The dynamic scaling of mesoscopically thick films (up to 10 4 atomic layers) grown with the Clarke-Vvedensky model is investigated numerically for broad ranges of values of the diffusion-to-deposition ratio R and lateral neighbor detachment probability ǫ, but with no barrier at step edges. The global roughness scales with the film thickness t as W ∼ t β / R 3/2 (ǫ + a) , where β ≈ 0.2 is the growth exponent consistent with Villain-Lai-Das Sarma (VLDS) scaling and a = 0.025. This general dependence on R and ǫ is inferred from renormalization studies and shows a remarkable effect of the former but a small effect of the latter, for ǫ ≤ 0.1. For R ≥ 10 4 , very smooth surfaces are always produced. The local roughness shows apparent anomalous scaling for very low temperatures (R ≤ 10 2 ), which is a consequence of large scaling corrections to asymptotic normal scaling. The scaling variable R 3/2 (ǫ + a) also represents the temperature effects in the scaling of the correlation length and appears in the dynamic scaling relation of the local roughness, which gives dynamic exponent z ≈ 3.3 also consistent with the VLDS class.

In the literature about field emission, finite elements and finite differences techniques are being increasingly employed to understand the local field enhancement factor (FEF) via numerical simulations. In theoretical analyses, it is usual to consider the emitter as isolated, i.e, a single tip field emitter infinitely far from any physical boundary, except the substrate. However, simulation domains must be finite and the simulation boundaries influences the electrostatic potential distribution. In either finite elements or finite differences techniques, there is a systematic error ( ) in the FEF caused by the finite size of the simulation domain. It is attempting to oversize the domain to avoid any influence from the boundaries, however, the computation might become memory and time consuming, especially in full three dimensional analyses. In this work, we provide the minimum width and height of the simulation domain necessary to evaluate the FEF with at the desired tolerance. The minimum width (A) and height (B) are given relative to the height of the emitter (h), that is, (A/h)min × (B/h)min necessary to simulate isolated emitters on a substrate. We also provide the (B/h)min to simulate arrays and the (A/h)min to simulate an emitter between an anode-cathode planar capacitor. At last, we present the formulae to obtain the minimal domain size to simulate clusters of emitters with precision tol . Our formulae account for ellipsoidal emitters and hemisphere on cylindrical posts. In the latter case, where an analytical solution is not known at present, our results are expected to produce an unprecedented numerical accuracy in the corresponding local FEF.

No abstract

This work presents an accurate numerical study of the electrostatics of a system formed by individual nanostructures mounted on support substrate tips, which provides a theoretical prototype for applications in field electron emission or for the construction of tips in probe microscopy that requires high resolution. The aim is describe the conditions to produce structures mechanically robust with desirable field enhancement factor (FEF). We modeled a substrate tip with a height h1, radius r1 and characteristic FEF γ1, and a top nanostructure with a height h2, radius r2 < r1 and FEF γ2, for both hemispheres on post-like structures. The nanostructure mounted on the support substrate tip then has a characteristic FEF, γC . Defining the relative difference ηR = (γC − γ1)/(γ3 − γ1), where γ3 corresponds to the reference FEF for a hemisphere of the post structure with a radius r3 = r2 and height h3 = h1 + h2, our results show, from a numerical solution of Laplace's equation using a finite element scheme, a scaling ηR = f (u ≡ λθ −1 ), where λ ≡ h2/h1 and θ = r1/r2. Given a characteristic variable uc, for u uc, we found a power law ηR ∼ u κ , with κ ≈ 0.55. For u uc, ηR → 1, which led to conditions where γC → γ3. As a consequence of scale invariance, it is possible to derive a simple expression for γC and to predict the conditions needed to produce related systems with a desirable FEF that are robust owing to the presence of the substrate tip. Finally, we discuss the validity of Schottky's conjecture (SC) for these systems, showing that, while to obey SC is indicative of scale invariance, the opposite is not necessarily true. This result suggests that a careful analysis must be performed before attributing SC as an origin of giant FEF in experiments.Producing nanostructures that allow one to amplify the applied electric field in their vicinity and which are mechanically stable remains an engineering challenge. This can be observed already a long time ago in the pioneer work by Gomer who discuss a method for growing metal whiskers in a modified field emission tube [1]. In fact, the issue of mechanical stability requires a solution for the degradation and failure of nanostructures that occurs during field electron emission at or near the substrate emitter contact [2] and for the self-mechanical oscillations that occur during field electron emission measurements [3,4] or from electrostatic interactions [5]. In particular, a method to study the self-oscillations of a nanostructure mounted on a macroscopic frame requires using a laser beam to excite the sample; subsequently, a second laser beam is then used to register the amplitude of vibrations at a certain point from the object [6].Applications of these nanostructures mounted on tip devices include carbon nanotubes (CNTs) mounted on a support tip, which can be used as an electron source in a high-resolution electron beam. The latter acquires properties such as a stable emitted current and high brightness [7]. Moreover, due to screening effects [8], there is a tendency to...

Descobertas recentes revelam que modelos matemáticos euclidianos, de há muito estabelecidos e que procuram reproduzir a geometria da natureza,às vezes se apresentam incompletos e, em determinadas situações, inadequados. Especificamente, muitas das formas encontradas na natureza não são círculos, triângulos, esferas, icosaedros ou retângulos. Enfim, não são simples curvas, superfícies ou sólidos, conforme definidos na geometria clássica de Euclides (300 a.C), cujos teoremas possuem lugar de destaque nos textos de geometria. Neste trabalho apresenta-se uma breve e elementar, mas que busca ser consistente, discussão sobre algumas definições e aplicações relacionadasà geometria fractal, em particular fractais ideais. Caracterizaremos alguns fractais auto-similares que, por sua importância histórica ou riqueza de características, constituem exemplos ilustrativos "clássicos" de propriedades de fractais, propriedades estas que muitas vezes aparecem dispersas numa literatura mais especializada. Mostra-se, por construção, que suas medidas de comprimento,área e volume, nas dimensões euclidianas usuais, dão margem a resultados contraditórios. Estes podem ser explicados pelo fato de que tais objetos só podem ser adequadamente mensurados em espaços de dimensão fracionária. Palavras-chave: fractais, auto-similaridade, dimensão fractal.Recent discoveries reveal that mathematical models, established a long time ago and searching to reproduce the nature's geometry, sometimes result being incomplete and even inadequate in some situations. Specifically, many of the forms found in the nature are not circles, triangles, spheres, icosahedrons or rectangles. Finally, they are not simple curves, surfaces or solids, as defined in the classical geometry of Euclides (300 b.C), whose theorems possesses a prominent place in the geometry texts. In this work a brief and elementary, although intended to be consistent, discussion about some definitions and applications related to the fractal geometry is presented. It is also presented properties of some fractals that, for its historical importance or wealth of characteristics, constitute "classical" illustrative examples of the fractals properties which, despite this, many times appear dispersed in the specialized literature. It is shown, by construction, that the measures of length, area and volume for these objects, within the usual Euclidean dimensions, lead to contradictory results. This can be explained by considering that these objects can be adequately measured using spaces of fractional dimensions.

This work investigates the scaled height distribution, ρ(q), of irregular profiles that are grown based on two sets of local rules: those of the restricted solid on solid (RSOS) and ballistic deposition (BD) models. At each time step, these rules are respectively chosen with probability p and r=1-p. Large-scale Monte Carlo simulations indicate that the system behaves differently in three succeeding intervals of values of p: I(B) ≈ [0,0.75),I(T) ≈ (0.75,0.9), and I(R) ≈ (0.9,1.0]. In I(B), the ballistic character prevails: the growth velocity υ(∞) decreases with p in a linear way, and similar behavior is found for Γ(∞) (p), the amplitude of the t(1/3)-fluctuations, which is measured from the second-order height cumulant. The distribution of scaled height fluctuations follows the Gaussian orthogonal ensemble (GOE) Tracy-Widom (TW) distribution with resolution roughly close to 10(-4). The skewness and kurtosis of the computed distribution coincide with those for TW distribution. Similar results are observed in the interval I(R), with prevalent RSOS features. In this case, the skewness become negative. In the transition interval I(T), the system goes smoothly from one regime to the other: the height distribution becomes apparently Gaussian, which motivates us to identify this phenomenon as a transition from Kardar-Parisi-Zhang (KPZ) behavior to Edwards-Wilkinson (EW) behavior back to KPZ behavior.

This work considers the timescales associated with the global order parameter and the interlayer synchronization of coupled Kuramoto oscillators on multiplexes. For the two-layer multiplexes with initially high degree of synchronization in each layer, the difference between the average phases in each layer is analyzed from two different perspectives: the spectral analysis and the non-linear Kuramoto model. Both viewpoints confirm that the prior timescales are inversely proportional to the interlayer coupling strength. Thus, increasing the interlayer coupling always shortens the transient regimes of both the global order parameter and the interlayer synchronization. Surprisingly, the analytical results show that the convergence of the global order parameter is faster than the interlayer synchronization, and the latter is generally faster than the global synchronization of the multiplex. The formalism also outlines the effects of frequencies on the difference between the average phases of each layer, and identifies the conditions for an oscillatory behavior. Computer simulations are in fairly good agreement with the analytical findings and reveal that the timescale of the global order parameter is at least half times smaller than timescale of the multiplex.

We present an optimal detrended fluctuation analysis (DFA) and applied it to evaluate the local roughness exponent in non-equilibrium surface growth models with mounded morphology. Our method consists in analyzing the height fluctuations computing the shortest distance of each point of the profile to a detrending curved that fits the surface within the investigated interval. We compare the optimal DFA (ODFA) with both the standard DFA and nondetrended analysis. We validate the ODFA method considering a one-dimensional model in the Kardar-Parisi-Zhang universality class starting from a mounded initial condition. We applied the methods to the Clarke-Vvdensky (CV) model in 2 + 1 dimensions with thermally activated surface diffusion and absence of step barriers. It is expected that this model belongs to the nonlinear Molecular Beam Epitaxy (nMBE) universality class. However, an explicit observation of the roughness exponent in agreement with the nMBE class was still missing. The effective roughness exponent obtained with ODFA agrees with the value expected for nMBE class whereas using the other methods it does not. We also characterized the transient anomalous scaling of the CV model and obtained that the corresponding exponent is in agreement with the value reported for other nMBE models with weaker corrections to the scaling.