Two-dimensional simulation of polysilicon etching with chlorine in a high density plasma reactor

Lymberopoulos, Dimitris P.; Economou, Demetre J.

doi:10.1109/27.467977

Cited by 71 publications

(60 citation statements)

References 11 publications

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“…They concluded that the effects of the gas flow on the global plasma parameters were secondary for low pressure systems. The Modular Plasma Reactor Simulator code of Lymberopoulos and Economou 22 recognized the importance of gas-plasma coupling but only included mass conservation for the neutrals, neglecting momentum and energy conservation. The convective term V .…”

Section: Introductionmentioning

confidence: 99%

Impact of gas heating in inductively coupled plasmas

Hash

Bose

Rao

et al. 2001

Journal of Applied Physics

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Recently it has been recognized that the neutral gas in inductively coupled plasma reactors heats up significantly during processing. The resulting gas density variations across the reactor affect reaction rates, radical densities, plasma characteristics, and uniformity within the reactor. A self-consistent model that couples the plasma generation and transport to the gas flow and heating has been developed and used to study CF 4 discharges. A Langmuir probe has been used to measure radial profiles of electron density and temperature. The model predictions agree well with the experimental results. As a result of these comparisons along with the poorer performance of the model without the gas-plasma coupling, the importance of gas heating in plasma processing has been verified.

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Section: Introductionmentioning

confidence: 99%

Impact of gas heating in inductively coupled plasmas

Hash

Bose

Rao

et al. 2001

Journal of Applied Physics

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“…In addition, one could neglect the crossderivative entries, such that the ow eld and the electric eld are updated in a decoupled manner, as was done for the Picard method. The following approximate Newton method is thus obtained: (25) where no approximations have been used in the residual expression R n Q . The approximate JacobianJ n Q is given bỹ…”

Section: Approximate Newton Methodsmentioning

confidence: 99%

Efficient Computational Model for Inductive Plasma Flows

Abeele

Degrez

2000

AIAA Journal

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A new approach to the numerical modeling of high-pressure inductive plasmas is presented. The governing magnetohydrodynamicequations are discretized in a second-order accurate nite volumemanner. A pressure-stabilized ow eld solver is introduced as an alternative to the staggered-mesh solvers used in traditional algorithms. It is argued that the widely used integral boundary formulation for the electric eld is computationally expensive and cannot be incorporated into an ef cient iterative solution procedure. A far-eld formulation of the electric eld is adopted instead, such that state-of-the-art iterative methods can be applied to speed up the calculation. The discretized equations are solved through a damped Picard method and an approximate and a full Newton method. Ef cient linear algebra methods are used to solve the linear systems arising from the iterative methods. An appropriate linearization of the (strongly positive) joule heating source term is found to be important for the convergence at the linear level. The new model is tested on a 3-species argon and an 11-species air inductive plasma computation. The proposed damped Picard iterative method is found to be very robust during the initial iterations, but does not converge well thereafter. The approximate Newton method converges substantially better. Finally, the full Newton method yields rapid quadratic convergence rates at the expense of an increase in storage. Taking into account both speed of convergence and memory use, the approximate Newton method is considered to be optimal. NomenclatureCFL = damping parameter similar to the Courant number D{J } = block diagonal of the Jacobian/Picard matrix J E = total electric-eld amplitude (complex quantity), V¢ m ¡ 1 ; global vector containing electric-eld variables E = total electric eld (radio frequency), V¢ m ¡ 1 E P = electric-eld amplitude generated by plasma (complex quantity), V¢ m ¡ 1 E V = electric-eld amplitude generated by inductor in absence of plasma (complex quantity), V¢ m ¡ 1 F r = radial component of Lorentz force per unit volume, N¢ m ¡ 3 F z = axial component of Lorentz force per unit volume, N¢ m ¡ 3 f = torch operating frequency, Hz f m = discrete mass ux in axial direction, kg¢ m ¡ 2 ¢ s ¡ 1 h = internal enthalpy per unit mass, J¢ kg ¡ 1 I C = current through outer inductor, A J E = electric-eld Jacobian matrix J P = Picard matrix J Q = total Jacobian matrix J U = ow eld Jacobian matrix n e = Number density of electrons, m ¡ 3 P joule = joule heating source term, W¢ m ¡ 3 P rad = radiative loss term, W¢ m ¡ 3 p = static pressure, N¢ m ¡ 2 Q = total solution vector ( ow eld plus electric eld) q z, r = conductive heat uxes,W¢ m ¡ 2 R = inner radius of quartz tube, m Re D z = Reynolds number based on cell length in z direction R E Steenweg-op-Waterloo; vdabeele@vki.ac.be. Student Member AIAA. † Professor, Department of Aeronautics and Aerospace, 72 Steenweg-opWaterloo. Senior Member AIAA. R Q= total residual R U = ow eld residual r = radial distance from torch axis, m T = temperature, K U = o...

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“…[3] [4] In these equations the subscript s denotes all ns species including electrons (s ϭ e), ions (s ϭ i), and all neutral species. The subscript r sums over all nr gas-phase chemical reactions.…”

Section: Modelmentioning

confidence: 99%

A Continuum Model for the Inductively Coupled Plasma Reactor in Semiconductor Processing

Bose

Govindan

Meyyappan

1999

J. Electrochem. Soc.

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The microelectronics industry is facing two major challenges simultaneously with migration toward 300 mm wafers and reduction of the feature size to 0.18 m. This places unprecedented stringent demands on processes and equipment. Recently computational modeling has shown some promise to reduce the number of trialand-error iterations in equipment design. Due to this promise and the challenges faced by equipment vendors, computational modeling is called upon, more than ever, to play a complementary role in equipment and process design.A reliable technology computer-aided design (TCAD) tool must be physically accurate and executable with reasonable computing resources. Plasma reactors can be classified into continuum models, particle or kinetic models, and hybrid models. Kinetic models, in general, are physically accurate but computationally very expensive. For example, a direct-simulation Monte Carlo technique historically used in rarefied gas-flow studies is suitable for plasma process modeling, but simultaneous simulation of neutrals, ions, and electrons is currently impossible. Multidimensional solution to the Boltzmann equation for practical reacting mixtures is also a difficult task. Therefore, such approaches do not offer hope of becoming TCAD candidates in the near future. In contrast, continuum models are computationally attractive. But there has been doubt in the modeling community about the validity of continuum models for low-pressure plasma-processing situations based on mean free path vs. characteristic dimension arguments. Reactor size is on the increase while the operating pressure in high-density reactors has remained the same (1-20 mTorr) through several reactor generations. The subgrid scale sheaths near plasma edges can be considered effectively using analytical sheath models 1 and integrated with reactor models. Hence, application of continuum models is a viable option to study highdensity-plasma reactors. Indeed, the most important aspect dictating the validity of continuum models is the description of electron impact reactions. In kinetic models, the nature of electron energy distribution function (eedf) is an explicit output. This along with the input of fundamental collision cross sections, does not leave any ambiguity in the treatment of inelastic collisions (although it may be difficult to statistically resolve the high energy tail of the distribution). In contrast, the information on eedf must be fed to the fluid model. Typically, rate constants for electron-impact reactions are derived from an assumed eedf and known cross sections. The usual practice is to assume a Maxwellian distribution and its validity is case-dependent. This would make the rate constants suspect and result in poor predictive capability if the deviation from Maxwellian is significant. In such cases, one may solve the Boltzmann equation locally (a zero-dimensional solution in space) and use the ensuing eedf to compute rate constants. Alternatively, a hybrid model combining the continuum and kinetic schemes may be used. ...

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Two-dimensional simulation of polysilicon etching with chlorine in a high density plasma reactor

Cited by 71 publications

References 11 publications

Impact of gas heating in inductively coupled plasmas

Impact of gas heating in inductively coupled plasmas

Efficient Computational Model for Inductive Plasma Flows

A Continuum Model for the Inductively Coupled Plasma Reactor in Semiconductor Processing

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