Plasma Modeling

There are three methods typically employed to model plasmas. All methods essentially involve solving an appropriate transport equation along with Maxwell’s equations.

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The Kinetic approach solves for the distribution function for the ions and electrons in a plasma by solving the Boltzmann equation. The distribution function can also be computed using a Lagrangian approach, where ordinary differential equations are solved for the electron position and velocity, as is the case in particle-in-cell (PIC) codes. The advantage of this approach is that it allows for the electron energy distribution function to take an arbitrary shape in energy space and can also show effects that are not captured by fluid models. Drawbacks include difficulties associated with implementing arbitrary plasma chemistries and high computational cost.

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The Fluid approximation describes the plasma in terms of macroscopic quantities by assuming a specific form of the distribution function and taking velocity moments of the Boltzmann equation. This results in a set of coupled partial differential equations that can be solved using the finite element method. The Plasma Module uses the fluid approximation. Advantages of the fluid approximation are that it is efficient to solve the resulting equations and straightforward to define arbitrarily complex plasma chemistries. It is also easy to couple the electron dynamics to the electromagnetic fields, which are also computed using the finite element method.

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Hybrid methods treat some of the components of a plasma as a fluid and others kinetically. Usually, the electrons are treated kinetically by solving the Boltzmann equation or using a Monte Carlo method, and the ions and other radical species are computed using a fluid model. In this sense, hybrid methods offer a compromise between the fidelity of kinetic models and the speed and convenience of fluid models.

Modeling the interaction between the plasma and an external circuit is an important part of understanding the overall characteristics of a discharge. This module has tools to add circuit elements directly to a 1D, 2D, or 3D model, or import an existing SPICE Netlist into the model. The plasma chemistry is specified either by loading in sets of collision cross sections from a file or by adding reactions and species to the Model Builder.

Physics Guide

The complexity of plasma modeling lies in the fact that it combines elements of reaction engineering, statistical physics, fluid mechanics, physical kinetics, heat transfer, mass transfer, and electromagnetics. The net result is a true multiphysics problem involving complicated coupling between the different physics. The module is designed to simplify the process of setting up a self-consistent model of a low-temperature plasma.

The physics interfaces include all the necessary tools to model plasma discharges, beginning with a Boltzmann Equation, Two-Term Approximation solver that computes the electron transport properties and source coefficients from a set of electron impact collision cross sections. This interface makes it possible to determine many of the interesting characteristics of a discharge by providing input properties such as the electric field and the electron impact reactions that make up the plasma chemistry, without solving a space-dependent problem.

For space-dependent models, the reactions and species which make up the plasma chemistry are conveniently managed in the Model Builder. When the fluid velocity and gas temperature are of interest, there are physics interfaces available for laminar flow and heat transfer. There are several options available when coupling the charged species transport to the electromagnetic fields.

Physics Interface List by Space Dimension and Preset Study Type


Physics interface		Icon	Tag	Space Dimension	Available Study Type
AC/DC
Electrical Circuit			cir	Not space dependent	stationary; frequency domain; time dependent; frequency domain; eigenfrequency
Electrostatics1			es	all dimensions	stationary; time dependent; stationary source sweep
Fluid Flow
	Single-Phase Flow
	Laminar Flow1		spf	3D, 2D, 2D axisymmetric	stationary; time dependent
Plasma
Boltzmann Equation, Two-Term Approximation			be	0D	mean energies; reduced electric fields; time dependent
Plasma, Time Periodic			ptp	1D, 2D axisymmetric	time periodic; time periodic to time dependent
Plasma			plas	all dimensions	time dependent; stationary
Inductively Coupled Plasma2, 4			—	3D, 2D, 2D axisymmetric.	frequency-transient; frequency-stationary
Inductively Coupled Plasma with RF Bias2, 4			—	2D, 2D axisymmetric.	Frequency-Time periodic; time periodic to time dependent
Microwave Plasma3, 4			—	3D, 2D, 2D axisymmetric	frequency-transient
	Electric Discharges
	Corona Discharge4		—	all dimensions	stationary
	Electrical Breakdown Detection		ebd	3D, 2D, 2D axisymmetric	time dependent
	Equilibrium Discharges
	Equilibrium Discharges, Out-of-Plane Currents2,4		—	2D, 2D axisymmetric	stationary; time dependent; frequency-transient; frequency-stationary
	Equilibrium Discharges, In-Plane Currents2,4		—	2D, 2D axisymmetric	stationary; time dependent; frequency-transient; frequency-stationary
	Equilibrium Discharges2,4		—	3D	stationary; time dependent; frequency-transient; frequency-stationary
	Species Transport
	Drift Diffusion		dd	all dimensions	time dependent
	Heavy Species Transport		hs	all dimensions	time dependent
	Charge Transport		ct	all dimensions	stationary
1 This physics interface is included with the core COMSOL package but has added functionality for this module. 2 Requires the addition of the AC/DC Module. 3 Requires the addition of the RF Module. 4 This physics interface is a predefined multiphysics coupling that automatically adds all the physics interfaces and coupling features required.

AC/DC Interfaces

The AC/DC Branch chapter describes the two physics interfaces available with this module under the AC/DC branch of the Model Wizard. Use the Electrostatics interface to compute the electrostatic field in the plasma caused by separation of space charge between the electrons and ions. The Plasma Module enhances the interface included with the basic COMSOL license.

Fluid Flow Interfaces

The Fluid Flow Branch describes the Laminar Flow interface, which has a few additional features available for this module compared to the basic license.

Boltzmann Equation, Two-Term Approximation Interface

The Boltzmann Equation, Two-Term Approximation interface

computes the electron energy distribution function (EEDF) from a set of collision cross sections for some mean discharge conditions. The interface can be used as a preprocessing stage before solving a full space dependent model. The main purpose of this interface is to compute electron source coefficients and transport properties.

Drift Diffusion Interface

The Drift Diffusion interface

is used to compute the electron density and mean electron energy for any type of plasma. A wide range of boundary conditions are available to handle secondary emission, thermionic emission, and wall losses. This interface rarely needs to be used by itself as it makes up part of the application specific interfaces described later.

charge transport

The Charge Transport interface computes the density of charge carriers in a background gas under the assumption that the transport is dominated by migration. This is typically only used as part of the Corona Discharge multiphysics interface (see below).

The Heavy Species Interface

The Heavy Species Transport interface

solves a mass balance equation for all nonelectron species. This includes charged, neutral, and electronically excited species. The interface also allows you to add electron impact reactions, chemical reactions, surface reactions, volumetric species, and surface species via the Model Builder. This interface rarely needs to be used by itself as it makes up part of the application specific interfaces described later.

electrical breakdown detection

The Electrical Breakdown Detection interfaces uses an approximate method to determine if electrical breakdown will occur in a given design by integrating Townsend growth coefficients along electric field lines.

Application Specific Interfaces

The Plasma Module provides four application specific multiphysics interfaces for modeling the most common types of discharge.

Plasma

The Plasma interface

can be used to model positive columns, DBD discharges, glow discharges, and corona discharges. The complicated coupling between the electron transport, heavy species transport, and electrostatic field is handled automatically by the software. Furthermore, the secondary emission flux from ion bombardment on an electrode is automatically computed and used in the boundary condition for electrons.

Plasma, Time Periodic

The Plasma, Time Periodic interface

can be used to model capacitively coupled plasmas. Instead of solving the problem in the time domain, the periodic steady state solution is computed. This avoids having to solve for tens or hundreds of thousands of RF cycles, which is typically how long it takes before the plasma reaches the periodic steady state solution. This approach maintains all the non–linearity of the model while dramatically reducing computation time. The physics interface accomplishes this by attaching an extra dimension to the underlying mathematical equations representing one RF cycle, and enforcing periodic boundary conditions in the aforementioned extra dimension.

Inductively Coupled Plasma

The Inductively Coupled Plasma interface

can be used to model discharges sustained through induction currents. These discharges typically operate in the MHz frequency range. Inductively coupled plasmas (ICP) are important in plasma processing and plasma sources because the plasma density can be considerably higher than in capacitively coupled discharges. Inductively coupled plasmas are also attractive from the modeling perspective because they are relatively straightforward to model, due to the fact that the induction currents can be solved for in the frequency domain. This means that the RF cycle applied to the driving coil does not need to be explicitly resolved when solving. As such, the quasi steady-state solution is reached in relatively few time steps.

Inductively Coupled Plasma with rf bias

The Inductively Coupled Plasma with Rf Bias interface

can be used to model discharges sustained through induction currents and that have a periodic RF biased electrode. This interface uses the Plasma, Time Periodic interface to solve the periodic steady state for the RF bias. The inductive currents are solved in the frequency domain as in the Inductively Coupled Plasma interface. This type of discharges are interesting because the plasma density and the ions flux at a surface can be controlled somewhat independently.

Microwave Plasma

The Microwave Plasma interface

can be used to model discharges which are sustained through heating of electrons due to electromagnetic waves. These discharges typically operate in the GHz frequency range. Wave–heated discharges usually fall into one of two categories: discharges with no external DC magnetic field and discharges with a high intensity static magnetic field. If a suitably high DC magnetic field is present then electron cyclotron resonance (ECR) can occur where electrons continually gain energy from the electric field over 1 RF period. Modeling microwave plasmas involves solving equations for the electron density, mean electron energy, heavy species, the electrostatic potential, and the high frequency electric field. The high frequency electric field is computed in the frequency domain and losses are introduced via a complex plasma conductivity.

Corona Discharge

The Corona Discharge interface uses a simplified charge transport model coupled with electrostatics to provide an approximate method of computing the charge density and the electrostatic field in corona discharges.

Equilibrium Discharges, Out-of-Plane Currents

The Equilibrium Discharges, Out-of-plane Currents

multiphysics interface, available in 2D and 2D axisymmetric, is used to study equilibrium discharges in a magnetohydrodynamics (MHD) framework where the currents are out-of-plane. This multiphysics interface adds three single physics interfaces: Magnetic Fields, Heat Transfer in Fluids, and Laminar Flow, together with several multiphysics coupling features. The multiphysics couplings add the MHD coupling between the Magnetic Fields and the Laminar Flow interfaces. The multiphysics couplings also add heating and cooling of the equilibrium plasma by enthalpy transport, Joule heating and radiation loss.

Equilibrium Discharges, In-Plane Currents

The Equilibrium Discharges, in-plane Currents

multiphysics interface, available in 2D and 2D axisymmetric, is used to study equilibrium discharges in a magnetohydrodynamics (MHD) framework where the currents are in-plane. This multiphysics interface adds three single physics interfaces: Magnetic and Electric Fields, Heat Transfer in Fluids, and Laminar Flow, together with several multiphysics coupling features. The multiphysics couplings add the MHD coupling between the Magnetic and Electric Fields and the Laminar Flow interfaces. The multiphysics couplings also add heating and cooling of the equilibrium plasma by enthalpy transport, Joule heating and radiation loss as well as special boundary conditions to model the ion and electron heating at the plasma boundaries.

Equilibrium Discharges

The Equilibrium Discharges

multiphysics interface, available in 3D, is used to study equilibrium discharges in a magnetohydrodynamics (MHD) framework. This multiphysics interface adds three single physics interfaces: Magnetic and Electric Fields, Heat Transfer in Fluids, and Laminar Flow, together with several multiphysics coupling features. The multiphysics couplings add the MHD coupling between the Magnetic and Electric Fields and the Laminar Flow interfaces. The multiphysics couplings also add heating and cooling of the equilibrium plasma by enthalpy transport, Joule heating and radiation loss as well as special boundary conditions to model the ion and electron heating at the plasma boundaries.

Limitations of the Plasma Module

The Plasma Module cannot currently model magnetrons and space thrusters due to the fact that the perpendicular versus cross field electron mobility and diffusivity can be on the order of 108. This poses severe numerical challenges and inevitably there is unphysical numerical diffusion across the magnetic field lines. As such, a truly self consistent model of a Hall thruster or magnetron is not currently possible, although it is still possible to gain valuable insight into the physics. The Plasma module also cannot model plasmas that are not collisional enough for the fluid-type equations used to be valid. For reactors with characteristic dimensions of 10 cm the lower possible pressure would be 20 mTorr.