COMSOL 6.4 - Global Model of a CF4/O2 Plasma Reactor

This tutorial studies the plasma chemistry of a CF4/O2 plasma at low pressure using a global model. The plasma chemistry is based on Ref. 1 and the electron impact reactions are taken from LxCat (Ref. 2, Ref. 3, and Ref. 4).

Model Definition

The model used in this work assumes that the spatial distribution of the different quantities in the plasma reactor can be treated as uniform. Without spatial derivatives, the numerical solution of the equation set becomes considerably simpler and the computation time is reduced. These advantages make a global model a good first approach to study a plasma reactor, especially when complex chemistries are involved.

When using a global plasma model the species densities and the electron temperature are treated as volume-averaged quantities. Detailed information on the global model can be found in the section Theory for Global Models in the Plasma Module User’s Guide. For heavy species the following equation is solved for the mass fraction

where ρ is the mass density (SI unit: kg/m3), wk is the mass fraction, wf,k is the mass fraction in the feed, mf and mo are the mass-flow rates of the total feed and outlet, and Rk is the rate expression (SI unit: kg/(m3·s)). The fourth term on the right-hand side accounts for surface losses and creation, where Al is the surface area, hl is a dimensionless correction term, V is the reactor volume, Mk is the species molar mass (SI unit: kg/mol) and Rsurf,k,l is the surface rate expression (SI unit: mol/(m2·s)) at a surface l. The last term is introduced because the species mass balance equations are written in the nonconservative form and it used the mass-continuity equation to replace for the mass density time derivative. In the last term Mf,l is the inward mass flux of surface l (SI unit: kg/(m2·s)). The sum in the last two terms is over all surfaces where there are surface reactions.

To take possible variations of the system’s total mass or pressure into account, the mass-continuity equation can also be solved:

The electron number density is obtained from electroneutrality:

Using the local energy approximation (LEA), the electron energy density nε (SI unit: V/ m3) is computed from

where Rε is the electron energy loss due to inelastic and elastic collisions, Pabs is the power absorbed by the electrons (SI unit: W), and e is the elementary charge. The last term on the right side accounts for the kinetic energy transported to the surface by electrons and ions. The summation is over all positive ions, εe is the mean kinetic energy lost per electron lost, εi is the mean kinetic energy lost per ion lost, and Na is Avogadro’s number. If using the local field approximation (LFA) the electron mean energy equation is not solved and the electron mean energy can be: (i) provided as a function of the electric field; or (ii) obtained by solving the Boltzmann equation in the two-term approximation.

The rate coefficients for electron impact reactions can be computed by appropriate averaging of cross sections over an EEDF. The EEDF can either be analytic or obtained by solving the steady-state Boltzmann equation in the two-term approximation coupled with the equation system (The Boltzmann Equation, Two-Term Approximation Interface in the Plasma Module User’s Guide). When solving for the EEDF, the coupling between the equations is as follows: (i) if the LEA is used, the electron mean energy obtained from the electron mean energy equation is given as input to the Boltzmann solver; (ii) if the LFA is used, the reduced electric field must be given as input to the Boltzmann solver and the electron mean energy comes from averaging over the computed EEDF.

This model uses the LEA and a Maxwellian EEDF, as in Ref. 1.

Plasma Chemistry

The plasma chemistry is based on Ref. 1. The electron impact cross sections used in this model are retrieved from different databases from LxCat: Ref. 2, Ref. 3, and Ref. 4. The data from Ref. 2 further refers to Ref. 5 and Ref. 6. The model includes 29 species: electrons, CF4, CF3, CF2, CF, CF3+, CF2+, CF+, F2, F2+, F, F+, F-, O2, O2+, O, O+, O-, O2*, O*, C, C+, CO2, CO2+, CO, CO+, COF, COF2, and FO.

Results and Discussion

The model contains two studies. In the first study, a base case is solved using a time-dependent solver for an input power of 120 W and an oxygen mole fraction of 0.01. In the second study, the power is kept at 120 W and the oxygen mole fraction is varied between 0 and 1 for pressures of 8, 15, and 25 mTorr.

Figure 1, Figure 2, Figure 3, and Figure 4 show the electron density, electron temperature, F number density, and O number density, respectively. In general, there is good agreement with the results from Ref. 1.