Measurements, simulations, and models of the point-spread function of electron-beam lithography

When a sample is exposed using electron-beam lithography, the electrons scatter deep and far in the substrate, resulting in unwanted deposition of dose at both the nano- and the microscale. This proximity effect can be mitigated by proximity effect correction provided that accurate and validated models of the point-spread function of the electron scattering are available. Most works so far considered a double-Gaussian model of the electron point-spread function, which is very inaccurate for modern electron-beam writers with high acceleration voltages. We present measurements of the process point-spread function for chemically semi-amplified resist on silicon and indium phosphide substrates using a 150 kV electron-beam lithography system. We find that the double-Gaussian model deviates from experiments by up to four orders of magnitude. We propose instead a model comprising the sum of a power-law and a Gaussian, which is in excellent agreement with simulations of the electron scattering obtained by a Monte Carlo method. We apply the power-law plus Gaussian model to quantify the electron scattering and proximity effect correction parameters across material stacks, processing, and voltages from 5 kV to 150 kV. We find that the power-law term remains remarkably constant, whereas the long-range dose contributions and the clearing dose are significantly affected by the substrate and the acceleration voltage.