Monte Carlo modelling of electron beam lithography: a scaling law

The Π theorem of Dimensional Analysis, usually applied to the inference of physical laws, is for the first time applied to the derivation of interpolation curves of numerical data, leading to a simplified dependence on a reduced number of arguments Π, dimensionless combination of variables. In particular, Monte Carlo modelling of electron beam lithography is considered and the backscattering coefficient η addressed, in case of a general substrate layer, in the elastic regime and in the energy range 5 to 100 keV. The many variables involved (electron energy, substrate physical constants and thickness) are demonstrated to ultimately enter in determining η through a single dimensionless parameter Π0. Thus, a scaling law is determined, an important guide in microsystem designing, indicating, if any part of the configuration is modified, how the other parameters should change (or "scale") without affecting the result. Finally, a simple law η=83 Π0 is shown to account for all variations of the parameters over all substrates of the periodic table.