Showerhead-Enhanced Inertial Particle Deposition in Parallel-Plate Reactors

Particle deposition is modeled in a parallel-plate reactor geometry characteristic of a wide range of semiconductor process tools: uniform downward flow exiting a perforated-plate showerhead separated by a small gap from a circular wafer resting on a parallel susceptor. Because the area available to flow is constricted inside the showerhead, high gas velocities within showerhead holes can accelerate particles and lead to inertia-enhanced particle deposition on the wafer below. Numerical models are presented for determining the extent of inertia-enhanced deposition in a parallel-plate geometry for particles originating upstream of the showerhead. The problem is treated in two steps, for which particle and fluid transport are determined: I) within a showerhead-hole, and 2) between the showerhead and susceptor. Analytic expressions are presented for the gas velocity in the showerhead holes. Particle transport analysis leads to an analytic expression for the dimensionless particle velocity at the exit of the showerhead as a function of the dimensionless showerhead thickness, the showerhead porosity, and the particle Stokes number. The fluid flow field in the inter-plate region is calculated by both an analytic I-D creeping-flow approximation and by a numerical 2-D finite element method; for Reynolds numbers less than about four, the two methods are found to be in good agreement. To calculate the net effect of inertia-enhanced deposition, the analysis of particle acceleration in the showerhead is coupled with that for interplate transport. The final result is the particle collection efficiency (fraction of particles entering the reactor that deposit on the wafer) as a function of the dimensionless parameters characterizing: showerhead porosity, showerhead thickness, inter-plate Reynolds number, dimensionless particle drift velocity (a measure of the strength of applied external forces), and particle Stokes number. Conditions are found under which particle acceleration in the showerhead greatly enhances particle deposition on wafers. Numerical predictions are presented for the critical Stokes number, which is the nondimensional particle size at which inertial effects lead to substantial particle deposition even in the absence of external forces. Specific guidelines for reducing inertia-enhanced particle deposition include: increase the number of and/or enlarge the size of the showerhead holes, make the showerhead very thin, keep the inter-plate gap large, apply an external force that resists deposition, use low mass flow rates, avoid low pressure, or use a high molecular weight gas. For Reynolds numbers less than four, particle deposition efficiencies calculated with the analytic inter-plate flow field approximation are found to be in good agreement with numerical 2-D simulations.