Simplified modeling of 13.5 nm unresolved transition array emission of a Sn plasma and comparison with experiment

Abstract

One key aspect in the drive to optimize the radiative output of a laser-produced plasma for extreme ultraviolet lithography is the radiation transport through the plasma. In tin-based plasmas, the radiation in the 2% bandwidth at 13.5 nm is predominantly due to 4d-4f and 4p-4d transitions from a range of tin ions (Sn7+ to Sn12+). The complexity of the configurations involved in these transitions is such that a line-by-line analysis is, computationally, extremely intensive. This work seeks to model the emission profiles of each ion by treating the transition arrays statistically, thus greatly simplifying radiation transport modeling. The results of the model are compared with experimental spectra from tin-based laser-produced plasmas.

Simplified one-dimensional calculation of 13.5 nm emission in a tin plasma including radiation transport

“…Furthermore, the wavelength at peak emission decreases as power density increases, as seen both experimentally 28 and in the steady-state analysis. 13 A calculated spectrum for the power density of maximum CE (0.8 x 10 11 W/cm 2 ) and corresponding experimental spectrum from Ref. 28 is shown in Fig.…”

“…11 Because of the large number of transitions, statistical methods were used to calculate the UTA mean and width, 12,18 which were then weighted by ion distribution within the LPP 14 to estimate the steady-state spectral profile. 13 For each ion, an UTA is characterized by position and width, 11,12 where the mean (µ 1 ) and noncentred variance (µ 2 ) are given by Eq. 1 and the area is equal to the summed weighted oscillator strength (Σgf).…”

“…In a previous paper 13 we calculated UTA statistics for the main contributing 13.5-nm tin ions in the optically thin regime, which showed a maximum in-band (13.5 nm ± 1%) emission from Sn 8+ -Sn 12+ for a plasma electron temperature of 40 eV. Weighted dipole oscillator strengths versus wavelength were calculated from a Hartree-Fock configuration interaction (HFCI) code 11 and weighted by fractional ion density using a steady-state plasma model.…”

“…Weighted dipole oscillator strengths versus wavelength were calculated from a Hartree-Fock configuration interaction (HFCI) code 11 and weighted by fractional ion density using a steady-state plasma model. 14 In another paper, 15 using the statistical representation of the ion mean energies, 13 and an energy functional method to interpolate between the l-degenerate population levels. 16 In this paper, the statistically simplified UTA parameters and the nl-level populations are used in radiation transport calculations to determine the resultant emission in an optically thick plasma, with significant computational savings.…”

“…[8][9][10] The complexity of the configurations is such that a line-by-line analysis is computationally extremely intensive and statistical methods can be used to reduce its complexity for computational purposes, [11][12][13] where the UTA is represented as a Gaussian profile characterised by three parameters: position (µ 1 ), width (σ), and summed weighted oscillator strength (Σgf).…”