A semi-empirical approach for predicting LER statistical distribution in advanced nodes

Given that the energy of photons in Extreme Ultraviolet (EUV) is significantly higher than in Deep Ultraviolet (DUV), EUV photoresists undergo exposure via photo-ionization. In this process, high-energy photons are absorbed by the photoresist, which ionizes the polymer and subsequently generates photoelectrons. These photoelectrons produce more secondary electrons through scattering. Furthermore, compared to DUV, the number of incident photons in EUV is fewer for the same exposure dose, leading to more significant stochastic effects, such as Line Edge Roughness (LER). Therefore, modeling these stochastic effects in EUV is a noteworthy issue. Due to the complexity of the secondary electron scattering, it is extremely challenging to establish a strict EUV stochastic model from first principles. The current approach to simulating the stochastic behavior in EUV photoresist is primarily conducted through Monte Carlo dynamics-based stochastic methods, which have been proven to be an accurate method. However, these approaches necessitate conducting repeated simulations to obtain the statistical distribution of LER, which can be time-consuming. Using a semi-empirical approach to equivalently process secondary electron scattering is a worthwhile method. In this paper, we employ the Gaussian function to equivalently process the energy distribution after the scattering of secondary electrons. Subsequently, based on the PEB model that incorporates physical mechanisms, the statistical distribution of LER is derived through the transmission of variance. The model is calibrated using the statistical results of the Monte Carlo dynamics-based stochastic methods. The result shows that this approach can quickly predict the statistical distribution of LER in EUV with high accuracy.