Anthony S. Geller

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

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 ...

Showerhead-Enhanced Inertial Particle Deposition in Parallel-Plate Reactors

ABSTRACT 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: 1) 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 exi...

Boundary element method calculations of the mobility of nonspherical particles—1. Linear chains

Abstract The utility of the boundary element technique is demonstrated on the problem of linear chains of spheres translating and rotating through a quiescent fluid. The method takes advantage of the linearity of the problem by using superposition of the general solution for the flow generated by point forces on the bounding surfaces of the fluid to satisfy the boundary conditions. The BEM results for translational drag of chains of spheres compare very well with published experimental and computational data. We also show that slender body theory provides an approximate analytic result that is useful in interpreting and correlating the BEM calculations. Slender body theory also revealed that the model of the particle as a single prolate spheroid with equal aspect ratio produced a result equal to that for the chain of spheroids correct to second order, while the model of the chain as a cylinder produced an upper bound on the drag. Slender body theory also gives a reasonable estimate for the rotational resistance for chains, which, together with the BEM results, are reported here for the first time.

Calculation of particle concentration around aircraft-like geometries

Abstract A computationally-efficient method is presented to calculate local particle concentration enhancements resulting from potential fluid flow around an idealized aircraft fuselage and wing. The geometries chosen for study are a 10:1 prolate ellipsoid at 0° angle of attack and a Joukowski airfoil at 0° and 5° angles of attack, for which potential flow analytic solutions are known. The collection efficiency of and surface concentration on a cylinder in potential flow are also considered for algorithm verification. Particle concentration is calculated along particle pathlines by a mixed Eulerian-Lagrangian technique developed by Fernandez de la Mora and Rosner (1981, Fernandez de la Mora, J. F. and Rosner, D. E., Physico Chem. Hydro . 2 , 1). Ordinary differential equations for particle position, velocity, and concentration are integrated numerically by a variable order, backward difference algorithm. The calculations show the creation of regions of increased concentration near objects, and of particle-free shadow zones downstream. The magnitudes of the concentration disturbances are greatest at intermediate Stokes numbers (0.1–1.0) where inertia and drag are equally dominant. Samplers placed in these regions of enhanced particle concentration may not provide accurate concentration measurements. Ultimately, this approach could be included with detailed flow solutions about specific aircraft geometries to provide guidance in locating samplers in regions of acceptably small concentration deviations.

Boundary element method calculations of the mobility of nonspherical particles—II. Branched chains and flakes

We combine a numerical, boundary element method with analytical techniques to predict the motion of isolated, nonspherical particles moving at low Reynolds number through a Newtonian fluid. The boundary element method is used to determine the constant components of the resistance matrix (a geometry-specific matrix relating a particles linear and angular velocities to the applied forces and torques). Once the resistance matrix has been constructed, direct simulation of the translation and rotation of the particle in streaming flow can be performed. Applications in aerosol characterization have led us to do this analysis for agglomerates of spheres, as well as flake-like particles. We obtain excellent agreement with the limited, published, experimental data.

Particle Deposition in Parallel-Plate Reactors: Simultaneous Diffusion and External Forces

Particle deposition resulting from uniform external forces and Brownian motion is modeled in a parallel-plate reactor geometry characteristic of a wide range of semiconductor process tools: uniform, isothermal, downward flow exiting a perforated-plate showerhead separated by a small gap from a parallel, circular wafer. Particle transport is modeled using a Eulerian approach neglecting particle inertia and interception. Particles are assumed to originate in a planar trap located between the plates, such as would result for particles released from a plasma-induced particle trap after plasma extinction. Flow between infinite parallel plates is described by an analytic quasi-one-dimensional creeping flow approximation, where the showerhead is treated as a porous plate. An analytic, integral expression for particle collection efficiency (fraction of particles that end up on the wafer) is derived as a function of four dimensionless parameters: the flow Reynolds number, a dimensionless trap height, a dimensionless particle drift velocity, and the particle Peclet number. Numerical quadrature is used to calculate particle collection efficiency in terms of the controlling dimensionless parameters for external forces, which either enhance or inhibit particle deposition. Example calculations of collection efficiency are also presented in dimensional terms for a representative set of process conditions. Strategies to reduce particle deposition include the use of a protective external force and manipulation of the trap to keep it as far from the wafer as possible.

Transport and Deposition of Aerosol Particles

Publisher Summary This chapter reviews the theoretical models available to describe particle transport in typical semiconductor processing environments. Particle concentrations are assumed to be low enough so that the influence of the particle on fluid transport can be neglected; particle–particle interactions are also neglected. Under this assumption, the fluid and thermal fields are calculated first (in the absence of particles), and then used as input for subsequent particle transport calculations. The theoretical underpinnings for both the Lagrangian approach (where individual particle trajectories are calculated) and the Eulerian approach (where the particle concentration field is modeled as a continuum) are presented in the chapter. The strength of the Lagrangian formulation is in predicting particle transport resulting from external forces including particle inertia; but the current implementation cannot describe the chaotic effect of particle Brownian motion (i.e., particle diffusion) on particle transport. On the other hand, the Eulerian formulation can describe particle transport resulting from applied forces and particle diffusion, but the current implementation cannot account for particle inertia.

Boundary element method calculations of the mobility of nonspherical particles—3. Parallel implementation for long chains

Abstract A boundary element formulation for implementation on computers with massively parallel architecture, which allows calculation of particle drag and torque for particle agglomerates, is described. Shape factors for long, linear chains of spheres translating through a quiescent fluid are calculated. These calculations show that the correlation presented in an earlier paper ( Geller et al., 1993 ), which was based on boundary element results for much shorter chains, continues to hold for chains of up to 700 constituent spheres; and, as expected, the slender body results, also presented by Geller and coworkers (1993), more closely agree with the numerical results for these longer chains.

Protection of extreme ultraviolet lithography masks. I. Thermophoretic protection factors at low pressure for diffusing nanoscale particles

Model calculations are presented for thermophoretic protection of an extreme ultraviolet (EUV) mask placed face down in an EUV mask inspection tool. The protection factors, defined as the ratio of challenge particles to deposited particles, are calculated for a variety of test conditions (temperature gradient, gas type, particle density, and particle position) for a reticle bathed in clean gas from a facing showerhead. Thermophoretic protection (in combination with gravity) provides robust protection for particle sizes greater than ∼20 nm. However, for particle sizes less than ∼20 nm, protection falters quickly and is severely degraded for highly diffusing 10 nm particles that are of concern for mask contamination. Estimates are made for the required level of particle protection in both EUV mask inspection and EUV projection lithography. When compared with these estimates for the required protection, it is clear that thermophoresis alone cannot successfully defend against particles smaller than ∼20 nm, an...

Protection of extreme ultraviolet lithography masks. II. Showerhead flow mitigation of nanoscale particulate contamination

An analysis is presented of a method to protect the reticle (mask) in an extreme ultraviolet (EUV) mask inspection tool using a showerhead plenum to provide a continuous flow of clean gas over the surface of a reticle. The reticle is suspended in an inverted fashion (face down) within a stage/holder that moves back and forth over the showerhead plenum as the reticle is inspected. It is essential that no particles of 10-nm diameter or larger be deposited on the reticle during inspection. Particles can originate from multiple sources in the system, and mask protection from each source is explicitly analyzed. The showerhead plate has an internal plenum with a solid conical wall isolating the aperture. The upper and lower surfaces of the plate are thin flat sheets of porous-metal material. These porous sheets form the top and bottom showerheads that supply the region between the showerhead plate and the reticle and the region between the conical aperture and the Optics Zone box with continuous flows of clean ...