Esgee Technologies will present “Computational Modeling of Microwave Interactions with Self-consistent Plasma” at the 70th Annual Gaseous Electronics Conference in Pittsburgh, PA on Nov. 7th, 2017. This talk will address current challenges in computational modeling of microwave-sustained plasmas. Examples include discharges sustained by surface wave propagation (SWP) along plasma- dielectric interface (SPP) and discharged sustained by SWP along the plasma-sheath interface (MVP). These are important processes for the creation and sustenance of plasmas in industrial scale plasma reactors for material processing applications. For more information, contact us at info@esgeetech.com.
This application note illustrates the transient development of microwave field in an air-filled waveguide with an L-bend geometric feature. The Time-Domain Electromagnetic Wave Solver Module of the VizEM Electromagnetics Modeling Software Package developed by Esgee Technologies Inc. is used for this problem.
The 2D planar waveguide geometry and the computational mesh used for the simulations of this problem are shown in Figure 1. The waveguide is 100 cm long in the horizontal direction (x-direction) and 55 cm in the vertical (y-direction). The waveguide channel width is 10 cm in along the horizontal length and 18 cm along the vertical direction. The geometry comprises a wave inlet to the right and perfect electric conducting side walls along the length of the waveguide. The waveguide is terminated with a perfect conductor. The computational mesh has approximately 6,900 cells and comprises a single sub-domain for the wave propagation. The maximum of the mesh size is determined by the frequency of the microwave. For this particular simulation, the drive frequency is 1.5 GHz with in a vacuum wavelength of about 20 cm. The mesh resolution is such that about 40 mesh cells (5 mm length) resolve a single wavelength. It is important to note that the problem geometry does not accommodate a perfect integer number of wave lengths. Hence a true standing wave pattern at the drive frequency of 1.5 GHz is not expected for this problem.
For the time-domain solver a time-step of 10-12 sec is used. This corresponds to an approximate CFL number of 0.3 for the time-step. The simulation is run to a final time of 84 ns.
A wave is launched from the right inlet boundary with a non-zero wave field component in the y-direction. The wave travels down the waveguide (in the –x direction) and negotiates the L-bend and further propagates up the waveguide in the +y direction. Finally the wave interacts with the perfect electric conductor termination and is reflected back down the waveguide to form a standing wave in the waveguide. As mentioned above the geometry is such that a stationary (steady) standing wave can never be achieved in this geometry. Figure 2 shows a snapshot of the wave at 0.6 ns after the wave is launched at the inlet (the initial conditions being a wave-free domain). The wavelength is observed to be about 20 cm and is exactly the vacuum wavelength for the 1.5 GHz excitation. At a perfect electric conducting wall the tangential component of the wave field is zero while the normal component is non-zero. This feature is clearly seen in the wave at this time.
Figure 3 shows the transient development of the wave field at various times after the wave is launched at the inlet. At 6.6 ns the wave has reflected off the termination boundary and formed a standing wave pattern within the wave guide. Note that the perfect conducting waveguide wall in the vertical section of the waveguide dictates that the y-component of the wave field is zero, which results in a wave pattern that looks different from the horizontal section of the waveguide.
As seen in Figure 3, the amplitude of the standing wave grows in time until about 13 ns following which the amplitude decreases to reach a minimum at about 60 ns. After 60 ns the amplitude of the standing wave starts increasing again and the pattern repeats itself. Essentially the transient snapshots in Figure 3 show the occurrence of a lower frequency modulation of the standing wave indicating that a stationary (steady) standing wave is not established in this geometry as mentioned earlier.
This problem demonstrates the importance of using a time-accurate (time-domain) simulation of the electromagnetic wave phenomena and ability to capture higher and lower harmonic transients that are not resolved by a frequency-domain solution.
The VizEM software package provides a very robust environment to solve such problems with quick turnaround. The VizEM software is available through the Overviz framework. This framework features an easy-to-use interface that provides utilities for problem set-up, problem solution, and post-processing of the solution. Once a mesh is available, the problem discussed in this note can be set-up within a matter of a few minutes.
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.
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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.
