [12][13][14], surface diffusion currents lead HAR nanostructures to spontaneously develop overhangs and complex surfaces that cannot be described by a single-valued function. Nevertheless, even in those situations where both GFS and SFS are suitable, the GFS expansion usually is the best option to describe a given pattern.…”

“…during its evolution) and, in addition, a non-exponential amplitude decrease takes place. In fact, in a recent paper, it was shown [13] that, when ∼ 1, the actual shape during the pattern decay is very accurately described by a vectorial function that we shall call r(p, t), whose parametric expression is given by…”

“…The expression in (4) was introduced [13] as an Ansatz to give an accurate mathematical description to the shapes found during decay processes, however, up to now, no deep discussion about the root of such expression was done. In particular, if we consider the expression in (4) we may ask ourselves: Why does a linear term appear in the parameter p in the first component?…”

Surface Relaxation of High-Aspect-Ratio Nanostructures: Theory and Experiments

“…Throughout this paper, we have chosen a simple model for the activation energies, in which a harmonic dependence of the energy as a function of the reaction degree is assumed, and the activation energies are simply obtained by considering the crossing of two of such harmonic potentials displaced in one lattice-constant unit. A more detailed description of such model can be found in references, ,, and here we shall restrict ourselves to give the final expression for the activation energy, namely E ini → fin act = β 2 ( E fin − E ini β + 1 2 ) 2 where E ini and E fin are the energies of the initial and final states, respectively. Also we called β = k a 2 , where k is the force constant of the harmonic wells and a is the lattice constant.…”

Estudio teórico y computacional de procesos de difusión superficial sobre micro/nano-estructuras de alta razón de aspecto

“…Como contraparte, en losúltimos años se ha comenzado a investigar sobre este tipo de sistemas desde una perspectiva teórica, procurando tanto describir matemáticamente los aspectos geométricos estáticos de las morfologías obtenidas como resultado de las transformaciones, como entender los aspectos cinéticos del proceso de transformación [16,36,37,38,39].…”