A new set of two-fluid equations that are valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates with no zero-order drifts, a set of moment equations describing plasma transport along the field lines of a space- and time-dependent magnetic field is derived. No restriction on the anisotropy of the ion distribution function is imposed. In the highly collisional limit, these equations reduce to those of Braginskii, while in the weakly collisional limit they are similar to the double adiabatic or Chew, Goldberger, and Low (CGL) equations [Proc. R. Soc. London, Ser. A 236, 112 (1956)]. The new set of equations also exhibits a physical singularity at the sound speed. This singularity is used to derive and compute the sound speed. Numerical examples comparing these equations with conventional transport equations show that in the limit where the ratio of the mean free path λ to the scale length of the magnetic field gradient LB approaches zero, there is no significant difference between the solution of the new and conventional transport equations. However, conventional fluid equations, ordinarily expected to be correct to the order (λ/LB)2, are found to have errors of order (λ/Lu)2 =(λ/LB)2/(1−M2)2, where Lu is the scale length of the flow velocity gradient and M is the Mach number. As such, the conventional equations may contain large errors near the sound speed (M≊1).
A new set of two-fluid heat transport equations that is valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates, a set of moment equations describing plasma energy transport along the field lines of a space- and time-dependent magnetic field is derived. No restrictions on the anisotropy of the ion distribution function or collisionality are imposed. In the highly collisional limit, these equations reduce to the classical heat conduction equation (e.g., Spitzer and Härm or Braginskii), while in the weakly collisional limit, they describe a saturated heat flux (flux limited). Numerical examples comparing these equations with conventional heat transport equations show that in the limit where the ratio of the mean free path λ to the scale length of the temperature gradient LT approaches zero, there is no significant difference between the solutions of the new and conventional heat transport equations. As λ/LT→1, the conventional heat conduction equation contains a significantly larger error than (λ/LT)2. The error is found to be O(λ/L)2, where L is the smallest of the scale lengths of the gradient in the magnetic field, or the macroscopic plasma parameters (e.g., velocity scale length, temperature scale length, and density scale length). The accuracy of the flux-limited model depends significantly on the value of the flux limit parameter which, in general, is not known. The new set of equations shows that the flux-limited parameter is a function of the magnetic field and plasma parameter profiles.
A plasma etching model for rf-discharge plasma reactors has been developed. The model considers the linear and nonlinear effects on plasma kinetics of atomic effects, ion chemistry, space-charge effects, and the plasma/surface interactions. The linear effects of physical and chemical etching and the nonlinear effects of the ‘‘enhanced’’ physical and chemical etching, which are due to the plasma/surface interactions, are also discussed. New generalized plasma transport equations are introduced. These equations are valid for collisional to weakly collisional plasmas [λ/L≲O(1)], where λ is the ion mean free path and L is the smallest of the scale lengths of the gradient in the electric field or in the macroscopic plasma parameters. The transport equations are used to calculate the magnitude and profile of the plasma particle and energy fluxes for the etching model. The model has shown good agreement with experimental etch rates of silicon dioxide, borophosphosilicate glass, and photoresist.
Recently, a set of two fluid equations has been derived that describes the transport of plasmas with anisotropic pressure along the lines of force and is valid for collisional to weakly collisional regimes [Phys. Fluids 29, 463 (1986); 31, 3280 (1988)]. These transport equations coupled with Poisson’s equation are used to study the structure of the nonquasineutral transition region between the plasma and the material wall. This transition region is defined as the electric sheath region. The Bohm sheath criteria has been derived by various authors by examining behavior of the solution to the plasma-sheath equation. These investigations have shown that only for a supersonic flow would an increasing electric potential (in magnitude) exist in the sheath region to accelerate the ions further toward the wall. For subsonic flows, the behavior of the electric potential is purely oscillatory, which is not physical (electrons would trap in these potential wells and wipe them out). This work shows that when a more accurate set of equations is used, the solution is purely oscillatory when the flow speed is below the ion ‘‘thermal speed.’’ For flow velocities in the range between the thermal and sound speed, there exist solutions where the electric field is positive definite and oscillatory but the potential is monotonic. Therefore, solutions to the sheath problem exist for subsonic flow and Bohm’s sheath criteria can be violated.
A generalized plasma etching model has been developed. The new model is a generalization of our previous model presented in a recent paper [E. Zawaideh and N. S. Kim, J. Appl. Phys. 62, 2498 (1987)]. The new model includes the effects of multi-ion and multineutral gas species. New generalized plasma transport equations are also introduced. These equations are derived for multi-ion species. No restrictions on the anisotropy of the ion distribution functions are imposed. The new generalized plasma transport equations are valid for collisional to weakly collisional plasma [λ/L≤0(1)], where λ is the ion mean free path and L is the smallest of the scale lengths of the gradient in the electric field or in the macroscopic plasma parameters. A new particle balance model has also been introduced which incorporates the effects of gas composition, gas flowrate, pumping rate, ion and neutral gas chemistry, and atomic reactions on the neutral gas and plasma parameters (e.g., densities and pressures of the various neutral gas and plasma species). As an example, silicon dioxide (SiO2) plasma etching using carbon tetrafluoride (CF4) gas is used to illustrate this new generalized model. The model has shown good agreement with the experimental etch rates of SiO2 for various plasma and reactor parameters (e.g., neutral gas pressure, CF4 flowrate, and rf power).
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