Roughness characteristic length scales of micro-machined surfaces: A multi-scale modelling

The MEMS structure integrity, their dynamic properties as well as their electrostatic characteristics, strongly depend on the achieved surfaces roughness produced by the micromachining process. It is therefore, not surprising that numerous works are devoted to propose relations between roughness and physical or mechanical properties in this field. Yet the issue is full of complexities since roughness parameters depend on the method used for their evaluation. This article introduces a new approach of the roughness characterization, based on the scaling analysis. Experimental results obtained on micro machined surfaces show that the range roughness amplitude depends on the scan length and that roughness amplitude follows three stages. The stage I is due to a smoothing effect of the surface induced by the tip radius of the profilometer, stage II presents a piecewise power-law roughness distribution until a critical length that characterises the fractal behaviour of the surface, and stage III is characterised by extreme values statistics. The fractal parameter, the extreme values estimators and the crossover between stages II and III are shown to be related to the micromachining process. As a result, an original probabilistic model based on the Generalized Lambda Distribution (GLD) is proposed to estimate the multi-scale roughness in the stage III. Finally, thanks to a Bootstrap protocol coupled with a Monte-Carlo simulation, the maximal roughness amplitude probability density function is estimated at a scale higher than the scanning length