Optimal detrended fluctuation analysis as a tool for the determination of the roughness exponent of the mounded surfaces

Abstract

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.

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension

“…Using the range 50 r 170, which corresponds to the plateau shown in Fig. 3(b), the exponent obtained with ODFA 2 is α (CV) loc = 0.96(1), in consonance with the normal scaling of the CV model observed in two dimensions [11,34]. Figure 2(c) shows the local interface roughness for the DT model without noise reduction, in which noticeable differences can be seen in ODFA 2 and DFA 2 methods.…”

“…An evaluation of α loc for CV model in two-dimensions, in agreement with the nMBE class, was recently reported [34]. The effective roughness exponent was calculated with the optimal detrended fluctuation analysis (ODFA), while the exponents obtained with other methods did not match with those of the nMBE class [34]. This result provided support for non-anomalous asymp-totic scaling in CV model, corroborating the claim that this transient effect is a consequence of large corrections to the asymptotic normal scaling [11].…”

“…Differences between the exponents yielded by DFA and ODFA methods were reported [34] in the kinetic roughening obtained for the deposition on initially mounded substrates. The second order ODFA 2 method allows to capture the expected universality class of the fluctuations at scales shorter than the average mound length, whereas DFA 2 underestimates the exponents [34]. In both cases, the main advantage of the second order methods with respect to the first order ones are more extended regions of scaling, represented by longer plateaus in the effective roughness exponent,…”

“…In the CV model, deposition occurs at a constant and uniform rate while the adatom diffusion rate is given by an Arrhenius law in the form D = ν 0 exp(−E/k B T ) where ν 0 is an attempt frequency, k B the Boltzmann constant, and E is an energy barrier for the hopping, which includes the contribution of the substrate (E S ) and lateral bonds (E N ) assuming the form E = E S + nE N . The ratio R = D 0 /F , in which D = D 0 ε n is the hopping rate if an adatom has n lateral neighbors, is a control parameter of the model [11,34]. In this work we use R = 10 and ε = 0.01, which leads to a large surface roughness, since it corresponds to a low temperature (low mobility) regime.…”

“…An evaluation of α loc for CV model in two-dimensions, in agreement with the nMBE class, was recently reported [34]. The effective roughness exponent was calculated with the optimal detrended fluctuation analysis (ODFA), while the exponents obtained with other methods did not match with those of the nMBE class [34].…”