Modelling of the X,Y,Z positioning errors and uncertainty evaluation for the LNE's mAFM using the Monte Carlo method

In order to evaluate the uncertainty budget of the LNE's mAFM, a reference instrument dedicated to the calibration of nanoscale dimensional standards, a numerical model has been developed to evaluate the measurement uncertainty of the metrology loop involved in the XYZ positioning of the tip relative to the sample. The objective of this model is to overcome difficulties experienced when trying to evaluate some uncertainty components which cannot be experimentally determined and more specifically, the one linked to the geometry of the metrology loop. The model is based on object-oriented programming and developed under Matlab. It integrates one hundred parameters that allow the control of the geometry of the metrology loop without using analytical formulae. The created objects, mainly the reference and the mobile prism and their mirrors, the interferometers and their laser beams, can be moved and deformed freely to take into account several error sources. The Monte Carlo method is then used to determine the positioning uncertainty of the instrument by randomly drawing the parameters according to their associated tolerances and their probability density functions (PDFs). The whole process follows Supplement 2 to 'The Guide to the Expression of the Uncertainty in Measurement' (GUM). Some advanced statistical tools like Morris design and Sobol indices are also used to provide a sensitivity analysis by identifying the most influential parameters and quantifying their contribution to the XYZ positioning uncertainty. The approach validated in the paper shows that the actual positioning uncertainty is about 6 nm. As the final objective is to reach 1 nm, we engage in a discussion to estimate the most effective way to reduce the uncertainty.