Microwave plasmas, or wave-heated discharges, find applications in many industrial areas such as semiconductor processing, surface treatment, and the abatement of hazardous gases. This blog post describes the theoretical basis of the Microwave Plasma interface available in the Plasma Module.

### Introduction

Microwave plasmas are sustained when electrons can gain enough energy from an electromagnetic wave as it penetrates into the plasma. The physics of a microwave plasma is quite different depending on whether the TE mode (out-of-plane electric field) or the TM mode (in-plane electric field) is propagating. In both cases, it is not possible for the electromagnetic wave to penetrate into regions of the plasma where the electron density exceeds the critical electron density (around 7.6×1016 1/m3 for 2.45 GHz). The critical electron density is given by the formula:

n_e = \frac{\epsilon_0 m_e \omega^2}{e^2}

where \epsilon_0 is the permittivity of free space, m_e is the electron mass, \omega is the angular frequency, and e is the electron charge. This corresponds to the point at which the angular frequency of the electromagnetic wave is equal to the plasma frequency. The pressure range for microwave plasmas is very broad. For electron cyclotron resonance (ECR) plasmas, the pressure can be on the order of around 1 Pa, while for non-ECR plasmas, the pressure typically ranges from 100 Pa up to atmospheric pressure. The power can range from a few watts to several kilowatts. Microwave plasmas are popular due to the cheap availability of microwave power.

### Microwave Discharge Theory

In order to understand the nuances associated with modeling microwave plasmas, it is necessary to go over some of the theory as to how the discharge is sustained. The plasma characteristics on the microwave time scale are separated from the longer term plasma behavior, which is governed by the ambipolar fields.

#### Electromagnetic Fields

In the Plasma Module, the electromagnetic waves are computed in the frequency domain and all other variables in the time domain. In order to justify this approach, we start from Maxwell’s equations, which state that:

(2)

\nabla \times \mathbf{\tilde{E}} = -\frac{\partial \mathbf{\tilde{B}}} {\partial t}

(3)

\nabla \times \mathbf{\tilde{H}} = \mathbf{\tilde{J}}_p+\frac{\partial \mathbf{\tilde{D}}} {\partial t}

where \mathbf{\tilde{E}} is the electric field (V/m), \mathbf{\tilde{B}} is the magnetic flux density (T), \mathbf{\tilde{H}} is the magnetic field (A/m), \mathbf{\tilde{J}_p} is the plasma current density (A/m2), and \mathbf{\tilde{D}} is the electrical displacement (C/m2). The tilde is used to denote that the field is varying in time with frequency \omega/2 \pi. The plasma current density can be approximated by this expression:

(4)

\mathbf{\tilde{J}}_p = -e n_e \mathbf{\tilde{v}_e}

where e is the unit charge (C), n_e is the electron density (1/m3), and \mathbf{\tilde{v}_e} is the mean electron velocity under the following two assumptions:

- Ion motion is neglected with respect to the electron motion on the microwave time scale.
- The electron density is assumed constant in space on the microwave time scale.

The mean electron velocity on the microwave time scale, \tilde{\mathbf{v}}_e, is obtained by assuming a Maxwellian distribution function and taking a first moment of the Boltzmann equation:

(5)

\frac{\partial \mathbf{\tilde{v}_e}}{\partial t} = -\frac{e}{m_e}\mathbf{\tilde{E}}-\nu_m \mathbf{\tilde{v}}_e

where m_e is the electron mass (kg) and \nu_m is the momentum transfer frequency between the electrons and background gas (1/s). As pointed out in, the equations are linear, so we can take a Fourier transform of the equation system. Taking a Fourier transform of equation (5) gives:

(6)

j \omega \mathbf{\bar{v}}_e+\nu_m \mathbf{\bar{v}}_e =-\frac{e}{m_e} \mathbf{\bar{E}}