Douglas W. Cooper

Particulate Contamination and Microelectronics Manufacturing: An Introduction

The manufacture of integrated circuits for computers and various microprocessors is of increasing importance. Higher speeds of operation and lower costs require higher circuit densities, thus furth...

Predicted Deposition of Submicrometer Particles Due to Diffusion and Electrostatics in Viscous Axisymmetric Stagnation-Point Flow

Various mathematical solutions to the convective-diffusion equation for noninteracting Brownian particles were carried out to predict deposition of submicrometer particles onto a flat surface in viscous, three-dimensional (axisymmetric) stagnation-point flow at clean-room velocities (∼ 100 cm/s). The particle deposition aspects modeled included electrostatics, inasmuch as both diffusion and electrostatics are the dominant mechanisms expected. The results were obtained in terms of dimensionless groups for deposition, convective-diffusion, and electrostatic attraction. It was found that the deposition velocity can be well approximated by a simple combination of the convective-diffusion velocity and the eletro-static velocity. These results are translated into practical terms, examples are given, and the predictions are compared with predictions made by other methods. A disk 20 cm in diameter charged to 2000-V potential is shown to attract a particle 0.1 μm in diameter so as to produce an electrostatic depos...

The effects of electrostatic and inertial forces on the diffusive deposition of small particles onto large disks: Viscous axisymmetric stagnation point flow approximations

Fine particles can deposit on microelectronic parts during their manufacture and cause damage. Prediction of the rates of deposition is important in planning to reduce such losses. Analytic expressions were developed and Monte Carlo simulations (Brownian dynamics) were carried out to predict deposition rates of sub-micron particles in viscous axisymmetric stagnation point flow. Diffusive, inertial and electrostatic deposition mechanisms were incorporated in the analysis. The electrostatic forces considered included the Coulombic force on a charged particle in an electric field and the image force on a charged particle near a conducting plane. Analytical solutions to the convective-diffusion equation, in the absence of inertial effects, showed that for a Boltzmann charge equilibrium, the electrostatic image force did not appreciably enhance deposition for a free-stream velocity of 50 cm s−1. However, the Coulomb electrostatic attraction of a charged particle towards a charged surface greatly enhanced deposition, assuming a field from the disk approximately equivalent to having a disk 20 cm in diameter (80 pF capacitance) at 2000 V. Some areas for future study are suggested by these results. Brownian dynamics simulations carried out for particles of the order of 1 μm in diameter at a much higher velocity (30 m s−1) demonstrated an interaction between inertial and diffusive deposition that produced inertially enhanced deposition at Stokes values below the critical value, St < 0.15. Where inertia and electrostatic effects were negligible, the simulations matched the known solution for stagnation flow deposition due to diffusion. In the Appendix is presented a derivation of the Langevin equation governing single-particle motion in a fluid that is not uniform in temperature as well as in velocity.

DEPOSITION OF SUBMICRON AEROSOL PARTICLES DURING INTEGRATED CIRCUIT MANUFACTURING: THEORY

ABSTRACT Submicron (≤1μm) particle contamination can produce unacceptably low yields in the manufacture of integrated circuits. Calculations were made to predict deposition velocities of 0·01-lOμm particles, incorporating gravitational, dlffusional, and electrostatic effects. The results were summarized in equations that correlate non-dimensional deposition (Sherwood number) with convective-diffusion (Peclet number) and with electrostatics (Boltzmann and Fuchs charge distributions). These equations were used In conjunction with particle size distributions to predict particle deposition. In a companion paper |25| the predictions were shown to compare well with limited experimental data. To reduce deposition product surfaces should not be electrically charged and, where possible, these surfaces should be at higher temperatures than the ambient gas. For quality control purposes, the deposition flux predictions could serve to link the specifications of gas cleanliness with the specifications of surface cleanl...

Surface cleaning by electrostatic removal of particles

Contamination of product surfaces by particles during microelectronics manufacturing can lead to significant losses of product yield and reliability. In many cases, cleaning by an effective dry process would be preferable to cleaning by an equally effective wet process. A new dry process for cleaning is described. The product surface to be cleaned is grounded or placed on a grounded surface. An insulating film, approximately 20 μm thick, is placed on the contaminated product surface. A counterelectrode is placed on the film, with enough direct current voltage, typically kilovolts, to create fields of approximately 1 MV/cm. When the film is removed from the contaminated product surface, particles as small as 1 μm are removed with the film. Particles deposited dry are substantially easier to remove than particles deposited wet and allowed to dry. Also very difficult to remove are particles that melt and then solidify on a surface. Various insulating films have shown successful particle removal, but all film...

Parking problem (sequential packing) simulations in two and three dimensions

Abstract The goal of the parking limit problem is to determine the mean fraction of a space that would be occupied by fixed objects, each of the same size, that are placed or created randomly in that space until no more can fit. The goal of the two-dimensional parking limit problem is to determine the fraction of an area that would be occupied by disks of one size; a three-dimensional parking limit problem involves creating spheres in a volume. We tried a simple numerical simulation algorithm combined with regression analysis to do the two-dimensional problem and found the parking limit to be 0.51 to 0.56 area fraction for disks on a plane, which is consistent with the results of others. We then used the three-dimensional version of this algorithm to obtain the parking fraction in a cubical region with penetrable walls. Our results can be multiplied by a geometrical factor to give the parking fraction for a cube with impenetrable walls and can be extrapolated to give the parking fraction for an infinite region. Regression equations were obtained for the effects of the ratio of the sphere radius to the side of the cubical volume and of the dimensionless number of attempts to park. We found the parking limit for an infinite volume to be 0.37 to 0.40 for spheres.

Evaluation of aerosol deconvolution algorithms for determining submicron particle size distributions with diffusion battery and condensation nucleus counter

Abstract Determining the particle size distribution of airborne particles is important in many contexts, including understanding and thus reducing the deposition of particles on micro-electronic components during their manufacture. Sizing of particles smaller than 0.1 μm is usually done with a diffusion battery, which requires use of a deconvolution algorithm to obtain particle size distributions. The following algorithms were evaluated: CINVERSE (Crump and Seinfeld, Aerosol Sci. Technol. 1 , 363, 1982). Twomeys iterative procedure (Twomey, J. Comput. Phys. 18 , 188, 1975), expectation maximization (Maher and Laird, J. Aerosol Sci. 16 , 557, 1985), constrained least-squares fit (Nelder and Mead, Comput. J. 7 , 308, 1965; Cooper and Spielman, Atoms. Envir. 10 , 1976). Expectation maximization and constrained least-squares fit are more suited to this use than are the other two. The non-monotonic response of the diffusion battery with respect to particle size cannot be corrected for by any such algorithm. One could modify the diffusion battery to prevent entrance of the larger particles or one could use an independent measurement of the larger particles to correct the diffusion battery data. Using the latter approach provided an improved estimate of the particle size distribution in a clean room.

The inversion matrix and error estimation in data inversion: application to diffusion battery measurements

Judicious selection of the measurement conditions and analysis methods to be used can make it less difficult to produce accurate data inversions in the presence of experimental error. The response (data) vector (b) of a multi-channel instrument, such as an optical particle counter or multi-stage impactor or diffusion battery, to an input distribution vector (x) can be modelled as a set of linear equations given by the vector-matrix equation b = Ax. For low resolution instruments, one of several methods of data inversion is usefully employed: simple inversion, least-squares inversion, various smoothing inversions and various non-linear approaches. One non-linear approach [Twomey, S. (1975) J. Comput. Phys.18, 188–200.] we found to be sensitive to starting conditions and to show cycling during iteration, similar to equations leading to ‘chaos’ [Wu, J. J. et al. (1989) J. Aerosol Sci.20, 477–482.]. Simple inversion, least-squares inversion and smoothing are alike in that they produce their solutions from x = Zb, where Z(i, j) are the elements of what one could call the ‘inversion matrix’, Z, a kind of transfer function. Z gives the sensitivity of the inferred values to changes (or errors) in the data values. A criterion for the best measurement instrument or measurement conditions could be the minimum largest absolute Z(i, j) or the mean absolute value or some other weighting. Propagation of error analysis indicates that another measure of Z(i, j) that would be useful would be its root mean square. The ‘condition number’ is another measure that has also been suggested [Cooper, D. W (1974) Ph.D. dissertation. Division of Engineering and Applied Science, Harvard University, Cambridge, MA, (1975) 68th Annual Meeting of the Air Pollution Control Assoc., Boston, MA; Yu, P.-Y. (1983) Ph.D. dissertation. Department of Chemical and Nuclear Engineering, College Park, MD; Farzanah, F. F. et al. (1984) Environ. Sci. Technol.19, 121–126; Hirleman, E. D. (1987) 1st Intl. Conf. on Particle Sizing, Rouen, France.]. Some comparisons of these measures are made. The inversion matrix gives the clearest indication of the relationship between the data and the results of inversion. We recommend that proposed experimental conditions should be adjusted based on inversion matrix studies in order to lessen ill-conditioning and the reliance on various data analysis methods to cope with ill-conditioned systems.

THEORETICAL STUDY OF A NEW MATERIAL FOR FILTERS: AEROGELS

ABSTRACT Aerogels are highly porous materials made by removing the liquid phase of a dilute suspension of solids by taking off the vapor under super-critical temperature and pressure conditions. The aerogel structure has a skeleton of nanometer-size elements forming pores that are tens of nanometers in dimension. In some cases the structure of the aerogel arises from clusters having fractal geometry. The collection efficiency and pressure drop for such material is predicted here from models in the literature. A quality factor is used to compare aerogel material with other filter materials. The quality factor relating gas filtration efficiency with pressure drop is predicted to be unusually high for these materials in comparison with other common fibrous filter media, predicting lower flow resistance for aerogels at equivalent particle collection efficiency levels.

The effects of electrostatic forces on the thermophoretic suppression of particle diffusional deposition onto hot surfaces

Abstract By raising the temperature of a surface above that of the ambient fluid, a temperature gradient that results in a thermophoretic force that repels particles from the surface is created. It has been recommended that this effect be used to reduce the contamination of product surfaces during manufacturing. It has been shown that even modest levels of electrostatic interactions between particles and surfaces can overwhelm the thermophoretic repulsion available by raising the temperature of the surface 10% above that of the ambient fluid. We present an analysis for axisymmetric, viscous stagnation-point flow, extending the work of Friedlander et al. (1988, J. Colloid Interface Sci.125, 351), and show the temperature differences that must be achieved to succeed in preventing particle deposition due to diffusion, gravitation, and electrostatic attraction. Simple algebraic relationships are established that allow for a ready determination of the temperature differences needed to suppress particle diffusional deposition onto surfaces under the action of uniform external forces, such as gravitational and coulombic forces, and a spatially dependent external force, which for this study we have chosen to be the electrostatic image force. These relationships are further verified through approximate analytical and “exact” numerical solutions also given here. Various results of practical interest are discussed. For example, for a flow intensity of 6.67 s−1 in air, an electric field as small as 100 V/cm makes it unlikely that particles with a minimal charge distribution (Boltzmann) will be kept from deposition on a surface by raising the surface temperature 10 or 20% above the ambient temperature. This emphasizes the importance of reducing the electric charge levels of product surfaces during manufacturing.