Stochastic Analysis and Regeneration of Rough Surfaces

We investigate the 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.

Figure 1

Test of Chapman-Kolmogorov equation for different values $\Delta h_1=-21\mathrm{n}\mathrm{m}$, $\Delta h_1=0$, and $\Delta h_1=21\mathrm{n}\mathrm{m}$. The bold and open symbols represent directly evaluated PDF and the integrated PDF, respectively. The length scales $\Delta x_1$, $\Delta x_2$, and $\Delta x_3$ are 180, 320, and 260 nm, respectively.

Figure 2

Drift and diffusion coefficients $D^1(\Delta h)$ and $D^2(\Delta h)$ are estimated from Eq. (3). The $D^1$ and $D^2$ present the linear and quadratic behavior, respectively.

Figure 3

Probability densities of the height difference $\Delta h=h(x+\Delta x)-h(x)$ for the length scales $\Delta x=200$, 300, and 500 nm (from top to bottom). The results obtains from the data analysis of the AFM image ($R$) and numerical integration of an effective Fokker-Planck equation ($E$), i.e., Eq. (2), respectively. The PDFs are shifted in vertical directions for convenience of presentation and $\Delta h$ are measured in units of the standard deviation of $\Delta h$ at $\Delta x=200\mathrm{n}\mathrm{m}$.

Figure 4

AFM and regenerated surface images (from top to bottom), where we have regenerated the rough surface using the Langevin equation for dynamics of $h(x)$, as drift term $D^{(1)}(h)=-0.01h$ and as diffusion term $D^{(2)}(h)=0.088–0.004h+5.19\times {10}^{-5}{h}^2$.

Figure 5

Log-log plot of the second moment of height difference vs $l$, for real sample and regenerated sample. The roughness exponents for real and regenerated are $0.83\pm 0.03$ and $0.83\pm 0.01$, respectively.