Stochastic Analysis and Regeneration of Rough Surfaces

Abstract

We investigate Markov property of rough surfaces. Using stochastic analysis we characterize the complexity of the surface roughness by means of a Fokker-Planck or Langevin equation. The obtained Langevin equation enables us to regenerate surfaces with similar statistical properties compared with the observed morphology by atomic force microscopy.Studying the growth, formation and morphology of interfaces has been one of the recent interesting fields of study because of its high technical and rich theoretical advantages [1]. One of the main problems in this area is the scaling behaviour of the moments of height difference ∆h = h(x 1 ) − h(x 2 ) and the evolution of the probability density function (PDF) of ∆h, i.e. P (∆h, ∆x) in terms of the length scale ∆x. Recently Friedrich and Peinke have been able to obtain a Fokker-Planck equation describing the evolution of the probability distribution function in terms of the length scale, by analyzing some stochastic phenomena, such as turbulent free jet, etc. [2][3][4]. They noticed that the conditional probability density of field increments (velocity field, etc.) satisfies the ChapmanKolmogorov equation. Mathematically this is a necessary condition for the fluctuating data to be a Markovian process in the length scales [5].In this letter using the method proposed by Friedrich and Peinke, we measure the Kramers-Moyal's (KM) coefficients for the fluctuating fields ∆h and h(x) of a deposited copper film. It is shown that the first and second KM's coefficients have well-defined values, while the third and fourth order coefficients tend to zero. Therefore, by addressing the implications dictated by the theorem [5] a Fokker-Planck evolution operator has been found. The Fokker-Planck equation for P (∆h, ∆x) is used to give information on changing the shape of PDF as a function of the length scale ∆x. By using this strategy the information of the observed intermittency of the height fluctuation is verified [6]. The first and second KM's coefficients for the fluctuations of h(x), enables us to write a Langevin equation for the evolution of height with respect to x. Using this equation we regenerate the surface with similar statistical properties, compared with the observed morphology by atomic force microscopy. The regeneration of a surface is known as the inverse method. There are other inverse method approaches introduced in the literature [13]. In the previous attempts, to regenerate the surface, an evolution equation for h(x, t) vs t has been evaluated. Here we do this by an evolution equation for h(x) vs x, for a certain time.For this purpose, a copper film was deposited on a polished Si(100) substrate by the resistive evaporation method in a high vacuum chamber. The pressure during evaporation was 10 −6 Torr. The thickness of the growing films was measured in situ by a quartz crystal thickness monitor. We performed all depositions at room temperature, with a deposition rate about 20 − 30nm/min. The substrate temperature was determined using a chromel/alumel thermocouple ...

Multiscale reconstruction of time series

“…Successful attempts have been made to describe these systems as processes in scale rather than time or space itself. Examples are the description of roughness of surfaces [1,2], turbulence [3,4,5], earthquakes [6] and finance [7,8]. These successful attempts are characterised by the fact, that they can correctly provide the joint probability density function p(y 1 (τ 1 ), ..., y n (τ n )) of the increments y(τ ) of the process variable x at different scales τ , y(t, τ ) := x(t) − x(t − τ ).…”

On the characterization of nanoporous materials by means of empirical and intrinsic transport coefficients

“…Illustration of the model: free diffusion within the channel and channel walls in y-and z-directions (reflective boundaries). Table 1 Applications of intrinsic transport coefficients Turbulence [3] Human tremor [7] Engineering [4] Surface sciences [8] Economics [5] Traffic [9] Time-delayed systems [6] Porous materials [2] Reaction-diffusion systems [10] that the definition of diffusion tensors by means of BD theory is not affected by the presence or absence of external or internal forces. More precisely, the evolution of the MSD components will of course depend for large times on the detailed forces acting on and between the fluid particles in a nanoporous material.…”

Etched glass surfaces, atomic force microscopy and stochastic analysis

“…We have shown that in the saturate state the structure of topography has fractal feature with fractal dimension D f = 1.30. In addition, Langevin characterization of the etched surfaces enable us to regenerate the rough surfaces grown at the different etching time, with the same statistical properties in the considered scales 15 .…”

“…P (h, x) in terms of the length scale x. Recently some authors have been able to obtain a Fokker-Planck equation describing the evolution of the probability distribution function in terms of the length scale, by analyzing some stochastic phenomena, such as rough surfaces 15,16,17 , turbulent system 18 , financial data 19 , cosmic background radiation 20 and heart interbeats 21 etc. They noticed that the conditional probability density of field increment satisfies the Chapman-Kolmogorov equation.…”