Application of the Π theorem of Dimensional Analysis to electron scattering in multi-component systems

Monte Carlo simulation of electron scattering in solids is considered and an extremely simplified picture of the process is demonstrated by a semi-empirical application of the Π theorem of dimensional analysis. The case of electron beam lithography is presented as an example. A set of parameters fully characterising the process is derived, and their dependence is investigated on electron energy, substrate thickness and the material physical constants, for a great number of single- and multi-component substrates. A single generalised argument is shown to be able to effectively describe the properties of forward and backward scattering of any substrate, in the elastic regime. The physical laws governing the process are approximated through analytical functions of this argument. These functions are theoretically deduced from very simple models and empirically demonstrated by fitting the numerical data. The efficacy of the proposed method, applicable to all elemental and composite substrate materials, is shown in the 5 to 150 keV energy range, with substrate thickness varying from 0.002 to 8 μm. The leading role of diamond as substrate material, emerging from such an analysis, is finally emphasised.