Comparison of electrostatic and electromagnetic simulations for very high frequency plasmas

Abstract

A two-dimensional self-consistent fluid model combined with the full set of Maxwell equations is developed to investigate an argon capacitively coupled plasma, focusing on the electromagnetic effects on the discharge characteristics at various discharge conditions. The results indicate that there exist distinct differences in plasma characteristics calculated with the so-called electrostatic model (i.e., without taking into account the electromagnetic effects) and the electromagnetic model (which includes the electromagnetic effects), especially at very high frequencies. Indeed, when the excitation source is in the high frequency regime and the electromagnetic effects are taken into account, the plasma density increases significantly and meanwhile the ionization rate evolves to a very different distribution when the electromagnetic effects are dominant. Furthermore, the dependence of the plasma characteristics on the voltage and pressure is also investigated, at constant frequency. It is observed that when the voltage is low, the difference between these two models becomes more obvious than at higher voltages. As the pressure increases, the plasma density profiles obtained from the electromagnetic model smoothly shift from edge-peaked over uniform to a broad maximum in the center. In addition, the edge effect becomes less pronounced with increasing frequency and pressure, and the skin effect rather than the standing-wave effect becomes dominant when the voltage is high.

Symmetry breaking in high frequency, symmetric capacitively coupled plasmas

“…[6][7][8][9][10][11] Results have also been obtained from more sophisticated simulations which solve Maxwell's equations in the time-domain and capture non-linear effects. [12][13][14][15][16] The main conclusion is that at higher frequencies and/or larger areas, the wavelengths of the EM surface waves in the plasma can become shorter than the reactor radius, leading to standing wave effects and consequent plasma non-uniformities. For intermediate frequencies, above the typical drive frequency of 13.56 MHz but well below the first spatial resonances of the waves, nonlinearly generated harmonics can also become resonant or near-resonant.…”

Electromagnetic effects in high-frequency large-area capacitive discharges: A review

“…These effects have been studied by using analytical EM models, 6,20 a more self-consistent transmission line approach, 19,23,25 numerical simulations, 15,17,[26][27][28] and experiments. 9,10,36,37 The three main EM effects, compromising the plasma nonuniformity, are the standing wave effect, the skin effect and the edge effect.…”

“…A similar model was developed by Zhang et al 27 to simulate the EM wave effect and its influence on the plasma uniformity in VHF capacitive discharges. They found that at 13.56 MHz, the plasma density exhibits a peak at the edge, which is caused by the electrostatic edge effect, due to the locally enhanced electrostatic field.…”

Two-dimensional particle-in-cell simulations of standing waves and wave-induced hysteresis in asymmetric capacitive discharges

“…However, electromagnetic or electrostatic effects at higher frequencies and large areas, such as standing waves and skin effects, come into play and can negatively affect the plasma uniformity [4,5]. To understand these effects, discharge models and simulations, treating the electromagnetic field both in the linearized frequency domain [6][7][8][9][10][11][12] and in the time domain [13][14][15][16][17][18], have been developed. The linear models initially considered uniform electron density in the bulk plasma and fixed sheath size, but linear (in frequency domain) transmission line models were later coupled to Child law sheaths and to particle and energy balance to obtain more self-consistent radial variations of the wave properties, sheath size, and electron density [7,[19][20][21][22][23].…”