Session RR3: Modeling & Simulation III

Kinetic modeling of Active Plasma Resonance Spectroscopy: Why ``Drude'' is not enough

The term active plasma resonance spectroscopy (APRS) denotes a class of related techniques which utilize, for diagnostic purposes, the natural ability of plasmas to resonate on or near the electron plasma frequency $\omega_{\mathrm{pe}}$: a radio frequent signal (in the GHz range) is coupled into the plasma via a probe, the spectral response is recorded, and a mathematical model is used to determine parameters such as the plasma density or the electron temperature. This contribution discusses the importance of choosing the proper physical theory for that mathematical model. The so-called Drude model (also known as cold plasma model) provides a relatively simple qualitative description, but is not suited for a quantitatively correct analysis. The physically superior kinetic theory is, unfortunately, mathematically more complex. The contribution discusses the difference between the two approaches and shows how the mathematical problems of the kinetic description can be overcome.   

Sheath Model Coupled Electromagnetic Simulation of Capacitively Coupled Plasma Reactors

Capacitively coupled plasmas (CCP) are widely used for microelectronics manufacturing. 3D electromagnetic (EM) simulations are used to exam EM field and power distribution in practical reactors. Bulk plasma is often treated as lossy media with complex conductivity in these simulations. However, plasma sheath plays an essential role in determining EM response of CCP reactors that needs to be considered. In this study, a simplified CCP reactor is simulated using a FDTD solver coupled to a RF sheath model, which is treated as a nonlinear distributed circuit consisting of a capacitor, a diode and a current source. The circuit components are self-consistently solved using sheath models [1,2]. The resulted EM field distribution shows higher harmonics due to the nonlinearity of the sheath. An electrical asymmetry is achieved using unequal electrode areas or modulated waveforms. The reactor impedance is calculated from lumped voltage and current, and is compared to the result of a fluid-based plasma model. The coupled model demonstrates the importance of plasma sheath to EM field distribution and provides a flexible way to characterize EM response of CCP reactors. 1. MA Lieberman, IEEE Trans. Plasma Sci. 16 638 (1988) 2. A Metze et al, J. Appl. Phys. 60 3081 (1986)   

Model-experiment comparison of radiofrequency phase resolved plasma parameters for moderate pressure capacitively coupled discharges

Spatial profiles of plasma parameters along with voltage and current characteristics in a parallel plate capacitively coupled discharge at moderate pressures are compared with 2-dimensional fluid plasma simulation results. Plasma parameters including electron density, effective collision frequency, and probe sheath thickness are measured with a hairpin resonator probe over different pressures and powers ranging from 1.3-266 Pa and RF voltage amplitude 80-400 V in Ar, He, and N$_{\mathrm{2}}$ plasmas driven at 13.56 MHz with a gap thicknesses of 2.54 cm. Spatial measurements are made in the axial and radial directions. Probe sheath thickness is determined using a time resolved system capable of 4 ns resolution. The high time resolution is leveraged to measure electron density and effective collision frequency in the rf cycle versus axial distance for a variety of conditions to explore powered sheath dynamics. Measurements show a region of strong electron density modulation close to the powered electrode corresponding to the rf sheath while also showing oscillations in the plasma bulk. Measurements using different floating probe geometries are compared and yield similar results, suggesting that the geometries used are sufficient to inhibit the formation of an RF sheath across relevant probe surfaces.   

Ideal Multipole Resonance Probe: a Spectral Kinetic Approach

Active Plasma Resonance Spectroscopy (APRS) denotes a class of industry-compatible plasma diagnostic methods which utilize the natural ability of plasmas to resonate on or near the electron plasma frequency. One particular realization of APRS with a high degree of geometric and electric symmetry is Multipole Resonance Probe (MRP). The Ideal MRP (IMRP) is an even more symmetric idealization which is suited for theoretical investigations. In this work, a spectral kinetic scheme is presented to investigate the behavior of the IMRP in the low pressure regime. The proposed kinetic model overcomes limitation of the cold plasma model and covers kinetic effects such as collisionless damping. Most importantly, it provides the possibility to calculate both the electron temperature and electron density from measured resonance curves.   

Control of electron, ion and neutral heating in radio-frequency hollow cathodes

Low power and compact hollow cathode plasmas are of significant interest for applications including materials surface processing and electric propulsion. To control the charged and neutral particle dynamics under collisional conditions, radio frequency (rf) excitation can regulate power delivery to them via the sheath motion and formation of a dc self-bias voltage. In this study, we investigate electron, ion and neutral-gas heating in an rf hollow cathode flow reactor operating in argon at ~1~Torr (133~Pa) with a fundamental applied-voltage frequency of 13.56~MHz. Two dimensional, fluid-kinetic simulations undertaken with the Hybrid Plasma Equipment Model corroborate measurements of the electron-impact excitation rate via phase-resolved optical emission spectroscopy for single frequency, dual frequency and tailored-waveform excitation. Structured ion energy distribution functions (IEDFs) are produced with a single voltage waveform as its frequency is increased at constant amplitude, and this is explained by a simultaneous increase in the Ar+ ion-neutral mean-free-path and decrease in the time-averaged extension of the sheath. Dual-frequency voltage waveforms are applied to regulate the ion-power fraction via the dc self-bias voltage, and thereby the temperature of the neutral gas as it travels between the plenum and expansion regions. Tailored voltage waveforms, generated through the superposition of multiple frequency components, induce temporal asymmetries in the electron heating. The resulting enhancement to the control of ion and neutral heating increases prospects for optimising particle energies and fluxes in intermediate-pressure, collisional plasma applications.   

Simulations of the real planar multipole resonance probe in electrostatic approximation

Active Plasma Resonance Spectroscopy (APRS) is an industry compatible plasma diagnostic method, which takes advantage of the ability of the plasma to resonate in the near of the electron plasma frequency. The planar multipole resonance probe (pMRP) is a specific design of APRS and a promising candidate to monitor plasma processes without perturbing them, due to its planar design, which is mounted inside of the chamber wall. Based on the cold plasma model an analytic solution of the response function for the ideal pMRP could be derived, which allowed to determine the resonance frequency of the probe plasma system. However, the geometry of the real pMRP is more complicated and requires a numerical model for full 3D electromagnetic simulation in CST. The calculated resonance frequencies of both models are qualitatively in excellent agreement, but differ in the exact position. This difference is dominated by the difference in the geometry, which cannot be taken into account in the analytic solution. Thus, a simulation of a more realistic geometry in electrostatic approximation will be presented in Comsol Multiphysics. The results will be compared to the former investigations to fully understand the influence of the geometry.   

Validation of the Open Source Multi-Fluid Plasma Code: Zapdos

Validation work was done through comparisons between results obtain from Zapdos and experimental and simulation efforts, this included both low pressure discharges and atmospheric devices (such as the COST plasma jet and GEC reference cell). This effort reveals good agreement between Zapdos and previous efforts in comparing electron density, electron temperature, mean electron energy, metastable densities, and electric potential. The Zapdos application is an open source finite element code for modelling plasmas using the multi-fluid method based in the MOOSE framework. The application includes the continuity equations for electrons, ions, and metastable populations. The mean electron energy was calculated using the energy balance equations. The electric field is determined using the electrostatic approximation and the source terms are determined by the chemistry application CRANE, another MOOSE based application. The modelling of RF CCP discharges at low pressures (around 1 mTorr) and atmospheric devices (such as the COST plasma jet) are compared to previously published experimental data, experimental efforts undertaken at NCSU, and simulation results from other vetted fluid models.   

Block matrix based LU decomposition to compute kinetic spectra of the Multipole Resonance Probe

The multipole resonance probe (MRP) is a diagnostic tool to measure electron densities and electron temperatures in low pressure plasmas. It is based on the concept of active plasma resonance spectroscopy and excites a resonance of the dipole mode, where the resonance frequency is proportional to the electron plasma frequency. From the half width of the resonance peak the electron temperature could be measured, but the peak is broadened by kinetic effects and requires a kinetic model to predict the correct half width. Such a model in electrostatic approximation based on functional analytic methods is derived and yields the admittance Y as response function of the probe-plasma system. To approximate specific spectra of Y functional analytic methods can also be applied. This approach leads to a huge system of linear equations, which is characterized with a block structured matrix. To solve this linear system of equations, a block matrix based LU decomposition can be applied to reduce the memory in use. The resulting spectra show a broadening of the resonance peak and will be investigated to determine a relation between the half width and the electron temperature.