COMSOL 6.4 - Global Model of an SF6/Argon Plasma

This tutorial studies the chemistry of a SF6/argon plasma in moderate pressures using a global model. The main goal is to show how to prepare a model with a mixture of different elements (in this case Ar and SF6) in which one of the species can dissociate by electron impact into many fragments (SF6 dissociates into SFx, F, and S) and where multiple negative ions exist.

A simplified plasma chemistry is used to discuss the main aspects of such discharges. It is important to keep in mind that a benchmark is not attempted and the idea is to provide a template that can be used to develop more complex and precise chemistries. In fact, quite probably it will be necessary to modify the data used and add more reactions to achieve experimental verification. The chemistry is based on Ref. 1, Ref. 2, Ref. 3, and Ref. 4.

Model Definition

The model used in this work considers that the spatial distribution of the different quantities in the plasma reactor can be treated as uniform. Without the 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 or the influence of the EEDF is to be studied.

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 work uses the LEA and a Maxwellian EEDF.

The present study also solves the gas heat equation

where Cp is the specific heat at constant pressure of the mixture, T is the gas temperature, hf,k is the enthalpy of species k in the feed, hk is the enthalpy of species k. The heat source (SI unit: W/m3) is given as:

where Δεj is the electron energy loss from reaction j (SI unit: V) and F is the Faraday constant (SI unit: C/mol). The last term in the equation above is only added for electron impact reactions to account for the energy loss or gain by the electron. Note that the electron enthalpy is set to zero and does not contribute to Hj. For electron impact reactions resulting in excitation and ionization 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. Heat losses by transport are including in a simplified form:

where k is the thermal conductivity of the mixture, TS is the surface temperature, and ΛS is the diffusion length.

Plasma Chemistry

The plasma chemistry is based on Ref. 1, Ref. 2 and Ref. 3. The electron impact cross sections for SF6 are from Ref. 4 and Ref. 6, and for F and F2 are from Ref. 5 and Ref. 7. Argon cross sections are from Ref. 8. For the rate constants of heavy species reactions were given generic order of magnitude values characteristic of the reaction type. The model includes 28 species: electrons, SFx, SFx+, SFx-, Fx, Fx+, Fx-, Ar, Ars, and Ar+.

This plasma chemistry is completely prepared in text data files and then imported using the Plasma Chemistry Import feature. Five reaction groups are created as listed in Table 1.

Table 1: Reaction group features included in this model.
Reaction group for	Type	No. of reactions included
SFx	Electron Impact Reactions	32
Fx	Electron Impact Reactions	14
Ar	Electron Impact Reactions	5
Ion–ion	Heavy Species Reactions	56
Neutral–neutral	Heavy Species Reactions	9

In addition to the volume reactions, 11 surface reactions are also implemented.

Results and Discussion

On this work, 3 studies are made. In the first study, a base case is solved using a stationary solver for an input power of 2000 W and an argon mole fraction of 0.1. In the second study, a power sweep is solved for the input power ranging from 2000 W down to 300 W. In the third study, the argon mole fraction is swept from 0.1 to 0.9. For all cases the pressure is kept constant at 20 mTorr, a Maxwellian electron energy distribution function is used, and the heavy species heat equation is solved for.

Figure 1 shows the number densities of negatively charged species as a function of input power. The values are in agreement with Ref. 1. The density of F- shows significant increase with power because of higher dissociation degree.

Figure 2 shows the electron temperature as a function of input power. The electron temperature decreases with power increase in agreement with Ref. 1 suggesting that the plasma is better confined at higher powers.

Figure 3 shows the gas temperature as a function of input power. The gas temperature increases significantly with power as expected. The values are in a reasonable range for the present operation conditions.

Figure 4 shows the number densities of positively charged species as a function of the argon mole fraction. As expected, the Ar+ density increases with the argon mole fraction as more argon is available to ionize.

Figure 5 shows the electronegativity as a function of argon mole fraction. As expected, the electronegativity drops as the mixture becomes lean in SF6 but the dominant charged species are still ions even at 90% argon.

Figure 1: Number densities of negatively charged species as a function of input power.

Figure 2: Electron temperature as a function of input power.

Figure 3: Gas temperature as a function of input power.

Figure 4: Number densities of positively charged species as a function of argon mole fraction.

Figure 5: Electronegativity as a function of argon mole fraction.

References

1. M. Mao, Y.N. Wang, and A. Bogaerts, “Numerical study of the plasma chemistry in inductively coupled SF6 and SF6/Ar plasmas used for deep silicon etching applications,” J. Phys. D: Appl. Phys., vol. 44, no. 43, p. 435202, 2011.

2. G. Kokkoris and others, “A global model for SF6 plasmas coupling reaction kinetics in the gas phase and on the surface of the reactor walls,” J. Phys. D: Appl. Phys, vol. 42, no. 5, p. 055209, 2009.

3. L. Lallement, and others, “Global model and diagnostic of a low-pressure SF6/Ar inductively coupled plasma,” Plasma Sources Sci. Technol., vol. 18, no. 2, p. 025001, 2009.

4. L.G. Christophorou and J.K. Olthoff, “Electron interactions with SF6,” J. Phys. Chem. Ref. Data, vol. 29, no. 3, pp. 267–330, 2000 ( www.lxcat.net, retrieved May 2024).

5. V. Gedeon and others, “B-spline R-matrix-with-pseudostates calculations for electron-impact excitation and ionization of fluorine,” Phys. Rev. A, vol. 89, no. 5, p. 052713, 2014.

6. Biagi database, www.lxcat.net, retrieved on 2024.

7. Morgan database, www.lxcat.net, retrieved on 2024.

8. Phelps database, www.lxcat.net, retrieved 2017.

Plasma_Module/Global_Modeling/sf6_argon_global_model

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

Add some parameters to be used in the model.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Qfeed	100	100	Feed mass flow in SCCM
Temp	315[K]	315 K	Gas temperature
pA	20[mTorr]	2.6664 Pa	Gas pressure
Pinp	2000[W]	2000 W	Power input from source
abs_to_inp	0.5	0.5	Power absorption ratio
Pabs	Pinp*abs_to_inp	1000 W	Power absorbed by plasma
xAr	0.1	0.1	Argon mole fraction
width	19[cm]	0.19 m	Chamber radius
height	17[cm]	0.17 m	Chamber height
lambda_diff	((pi/height)^2+(2.405/width)^2)^-0.5	0.044644 m	Diffusion length for heat

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Set the domain dimensions. The volume and surface areas used in the global model of the reactor are obtained automatically from this geometry.

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type width.

In the text field, type height.

Click

Plasma (plas)

Choose to solve for a global model of a constant pressure reactor and include the heavy species energy equation.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. Select the checkbox.

Click to expand the section. Select the checkbox.

Plasma Model 1

Set the temperature, pressure, mass flow, power absorbed by the electrons, mean kinetic energy lost per electron lost, an estimation of the plasma sheath voltage drop (for the mean kinetic energy lost per ion lost), surface temperature, and the diffusion length for the heat equation.

In the window, under > click .

In the window for , locate the section.

In the T0 text field, type Temp.

In the pA text field, type pA.

Locate the section. In the Qsccm text field, type Qfeed.

In the Tf text field, type Temp.

Locate the section. In the Pabs text field, type Pabs.

In the εe text field, type 2*plas.Te.

In the εi text field, type 10[V].

Locate the section. In the TS text field, type Temp.

In the ΛS text field, type lambda_diff.

The Plasma Chemistry Import feature

The next steps have instructions to use the feature to import a file that automatically creates the SF6/Ar plasma chemistry.

The following is set or created automatically:

Species properties

Reaction group features for SF6, F2 and Ar

Surface reactions

The documentation accompanying the feature contains more information about the file structure and what can be set automatically.

Plasma Chemistry Import 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file SF6_Ar_plasma_chemistry.txt.

Click

Set properties for some species.

Set SF6 to be the species that the mass fraction is found from mass constraint.

Species: SF6

In the window, expand the > > node, then click .

In the window for , locate the section.

Select the checkbox.

Locate the section. In the xfeed text field, type 1-xAr.

Species: Ar

In the window, click .

In the window for , locate the section.

In the xfeed text field, type xAr.

Species: F2+

In the window, click .

In the window for , click to expand the section.

In the Δh text field, type 15.69.

Species: F+

In the window, click .

In the window for , locate the section.

In the Δh text field, type 17.687.

Species: Ars

In the window, click .

In the window for , locate the section.

In the Δh text field, type 11.50.

Species: Ar+

In the window, click .

In the window for , locate the section.

In the Δh text field, type 15.80.

Group - Species

In the window, collapse the > > node.

The surface reactions used in the model were created automatically but it is still necessary to specify the boundaries where they take place.

Surface Reactions - Neutrals

In the window, click .

In the window for , locate the section.

From the list, choose .

Surface Reactions - Ions

In the window, click .

In the window for , locate the section.

From the list, choose .

In the following prepare a base study with an input power of 2000 W, the results of this base study can be used as the initial values for the following parametric studies.

Study 1 - Base Case

In the window, click .

In the window for , type Study 1 - Base Case in the text field.

Locate the section. Clear the checkbox.

In the toolbar, click

In the following measure number densities of all the species, which can be used to benchmark with reference results.

Results

Global Evaluation 1

In the window, expand the node.

Right-click > and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
plas.n_wSF6	1/m^3	SF6
plas.n_wSF5	1/m^3	SF5
plas.n_wSF4	1/m^3	SF4
plas.n_wSF3	1/m^3	SF3
plas.n_wSF2	1/m^3	SF2
plas.n_wSF	1/m^3	SF
plas.n_wS	1/m^3	S
plas.n_wF	1/m^3	F
plas.n_wF2	1/m^3	F2
plas.n_wSF5_1p	1/m^3	SF5+
plas.n_wSF4_1p	1/m^3	SF4+
plas.n_wSF3_1p	1/m^3	SF3+
plas.n_wSF2_1p	1/m^3	SF2+
plas.n_wSF_1p	1/m^3	SF+
plas.n_wS_1p	1/m^3	S+
plas.n_wF_1p	1/m^3	F+
plas.n_wF2_1p	1/m^3	F2+
plas.n_wSF6_1m	1/m^3	SF6-
plas.n_wSF5_1m	1/m^3	SF5-
plas.n_wSF4_1m	1/m^3	SF4-
plas.n_wSF3_1m	1/m^3	SF3-
plas.n_wSF2_1m	1/m^3	SF2-
plas.n_wF_1m	1/m^3	F-
plas.n_wF2_1m	1/m^3	F2-
plas.ne	1/m^3	e

Click

In the following prepare a parameterization of the input power from 2000 to 300 W with a step of 100 W.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 2 - Power Sweep

In the window for , type Study 2 - Power Sweep in the text field.

Locate the section. Clear the checkbox.

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Find the subsection. From the list, choose .

From the list, choose .

Click to expand the section. Select the checkbox.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
Pinp (Power input from source)		W

In the table, click to select the cell at row number 1 and column number 2.

Click

In the dialog, type 2000 in the text field.

In the text field, type -100.

In the text field, type 300.

Click .

In the toolbar, click

In the following create plots for the number densities of neutral, positive, and negative species respectively as a function of input power. The electron temperature, electronegativity and gas temperature are also plotted with respect to input power.

Results

Species densities vs. Power - Neutral

In the toolbar, click

In the window for , type Species densities vs. Power - Neutral in the text field.

Locate the section. From the list, choose .

Click to expand the section. From the list, choose .

Locate the section.

Select the checkbox. In the associated text field, type Power (W).

Select the checkbox. In the associated text field, type Number density (1/m<sup>3</sup>).

Locate the section. Select the checkbox.

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
plas.n_wSF6	1/m^3	SF6
plas.n_wSF5	1/m^3	SF5
plas.n_wSF4	1/m^3	SF4
plas.n_wSF3	1/m^3	SF3
plas.n_wSF2	1/m^3	SF2
plas.n_wSF	1/m^3	SF
plas.n_wS	1/m^3	S
plas.n_wF2	1/m^3	F2
plas.n_wF	1/m^3	F
plas.n_wAr	1/m^3	Ar
plas.n_wArs	1/m^3	Ars

Click to expand the section. Find the subsection. Clear the checkbox.

In the toolbar, click

Species densities vs. Power - Positive

In the window, right-click and choose .

In the window for , type Species densities vs. Power - Positive in the text field.

Locate the section. From the list, choose .

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

Click

In the table, enter the following settings:


Expression	Unit	Description
plas.n_wSF5_1p	1/m^3	SF5+
plas.n_wSF4_1p	1/m^3	SF4+
plas.n_wSF3_1p	1/m^3	SF3+
plas.n_wSF2_1p	1/m^3	SF2+
plas.n_wSF_1p	1/m^3	SF+
plas.n_wS_1p	1/m^3	S+
plas.n_wF2_1p	1/m^3	F2+
plas.n_wF_1p	1/m^3	F+
plas.n_wAr_1p	1/m^3	Ar+

In the toolbar, click

Species densities vs. Power - Negative

In the window, right-click and choose .

In the window for , type Species densities vs. Power - Negative in the text field.

Locate the section. From the list, choose .

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

Click

In the table, enter the following settings:


Expression	Unit	Description
plas.ne	1/m^3	e
plas.n_wF2_1m	1/m^3	F2-
plas.n_wF_1m	1/m^3	F-
plas.n_wSF2_1m	1/m^3	SF2-
plas.n_wSF3_1m	1/m^3	SF3-
plas.n_wSF4_1m	1/m^3	SF4-
plas.n_wSF5_1m	1/m^3	SF5-
plas.n_wSF6_1m	1/m^3	SF6-