# EUV-Induced Plasma: A Peculiar Phenomenon of a Modern Lithographic Technology
## 2 EUV-Induced Plasma
### Recent Numerical Work on EUV-Induced Bulk Plasmas  
The usual choices for modelling of EUV-induced plasmas are kinetic methods from the particle-in-cell (PIC) family [19]. The PIC method allows for accommodating an arbitrary shape of the energy distribution function against reasonable computational costs. To the authors' knowledge, the first PIC modelling of EUV-induced plasmas was performed by [PERSON], [PERSON] and [PERSON] [20]. These authors used one-dimensional (1D) particle-in-cell codes and estimated the damage to a multilayer mirror (MLM) due to physical sputtering. The comparison between numerical outputs and experiments, and validation was limited due to the 1D nature of the codes and the non-1D geometry of experimental systems.  
The next steps included the development of an EUV-induced plasma tailored two-dimensional particle-in-cell code, which several experiments validated [21; 22]. These efforts enabled modeling plasmas induced by irradiation with one single pulse of EUV radiation. It was found that an accurate set of differential cross-sections was needed for the description of the ignition of such an EUV-induced plasma, since the models' results were very sensitive to the energy distribution between the two electrons that were produced after ionization [21]. This conclusion correlates with the work of [PERSON] and [PERSON] [23], who found as well that differential cross-sections were crucial for an accurate description of ionization coefficients.  
Although single pulse modelling matches reasonably well with experiments (see Figure 3) [21], these models were not sufficiently developed to accurately describe the full situation under scanner conditions. This is because, typically, tin-based LPP sources in the field use repetition rates up to 50 kHz [7; 15]. According to these modeling efforts, the electron temperature in an EUV-induced plasma in low pressure H\({}_{2}\) gas decreases to 0.1 eV within a few \(\upmu\)s after plasma initiation. Hence, for the enclosing chamber with characteristic dimensions mentioned before, the accumulation of plasma would occur at increased repetition rates. However, it is hardly possible to describe this situation with solely a PIC approach due to the very large simulation time that is needed.  
Figure 2: Particle-in-cell simulated electron energy distribution functions (EEDFs) in an EUV-induced plasma during the EUV pulse (purple curve, at \(t=0.04\)\(\upmu\)s the EUV intensity has its maximum value), at the end of the EUV pulse (red curve) and at different moments in time after the EUV pulse (\(t=0.4\)\(\upmu\)s and \(t=1\)\(\upmu\)s are approximately 0.2 \(\upmu\)s and 0.8 \(\upmu\)s after the end of the EUV pulse, respectively). The colored dashed-lines represent Maxwellian EEDFs with a temperature corresponding to the calculated mean electron energy.  
To that end, an attempt was recently made to estimate the accumulation effects via auxiliary diffusion simulations, in which all the complex effects are summed to an effective source term that was computed by a PIC code. The effects of EUV-induced plasma ignition in pre-existing plasmas were estimated by additional PIC simulations. This approach has shown reasonable agreement with experiments in the cases where plasma accumulation is large (e.g., \(>\)100 pulses are accumulated) [24]. Nevertheless, a better solution is needed, especially when simulating accumulation over fewer pulses (e.g., \(<\)10 pulses), since the injection of electrons into the plasma can increase fluxes of plasma-created species to the walls [25].