Kinetic PIC Modeling of Ion Beam Neutralization

Kinetic PIC Modeling of Ion Beam Neutralization

Ion beam neutralization is a significant challenge in electric propulsion and is needed to reduce beam electric fields, manage space charge, and reduce ion sputter of the spacecraft. To address these challenges, simulation can be used to investigate ion/electron interactions, predict space charge distributions, and optimize system properties, such as cathode location, beamlet current, ion density, and electron temperature. However, beam neutralization can be a challenging problem to simulate due to the extreme difference in mass between ions and electrons as well as the complex interaction of electromagnetic forces.

VizGrain is used to model the beam neutralization using a full, kinetic particle-in-cell (PIC) modeling approach. Both ions and electrons are modeled as kinetic particles. Typical simulation approaches in literature involve initializing both ion and electron beams from a single, pre-mixed source. This example simulates a configuration in which the electron beam source is located outside of the ion beam. This allows us to investigate the initial mixing and entrainment of the electrons.

The 2D simulation geometry is shown in the Figure 1.

Figure 1: Beam neutralization geometry and simulation setup

The spacecraft, shown in green, is set to a reference of 0 V. The electron beam, shown in blue, is injected with a temperature of 2 eV. The ion beam, shown in red, has a beamlet current of 5mA. To adequately resolve the electrostatic potential required to model ion and electron interaction, the Debye length must be resolved in the mesh.  The resulting mesh is ~400K cells using a structured/unstructured mixed meshing approach.

Animations of the results are shown in Figures 2 and 3. Figure 2 shows the electrostatic potential with the interaction of ions (red) and electrons (blue). Figure 3 colors the particles by velocity.

Figure 2: Beam neutralization animation of electrostatic potential, ions = red, electrons = blue

Figure 3: Beam neutralization animation with particles colored by velocity

The electrostatic potential of the ion beam attracts and entrains the electrons. As shown in the velocity plot, the electrons are accelerated as they enter the electrostatic potential well created by the ion beam, then slow down as they exit the beam. Additionally, an instability can be observed in the electron beam in which the electrons begin to oscillate around the ion beam. The magnitude of the oscillations will likely decrease as the neutralization approaches steady state.

Finally, Figure 4 compares the electrostatic potential with and without electron neutralization.

Figure 4:  Electrostatic potential with and without electron neutralization

As expected the electron beam greatly reduces the potential, effectively neutralizing the beam. Note that the minimal ion beam divergence for the case without neutralization is attributed to the low beamlet current.

This example demonstrates VizGrain’s PIC modeling capability for electric propulsion applications. VizGrain is the 1D/2D/3D kinetic particle module within the OverViz Simulation Suite that provides scalable parallel simulation for large, complex problems. OverViz is an industrial multiphysics framework for performing hybrid plasma, fluid flow, electromagnetic, particle simulations. For more information, please contact us at


ChemZone: Simple Platform for Complex Chemistries

ChemZone: Simple Platform for Complex Chemistries

Our flagship product, VizGlow, is a powerful, multi-dimensional software tool for simulating industrial, non-equilibrium plasmas. VizGlow can resolve spatial and time accurate plasma discharges with complex reactor geometries in three dimensions using a parallel simulation framework. While it is extremely robust, it is not always the best place to start when implementing a new chemistry mechanism or optimizing a plasma process. For these types of problems, we recommend starting with ChemZone.

ChemZone is a fast, 0-D reactor simulation tool for finite-rate chemistry mechanism and plasma process development. It is based on a well-stirred assumption in which the reactor is represented by a 0-D control volume with a single value for all discharge parameters. The main benefit of this approach is the very fast execution time. A single VizGlow simulation can take hours to days to complete, while a single ChemZone simulation can be completed in seconds. This allows for rapid evaluation of large parametric spaces, sensitivity studies, and uncertainty analyses for complex chemistries requiring thousands of simulations. And these exhaustive simulation approaches are becoming more relevant as semiconductor patterns shrink and process optimization becomes more challenging.

The recommended approach is to first use ChemZone to develop and refine reaction chemistries and explore process sensitivities. Once the mechanism is optimized, VizGlow can be used to spatially resolve the discharge and investigate other performance attributes, such as ion energy and angular distributions at substrates surfaces. This approach improves the accuracy and efficiency of the chemistry mechanism and is ideal for plasma process engineers developing and optimizing reactor processes.

A few examples of how plasma process engineers have used ChemZone for plasma process development include:

  • Perform simulated design of experiments to refine process controls (e.g. flow rate, pressure, power deposition)
  • Improve etch recipe through sensitivity analysis of process gas constituents
  • Explore effects of surface chemistries and heat transfer mechanisms to reduce process variability

ChemZone simulates the generic reactor configuration shown in Figure 1 using a multi-species, multi-temperature, spatially uniform gas mixture formulation. The configuration includes an inlet port with mixture gas, outlet port, volumetric energy source for bulk gas and electrons, volumetric energy loss, and an arbitrary number of surfaces consisting of different areas, materials, and bias voltages. The governing equations consist of species density, bulk temperature, electron temperature, and surface site fractions. Effects of parameters such as flow residence time, initial species concentrations, and external heat loss can also be captured.

Figure 1. Generalized model configuration for simulating plasma reactors in ChemZone

Key model outputs include chemical species densities for ions, electrons, neutrals, electron temperature, bulk gas temperature, net species production rates, forward and backward reaction rate coefficients, and contribution of different reactions to the net production of each species.


Mechanism Evaluation Example

Consider the mechanism evaluation of SF6/O2 plasma chemistry with 35 species and 147 reactions at 25mTorr. Figure 2 shows the transient evolution of dominant charged species and radicals until steady state is reached. F- is the dominant negative species and O2+ is the dominant positive species. The high concentration of negative ions (electronegative plasma) can significantly influence the discharge characteristics and therefore surface processes. Surface biasing or modification to the operating conditions could be used to enhance or suppress negative ion formation.

ChemZone Plasma Species Densities

Figure 2. Generalized model configuration for simulating plasma reactors in ChemZone

Next, Figure 3 shows the contribution to net electron production rates from all 147 reactions at steady state. From this plot, we are able to identify which specific reactions (reactions 16 and 29) contribute more to the production of electrons. To increase production, constituents contained within those reactions can be modified or increased. Reactions can also be enabled/disabled to explore their effects on the production rates. This can be rapidly evaluated through simulation.

ChemZone Plasma Production Rates

Figure 3. Contribution to net electron production rates from all 147 reactions at steady state

Furthermore, the information can be used to design a reduced mechanism.  For example, if certain species or reactions are shown to play a negligible role, they can sometimes be excluded from the mechanism. The reduced mechanism can be then used in VizGlow for a full spatially accurate simulation. A reduced mechanism will run much faster than the full mechanism, depending on the total number of reduced species.


Sensitivity Analyses

ChemZone can perform automated sensitivity analyses for reactor quantities of interest, such as species mass fraction, number densities, wall flux, temperature, and pressure. A user may simply check quantities of interest within the graphical user interface to enable them in the sensitivity study. When ChemZone is executed, the solver will systematically perturb the reaction rate coefficient of each reaction and quantify its effect on the mechanism.

Continuing with the SF6/O2 example, Figure 4 shows a sensitivity analysis of the one-directional wall flux of SF+ ions to the reaction rate coefficients at the wafer surface. The one-directional wall flux of SF+ is dominated by only a few reactions, while the rest play a negligible role. The first dominant reaction, reaction 5, is the electron impact ionization of SF forming SF+. The second, reaction 60, is the collision exchange between SF and O2+ forming SF+ and O2. These reactions are the dominate factors that create SF+ availability for interacting with the wafer surface. A similar investigation can be performed for any quantity of interest.

ChemZone Plasma Wall Flux

Figure 4. Sensitivity of the one-directional wall flux of SF+ ions to the reaction rate coefficients at the wafer surface


ChemZone is a fast reactor analysis tool for developing and optimizing chemistry mechanisms and plasma processes. ChemZone can also be used for applications in which the gas is not ionized, such as hydrocarbon combustion in internal combustion engines, gas turbines, and ignition delay timing. The primary benefit of ChemZone is its fast execution, allowing for rapid evaluation of large sensitivity studies, mechanism reductions, forward analyses, and process optimizations. ChemZone is a module within the OverViz Simulation Suite. OverViz is an industrial multiphysics framework for performing hybrid plasma, fluid flow, electromagnetic, particle simulations. For more information, please contact us at

Kinetic PIC Modeling of Ion Beam Neutralization

Dust Particle Charge-up in a Corona Discharge

This example demonstrates charge-up of initially charge-free dust particles as they move through a pin-corona plasma. Two sets of particles move through the corona with different initial velocities. The dust particles are subject to Lorentz forcing in the electric field and their resulting trajectories are compared. The macroscopic particle dynamics is modeled in VizGrain and coupled to a background plasma solution modeled in VizGlow.

The 2D planar pin geometry and mesh is shown in Figure 1. The pin is 5mm long with a base diameter of 3mm. The pin is modeled with a sharp tip. The geometry is discretized using a fully structured mesh with 79,050 cells. A voltage of -10kV is applied to the pin with a background gas of pure argon at 1atm.

Two sets of dust particles enter from the left boundary with an initial velocity in the horizontal direction. The dust particles have a mass of 1.4×1017 kg and a diameter of 300nm. The first set of particles has an initial, horizontal velocity of 150m/s. The second set of particles has an initial velocity of 50m/s. All particles are initially uncharged.

First, the background plasma and electric field is solved in VizGlow. Results from the plasma solution is shown in Figure 2.


The maximum electric potential is observed in the steady state solution around the pin geometry. The negative corona discharge is formed from the electron avalanche extending tip of the pin into the volume. The electron number density reaches a maximum of 3.6×1020 very close to pin tip. The electron temperature reaches a maximum of 33,500K or approximately 3eV.

Next, VizGrain is used to solve the dust particle motions over the background plasma. This uses a one-way coupling of the electrostatic potential and Lorentz forces with the dust particle dynamics. The resulting VizGrain solution is shown in the animation in Figure 3.


Figure 3. Dust particle trajectories through pin-corona discharge. First column of particles has an initial velocity of 150m/s. Second column of particles has an initial velocity of 50m/s.


Both sets of particles charge-up as they move to the right. Charge-up is caused by the flux of ions and electrons on the particle surface. The flux is a function of dust particle size, as well as the ion and electron number densities, masses, and temperatures. This flux is also dependent on the floating potential of particles, obtained from the VizGlow plasma solution.

The first set of particles, with an initial velocity of 150m/m, has sufficient momentum to pass through the corona as they charge-up. The second set of particles, with an initial velocity of 50m/s, does not have sufficient momentum to pass through the corona and are deflected as they charge-up in the corona.

This demonstrates the effects of dust particle charge-up and Lorentz forcing using a one-way coupled solution between VizGlow and VizGrain. VizGlow and VizGrain are both modules within the OverViz Simulation Suite. OverViz is an industrial multiphysics framework for performing hybrid plasma, fluid flow, electromagnetic, particle simulations. For more information, please contact us at

Kinetic PIC Modeling of Ion Beam Neutralization

Fluid vs Hybrid Plasma Formulations for IEADFs in Semiconductor Reactors

Ion energy and angular distribution functions (IEADFs) characterize the anisotropic impact behavior of ions onto wafer surfaces in cold plasma reactive ion etchers (RIE). IEADFs can be used to quantify the etching performance and guide reactor design in the semiconductor fabrication process. Reducing the angles of ion impact at the wafer surface sharpens the resolution of the etching process, enabling the production of smaller etched features. Controlling the ion energy distributions also offers control over etching rates, shape, and consistency of etched trenches. Improving the ability to quantify IEADFs and their relationships to reactor system parameters will serve the industry’s rapid progression towards fabricating smaller features with faster throughput.

Two modeling approaches are compared to quantify IEADFs for a generic capacitively coupled plasma (CCP) reactor. The differences and advantages are highlighted. The first approach uses a purely fluid/continuum formulation of the plasma governing equations for the non-equilibrium plasma. This includes continuity, species, momentum, and energy. A self-consistent plasma formulation of the electrostatic potential is used to resolve the plasma sheaths. Ion momentum and energies and solved continuously. An external circuit model is used to supply radio-frequency power to the CCP. Boundary surface charging is also included to capture charge build-up on the wafer and dielectric surfaces.

The second approach uses a hybrid, particle-fluid approach that simulates individual ion particle motions over a fluid/continuum background fluid. Particles are dynamically generated and tracked as ions are produced in the plasma. Ion particle motions are solved using Newton’s law and include the Lorentz effects of the local electric and magnetic fields. Particle collisions are also modeled probabilistically using a Direct Simulation Monte Carlo approach.

For this comparison, a generic CCP is assumed. The CCP uses pure argon gas and is operated at 13mTorr. It is powered using a three-circuit fixed voltage waveform of 100V with circuit frequencies of 60MHz, 30MHz, and 2MHz. A multiple frequency design allows higher frequencies to control the plasma while the lower frequency serves to accelerate ions in the sheath. This CCP example is modeled using a 2D axisymmetric geometry. The geometry includes the gas/plasma volume, wafer, and dielectric volumes. The plasma gas phase chemistry is represented by 4 species (electrons E, argon ions Ar+, argon metastable Ar*, and argon neutrals Ar). The 2D finite volume mesh and key model features are shown in Figure 1. The left image shows a model schematic for the fluid approach. The right image shows a model schematic for hybrid approach. Both models used the same axisymmetric structured mesh consisting of approximately 40,000 quadrilateral cells.

Simulations and IEADF calculations were performed using modules within the OverViz Simulation Suite. The self-consistent, non-equilibrium plasma model within VizGlow was used for the fluid/continuum plasma modeling. VizGrain was used for the kinetic particle modeling.

Continuum results for the plasma simulation are given in Figure 2. The cycle averaged electrostatic potential is uniform throughout the plasma with a potential drop across the sheaths. The cycled averaged temperature is also uniform throughout the reactor at approximately 5eV. Adjacent to the powered electrode and wafer surface, the electron temperature reaches 8eV. The cycle averaged electron number density E and cycled averaged ion density AR+ contours (right side of Figure 2) match fairly closely. The plasma structure is uniform across the wafer radius with a peak near the wafer edge.

Figure 3 shows the electron potential with particles for the hybrid approach. A close-up of the wafer edge and sheath region is also shown. Approximately 600,000 particles were simulated.

Figure 4 shows a comparison between the fluid model (left) and the hybrid model (right). Both ion impact energy (top) and impact number flux (bottom) are compared. The ion impact number density agrees well between the two approaches. Both approaches also show a uniform impact energy distribution across the radius with a sharp decrease in energy close to the wafer edge. However, the fluid approach predicts significantly lower impact energy magnitudes. The fluid model is only able to capture average values of the plasma quantities and must assume a velocity distribution for the represented particles. The fluid model accuracy degrades as this assumption breaks down, particularly for low pressure systems where the length scale of the particle mean free path approaches the relevant length scale of the system. The is particularly relevant to the sheath region within this low pressure CCP reactor. Unlike the fluid approach, the hybrid approach correctly represents the statistical velocity distributions within the sheath. This is important for plasma processing applications where wafer surface treatment is dominated by the distributions if ion behavior.

Next, the impact energy and angular distribution functions are explored using the hybrid approach. The impact energy distributions along the wafer radius is shown in Figure 5. A top view of the distribution functions is given on the left side of the figure and a perspective view of the distributions function is given on the right side. A typical bimodal distribution is evident along the wafer radius with a lower peak at approximately 120eV and the higher peak around 200eV. The energy distribution is consistent along the radius. Close to the wafer edge, the impact energies decrease and becomes more diffuse.

The impact angle distributions along the wafer radius is shown in Figure 6, similarly with top and perspective views. The incident angle is close to zero and fairly uniform across the wafer radius with a slight small positive incidence. A more substantial spread in impact angle is found near the wafer edge due to the potential gradient edge effects.

Finally, Figure 7 shows impact energy and angular distributions at five locations along the wafer radius. At the first four locations, the impact bands are consistent and distributed close to 0°. Edge effects are prominent at the fifth location where the impact band becomes wider and less focused with a decrease in impact energies.

Fluid models are typically implemented when the particle mean free path in a plasma is much less than the relevant length scale of the system. At lower pressures, the length scales of the particle motions begin to approach those of the system. In these cases, kinetic particle models are considered more representative. However, kinetic models are considerably more computationally expensive than an equivalent fluid model. Hybrid models offer a compromise between the two, improving the accuracy of fluid simulations where particle length scales become relevant, while requiring less resources than a full kinetic model. The use of these models can help guide plasma reactor design, refine etch process parameters, investigate gas composition effects, explore process anomalies, and offer enhanced engineering insight into physical processes beyond traditional measurement techniques.

Analyses in this study were performed using the OverViz Simulation Suite. OverViz is a multiphysics framework for performing hybrid plasma, fluid flow, electromagnetic, particle simulations. For more information contact us at


Ion Momentum and Drift Diffusion Approximation in VizGlow

Ion Momentum and Drift Diffusion Approximation in VizGlow

When setting up a new non-equilibrium plasma simulation in VizGlow, the user must select an ion momentum model. Two available options in VizGlow are “Solve ion momentum equation” and “Use drift-diffusion approximation.” This discussion is intended to help a new user understand the differences between these options and to determine which selection is appropriate for their problem.

The drift-diffusion relationships are derived from the species momentum equations and characterizes the spatial transport of particles. Drift refers to a velocity induced by the presence of an electric field. Diffusion refers to velocity induced by a density or concentration gradient.  Both drift and diffusion are collisional in nature, e.g. both are governed by an average motion due to a driving electric field or density gradient force along with numerous collisions with a dominant background species.

The first option, “Solve ion momentum equation,” solves a separate species momentum equation for each ion, while using the drift-diffusion approximation for neutral species and electrons. The second option “Use drift-diffusion approximation” uses the simplified drift-diffusion approximation for all ions, neutral species, and electrons.

We first consider the species momentum equation:

The left side describes the time dependent and convective accelerations. The right side describes the forces acting on a particle that includes electric field, pressure gradient, and particle collisions, respectively.

The drift-diffusion approximation is a simplification of the momentum equation which assumes that collisional processes (the terms on the right-hand side) are dominant.  It can be derived from the momentum equation as follows.  First, ion inertial effects are neglected.  Second, assume that time variations are slow such that   term can be neglected.  Pressure can be expressed using the ideal gas equation of state.  If time variations are slow, then temperatures will have time to equilibrate with their surroundings and T can be assumed to be constant (isothermal).

With these simplifications, the momentum equation becomes

This results in the following relationship such that drift-diffusion species flux  is function of electric field strength and density gradient.


Note that the background collision frequency appears in both the drift and diffusion variables which confirms that they are collisional dependent processes.

Because this equation is an algebraic function rather than a PDE, it is far less computationally expensive to solve (and also tends to be slightly more stable in practice from a numerical standpoint).

The need for solving the full ion momentum equation for ions arises in low pressure discharge (P < 100’s mTorr) or in small-scale micro-discharges despite the high pressures. In these situations, the number of collisions is reduced and ion inertial effects between collisions should be considered.

The drift-diffusion approximation is applied to both electrons and neutral species for either of the above options. The neutrals are not influenced by electric fields therefore the drift-diffusion approximation remains valid. The approximation is valid for electrons due to their very small mass and large thermal energies, even in strong electric field regions of a discharge.

Selections for a few typical non-equilibrium plasma models:

  • Low pressure CCP (P < 100 mTorr)  –  Solve ion momentum
  • High pressure CCP (P > 100 mTorr)  –  Drift-diffusion approximation
  • ICP coupled plasmas  –  Drift-diffusion approximation
  • Atmospheric plasma jet  –  Drift-diffusion approximation
  • High pressure plasma streamer  –  Drift-diffusion approximation
  • Low pressure magnetron reactor  –  Solve ion momentum
  • Dielectric barrier discharge  –  Drift-diffusion approximate
  • Micro discharge  –  Solve ion momentum

The important factor to consider is the influence of ion inertial effects and collisions to the particle momentum. If you have any further questions, please contact us at



Kinetic PIC Modeling of Ion Beam Neutralization

Plasma Simulations Strategies for Semiconductor Applications

The semiconductor industry is one of the most competitive global markets experiencing consistent month-to-month growth with annual global sales approaching $400B. There is a strong demand for integrated circuit technologies across growing markets, such as mobile devices and automotive electronics, with fierce international competition. This is driving an innovation race for enhanced process technologies, increased automation, and reduced product development timelines, in light of increasing development and fabrication costs. A new integrated circuit design can cost tens to hundreds of millions of dollars to develop. A new foundry can cost upwards of $15B to design and build. The need for innovation at a competitive cost presents a significant challenge to the semiconductor equipment and chip makers. This article will discuss how advancements in computational modeling and simulation can help meet some of these challenges.

Plasma reactors offer precise control over processing steps including etching, deposition, cleaning, doping, and surface activation of materials in semiconductor integrated circuit manufacture.  All of these rely on complex, highly nonlinear behavior of the plasma and its interaction with surfaces with limitless designs and configurations. They are powered by a variety of excitation sources (e.g. microwave, RF, hybrid DC/RF) and can be driven at one or more frequencies to form and control plasma discharges. The number and complexity of process steps required for a single chip are increasing; upwards of 500 steps in the most advanced memory fabrication. This further drives the need for faster and cheaper methods to evaluate and refine these processes.

Simulation has gained traction to support these efforts, but is underutilized as the industry still relies heavily on iterative testing and engineering judgment. With the continual decline in computational costs, the semiconductor industry stands much to gain from a robust, high-fidelity, scalable industrial simulation solution.

A few applications where modeling and simulation can be leveraged to support plasma reactor design includes:

  • Refine processes for improved uniformity of ion densities and surface fluxes.
  • Optimize process design with respect to etch uniformity and rate while controlling temperature to prevent wafer damage.
  • Select operating conditions to control gas and surface reactions for various materials.
  • Explore effects of varying feed gas mixtures and flow rates.
  • Investigate chamber contaminants with simulated particle source methods.
  • Provide better engineering insight into reactor performance through high fidelity simulation outputs that are not possible through experimental measurement techniques.

This article will discuss three simulation approaches that offer unique solutions to these challenges for a variety of semiconductor applications.

OverViz Simulation Suite

The OverViz Simulation Suite is an industrial, multiphysics framework with coupled plasma-fluid-electromagnetic-particle modeling capabilities with a comprehensive library of process chemistries. Three modeling approaches within OverViz are discussed: (1) reactor-scale modeling, (2) feature-scale modeling, and (3) fast-reactor-scale modeling. Each approach has unique benefits and applications.

These approaches use a low-temperature, non-equilibrium formulation for multi-species, multi-temperature, self-consistent (and quasi-neutral) plasma systems. The relevant physics that is modeled is shown in Figure 1. A reactor simulation may involve the coupling of non-equilibrium plasma formation, gas chemistries, surface chemistries, compressible gas flow, magnetic field effects, microwaves, external circuit dynamics, ICP induced electromagnetic fields, and kinetic particles. These may be selected and configured to represent a large range of plasma reactor applications, such as a Capacitively Coupled Plasma (CCP) Reactive Ion Etcher, Inductively Coupled Plasma (ICP), and DC magnetron sputtering reactor. The three approaches for these applications is discussed next.

Figure 1. Physics representations for non-equilibrium plasma reactors in OverViz


Reactor-Scale Model

The reactor-scale model is a high-fidelity representation of the reactor system, as shown in Figure 2. Physical features in this reactor model are on the order of centimeters and millimeters. Features on the order of 10’s of nanometers, such as etched trenches, are not represented in this model, but are represented in a separate feature-scale model. While it may be possible to represent both scales in a single model, it is far more computationally efficient to separate these into two scales. The reactor-scale is discussed first.

VizGlow is used for performing reactor-scale simulations. The reactor is modeled using a finite volume representation that is solved using a well-established continuum/fluid approach. A time accurate formulation for the conservation equations is solved in 1D/2D/3D. Governing equations include species continuity, ion momentum, particle momentum, bulk energy, electron energy, electrostatic potential, electromagnetic wave, surface charge density, bulk flow, and external circuit. Both self-consistent (for resolving plasma sheaths) as well as quasi-neutral (high-density plasmas with negligible sheaths) may be represented. A drift diffusion approach is used for all species with an optional full momentum formulation for ions.

Figure 2. Example simulation domain for reactor-scale model of a CCP. Electron number density shown.

In addition to a continuum/fluid approach, the reactor-scale simulation may be solved using a hybrid approach that includes kinetic particles. Hybrid models combine a fluid model representation for bulk gas characteristics with a particle model representation of ion kinetics. With some additional computational expense, hybrid models provide better physical representation of ion behavior in the sheath region (particularly for lower pressures). These models provide the framework for performing reactor-scale simulations.

A reactor-scale simulation takes of the order of hours to days to run, depending on the reactor complexity and the hardware that is used to execute the simulation (though large simulations can be parallelized offering significant speedup). Model inputs may be varied parametrically to explore system performance or to optimization a design. Sample parametric inputs include:

  • Reactor geometry
  • Operating pressure
  • Gas composition
  • Process chemistries
  • Feed gas flow rate
  • Material properties
  • Input power
  • Input voltage
  • Excitation frequency or frequencies
  • Bias voltage
  • External magnetic field strength
  • Generation of contaminant (dust) particles

Example results that may be explored in these parametric simulations include:

  • Power deposition
  • Uniformity of species densities / fluxes
  • Electron energies and densities
  • Plasma chemistry reactions
  • Surface chemistry interactions
  • Reactor skin depth
  • Wafer heating
  • Etch uniformity and rate
  • Etch selectivity
  • Transport of dust particles
  • Ion energy and angular distribution functions

A simple example of a parametric sensitivity to input power and presence of an external magnetic field is shown in Figure 3. The magnetic field provides higher densities with improved electron confinement.

Figure 3. Example parametric simulations for plasma reactor operating conditions. Electron density shown.

Another set of important outputs from the reactor-scale simulation are ion energy and angular distribution functions (IEADFs). The distribution and angle of ion impact determine how etched features form and evolve. Being able to quantify and control this behavior offers control over the etching process. For example, narrowing the angular distributions to have most ions impact the surface with normal incidence allows for high aspect ratio trench and via features to be etched.  Simulated IEADFs at seven locations along a wafer surface are shown in Figure 4. The IEADFs at the first six locations are tightly distributed around 0° (normal incidence). Edge effects become very prominent at the seventh location where significant fraction of the ion impacts with large off-normal angles of incidence, thus compromising the process integrity at these locations.

Figure 4. Ion energy and angular distribution functions along wafer radius.

Feature-Scale Model

The feature-scale model is a sub-model of the reactor-scale model that locally resolves the physics around small features in the plasma sheath, such as etched trenches, which can be on the order of tens to hundreds of nanometers. Here mean free paths of the ions and electrons are large compared to the feature dimensions.  Ion and electron motion must be represented using particle kinetics. IEADFs are therefore extracted from the reactor-scale model in the region surrounding the small features of interest and then used as boundary conditions for the feature-scale simulation. An example of this is shown in Figure 5. Feature-scale simulation runtime is similar to reactor-scale simulation runtime.

Figure 5. Feature-scale model uses local solution from reactor scale model to resolve features on the order of nanometers.

Feature-scale modeling can be useful to provide design insight into:

  • IEADF relationship to etched feature geometries (size and aspect ratio)
  • Surface charge-up and propensity of damage around the etched features
  • Effects of material properties
  • Overall relationship between bulk reactor operating conditions and local feature scale processing

Example results from a feature-scale simulation for a dielectric surface exposed to a bias is shown in Figure 6. Ions approach the feature with very narrow off-normal angles and can therefore deposit directly at the bottom surface of the features, thereby imparting a net positive charge at the bottom surfaces.  After passing through the sheath, electrons arrive at the trench with low energy and large off-normal incidence angles. Consequently, they deposit a negative charge at the corners of the feature. The differential charge deposition across the feature develops self-induced electric fields that can lead to dielectric breakdown damage.

Figure 6. Example electrostatic potential solution from feature scale model of etched trench.

Reactor-scale models resolve system performance on the length scale of the reactor, millimeters to centimeters. Feature-scale models resolve system performance on the length scale of the surface features, tens to hundreds of nanometers. Next, the fast-reactor-scale model is discussed.

Fast-Reactor-Scale Model

The fast-rector-scale model is a 0-D global representation of the reactor-scale model. ChemZone is used to perform fast-reactor simulation within the OverViz Simulation Suite. It solves the same governing equations for low-temperature, non-equilibrium, multi-species, multi-temperature plasma systems, including chemical reactions. However, the fast-reactor-scale model assumes that the reactor is well stirred, such that reactor gases are perfectly mixed and homogeneous. Reactor properties and constituents are assumed uniform through the reactor, allowing for a 0-D control volume representation. A schematic of a fast-reactor-scale model is given in Figure 7. The reactor is defined by volume V with homogeneous representation of bulk temperature, electron temperature, pressure, and species densities. It can also include mass inflow, mass outflow, energy source term, energy loss, and independent surface reactions. The fast-reactor-scale model provides detailed views of chemistry parameters, such as reaction rates and contributions of individual reactions to specific species. Very large chemistries can be represented, such as plasma discharge, neutral reactions, combustion, and chemical vapor depositions.

Figure 7. Schematic for fast-reactor-scale simulation including inflow, outflow, and surface reactions.

The primary benefit for the fast-reactor-scale modeling approach is its extremely fast runtime, requiring only seconds to minutes for full transient reactor simulations. This allows complex process design studies that involve hundreds or thousands of independent simulations that would not be possible with the standard reactor-scale model.

Specific examples of faster-reactor-scale applications include:

  • Large parametric studies of process formulations
  • Design of experiments (DOE) for new fabrication processes
  • Sensitivity analyses for reaction mechanisms to identify which species/ reactions are dominant
  • Process optimization for large numbers of chemistries
  • Troubleshooting and reverse engineering of process issues and defects

This benefits semiconductor design and process engineers that may be responsible for executing extensive DOE experimental studies. What could take months to complete in the lab could be performed on a laptop in minutes. While experiments cannot be completely eliminated (and should not be eliminated, as experiments are critical to validating simulation results and verifying design performance), this provides an upfront analysis capability that could greatly reduce the time and costs required for fabrication process testing.

Sample results for a fast-reactor-scale sensitivity analysis is shown in Figure 8. This study is used to identify which reactions dominate a reactor process. The time evolution of species densities is also shown.

Figure 8. Example outputs for fast-reactor-scale sensitivity analysis.

For well-stirred plasma reactors, the fast-reactor-scale model offers rapid process simulations for reactive flows and finite-rate chemical reaction mechanism development.


Reactor-scale, feature-scale, and fast-reactor-scale modeling are three approaches that can be leveraged for different aspects of semiconductor processing. These simulation approaches can be used very effectively to supplement experiments to reduce time and cost associated with hardware iterations. They can also be used upfront in the design process to refine the design space before executing a single experiment. This accelerates the design and development lifecycle reducing cost. The simulations can also be used to troubleshoot issues in the fabrication process reducing downtime.

Validated simulation can be an important tool in exploring innovation and shortening the development lifecycle. This is particularly useful in the semiconductor industry where the need for innovation and cost of development are growing as fast as the demand for their products.

For more information about these modeling approaches or the OverViz Simulation Suite, please contact us at