Massively parallel simulations of Brownian dynamics particle transport in low pressure parallel-plate reactors

Abstract

An understanding of particle transport is necessary to reduce contamination of semiconductor wafers during low-pressure processing. The trajectories of particles in these reactors are determined by external forces (the most important being neutral fluid drag, thermophoresis, electrostatic, viscous ion drag, and gravitational), by Brownian motion (due to neutral and charged gas molecule collisions), and by particle inertia. Gas velocity and temperature fields are also needed for particle transport calculations, but conventional continuum fluid approximations break down at low pressures when the gas mean free path becomes comparable to chamber dimensions. Thus, in this work we use a massively parallel direct simulation Monte Carlo method to calculate low-pressure internal gas flow fields which show temperature jump and velocity slip at the reactor boundaries. Because particle residence times can be short compared to particle response times in these low-pressure systems (for which continuum diffusion theory fails), we solve the Langevin equation using a numerical Lagrangian particle tracking model which includes a fluctuating Brownian force. Because of the need for large numbers of particle trajectories to ensure statistical accuracy, the particle tracking model is also implemented on a massively parallel computer. The particle transport model is validated by comparison to the Ornstein–Furth theoretical result for the mean square displacement of a cloud of particles. For long times, the particles tend toward a Maxwellian spatial distribution, while at short times, particle spread is controlled by their initial (Maxwellian) velocity distribution. Several simulations using these techniques are presented for particle transport and deposition in a low pressure, parallel-plate reactor geometry. The corresponding particle collection efficiencies on a wafer for different particle sizes, gas temperature gradients, and gas pressures are evaluated.