Modeling of protection schemes for critical surfaces under low pressure conditions: Comparison between analytical and numerical approach

Nanoparticle contamination is one of the crucial issues for the semiconductor industry on the move towards structure sizes of $50nm$ and below. In extreme ultraviolet lithography (EUVL), a likely successor to optical lithography, the masks cannot be protected by common pellicles. Different protection methods, such as a “thermophoretic pellicle” [L. Klebanoff and D. J. Rader, US Patent No. 6,153,044 (2000) and US Patent No. 6,253,464 B1 (2001)] have therefore been proposed to protect a face-down mask in an EUV scanner, which might be operated at $50mTorr$ $(6.7Pa)$⁠. In order to quantify the effectiveness of such protection schemes, we developed an analytical model that allows simple determination of the particle stopping distance as a function of particle and gas properties as well as a thermal gradient that might be employed to make use of a thermophoretic force in order to protect the mask. The analytical results indicate that drag force is most effective in slowing down particles, traveling at high initial velocities. Thermophoresis can add effective protection to particles traveling at low velocities and therefore decrease diffusional deposition. The results from the analytical model were used to check the accuracy of the discrete phase model in FLUENT for particle diameters between 100 and $500nm$ and pressure levels between $10mTorr$ $(1.3Pa)$ and $500mTorr$ $(66.7Pa)$ (corresponding Knudsen numbers $407≤Kn≤102000)$⁠. The comparison results indicate that with no thermal gradient, the results agree very well with less than 2% deviation. If thermophoresis is included, the absolute deviation generally increases with increasing Knudsen number and with increasing temperature gradient. For a temperature gradient of $10K∕cm$ and a Knudsen number of $102000$ $(p=10mTorr,dp=100nm)$⁠, the deviation reaches almost 50% for $2m∕s$ initial particle velocity.