Characterization of the anisotropy of rough surfaces: Crossing statistics

In this paper, we propose the use of crossing statistics and its generalizations as a new framework to characterize the anisotropy of a 2D rough surface. The proposed method is expandable to higher dimensions. By measuring the number of up-crossing, ν⁺ [crossing points with a positive slope at a given threshold of height (α)], and the generalized roughness function, N_tot, it is possible to characterize the nature of an anisotropy, rotational invariance, and Gaussianity of any given surface. In the case of anisotropic correlated self- or multi-affine surfaces, even with different correlation lengths in different directions and/or directional scaling exponents, we examine the relationship between ν⁺ and N_tot, and corresponding scaling parameters analytically. The method identifies the direction of anisotropy through the systematic use of P-value statistics. After applying the common methods in determining the corresponding scaling exponents in the identified anisotropic directions, we are able to determine the type and the ratio of the involved correlation lengths. To demonstrate capability and accuracy of the method, as well as to validate the analytical calculations, we apply the proposed measures on synthetic stochastic rough interfaces and rough interfaces generated from the simulation of ion etching. There is a good agreement between analytical results and the outcomes of the numerical models. The proposed algorithm can be implemented through a simple software in various instruments, such as AFM and STM, for surface analysis and characterization.