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

We present an optimal detrended fluctuation analysis (DFA) and apply it to evaluate the local roughness exponent in nonequilibrium 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 curve 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-Vvedensky (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 the nMBE class, whereas using the other methods it does not agree. We also characterize 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.

Figure 1

Schematic representation of ${\mathrm{DFA}}_1$ and ${\mathrm{ODFA}}_1$ methods. The dashed line represents the linear regression used to detrend the interface, and the solid line is parallel to the substrate.

Figure 2

Distributions of the local slope for different window sizes in a surface obtained with the CV model with $L=1024, t=100$, using parameters $R=10$ and $\epsilon ={10}^{-2}$.

Figure 3

(a) Mounded initial condition and the (shifted) profile after a deposition time $t={10}^3$ using the etching model. The corresponding correlation functions are shown in the inset. (b) Effective local roughness exponent as a function of the window size for a deposition time $t={10}^3$ using different methods. The horizontal dashed line indicates the value of the KPZ roughness exponent ${\alpha }_{{}_{\mathrm{KPZ}}}=1/2$.

Figure 4

(a) Typical morphology of a surface generated with the CV model, at deposition time $t={10}^6$, showing the formation of mounds. (b) Scaled autocorrelation function $\mathrm{\Gamma }(r,t)/\mathrm{\Gamma }\left(0\right)$ vs distance $r$. The inset shows the time evolution of the characteristic length given by the minimum of the autocorrelation function. The dashed line has a slope 0.30.

Figure 5

(a) Local roughness using different methods as a function of the window size for different times. (b) Effective local roughness exponent $[{\alpha }_{\mathrm{eff}}\equiv dln\left({\omega }^{\left(n\right)}\right)/dln\left(l\right)]$ analysis for different methods at a deposition time $t={10}^6$. The horizontal dashed line indicates the value of the nMBE roughness exponent ${\alpha }_{{}_{\mathrm{nMBE}}}=2/3$. The inset shows the time evolution of the global roughness obtained with ${\mathrm{ODFA}}_1$, and the dashed line has a slope 0.20.

Figure 6

Transient anomalous scaling analysis for the CV model using different methods: ${\mathrm{DFA}}_0, {\mathrm{DFA}}_1$, and ${\mathrm{ODFA}}_1$ with $y=0.09$, 0.12, and 0.23, respectively. The window size used to compute the local roughness is $l=5$. The inset represents a zoom of the vertical scale improving the visibility of the ${\mathrm{ODFA}}_1$ curve.