Alpha to Gamma Transition

Capacitively coupled RF discharges can operate in two distinct regimes depending on the discharge power. In the low power regime, known as the α regime, the electric field oscillations heats and creates electrons. In the high power regime, known as γ regime, the discharge is sustained primarily by electron avalanche within the plasma sheath. This is initiated by secondary electrons emitted due to ion bombardment of the electrodes.

The two regimes present fundamental differences that can have an important impact in plasma applications. In the following, results are presented for the electron creation and power absorbed by the electrons that illustrate the main features of the two regimes.

The reader can found more information about this subject in Ref. 1 and in the references therein.

Model Definition

The model presented here is based on the one reported in Ref. 1 and simulates an atmospheric pressure argon plasma sustained by a periodic electric excitation at 13.56 MHz. The model is 1-dimensional and describes the space- and time-periodic evolution of several macroscopic properties of the plasma sustained within a 2 mm gap.

The electron mobility and other electron transport properties are automatically computed from the electron impact reactions. For the ions, the mobility is given as a function of the reduced electric field using a lookup table. The ion diffusivity is obtained from the Einstein relation where the ion temperature comes from a local field approximation. It is assumed that the molecular ion has the same mobility as the atomic ion.

Electric excitation

The driven electrode has a fixed power. This corresponds to the following expression and constraint on the electric potential:

(1)

(2)

The constraint in Equation 2 is used to compute the RF potential, Va such that a fixed amount of power is deposited into the plasma.

plasma chemistry

Argon plasmas have one of the simplest reactions schemes. The electronically excited states can be lumped into single states. At atmospheric pressure and low gas temperature three–body reactions are important and lead to the creation of dimers. We use a simplified plasma chemistry similar to the one in Ref. 1 that comprises 9 volumetric reactions involving electrons (electron impact cross-section are obtained from Ref. 2), atomic and molecular ions, and a lumped level representing the argon 4s.

Table 1: table of collisions and reactions modeled.
reaction	formula	type
1	e+Ar=>e+Ar	Elastic	0
2	e+Ar=>e+Ars	Excitation	11.5
3	e+Ars=>e+Ar	Superelastic	-11.5
4	e+Ar=>2e+Ar+	Ionization	15.8
5	e+Ars=>2e+Ar+	Ionization	4.24
6	e+Ar2+=>Ars+Ar	Excitation	-2.5
7	Ars+Ars =>e+Ar+Ar+	Penning ionization	-
8	Ars => Ar	Spontaneous emission	-
9	2Ar+Ar+=>Ar+Ar2+	3-body Ar2+ creation	-

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

Table 2: table of surface reactions.
reaction	formula	Sticking Coefficient	Secondary emission coefficient	Mean energy of secondary electrons (V)
1	Ars=>Ar	1	0	0
3	Ar+=>Ar	0	0.1	5.8
4	Ar2+=>2Ar	0	0.1	5.8

When excited states make contact with the wall, they revert to the ground state argon atom with some probability. The ions use their internal energy to extract one electron from the wall with a probability of 0.1 and a mean energy of 5.8 V. For the ions, the sticking coefficient is zero, meaning that losses to the wall are assumed to be due to migration only.

Results and Discussion

The temporal evolution of the one-dimensional electron density is conveniently represented by extruding the solution into two dimensions. The extra dimension represents the period fraction. In COMSOL Multiphysics this is accomplished by adding a Parametric Extrusion 1D dataset.

Figure 1and Figure 2 present, for the α regime, the spatial distribution of the electron number density and electron temperature (along the horizontal axis) extruded in time for one period of the external excitation along the vertical axis. These results show that this discharge is similar to lower pressure RF capacitively coupled discharges. The electron density is static in time in the discharge center. However, near the wall the electron density evolves in time driven by the applied time–varying electric field. The electron temperature near the wall is also strongly time modulated attaining peak electron temperatures of 3.5 eV. In the plasma bulk the electron temperature is much cooler than near the walls and is much less modulated since the electric field is much less intense than in the sheaths.

Figure 3 shows the power absorbed by the plasma and the averaged electron density as a function of the voltage amplitude. It is possible to identify two regions: (i) a low power region, corresponding to the α regime, where larger voltages amplitudes are needed to increase the power absorbed by the discharge; (ii) and a high power region, corresponding to the γ regime, where high discharge powers are attained for progressively lower voltages amplitudes. The electron density always increases with the applied voltage, however, it increases at a faster rate in the γ regime.

The different mechanisms for electron heating and electron creation for both the α and γ regimes are shown in Figure 4 through Figure 7. Figure 4 and Figure 5 show the power absorbed by the electrons. Figure 6 and Figure 7 show the electron production rate. The white line marks a region of constant charge density that roughly corresponds to the plasma-sheath transition. In the α regime a great part of the absorbed power and most of electron creation is localized along the plasma-sheath transition with the peak values occurring during the sheath expansion. In the γ regime the majority of the absorbed power and ionization occur at the sheath maximum expansion.

Figure 1: Electron number density spatial and time evolution in the α regime (100 W).

Figure 2: Electron temperature spatial and time evolution in the α regime (100 W).

Figure 3: Power absorbed by the plasma and average electron density as a function of the source voltage amplitude.

Figure 4: Power absorbed by the electrons in the α regime (100 W). The white line is a counter of constant charge density (0.01 C/m3) that roughly represents the boundary between plasma and sheath.

Figure 5: Power absorbed by the electrons in the γ regime (630 W). The white line is a counter of constant charge density (0.01 C/m3) that roughly represents the boundary between plasma and sheath.

Figure 6: Electron creation rate in the α regime (100 W). The white line is a counter of constant charge density (0.01 C/m3) that roughly represents the boundary between plasma and sheath.

Figure 7: Electron creation rate in the γ regime (630 W). The white line is a counter of constant charge density (0.01 C/m3) that roughly represents the boundary between plasma and sheath.

References

1. N Balcon, G.J.M. Hagelaar, and J.P. Boeuf, “Numerical model of an argon atmospheric pressure RF discharge”, IEEE Trans. Plasma Sci., vol. 36, p. 2782 (6pp), 2008.

2. Phelps database, www.lxcat.net, retrieved on 2017.

3. M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, John Wiley & Sons, 2005.

Plasma_Module/Capacitively_Coupled_Plasmas/alpha_to_gamma_transition

Modeling Instructions

In the following are the instructions to create a 1D model of a RF capacitively coupled plasma that is used to demonstrate the α to γ transition in an argon discharge. The transition is represented in the form of absorbed power as a function of applied voltage. To obtain such plot, a study is performed where power is swept from 50 to 700 W. After that study, two studies are performed to convert the periodic steady-state solutions from two different powers (from the first study) to the time domain. The conversion to the time domain is necessary in order to create surface plots representing the spatial and time variation of different quantities.

From the menu, choose .

New

In the window, click

Model Wizard

In the window, Select the interface and the study to compute the periodic steady state solution of the plasma.

click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Create the 2 mm gap where the plasma is sustained.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Interval 1 (i1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Coordinates (mm)
0
2

Add some parameters for the power and excitation frequency.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
f0	13.56[MHz]	1.356E7 Hz
Prf	1[W]	1 W

Add a nonlocal average coupling to compute the average electron density in postprocessing.

Definitions (comp1)

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Set the area perpendicular to the solved dimension and set the period and the number of elements in the extra dimension.

Plasma, Time Periodic (ptp)

In the window, under click .

In the window for , locate the section.

In the A text field, type 80[cm^2].

Locate the section. In the Pxd text field, type 1/f0.

In the N text field, type 30.

Set the temperature and use the default pressure of 1 atm for the background gas. The electron mobility and other transport parameters are by default to automatically computed from the set of electron impact reactions.

Plasma Model 1

In the window, under click .

In the window for , locate the section.

In the T text field, type 300[K].

Initial Values 1

In the window, click .

In the window for , locate the section.

In the ne,0 text field, type 1E15[1/m^3].

In the ε0 text field, type 2[V].

Import cross section data for argon.

Cross Section 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 Ar_xsecs.txt.

Click

Add other reactions to complete the plasma chemistry.

Electron Impact Reaction 6

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type e+Ar2+=>Ars+Ar.

Locate the section. From the list, choose .

In the Δε text field, type -2.5[V].

Locate the section. In the kf text field, type 7e-13[m^3/s]*N_A_const*(300/(ptp.Te*11600))^0.5.

Reaction 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type Ars+Ars=>e+Ar+Ar+.

Locate the section. In the kf text field, type 1.2e-15[m^3/s]*N_A_const.

Reaction 2

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type Ars=>Ar.

Locate the section. In the kf text field, type 5e5.

Reaction 3

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type Ar+Ar+Ar+=>Ar2++Ar.

Locate the section. In the kf text field, type 2.5e-43[m^6/s]*N_A_const^2.

Species: Ar

In the window, click .

In the window for , locate the section.

Select the check box.

Locate the section. From the list, choose .

Species: Ars

In the window, click .

In the window for , locate the section.

From the list, choose .

The electric fields generated will be rather high in the sheath, so use the local field approximation for the ion temperature, and a lookup table for the ion mobility.

Species: Ar+

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the list, choose .

Locate the section. From the list, choose .

The mobility of the molecular ion is assumed equal to the atomic ion for simplicity.

Species: Ar2+

In the window, click .

In the window for , locate the section.

Select the check box.

Locate the section. In the Mw text field, type 0.08[kg/mol].

Locate the section. From the list, choose .

From the list, choose .

Locate the section. From the list, choose .

Define surface losses for ions and argon excited state. For the ions, set the to zero to assume that ion losses are due to migration only, and set the to 0.1 to match the work that it is being followed.

Surface Reaction 1

In the toolbar, click