This VizGlow example describes a self-consistent plasma simulation of an inductively coupled plasma (ICP) with a bias voltage.
Typically, a quasi-neutral formulation is preferable over self-consistent for high density plasma discharges where Debye lengths become very small relative to geometric dimensions in the discharge. Sheath thickness become negligible compared to the physical system and can be ignored. This is common in ICP and microwave plasmas and under these conditions a quasi-neutral approximation may be made. This approximation is similar to the approximation in high Reynolds number flows in fluid mechanics where the Euler equations are used and the thin boundary layers are neglected. The quasi-neutral formulation is easier to solve numerically compared to self-consistent, numerically less stiff, and allows for larger time-steps.
In this example however, resolving the sheath becomes important with the presence of an RF bias. Bias effects are used to enhance and control the ion energy distributions at substrate surfaces within the sheath. Capturing the sheath in these applications is necessary, therefore the self-consistent formulation is used.
The geometry and mesh for the plasma discharge simulation are shown in the following figure. The axisymmetric discharge comprises of a coilcage with seven coils, dielectric window, gas (plasma region), and electrodes separated by a dielectric. The coils are driven by radio-frequency (RF) currents to generate non-equilibrium plasma within the tube with an RF frequency of 13.56MHz and 1000W cycle-averaged power. The reactor is simulated at 50mTorr with pure Argon gas. The electrode bias voltage is -100V.
The coupled model is solved including electrostatic potential, species density, electron energy, bulk energy, electromagnetics in the frequency domain, and the external circuit model. First, volume traces are used to track changes in number densities to ensure a converged solution. The following plot shows number density of the charged species (Ar=red, electrons=blue) vs. time. The left plot tracks number density on one of the electrode surfaces. The right monitor tracks number density in the volume directly between two electrodes. Notice that the traces have all flattened out. This demonstrates steady state convergence of the simulation.
Results for the converged solution are shown below, including EM wave (real and imaginary), AR+ number density, and electron number density. The EM waves deposit power into the gas through the dielectric window on the coilcage side. The EM power deposition causes a localized plasma skin depth effect; however, the presence of bias enhances skin depth and power deposition. The plasma generation is centered between the sets of coils and extends through the region between the window and the electrodes. Peak plasma number density is approximately 2.0E+16 #/m3.
This example uses the self-consistent formulation to resolve the sheaths for the ICP with a bias voltage. VizGlow provides a wide range of physical models and options that allows one to simulate complex coupled phenomena that are encountered in most plasma discharges. It is part of the OverViz Simulation Suite which provides a variety of modules for plasma-fluid-electromagnetic-particle simulations with an intuitive interface. VizGlow is provably fast, robust, easy-to-use, and currently a leading industrial plasma simulation tool.
Plasma processing reactors are used to accomplish a variety of unit steps in a semiconductor integrated circuit manufacture. In most cases, complex feed gas mixtures are used in a plasma reactor to realize precise etch, deposition, doping, cleaning and other types of processes. The VizGlow Plasma Modeling Software Package provide capability for modeling plasma reactors with complex reactive plasma chemistries in the gas phase and at reactive surfaces. This technical note discusses the simulation of a hydrogen bromide (HBr) plasma in a dome-shaped Inductively Coupled Plasma (ICP) reactor representative of several real industrial reactor systems. HBr gas and mixtures of HBr with other gases are used commonly in the anisotropic etching of silicon-based material surfaces.
Figure 1 shows the geometry and mesh for the plasma reactor simulation discussed in this note. The geometry is represented by6 subdomains (plasma, dome-shaped quartz wall, inductive coils, external Faraday cage in which the coils are present, the wafer holder, and a wafer edge focus ring). The radius of the reactor quartz dome wall is 20 cm with a 200 mm dia. wafer. The inductive coil system comprises 20 turns. The mesh comprises a total of about 28,000 cells in the domain.
The pure HBr plasma is generated by driving 2 kW of power through the inductive coils with 460 kHz radio-frequency (RF) excitation. The HBr plasma is represented by 15 species (E, Br+, Br2+, HBr+, H+, H2+, Br–, Br, Br2, Br*, Br2*, H, H2, H*, HBr)and 51 reaction. The reaction mechanism includes electron impact dissociation, excitation, and ionization reactions, attachment reactions for negative ion formation, ion-neutral reactions, ion-ion reactions, and reactive neutral chemistries. The surface chemistry is represented by simple quench chemistries for all reactive ions and radicals to form their corresponding stable neutral species which are then injected back into the plasma.A pressure of 5 mTorr is assumed for the simulations.
The complete model representation of the reactor is provided by inductive wave physics, plasma dynamics and reactive plasma chemistries. The electromagnetic (EM) wave frequency domain solver from the VizEM Electromagnetics Modeling Package is solved in a coupled fashion with the core VizGlow package. The EM wave generated by the coils is Eq polarized, meaning that the only non-zero component of the wave E field is in the direction perpendicular to the domain representation. The high-density HBr plasma is represented using the quasi-neutral formulation. A flow residence time formulation is used to represent the effect of gas flow in the reactor. The plasma gas-phase and surface chemistries are managed by the Gasact and Surfact modules within VizGlow. Results at steady-state are reported here. A steady-state is found to be achieved in about 2 milliseconds of physical time in the simulation. This time corresponds to the ion ambipolar diffusion time scale for the reactor geometry. Figure 2 shows the real and imaginary parts of the magnetic vector potential of the Eq polarized EM wave.
Figure 3 shows the EM wave power absorbed by the plasma. The wave power absorption is determined by the plasma conductivity (determined by plasma electron density and electron temperature) and the magnitude of the wave electric field. For the relatively low excitation frequency of 460 kHz, the skin depth of the plasma is sufficiently large that EM wave power is absorbed relatively deep within the plasma, away from the quartz dielectric wall.
Figure 4 shows the structure of the plasma as identified by the dominant charges species in the plasma. Under the condition chosen for the simulation the plasma is largely electropositive (meaning that the negatively charged ion densities are much smaller than positively charges ion densities). The peak electron (E) densities are located on the axis of the reactor and have a magnitude of 5×1016 m-3. The dominant ion is Br+ which has a spatial profile and magnitude similar to the electrons. The HBr+ ions are also shown in the figure and have a peak profile that is off the reactor axis. Note that the electron and other charges species peak at locations that are far from the location of the EM wave power absorption peak which occurs close to the quartz dielectric wall (see Fig. 3). This has to do with the non-local nature of the plasma at the very low pressure of 5 mTorr.
Figure 5 shows the radial profiles for the dominant charged species fluxes to the wafer surfaces. This information is important to determine the uniformity of the process at the wafer. The dominant ion is Br+ whose radial profile is relatively uniform throughout the radius of the wafer with slight non-uniformity seen at the edge of the wafer. Other ions such as HBr+ have much lower flux to the wafer surface.
In summary, the VizGlow Plasma Modeling Software Package provides a wide range of physical models and options that allows one to simulate complex coupled phenomena that are encountered in most plasma discharges. The VizGlow Plasma Modeling Software Package is part of the Overviz framework suite which provides an intuitive interface to set-up a project to be solved using VizGlow, manipulate multiple projects for parametric studies. VizGlow is provably fast, robust, and easy-to-use software and currently a leading industrial plasma simulation tool.
This example application simulates a steady microwave field in a three-dimensional plasma reactor using in semiconductor materials processing. The Frequency-Domain Electromagnetic Wave Solver Module of the VizEM is used for this problem.
The geometry for the simulation is shown in Figure 1 and comprises a cylindrical processing reactor with an air-filled waveguide port in the top center of the reactor. The waveguide is rectangular in cross section and comprises an L-bend with a 45o mirror surface at the bend. The waveguide connects to a top dielectric window surface through which the wave is launched into the main reactor volume. The top dielectric window is made of quartz with a relative dielectric permittivity of 4 and the main reactor volume has a diameter of 40 cm. The bottom of the reactor has a wafer holder pedestal. The distance between the top dielectric window and the pedestal is 10 cm.
The entire reactor geometry is meshed with a 3D unstructured mixed mesh comprising a combination of tetrahedral, prismatic, pyramidal, and brick cell volumes that are automatically generated using third-party meshing software. The mesh contains over 1.3 million cells with 6 unknown in each cell (3 real and 3 imaginary components of the wave field) for a total of 7.8 million unknowns in the problem. The overall mesh count, quality, and resolution is determined by not just the geometry, but also the characteristics of the wave. In this case, the wave has a frequency of 2.45 GHz implying that the vacuum wavelength is about 12 cm and wavelength in quartz is about 6 cm. The resolution of the mesh must therefore be such that at least 20 mesh cells resolve a wavelength (a rule-of-thumb); which means a typical mesh cell dimension about 3 mm or less.
The geometry is divided into 5 physical sub-domains: the waveguide, the top dielectric window, the gas, the wafer and the pedestal. The bottom panel in Figure 1 shows a 90o cut through the reactor to expose the various components in the reactor. All material boundaries in the geometry are modeled as perfect electric conductors.
A uniform microwave field of frequency of 2.45 GHz is launched at the inlet of the waveguide. The wave is polarized with a single non-zero wave component in the z-direction (direction along axis of reactor). The wave travels horizontally in the waveguide and reflects of the mirror surface following which it propagates vertically down the reactor axis to the top dielectric window. The wave then travels radially along the thickness of the quartz dielectric and finally launches into the reactor volume, as seen in figure 2. The high dielectric permittivity of the quartz dielectric window “slows” the wave resulting in a lower wavelength in the window (from 12 cm to 6 cm, as mentioned earlier) thereby improving uniformity of the wave field as it is launched into the main reactor volume.
Figure 2 shows the three components of the microwave field. The wave reflection at the waveguide mirror surface at the L-bend produces a y-component of the wave field. The wave reflects off of the outer walls of the reactor to produce non-zero x-components of the wave field. The three-dimensional geometry therefore results in all three components of the wave field becoming active in the reactor even though only a single non-zero wave field component is launched in the reactor.
VizEM provides a very robust environment to solve such problems with quick turnaround. The different modules within the VizEM are seamlessly integrated with the other OverViz software packages. For example, the Frequency-Domain Electromagnetic Wave Solver Module used in the above problem can be called by VizGlow to solve coupled electromagnetic wave—plasma problems, such as in microwave reactors and inductively coupled plasma reactor.
Capacitively Coupled Plasma (CCP) discharges in parallel-plate configuration are commonly used in semiconductor and other materials processing applications. These discharges provide a compact platform in which a plasma can be generated to process a flat wafer surface. The highly directional ion impact at the wafer surface with high ion impact energies is beneficial to a number of wafer processes; in particular for etch processes. This application note discusses the simulation of a typical CCP reactor used in the semiconductor process industry. The VizGlow Plasma Modeling Software Package is used.
The geometry and computational mesh for this problem is shown in Fig. 1. The mesh contains about 21,000 cells, and is divided into 6 physical subdomains. The geometry includes a gas subdomain where the plasma is formed, a top grounded electrode, a bottom powered electrode that holds a wafer, a top dielectric that isolated the top electrode from the grounded reactor wall, a bottom dielectric that isolates the bottom powered electrode from the reactor walls, and a focus ring that defines the edge geometry for the wafer and the powered electrode. The mesh comprises a mix of quadrilateral and triangular cells in domain. Specifically the plasma (gas) subdomain is meshed with quadrilateral cells that have non-uniform resolution to capture the sheaths at wall and wafer boundaries accurately.
The wafer has a standard 300 mm dia. The wafer is assumed to be made of silicon, the dielectrics are made of alumina, and the focus ring is made of quartz. The powered electrode is driven by a 100 MHz excitation frequency that delivers 300 W radio-frequency (RF) power to the plasma. A pure argon gas at 50 mTorr pressure is assumed.
The plasma gas phase chemistry is represented by 4 species (electrons E, argon ions Ar+, argon metastables Ar*, argon neutrals Ar). The plasma chemistry comprises 6 reaction including electron impact excitation and direct ionization, stepwise ionization, Penning ionization, and metastable quenching. The simulations are run to a periodic steady state. The VizGlow Plasma Modeling Software Package is used in self-consistent mode to solve for the CCP plasma phenomena. These simulations are performed without including electromagnetic (EM) wave effects that are known to redistribute power deposition in the reactor.
Figure 2 shows a cycle-averaged electron temperature in the reactor. The electron temperature is nearly uniform in most of the reactor volume with a value of about 20,000 K (~2 eV). In the vicinity of the powered electrode/wafer the averaged electron temperature is about 150,000 K (~ 15 eV). The electron temperature is non-uniform along the radial length of the wafer and peak near the radial edge of the wafer.
Figure 3 shows the cycle-averaged electron density profile in the reactor. The electron densities are seen to be quite non-uniform with a wafer edge peak profile that corresponds to the dominant edge power deposition resulting for wafer edge electrostatic field focusing. Peak electron densities of ~ 3×1016 m-3 are seen. The argon ion densities correspond closely to electron density profiles.
Figure 4 shows the argon metastable species density profiles in the reactor. An off-axis metastable density peak at a similar radial location to the electron density peak is seen. The metastable densities are however peaked close to the wafer surface close to the regions where the electron temperatures are high.
Figure 5 shows the ion energy distribution function (IEDF) for ion impact at two locations on the wafer surface. The IEDF information is generated automatically by VizGlow and is a useful metric for process behavior at the wafer surface. IEDFs for the wafer center and edge are shown. Ions impact the wafer surface with a range of energies depending on the excitation frequencies, mass of ions, and the sheath thickness. A classical bi-modal IEDF is seen. For the reactor condition used in these simulations, the lowest ion impact energies are between 15 and 20 eV and the highest are between 30 to 35 eV. The IEDF width is nearly the same for both the wafer center and edge locations. The wafer edge IEDF is shifted to a slightly higher energy compared to the wafer center.
Finally, we’d like to provide an addition comment on the numerical convergence for these types of simulations. CCP simulations can be time consuming and the best judgment for simulation convergence is provided by placing trace points several locations in the simulation domain and tracking the time evolution of various important plasma properties. For example Fig. 6 shows the time evolution of electron and ion species densities and the argon metastable species density at the reactor center line (axis) mid-gap between the electrodes. Although the simulation was run 100 microseconds of simulated physical time, the time evolution of the species densities indicates that a numerically converged solution is obtained in about 40 microseconds.
VizGlow provides a wide range of physical models and options that allows one to simulate complex coupled phenomena that are encountered in most plasma discharges. VizGlow is provably fast, robust, and easy-to-use software and currently a leading industrial plasma simulation tool.