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Plasma Modeling with COMSOL Multiphysics®

Why COMSOL Multiphysics?

• Multiphysics– Coupled phenomena

• Single physics– One integrated environment – different

physics and applications• Adaptable, no need for user-subroutines

– Create your own multiphysics couplings– Type in nonlinear expressions, look-up tables,

or function calls– Optional user-interfaces for working directly

with equations: algebraic, PDEs, and ODEs– Parameterize on anything

• High-Performance Computing (HPC):– Multicore & Multiprocessor– Clusters

COMSOL Multiphysics 4.4 Product Line

COMSOL Multiphysics Plasma Module

What is a Plasma?

• Definition• Plasmas are conductive assemblies of charged particles, neutrals and fields that exhibit collective effects.

• Industries• Lighting• Semiconductor• Military• Coating• …

• The following are the most common types of plasmas:

• Inductively coupled plasmas (Easy)• DC discharges (Easy, Magnetic Field enhanced Hard)• Microwave plasmas (Medium, ECR Hard)• Electrical breakdown (Hard)• Capacitively coupled plasmas (Hard)• Combined ICP/CCP reactor ( )• In each of the above, the mechanism of energy transfer from the electromagnetic fields

to the electrons is different.

Increasing difficulty to

model

Types of Plasma

The Plasma Module is designed for non-nuclear, low temperature plasmas (non-equilibrium discharges)

Components of a plasma

• A plasma consists of:– Electromagnetic fields (instantaneous)

– Electron energy (<nsec)

– Electron transport (nsec)

– Ion transport (usec)

– Excited species transport (0.1msec)

– Neutral gas flow and temperature (msecs)

• The broad range of characteristic timescales for the different components which make up the plasma creates computational difficulties.

• Plasma Models are computationally stiff in time.

Additional difficulties with plasma modeling

• Stiff in space (space charge separation needs to be resolved).

• Large number of degrees of freedom (many species).

• Strong coupling between electron energy and electromagnetic fields.

• Plasma chemistry data can be difficult to find or not exist at all.

• The COMSOL Multiphysics Plasma Module makes it easier to set up a plasma model, but some level of expertise is still required.

Plasma Modeling Physics Interfaces

• Drift Diffusion– Interface to compute the electron density and mean electron energy for any type of plasma.

• Heavy Species Transport– A mass balance interface for all non-electron species. This includes charged, neutral, and electronically

excited species.

• Electrostatics– Interface to compute the electrostatic field in the plasma caused by separation of space charge between

the electrons and ions.

• Boltzmann Equation, Two-Term Approximation– This interface allows you to compute the electron energy distribution function.

• Electrical Circuits– Interface to add an external electrical circuit to the plasma model.

Application Specific Interfaces

• Inductively Coupled Plasma– Used for studying discharges which are sustained by induction currents. The induction currents are solved

for in the frequency domain.

• DC Discharge– Used for studying discharges that are sustained by a static electric field.

• Microwave Plasma– Used for studying discharges that are sustained by electromagnetic waves.

• Capacitively Coupled Plasma– Used for studying discharges that are sustained by a time-varying electrostatic field.

Discretization

• Finite element is the default discretization for all plasma physics interfaces.

• A finite volume discretization is available for:

– DC discharge.– Capacitively Coupled Plasma.

• The option is available in the Advanced Properties section.

Theory

Electron Transport

• COMSOL solves a pair of drift diffusion equations for the electron density and electron energy density.

• The transport properties may be tensors and functions of the mean electron energy and a DC magnetic flux density.

Tensor Electron Transport Properties

• The Plasma Module allows you to use tensor’s for the electron mobility, diffusivity, energy mobility and energy diffusivity.

• This allows for example Hall thrusters to be modeled.

Plot of the electron mobility vs the components of the magnetic flux density

Electron Transport Boundary Conditions

• There are a variety of boundary conditions available for the electrons:

– Wall which includes the effects of:• Secondary electron emission.• Thermionic emission.• Electron reflection.

– Flux which allows you to specify an arbitrary influx for the electron density and electron energy density.

– Fixed electron density and mean electron energy (not recommended).

– Insulation.

Heavy Species Transport

• Transport of the heavy species (non-electron species) is determined from solving a modified form of the Maxwell-Stefan equations.

where:

• An integrated reaction manager is required in order to keep track of the electron impact reactions, reactions, surface reactions and species.

• The neutral gas flow is determined by the Compressible Navier-Stokes equations with a modified heat source.

• The last term on the right hand side of the energy equation can lead to substantial gas heating for molecular gases at higher pressures.

Bulk Gas Flow Transport

Surface Reactions and Species

• Surface reactions can be specified in terms of rate or sticking coefficients.

• The surface rate constant and sticking coefficient are given by:

• Surface adsorbed species and bulk species may be included to model deposition processes.

Plasma Chemistry

Plasma Chemistry

• A large amount of data needs to be assembled about the chemical processes which occur in a plasma, before you start the modeling process.

– A set of electron impact reactions and the corresponding cross section data Data can be found at http://fr.lxcat.net/data/set_type.php

– List of all the gas phase reactions which occur and the rate coefficients for each reaction

– List of all the surface reactions which occur in the system along with the rate coefficients, sticking coefficients and secondary electron emission probability

– Molecular weight, potential characteristic length and potential energy minimum for each species• There is predefined data for the most commonly encountered species in COMSOL

– Thermodynamic property data for each species if you are computing the gas temperature

Chemical Mechanisms

• The behavior of the plasma is largely determined by the plasma chemistry.• Argon is the simplest of all plasma chemistries; there are only 7 reactions and 4

species.• Plasma chemistries can be much more complex, air chemistry is over 300 reactions

and 30 species.

• Always use argon first when starting a new model!

Cross Section Data

• Cross section data is vital piece of data required to perform a plasma simulation.

• Cross section data allows the rate coefficient for a given reaction to be computed based on the EEDF using:

Plot of a set collision cross sections for molecular oxygen

Electromagnetics

Electrostatic fields

• The plasma potential is computed from Poisson’s equation.

• The space charge is computed from the number density of electrons and other charged species.

Electrostatic boundary conditions

• Due to the different transport timescales for ions and electrons, a surface charge can accumulate on dielectric surfaces:

• The surface charge is used as a boundary condition in the electrostatics physics interface:

• For inductive discharges we solve for the magnetic vector potential in the frequency domain:

• For microwave plasmas, we solve for the electric field in the frequency domain:

• If a static magnetic field is present the plasma conductivity may be a full tensor:

Electromagnetic fields

Shielding

Shielding Electrostatic Surface InductiveICP X X √DC √ X XMWP X X XBreakdown √ √ XCCP √ √ XCombined ICP/CCP √ √ √

Electron Cyclotron Resonance

• In Electron Cyclotron Resonance (ECR) the plasma conductivity is a highly non-linear function of the DC magnetic flux density.

• At the resonant flux density, Bres, electrons continually gain energy from the magnetic and electric fields.

Plot of the plasma conductivity vs the components of the magnetic flux density on a log scale

Example - GEC ICP reactor

5 turn coil, 1500Watts, 13.56MHz

Wafer pedestal

Dielectric material

Plasma forms here

A demo of this model can be found at: http://www.comsol.com/products/plasma/

Example

• The GEC ICP reactor is modeled in COMSOL Multiphysics.• The GEC cell is a standard reference cell designed by NIST for studying plasmas

and benchmarking simulations.• The gas is Argon, and the pressure is 20mtorr.• The following chemical reactions are considered:

Step 1 – Select Physics Interface

• Select the appropriate physics interface from the Model Wizard.

• In this example we are modeling an Inductively Coupled Plasma.

• Additional interfaces for capacitively coupled plasmas, direct current discharges and microwave plasmas.

Step 2 – Draw or Import the Geometry

Step 3 – Import Cross Section Data

Reactions and species automatically appear in the model tree

Import cross section data for the electron impact reactions from file

Step 4 – Define volume and surface reactions

Step 5 – Define the coil domains and current

Step 6 – Boundary Conditions

Step 7 – Mesh the geometry

Boundary layer meshing on the plasma volume allows us to resolve separation of space charge

Step 8 – Compute the solution

Step 9 – Examine the Results

Results

• The results agree well with experimental data for the electron density, electron temperature and plasma potential.

Ref: “An Inductively Coupled Plasma Source for the Gaseous Electronics Conference RF Reference Cell, J. Res. Natl. Inst. Stand. Technol. 100, 427 (1995)”

Model Library

• Product ships with 19 example models, all complete with documentation and step-by-step instructions.

• Example models for:– Capacitively coupled plasmas

– Chemical vapor deposition

– Direct current discharges

– Inductively coupled plasmas

– Solving the two-term Boltzmann equation

– Wave heated discharges

Inductively Coupled Plasma

• An electrodeless lamp has no electrodes and thus a long life.• Plot of the electron density (left) and ground state Mercury (right).• This model has 12 species and 96 reactions.

Electron number density in an electrodeless lamp Mole fraction of ground state Hg in an electrodeless lamp

Ion Energy Distribution Function

• The ion energy distribution function (IEDF) and angular distribution functions can be computed with the Particle Tracing Module.

• Plots below are for an inductively coupled plasma.

Ion energy distribution function on the wafer Ion angular distribution function on the wafer

Dielectric barrier discharge

• Two dielectric plated are separated by a small gap (0.1mm).• A sinusodial voltage is applied to one of the plates, the other is grounded.• A plasma periodically forms in the gap – the gap transitions from being an insulator

to a conductor.

Dielectric barrier discharge

• Extruded plots of electron current density (left) and ion current density (right).

• The y-axis represents time and the x-axis represents space.

Direct Current Discharge

• A direct current discharge is sustained through secondary emission of electrons from the cathode.

• The electric potential is close to uniform everywhere except in the cathode fall region where it decreases very rapidly.

Microwave plasma

• A microwave discharge is sustained when an electromagnetic wave is absorbed by the plasma.

• The wave can’t penetrate into regions where the critical electron density is exceeded.

Gas flow

Wave

Boltzmann Analysis, Argon

• Metastables tend to produce a high energy tail in the electron energy distribution function.

• This results in an increase in the ionization rate coefficient for the same mean electron energy.

END