Plasma Modeling with COMSOL Multiphysics Copyright 2014 COMSOL. Any of the images, text, and equations here may be copied and modified for your own internal use. All trademarks are the property of their respective owners. See www.comsol.com/trademarks.
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
Product Suite COMSOL 5.0
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
Types of Plasma 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 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.
New in V5.0 Boundary Conditions There are special boundary conditions for the application specific interfaces: Metal Contact. Used for the interaction between the plasma and a metal wall. The terminal can be driven with a fixed voltage, fixed current or connected to a circuit. Dielectric Contact. Used for modeling the interaction of the plasma and a dielectric surface. Settings window for the Metal Contact feature
New in V5.0 Discretization Finite element is the default discretization for all plasma physics interfaces. A face centered finite volume discretization with Scharfetter- Gummel upwinding is also available. Discretization settings available in the application specific interfaces This was available as a beta version in V4.4, but is fully available in V5.0. The option is available in the Discretization section.
New in V5.0 Equilibrium Discharges The Plasma Module has added support for modeling discharges in local thermodynamic equilibrium (LTE) at atmospheric pressure. These interfaces do not solve for the electron, ion and neutral species, only the heat transfer, fluid flow and electromagnetics. They can be used to model things like atmospheric pressure plasma torches. Model Builder for an Equilibrium Inductively Coupled Plasma. The model consists of the standard Magnetic Fields, Heat Transfer in Fluids and Laminar Flow interfaces. In addition, a Plasma Heat Source multiphysics feature computes the plasma conductivity, heat generation and volumetric radiative loses.
New in V5.0 Equilibrium Discharge Interfaces There are 3 multiphysics interfaces available for modeling equilibrium discharges: Equilibrium DC Discharge (the equivalent of the DC Discharge interface) Equilibrium Inductively Coupled Plasma (the equivalent of the Inductively Coupled Plasma interface) Combined Inductive/DC Discharge (for discharges driven by induction currents and electric currents) Equilibrium Discharge interfaces can be found in the Plasma physics area.
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. Plot of the electron mobility vs the components of the magnetic flux density This allows for example Hall thrusters to be modeled.
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.
Bulk Gas Flow Transport 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.
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.
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:
Electromagnetic fields 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:
Shielding Shielding Electrostatic Surface Inductive ICP X X DC X X MWP X X X Breakdown X CCP X Combined 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. Plot of the plasma conductivity vs the components of the magnetic flux density on a log scale At the resonant flux density, Bres, electrons continually gain energy from the magnetic and electric fields.
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. Plot of a set collision cross sections for molecular oxygen Cross section data allows the rate coefficient for a given reaction to be computed based on the EEDF using:
Example - GEC ICP reactor 5 turn coil, 1500Watts, 13.56MHz Dielectric material Wafer pedestal 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 stepby-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. Wave Gas flow
New in V5.0 Inductively Coupled Plasma Torch An atmospheric argon plasma is sustained through induction currents. The temperature of the gas becomes very high, over 10,000[K]. Total power input in this model is 11[kW].
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