Two-Dimensional Particle-in-Cell Simulation of a Micro RF Ion Thruster

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1 Two-Dimensional Particle-in-Cell Simulation of a Micro RF Ion Thruster IEPC--7 Presented at the nd International Electric Propulsion Conference, Wiesbaden Germany September 5, Yoshinori Takao, Koji Eriguchi, and Kouichi Ono Kyoto University, Kyoto, -5, Japan Abstract: A two-dimensional particle-in-cell with a Monte Carlo collision (PIC-MCC) calculation has been conducted to investigate the frequency dependence of a micro inductively coupled plasma (ICP) source for a better performance of a micro RF ion thruster. The working gas is Xe and the micro ICP source size is mm in radius and 7 mm in length at the Xe pressure of. mtorr. For the frequency range examined (5 5 MHz), the PIC-MCC results have indicated that lower RF frequencies lead to higher plasma densities and more uniform plasma profiles. Nomenclature B = magnetic field E = electric field i = square root of j = current density m = mass q = charge r, θ, z = cylindrical coordinate t = time v = velocity ε = vacuum permittivity μ = vacuum permeability ν m = electron momentum transfer collision frequency ρ = charge density = electrostatic potential ω = angular frequency of the RF coil current = electron plasma frequency ω pe I. Introduction N recent years, there has been a growing interest in microspacecraft. The weight of microspacecraft can be less Ithan ten kilograms by using microelectronics and MEMS technology. Such microspacecraft can reduce total mission costs because of their low-mass structure and short-development period. They can also have a high reliability if a mission is composed of several microspacecraft. Microspacecraft require micro thrusters for the station keeping. Such microthrusters have to be simple, light and low-power consumption. It is desirable to use easyto-handle propellant. In addition, high precision control of thrust performance is required. Assistant Professor, Department of Aeronautics and Astronautics, Yoshida-Honmachi, Sakyo-ku, Kyoto -5, takao.yoshinori.7a@kyoto-u.ac.jp Associate Professor, Department of Aeronautics and Astronautics, eriguchi@kuaero.kyoto-u.ac.jp Professor, Department of Aeronautics and Astronautics, ono@kuaero.kyoto-u.ac.jp The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

2 Ion density ( cm - )..x -.x -.x -.x + Discharge Vessel RF Coil Grid System r (mm).. Xe Matching Network plasma Xe + e Neutralizer r(mm)... Ion energy (ev) r (mm) z(mm) V.x + 7.x +.x +.x +.x +... RF Source z(mm) z(mm) Figure. Schematic of the micro RF ion thruster. Figure. Distribution of ion beam profiles at the Ar pressure of mtorr and absorbed power of mw. We have developed an electrothermal type microthruster using microwave-excited plasmas, and analyzed the plasma characteristics and thruster performance both numerically and experimentally. - The thrust obtained was around mn using Ar at microwave powers below W, which is applicable to nanosatellites (< kg) for the station keeping, while the specific impulse obtained was below s. On the other hand, a microthruster with a high specific impulse, such as an ion thruster, is also required. The micro RF ion thruster presented here uses a cylindrical micro inductively coupled plasma (ICP) source. The micro ICP requires no magnets or electrodes, so that light structure and high reliability could be obtained. In addition, the plasma source can produce higher plasma densities compared with unmagnetized DC plasmas, which leads to higher efficiency. Figures shows the schematic of the micro RF ion thruster. The thruster consists of a micro ICP source, grid electrodes for ion beam extraction, and a neutralizer. Although some experimental studies on micro RF ion thrusters were performed, 7, there are few papers on numerical analyses, particularly on particle simulations. Since the spatial distribution of plasma parameters is not easily available in experiments for a small space less than a few millimetres, numerical simulations can be useful. We have developed a two-dimensional particle model for the micro RF ion thruster and investigated the plasma parameters and ion beam profiles as shown in Fig., where a planar spiral coil was employed for the plasma source and the inner radius and the length were and mm, respectively. 9 Since the coordinate system is axisymmetric, we used a ring-shaped slit for ion beam acceleration. The potential of the screen grid and acceleration grid is set at. kv and V, respectively. The ion beam trajectory seemed to fit the ring-shaped slit, where the thrust and specific impulse obtained were 9. N and 7 s, respectively, at the gas pressure of mtorr and absorbed power of mw, using argon as a propellant. The calculations we conducted above were based on the collisional heating model, or cold-plasma approximation, so that the only ohmic heating was considered as the mechanism of electron heating. However, if the electron mean free path is sufficiently long, that is, thermal effects are important, the cold-plasma approximation is no longer valid. These conditions produce nonlocal effects. In such a situation, we have to take non-collisional heating into account in the model, or a warm plasma model. In the present work, we have focused on the analysis of the micro ICP source using the warm plasma model and investigated the RF frequency dependence of the plasma characteristics, where xenon is used as a propellant gas. We describe the particle model developed in Sec. II. Results and discussion are then presented in Sec. III. Finally, conclusions are drawn in Sec. IV. II. Numerical Model A. Assumption To investigate the characteristics of the plasma source, we have conducted particle-in-cell simulations with a Monte Carlo collision calculation (PIC-MCC), - assuming the following conditions. The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

3 (i) Only Xe ions and electrons are treated as particles, and the ion species of interest is singly-ionized Xe + only. (ii) Neutral particles have Maxwellian velocity distribution at the gas temperature of K (=. ev). (iii) The reactions taken into account are elastic scattering, excitation, and ionization for electrons, and elastic scattering and charge exchange for ions, as below. (a) e + Xe e + Xe (Elastic Scattering) (b) e + Xe e + Xe * (Excitation) (c) e + Xe e + Xe + + e (Ionization) (d) Xe + + Xe Xe + Xe + (Charge Exchange) (e) Xe + + Xe Xe + + Xe (Elastic Scattering) (iv) The motion of excited-state atoms is not considered. (v) Coulomb collisions are not taken into account. (vi) The coordinate system is axisymmetric, in which a number of simulated particles (or superparticles for ions and electrons) are loaded in a two-dimensional spatial mesh (r, z), along with three velocity components (v r, v θ, v z ). (vii) Capacitive coupling from the RF antenna is not treated. B. Electrostatic Field The electrostatic field is given by. () The potential is derived from the space charge of charged particles. The Poisson equation is given by,. () Equation () is discretized in the same manner as described in Ref. The electrostatic field E is determined by the central difference from the potential. Here, to eliminate a systematic error in charge density on the axis (r = ) for cylindrical coordinates, we employ a general weighting scheme presented by Verboncoeur. We also apply a digital smoothing algorithm to the space charge in order to decrease the numerical noise owing to the limited number of superparticles. 5 C. Electromagnetic Field The micro ICP source is driven by the RF current applied to the antenna. All waves quantities, such as the electromagnetic field and current density, are assumed to vary harmonically in time with an dependence. We also assume that the electric field has only the azimuthal component. Then the complex amplitude of the electromagnetic fields is obtained from the following equation.. () The boundary conditions are zero at the metal walls and on the axis. On the dielectric window, the electric field is the sum of the field by the external coil and the field due to the plasma current in the source, and can be obtained analytically. - The magnetic field is then obtained from Faraday's law with the electric field determined by Eq. (). D. Motion and Collisions of Charged Particles Using the electrostatic field and electromagnetic field obtained above, we move charged particles by integrating the equation of motion:. () Equation () is solved by Buneman-Boris method with a coordinate rotation for the position advance. Motion and collisions of charged particles can be treated separately by the principle of decoupling if a chosen time step gives a small collision probability. 9 To reduce the cost of calculation time, we employ the null-collision method in MCC with cross sections for electrons - and ions. The postcollision velocities of electrons and ions are determined by the use of the conservation equations for momentum and energy. The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

4 Peak Electron Density ( cm - ) 5 Ar, cold unsustainable Xe, warm Ar, warm Pressure (mtorr) Figure. Peak Ar and Xe electron density as a function of neutral gas pressure at an absorbed power of mw. Calculations were conducted for both the cold and warm plasma models. 5 Peak Electron Density ( cm - ) r mm -turn Coil Simulation Area 7 mm Dielectric Meta Figure. Simulation area for the analysis of the micro ICP source. z Table. Calculation conditions for the micro ICP source. Chamber length Pressure Absorbed power Coil position l (mm) p (mtorr) P abs (mw) z (mm). 7...,.5, , 5.,.. Chamber radius r (mm) Dielectric thickness (mm) E. Power Deposition in the Plasma To solve Eq. (), a number of approximations are generally applied to relate j θ and E θ. The cold-electron approximation is commonly employed, so that the current is given by j θ = σ p E θ, where σ p is the plasma conductivity and defined as follows.. (5) Ion current can be ignored owing to the low mobility and we can solve Eq. () using Eq. (5). The time-averaged collisional power deposition per unit volume Q c is then given by Re. () Here, we refer to the analysis based on the cold plasma approximation as the cold plasma model. In the particle model, one can derive the plasma current density directly by following electron trajectories. This calculation is fully kinetic and no assumptions are required about the mechanism of electron heating. The kinetic plasma current density is obtained from the equation below. exp. (7) where V g is the cell volume centered at a grid point, W e is the weight of a electron superparticle, v is the amplitude of the azimuthal component of the electron velocity at the fundamental frequency, is the summation of all the electron superparticles in the volume V g, and is the phase difference between the RF coil current and j. The time-averaged kinetic power deposition per unit volume Q k is calculated from cos Δ. () The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

5 This implementation enables us to self-consistently take into consideration the effect of noncollisional heating, which is important to low pressure RF plasmas, and then we refer to the analysis based on it as the warm plasma model. In the simulation, the power deposition P abs is used as an input parameter, which is obtained by integrating Eq. () or () over the entire simulation area. Thus, we rescale the RF coil current and E to yield the specified absorbed power in the plasma until the steady state solution is obtained. As stated in Sec. I, if the thermal effects of electrons are important, the cold-plasma approximation is no longer valid and we have to take non-collisional heating into account in the model, or the warm plasma model as shown in Fig.. For the cold plasma model, the plasma density RF Frequency (MHz) Figure 5. RF frequency dependence of the volumeaveraged electron density under the conditions listed in Table. erroneously increases as the pressure decreases. On the contrary, the plasma density decreases with the pressure for the warm plasma model. Moreover, the peak plasma density is ten times lower than that for the cold plasma model. Volume-Averaged Plasma Density ( 9 cm - ) 9 7 r=5mm r=mm Azimuthal Electric Field: E (V/cm) Absorbed Power Density: Q k (W/cm ) (a) E, 5MHz (d) Qk, 5MHz Radial Distance (mm) (b) E, 5MHz 5 7 (e) Qk, 5MHz (c) E, 5MHz (f) Qk, 5MHz Axial Distance (mm) Figure. Two-dimensional distributions of the amplitude of the azimuthal electric fields E and the timeaveraged absorbed power density Q k in the micro ICP source, calculated at the Xe pressure p =. mtorr and absorbed power P abs = mw (W abs =. W/cm ) for f = 5, 5, and 5 MHz. 5 The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

6 . Ion Density: n i ( cm - ) Electron Density: n e ( cm - ) (a) ni, 5MHz (d) ne, 5MHz Radial Distance (mm). (b) ni, 5MHz (e) ne, 5MHz (c) ni, 5MHz Axial Distance (mm) Figure 7. Two-dimensional distributions of the time-averaged densities of ion n i electron n e in the micro ICP source, calculated under the same conditions as in Fig.. The behavior of the plasma density that the warm plasma model predicted is also shown in low-pressure large ICP sources. Hence, we use the warm plasma model in the following section. III. Results and Discussion Figure shows the simulation area for the analysis of the micro ICP source. The simulation area is divided at regular intervals with the grid spacing of. mm. The particle simulations in the micro ICP source were conducted under the conditions listed in Table, where the conditions were determined by scaling laws, which were obtained in experiments. 7 To investigate the frequency dependence, the RF frequency f was set at 5 5 MHz. As described in Sec. II, the absorbed power P abs is an input parameter for our calculations. The absorbed powers were obtained in such a way that the volume-averaged power density W abs was. W/cm for both chamber sizes, which is comparable to that for conventional ICP sources. 5 Figure 5 shows the RF frequency dependence of the volume-averaged plasma density. The plasma density increases monotonically with a decrease in RF frequency. For r = mm, the plasma densities obtained were 7. 9 and. 9 cm at f = 5 and 5 MHz, respectively, while those were. and. cm for r = 5mm. Since the ratio of the chamber surface to its volume decreases with an increase in radius, the plasma density increases with increasing radius at the same absorbed power density of. W/cm. The increasing rate is almost corresponding to the decreasing rate of the surface-to-volume ratio because the plasma loss mechanism is due to only the ambipolar diffusion to the chamber walls in our calculations. Based on the scaling laws, the optimum RF frequency is determined to be.5 and 7.5 MHz for r = and 5 mm, respectively, which is comparable to the The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5, (f) ne, 5MHz

7 Electron Temperature: T e (ev) (a) Te, 5MHz Potential: (V) (d), 5MHz Radial Distance (mm) (b) Te, 5MHz (e), 5MHz 5 7 (c) Te, 5MHz (f), 5MHz Axial Distance (mm) Figure. Two-dimensional distributions of the time-averaged electron temperature T e and potential in the micro ICP source, calculated under the same conditions as in Fig.. electron collision frequency at the corresponding pressure. In view of the fact that the frequency curve shows a broad maximum, the simulation result is in a reasonable agreement with experimental data. 7 It should be noted that the plasma was unsustainable below f = MHz and all the charged particles disappeared in the simulation area under the conditions in Table. Figures 9 show the two-dimensional distributions of the plasma parameters for different RF frequencies and chamber sizes. Notice that the plasma parameters were determined by averaging over RF cycles after the steady state was reached, except for E, which was obtained from Eq. (). The amplitude of the azimuthal electric field E induced by the RF coil current has a peak value on the surface of the dielectric window at the center coil and decreases monotonically into the plasma. Since the chamber size is smaller than the skin depth, the profile of the inductive electric fields is determined by the boundary conditions of the simulation area rather than the plasma condition. As stated in the assumption, capacitive coupling from the RF coil was not taken into account in the particle model developed, so that the absorbed power density was calculated to be the product of the azimuthal electric field and plasma current density with the phase difference between them as Eq. (). As shown in Figs. (d) (f), the peak of the power density was obtained at the core of the simulation area except for f = 5 MHz, where three peaks are exhibited and the negative power absorption can be seen. Godyak and Kolobov demonstrated in an ICP experiment that the RF current transferred by thermal electron motion can be opposite in phase to the local electric field resulting in a local negative energy transfer from the RF field to electrons in the area away from the skin layer near the dielectric. At f = 5 MHz and p =. mtorr, the angular frequency is much larger than the electron 7 The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

8 .. (a) E (V/cm), 5MHz (b) Qk (W/cm ), 5MHz Radial Distance (mm) (c) ni ( cm - ), 5MHz (d) ne ( cm - ), 5MHz (e) Te (ev), 5MHz (f) (V), 5MHz.. Axial Distance (mm) Figure 9. Two-dimensional distributions of (a) the amplitude of the azimuthal electric fields E and the time-averaged (b) absorbed power density Q k, (c) ion density n i, (d) electron density n e, (e) electron temperature T e, and (f) potential in the micro ICP source, calculated at the Xe pressure p =.5 mtorr and absorbed power P abs = 7 mw (W abs =. W/cm ) for f = 5 MHz. momentum transfer collision frequency m, i.e., collisionless effects caused by electron thermal motion are dominant. Therefore, the negative power absorption occurred even in the skin layer at f = 5 MHz. For f = 5 and 5 MHz, the collisionless effects are not so significant, and thus, the peak absorption power is obtained where there are enough electrons away from the sheath. Figure 7 indicates that there is a relatively thick sheath structure, which is almost mm, so that the plasma density is non uniform in the radial direction. As shown in Fig.7, the peak plasma density is almost constant among the three RF frequencies of 5, 5 and 5 MHz. On the other hand, more uniform density profile is obtained as the frequency decreases. Therefore, the volume-averaged plasma density increases with decreasing frequency as shown in Fig. 5. While the RF power deposition occurs near the coil, the maximum plasma density is obtained at the centerline of the plasma source, where the peak ion density obtained is about.5 cm. The distribution of the electron density is quite similar to that of the ion density, so that quasi-neutrality is confirmed. Whereas similar potential profiles are obtained among the three RF frequencies, the profile of the electron temperature is strongly dependent on the frequency. Since the azimuthal electric fields are higher at higher frequency as shown in Figs. (a) (c), higher electron temperature is obtained. The Debye length is proportional to the square root of the electron temperature, so that the slightly thicker sheath structure is obtained with increasing RF frequency, which also results in a slight increase in the potential as shown in Figs. (d) (e). The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,

9 The amplitude x of the displacement of electrons by the RF electric fields is generally derived from the following equation.. (9) ω ω ν Equation (9) indicates that the displacement of electrons significantly decreases with increasing RF frequency. The electrons can get the energy of the azimuthal electric fields near the dielectric boundary before they diffuse to the dielectric wall, so that the peak electron temperature is obtained thereat for f = 5 MHz. On the other hand, most of the electrons reach the dielectric walls before they get enough energy from the RF fields to be heated at lower frequency, so that the peak electron temperature is obtained a little away from the dielectric boundary where the peak absorbed power densities are taken as shown in Figs. (d) and (e). The surface-to-volume ration decreases with an increase in chamber radius, and thus, more uniform plasma density profiles are obtained for r = 5 mm. At a fixed volume-averaged power density of W abs =. W/cm, the peak absorbed power density and the azimuthal electric filed decreases for r = 5 mm, owing to the larger volume of the bulk plasma. The reduced electric fields result in a lower electron temperature and then the lower potential. Figure shows the two-dimensional distributions of the absolute value of the plasma current density, together with the current directions indicated by streamlines with arrows. It should be noted that the current density consists only of the electron current here due to the slow motion of ions and the values are obtained by averaging over RF cycles. Since the azimuthal current density should be zero after averaging over RF cycles, the absolute value contains of only the radial and axial directions of the current. At f = 5 MHz, all arrows point from the wall to the center of the cylindrical axis. Therefore, electrons are lost by ambipolar diffusion from the bulk plasma Radial Distance (mm) Axial Distance (mm) Figure. Two-dimensional distributions of the timeaveraged absolute value of the current density j p in the micro ICP source, calculated under the same condisions as in Fig. for f = (a) 5, (b) 5, and (c) 5 MHz. The current directions are shown by streamlines with arrows. regions to the chamber walls, particularly to the dielectric wall owing to the larger surface area than the side walls. The magnitude of the current density is about ten times larger than those for f = 5 and 5 MHz since the electrons diffuse to the wall before they are heated by the azimuthal electric fields as indicated by Eq. (9). With increasing frequency, the streamlines become complicated and some vortices structure can be seen. The reason of the complexity of the flows may be related with the thermal electron motion in the collisionless regime although it is still unclear and left for future work. IV. Conclusions A two-dimensional PIC-MCC simulation has been conducted to investigate the frequency dependence of a micro ICP source for a better performance of a micro RF ion thruster. The numerical model is based on a warm plasma model, i.e., collisionless heating is taken into account. The working gas is Xe and the size of the micro ICP source is 9 The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5, Current Density: j p (ma/cm ) (a) 5MHz (b) 5MHz (c) 5MHz

10 mm in radius and 7 mm in length at the Xe pressure of. mtorr. For comparison, we also carried out the simulation of the micro ICP souse 5 mm in radius and mm in length at the pressure of.5 mtorr. For the frequency (5 5 MHz) ranges examined, the PIC-MCC results have indicated that the plasma density increases monotonically with decreasing RF frequency and a more uniform profile of the plasma density is obtained, implying that ion beams can be extracted efficiently at lower RF frequencies. This tendency is in a reasonable agreement with experimental data. Acknowledgments This work was financially supported in part by a Grant-in-Aid for Young Scientists (B) (779) from Japan Society for the Promotion of Science. References Micci, M. M., and Ketsdever, A. D., Micropropulsion for Small Spacecraft, American Institute of Aeronautics and Astronautics, Reston,. 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11 Rauf, S., and Kushner, M. J., Model for noncollisional heating in inductively coupled plasma processing sources, Journal of Applied Physics, Vol., No. 9, 997, pp Takekida, H., and Nanbu, K., Self-consistent particle modeling of inductively coupled CF discharges and radical flow, IEEE Transactions on Plasma Science, Vol., No.,, pp Godyak, V. A., and Kolobov, V. I., Negative Power Absorption in Inductively Coupled Plasma, Physical Review Letters, Vol. 79, No., 997, pp The nd International Electric Propulsion Conference, Wiesbaden, Germany September 5,