When the thickness of the layer is smaller than the electrons mean free path, the morphology affects the conductivity directly based on the layer thickness. This issue provides basis in order to estimate the thickness of the layer by understanding the morphology and the value of the conductivity. This method is an inverse approach on thickness estimation and is applied to various samples. The comparison of the results with other thickness estimations shows good consistency. The benefits of this approach is that the only parameter that needs to be measured is the conductivity, which is quite trivial. Despite the simplicity of this approach, its results would prove adequate to study both the material properties and the morphology of the layer. In addition, the possibility of repeating the measurements on thickness for AC currents with various frequencies enables averaging the measurements in order to obtain the most precise results.
A general form for the surface roughness effects on the capacitance of a capacitor is proposed. We state that a capacitor with two uncoupled rough surfaces could be treated as two capacitors in series which have been divided from the mother capacitor by a slit. This is in contrast to the case where the two rough surfaces are coupled. When the rough surfaces are coupled, the type of coupling decides the modification of the capacitance in comparison to the uncoupled case. It is shown that if the coupling between the two surfaces of the capacitor is positive (negative), the capacitance is less (higher) than the case of two uncoupled rough plates. Also, we state that when the correlation length and the roughness exponent are small, the coupling effect is not negligible
The scaling behavior of friction between rough surfaces is a well-known phenomenon. It might be asked whether such a scaling feature also exists for friction at an atomic scale despite the absence of roughness on atomically flat surfaces. Indeed, other types of fluctuations, e.g., thermal and instrumental fluctuations, become appreciable at this length scale and can lead to scaling behavior of the measured atomic-scale friction. We investigate this using the lateral force exerted on the tip of an atomic force microscope (AFM) when the tip is dragged over the clean NaCl (001) surface in ultra-high vacuum at room temperature. Here the focus is on the fluctuations of the lateral force profile rather than its saw-tooth trend; we first eliminate the trend using the singular value decomposition technique and then explore the scaling behavior of the detrended data, which contains only fluctuations, using the multifractal detrended fluctuation analysis. The results demonstrate a scaling behavior for the friction data ranging from 0.2 to 2 nm with the Hurst exponent H = 0.61 ± 0.02 at a 1σ confidence interval. Moreover, the dependence of the generalized Hurst exponent, h(q), on the index variable q confirms the multifractal or multiscaling behavior of the nanofriction data. These results prove that fluctuation of nanofriction empirical data has a multifractal behavior which deviates from white noise.
Friction force at the nanoscale, as measured from the lateral deflection of the tip of an atomic force microscope, usually shows a regular stick-slip behavior superimposed by a stochastic part (fluctuations). Previous studies showed the overall fluctuations to be correlated and multi-fractal, and thus not describable simply by e.g. a white noise. In the present study, we investigate whether one can extract an equation to describe nano-friction fluctuations directly from experimental data. Analysing the raw data acquired by a silicon tip scanning the NaCl(001) surface (of lattice constant 5.6 Å) at room temperature and in ultra-high vacuum, we found that the fluctuations possess a Markovian behavior for length scales greater than 0.7 Å. Above this characteristic length, the Kramers-Moyal approach applies. However, the fourth-order KM coefficient turns out to be negligible compared to the second order coefficients, such that the KM expansion reduces to the Langevin equation. The drift and diffusion terms of the Langevin equation show linear and quadratic trends with respect to the fluctuations, respectively. The slope 0.61 ± 0.02 of the drift term, being identical to the Hurst exponent, expresses a degree of correlation among the fluctuations. Moreover, the quadratic trend in the diffusion term causes the scaling exponents to become nonlinear, which indicates multifractality in the fluctuations. These findings propose the practical way to correct the prior models that consider the fluctuations as a white noise.
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