Kinetic modeling and simulation of the planar multipole resonance probe
Final Report Abstract
The planar multipole resonance probe (pMRP) is an innovative sensor for measuring the electron density and the electron temperature of technical plasmas. Their mode of operation is based on the principle of active plasma resonance spectroscopy (APRS): An electromagnetic signal in the microwave range is coupled into the plasma via a probe and its resonance response is evaluated using a mathematical model. In particular, the electron density can be determined from the frequency of the measured resonance and the electron temperature from its width. In contrast to other realizations of this measuring principle, the pMRP does not disturb the plasma because it is coplanarly integrated into the reactor chamber wall. It is therefore particularly suitable for monitoring and controlling industrial plasma processes. This research project dealt, in two work packages, with the further development of the mathematical model required for the evaluation of the signal. In work package AP1, the focus was on the application of functional analytical methods. The signal was first formulated abstractly as a matrix element of the resolvent of the dynamic operator and then approximated in a finite basis. First, a pMRP of idealized geometry was analyzed with the fluid dynamic Drude model. At the same time, numerical simulations were carried out for a real geometry. A comparison revealed considerable differences in the position and width of the resonances. It turned out that these differences were caused by the idealization of the geometry; they could then be corrected. The understanding of the influence of the geometry was particularly important because the kinetic approaches to the pMRP are based on the same idealization. Next, a kinetic model of the pMRP was analyzed, initially again using functional analysis. Analytical expressions for the calculation of the matrices were derived and the first spectra were calculated. Howe- ver, sufficient accuracy required a great deal of effort. As an alternative, a spectral kinetic simulation was implemented. Here, the potential calculation was adopted, but the kinetic equation was solved using a particle-based simulation. First simulation results showed the expected resonance behavior and a good agreement with the results from work package AP2 of this project. In order to shorten the simulation time – currently several weeks – the corresponding code is currently being parallelized. In AP2, a complementary approach was chosen which combines analytical and numerical methods. In a first step, the linearized kinetic equation and Poisson’s equation were reduced to a one-dimensional integral equation for the disturbance potential δΦ using analytical methods. This equation was then dis- cretized in a second step and solved numerically. First, collision-free electron dynamics was assumed, approximately valid for low-pressure plasmas in the range of less than one Pascal. In the calculated spectra, the resonance and its broadening due to collision-free kinetic damping are clearly visible. With increasing electron temperature, the kinetic damping becomes stronger. In order to also cover a higher pressure range, electron impacts were also taken into account; the model contains both collision-free kinetic damping and collision damping. The kinetic models offer the possibility to obtain the electron density and the electron temperature from measurements. Further work is needed to reduce the simula- tion time of the collision kinetic model and then to apply both models to another idealized geometry, where the grounded chamber wall is replaced by the idealized dielectric. The understanding of the planar multipole resonance probe could be significantly deepened through the theoretical investigations carried out in this project. The mathematical models for describing the probe signals have been significantly improved and thus made suitable for practical use. After the completion of the parameter studies currently underway, reliable comparison spectra will be available for the evaluation of experiments.
Publications
- Planar Multipole Resonance Probe: A kinetic model based on a functional analytic description, 70th Annual Gaseous Electronics Conference, Bulletin of American Physical Society Volume 62, Number 10 (2017)
M. Friedrichs, R.P. Brinkmann, J. Oberrath
- Planar multipole resonance probe: Comparison of a functional analytic approach and full 3D electromagnetic field simulations, Verhandlungen der Deutschen Physikalischen Gesellschaft, Bremen 2017 (2017)
M. Friedrichs, C. Schulz, I. Rolfes, R.P. Brinkmann, J. Oberrath
- Planare Multipol-Resonanz-Sonde: Industrietaugliche Überwachungsdiagnostik, URSI Kleinheubacher Tagung, Miltenberg, Germany (2017)
M. Friedrichs, J. Oberrath
- A stacked planar sensor concept for minimally invasive plasma monitoring, 2018 Asia-Pacific Microwave Conference, 1315 (2018)
D. Pohle, C. Schulz, M. Oberberg, M. Friedrichs, A. Serwa, P. Uhlig, J. Oberrath, P. Awakowicz, I. Rolfes
(See online at https://doi.org/10.23919/apmc.2018.8617552) - Kinetic damping in the spectra of the spherical impedance probe, Plasma Sources Sci. Technol. 27, 045003 (2018)
J. Oberrath
(See online at https://doi.org/10.1088/1361-6595/aab745) - The planar multipole resonance probe: a functional analytic approach, EPJ Techn Instrum 5, 7 (2018)
M. Friedrichs, J. Oberrath
(See online at https://doi.org/10.1140/epjti/s40485-018-0049-x) - Kinetic investigation of the planar multipole resonance probe, 72nd Annual Gaseous Electronics Conference, Bulletin American Physical Society 64, 10 (2019)
C. Wang, M. Friedrichs, J. Gong, J. Oberrath, R.P. Brinkmann
- Kinetic modelling and simulation of the planar multipole resonance probe, Proceedings of the 13th Frontiers in Low-Temperature Plasma Diagnostics & 1st Frontiers in Low- Temperature Plasma Simulations, Bad Honnef, Germany (2019)
C. Wang, M. Friedrichs, J. Oberrath, R.P. Brinkmann
- Planar multipole resonance probe: Comparison of full-wave electromagnetic simulation and electrostatic approximation, 2019 Kleinheubach Conference, 1 (2019)
M. Friedrichs, D. Pohle, I. Rolfes, J. Oberrath
- Analysis of spectra of the pMRP using a spectral kinetic model, 73nd Annual Gaseous Electronics Conference, Bulletin American Physical Society 65, 10 (2020)
M. Friedrichs, J. Gong, R.P. Brinkmann, J. Oberrath
- Planar multipole resonance probe: Approximation of the ideal analytic solution adapted to the real geometry, 47th International Conference on Plasma Science (2020)
M. Friedrichs, D. Pohle, I. Rolfes, J. Oberrath
- The spherical multipole resonance probe: kinetic damping in its spectrum, Plasma Sources Sci. Technol. 29, 055005 (2020)
J. Oberrath
(See online at https://doi.org/10.1088/1361-6595/ab759f) - Kinetic investigation of the planar multipole resonance probe for arbitrary pressure
C. Wang, M. Friedrichs, J. Oberrath, R.P. Brinkmann
(See online at https://doi.org/10.48550/arXiv.2112.14190) - Kinetic investigation of the planar multipole resonance probe in the low-pressure plasma, Plasma Sources Sci. Technol. 30, 105011 (2021)
C. Wang, M. Friedrichs, J. Oberrath, R.P. Brinkmann
(See online at https://doi.org/10.1088/1361-6595/ac27bb) - On the multipole resonance probe: Current status of research and development, IEEE Transactions on Plasma Science 49, 3293 (2021)
J. Oberrath, M. Friedrichs, J Gong, M. Oberberg, D. Pohle, C. Schulz, C. Wang, Peter Awakowicz, R.P. Brinkmann, M. Lapke, T. Mussenbrock, T. Musch, I. Rolfes
(See online at https://doi.org/10.1109/tps.2021.3113832)
