In-Plane Microwave Plasma

Wave-heated discharges can be very simple, such as when a plane wave is guided into a reactor using a waveguide, or very complicated, such as in the case with ECR (electron cyclotron resonance) reactors. In this simple example, a wave is launched down a waveguide where it intersects a flowing gas at low pressure, resulting in formation of an argon plasma. Microwave plasmas typically have high number density without requiring significant power absorption. The plasma potential is also quite low compared to capacitive or DC discharges. Therefore, microwave plasmas share many of the characteristics of inductive discharges.

Figure 1: Diagram of geometry modeled. A TE or TM mode wave enters from the top port and intersects the gas flow leading to the formation of a plasma.

The model requires the Plasma Module and RF Module.

Model Definition

The electron density and mean electron energy are computed by solving a pair of drift-diffusion equations for the electron density and mean electron energy. Convection of electrons due to fluid motion is neglected. For detailed information on electron transport, see Theory for the Drift Diffusion Interface in the Plasma Module User’s Guide.

The electron source Re and the energy loss due to inelastic collisions Rε are defined later. The electron diffusivity, energy mobility, and energy diffusivity are computed from the electron mobility using:

The source coefficients in the above equations are determined by the plasma chemistry using rate coefficients. Suppose that there are M reactions that contribute to the growth or decay of electron density and P inelastic electron-neutral collisions. In general, P >> M. In the case of rate coefficients, the electron source term is given by:

where xj is the mole fraction of the target species for reaction j, kj is the rate coefficient for reaction j (SI unit: m3/s), and Nn is the total neutral number density (SI unit: 1/m3). The electron energy loss is obtained by summing the collisional energy loss over all reactions:

where Δεj is the energy loss from reaction j (SI unit: V). The rate coefficients can be computed from cross section data by the following integral:

where γ = (2q/me)1/2 (SI unit: C1/2/kg1/2), me is the electron mass (SI unit: kg), ε is energy (SI unit: V), σk is the collision cross section (SI unit: m2), and f is the electron energy distribution function. In this case, a Maxwellian EEDF is assumed.

For nonelectron 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 a microwave reactor the high frequency electric field is computed in the frequency domain using the following equation:

The relationship between the plasma current density and the electric field becomes more complicated in the presence of a DC magnetic field. The following equation defines this relationship:

Here, σ is the plasma conductivity tensor, which is a function of the electron density, collision frequency, and magnetic flux density. Using the definitions:

where q is the electron charge, me is the electron mass, ne is the collision frequency, and ω is the angular frequency of the electromagnetic field. In this example, the inverse of the plasma conductivity is diagonal because there is no external DC magnetic field:

The gas flow is modeled assuming a constant velocity in the x-direction.

plasma chemistry

The chemical mechanism for the plasma consists of only 3 species and 7 reactions (electron impact cross-section are obtained from Ref. 2):

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	Ars+Ars=>e+Ar+Ar+	Penning ionization	-
7	Ars+Ar=>Ar+Ar	Metastable quenching	-

Stepwise ionization (reaction 5) can play an important role in sustaining low pressure argon discharges. Excited argon atoms are consumed via superelastic collisions with electrons, quenching with neutral argon atoms, ionization or Penning ionization where two metastable argon atoms react to form a neutral argon atom, an argon ion and an electron. Reaction number 7 is responsible for heating of the gas. The 11.5 eV of energy which was consumed in creating the electronically excited argon atom is returns to the gas as thermal energy when the excited metastable quenches. In addition to volumetric reactions, the following surface reactions are implemented:

Table 2: table of surface reactions.
reaction	formula	Sticking Coefficient
1	Ars=>Ar	1
2	Ar+=>Ar	1

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

electrical excitation

The plasma is sustained through absorption of electromagnetic waves. The Port boundary condition is used to excite the plasma. A total absorbed power of 30 W is fed into the port.

In a second study, the electrical excitation is changed to the TM mode, where the electric field has only an in-plane component. The total absorbed power is the same as the TE mode case.

Results and Discussion

The electron density is plotted in Figure 2 and peaks slightly downstream of the crossing point. The electron density is also slightly asymmetric in the y-plane due to the fact that the electromagnetic waves are absorbed asymmetrically. The electron “temperature” is plotted in Figure 3. The electron temperature is relatively low everywhere, in part due to the high operating pressure (1 Torr). The electron “temperature” peaks directly underneath the waveguide where the wave is absorbed. The norm of the electric field can be seen in Figure 5. The electric field is high inside the waveguide and there are no losses. Once the wave is exposed to the plasma, the energy is absorbed by the electrons, raising the electron temperature enough to generate new electrons through ionization. The ionization rate is high enough to sustain the plasma. The contour of the critical plasma density is also plotted in Figure 5. The electromagnetic wave cannot penetrate into regions exceeding the critical plasma density. Since the electron “temperature” is relatively low, one would expect the plasma potential to be low. The plasma potential is plotted in Figure 4 and is only around 10 volts.

In the TE mode, electrons do not experience any change in the high-frequency electric field during the microwave time scale. This means that the phase coherence between the electrons and electromagnetic waves is only destroyed through collisions with the background gas. The loss of phase coherence between the electrons and high-frequency fields is what results in energy gain for the electrons. Therefore, the momentum collision frequency is simply given by:

where νe is the collision frequency between the electrons and neutrals.

When switching to the TM mode, the electron density, electron “temperature” and plasma potential are quite similar to the TE mode case. This can be seen in Figure 7, Figure 8 and Figure 9. The electric field is very different however, Figure 9. The electric field cannot penetrate past the contour of critical electron density, and has its greatest magnitude in this location. The power deposition, Figure 10 is also highly localized to the contour of critical electron density. The TM mode causes in-plane motion of the electrons on the microwave time scale, so in regions where the high-frequency electric field is significant (the contour where the electron density is equal to the critical density), the time-averaged electric field experienced by the electrons may be nonzero. This destroys the phase coherence between the electrons and the fields, causing the electrons to gain energy. This is an example of a nonlocal kinetic effect, which is difficult to approximate with a fluid model. However, since this effect is similar to collisions with a background gas, the nonlocal effects can be approximated by adding an effective collision frequency to the momentum collision frequency:

where νeff is the effective collision frequency to account for nonlocal effects. In this example, since the Doppler broadening parameter is set to 20, this corresponds to an effective collision frequency of:

Figure 2: Plot of the electron density.

Figure 3: Plot of the electron temperature in the reactor.

Figure 4: Plot of the plasma potential in the reactor.

Figure 5: Plot of the electric field norm. The white contour represents the critical plasma density, where the electron density is equal to 7.6E16[1/m3].

Figure 6: Plot of the power deposition into the plasma. The white contour represents the critical plasma density, where the electron density is equal to 7.6E16 1/m3.

Figure 7: Electron density for the TM mode case.

Figure 8: Plot of the electron temperature for the TM mode case.

Figure 9: Close up of the high frequency electric field norm for the TM mode case.

Figure 10: Close up of the power deposition into the plasma for the TM mode case.

Reference

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

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

Plasma_Module/Wave-Heated_Discharges/inplane_microwave_plasma

Modeling Instructions

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Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
P0	30[W]	30 W	Absorbed power

Geometry 1

Rectangle 1 (r1)

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Definitions

Walls

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Plasma (plas)

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Cross Section Import 1

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

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Reaction 1

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

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Reaction 2

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Species: Ar

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Species: Ar+

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Surface Reaction 1

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Surface Reaction 2

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Plasma Model 1

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10	x
0	y

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Initial Values 1

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Wall 1

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Ground 1

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Electron Outlet 1

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Species: Ars

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Outflow 1

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Species: Ar+

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Outflow 1

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Electromagnetic Waves, Frequency Domain (emw)

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Port 1

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Materials

Material 1 (mat1)

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In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	5	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic