COMSOL 6.4 - Benchmark Model of a Capacitively Coupled Plasma

Benchmark Model of a Capacitively Coupled Plasma

The underlying physics of a capacitively coupled plasma is rather complicated, even for rather simple geometric configurations and plasma chemistries. This model benchmarks the Plasma, Time Periodic interface against many different codes, the results of which are taken from Ref. 1.

Model Definition

The model geometry consists of a 1D gap of 0.067 m. A plasma forms in the gap provided the driving voltage and fill pressure are high enough. The driving frequency in this model is 13.56 MHz. Helium chemistry is used, as was the case in Ref. 1.

Electron transport is modeled by solving the continuity equation, the momentum equation under the drift-diffusion approximation, and the mean electron energy equation (for detailed information on electron transport, see Theory for the Drift Diffusion Interface in the Plasma Module User’s Guide)

The source coefficients in the above equations are determined by the plasma chemistry. The electron rate expression is defined as

where νe,j is the stoichiometric coefficient, and the reaction rate is defined as

where kjf is the forward rate constant and kjr is the reversed rate constant. Both the Electron Impact Reaction feature and Reaction feature can contribute to the electron rate expression. However, when using the Reaction feature it is important to note that the associated electron energy gain or loss is not included in the source term of the electron mean energy equation.

The rate constants can be computed from electron impact cross-section data

where γ = (2q/me)1/2 (SI unit: C1/2/kg1/2), me is the electron mass (SI unit: kg), ε is the electron energy (SI unit: V), σ is the electron impact collision cross section (SI unit: m2), and f is the electron energy distribution function.

When Townsend coefficients are used, the reaction rate is defined as

where αj/Nn is the reduced Townsend coefficient for reaction j (SI unit: m2) and Γe is the electron flux as defined above (SI unit: 1/(m2·s)). Townsend coefficients can increase the stability of the numerical scheme when the electron flux is field driven as is the case with DC discharges.

The total electron energy loss or gained is calculated by summing the collisional energy changes from all reactions defined with the Electron Impact Reaction feature as

where Δεj is the energy loss from reaction j (SI unit: V) and F is the Faraday constant (SI unit: C/mol). For excitation and ionization collisions Δεj corresponds to the energy of the excited state being excited/deexcited or ionized, for attachment Δεj is set to zero, and for elastic collisions

where me and mk are the electron and heavy species mass in kg, Te is the electron temperature in eV, and Tgas is the gas temperature in K.

For heavy species, the following equation is solved for the mass fraction of each species (for detailed information on the transport of the nonelectron species, see Theory for the Heavy Species Transport Interface in the Plasma Module User’s Guide):

The electrostatic field is computed using the following equation:

The space charge density ρ is automatically computed based on the plasma chemistry specified in the model using the formula

For detailed information about electrostatics see Theory for the Electrostatics Interface in the Plasma Module User’s Guide.

In Ref. 1 the ion diffusivity is defined as:

and the mobility, μ as:

where a = qE/m and λ is the charge exchange mean free path. The same ion mobility and diffusivity models can be set in COMSOL. With that objective, choose and in the section on the ion species node. After that, go to the and choose the ion mobility model with a σ equal to 3.10−19 m2 to reproduce the same ion mobility and diffusivity as in Ref. 1.

boundary conditions

Electrons are lost to the wall due to random motion within a few mean free paths of the wall and gained due to secondary emission effects, resulting in the following boundary condition for the electron flux

(1)

and the electron energy flux

(2)

In order to make the COMSOL Multiphysics implementation of the electron losses to the wall consistent with the reference, the value of r must be set to 5/11. The second term on the right-hand side of Equation 1 is the gain of electrons due to secondary emission effects, γp being the secondary emission coefficient. The second term in Equation 2 is the secondary emission energy flux, εp being the mean energy of the secondary electrons. For the helium ions are lost to the wall due to surface reactions and the fact that the electric field is directed toward the wall:

Plasma Chemistry

The reference paper suggests a simplistic plasma chemistry for helium consisting of only 3 reactions and 4 species (electron impact cross sections are obtained from Ref. 2):

Table 1: table of collisions and reactions modeled.
reaction	formula	type
1	e+He=>e+He	Elastic	0
2	e+He=>e+Hes	Excitation	19.5
3	e+Hes=>e+He+	Ionization	24.5

In addition to volumetric reactions, the following surface reactions are implemented:

Table 2: table of surface reactions.
reaction	formula	Sticking Coefficient
1	Hes=>He	1
2	He+=>He	0

When a metastable helium atom makes contact with the wall, it reverts to the ground state helium atom with some probability (the sticking coefficient).

Results and Discussion

The time averaged ion current density is plotted in Figure 1. The peak ion current density occurs on the electrodes, at a value of around 0.2 A/m2. At lower pressures the ion current density profile is more smooth across the gap. At 300 mtorr there is a pronounced flattening off of the ion current density in the plasma sheath. These results agree well with those presented in Figure 9 of Ref. 1.

Figure 1: Plot of the period averaged ion current as a function of the distance from the left electrode.

The period averaged excitation and ionization rates are plotted in Figure 2 and Figure 3.

Figure 2: Plot of the period averaged ionization rate as a function of the distance from the left electrode.

Figure 3: Plot of the period averaged excitation rate as a function of the distance from the left electrode.

The results for the ionization and excitation rates agree well with Ref. 1 in both absolute value and spatial distribution. The period averaged electron power deposition is plotted in Figure 4. The COMSOL results are again, in good agreement with the reference paper.

Figure 4: Plot of the period averaged electron power deposition as a function of the distance from the left electrode.

Figure 5 and Figure 6 plot the period averaged electron and ion density at different operating pressures. The electron and ion density is the same in the plasma bulk but the ion density is higher in the plasma sheath. This creates a net positive space charge density in the plasma sheath which tends to hold electrons in the plasma and accelerate ions toward the wall via the plasma potential.

Figure 5: Plot of the period averaged electron density at different operating pressures.

Figure 6: Plot of the period averaged ion density for different operating pressures.

Ref. 1 tabulates a wide range of lumped parameters for the discharge. These same parameters for the COMSOL model are shown in Table 3. There is good agreement between the COMSOL model and the values quoted in Ref. 1.