2780 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 42, NO. 10, OCTOBER 2014 Complex Electron Heating in Capacitive Multi-Frequency Plasmas Julian Schulze, Edmund Schüngel, Aranka Derzsi, Ihor Korolov, Thomas Mussenbrock, and Zoltán Donkó Abstract — Complex spatio-temporal electron heating dynamics are observed in kinetic simulations of geometrically symmetric low-pressure capacitive argon plasmas driven by multiple consec- utive harmonics of 13.56 MHz. These dynamics are caused by an electrically induced asymmetry that leads to the self-excitation of plasma series resonance oscillations of the current. Such oscillations cause a nonsinusoidal movement of the boundary sheath edges and multiple phases of fast sheath expansions. These expansion phases lead to the generation of negative space charges that propagate into the bulk, where they affect the heating rate significantly and relax quickly. Index Terms— Plasma sheaths, plasma simulation, plasma sources, plasmas. L OW-PRESSURE capacitive radio frequency (RF) plas- mas are frequently used for etching and sputtering of dielectric substrates. In order to optimize these applications, customized ion flux energy distributions are required. A novel concept to realize such distributions is voltage waveform tailoring [1]: N consecutive harmonics of a fundamental frequency, f , with adjustable harmonics amplitudes, φ k , and phases, θ k , are applied to one electrode, whereas the other is grounded, i.e., the driving voltage waveform is ˜ φ(t ) = N  k=1 φ k cos(2π ft + θ k ) with φ tot = N  k=1 φ k. (1) Here, we investigate the spatio-temporal electron heating dynamics in a geometrically symmetric argon plasma by 1d3v particle in cell/Monte Carlo collisions (PIC/MCC) simulations. One of two plane parallel electrodes is driven by N = 4 consecutive harmonics of f = 13.56 MHz according to equation (1). All phase shifts are set to 0° and φ tot = 800 V. The harmonics amplitudes are φ 1 = 320 V, φ 2 = 240 V, φ 3 = 160 V, and φ 4 = 80 V according to a criterion defined in [2]. Manuscript received November 2, 2013; revised January 23, 2014; accepted February 9, 2014. Date of publication March 4, 2014; date of current version October 21, 2014. This work was supported in part by the Hungarian Scientific Research Fund under Grant OTKA-K-105476 and Grant NN-103150, and in part by DFG under Grant SFB TR 87. J. Schulze and E. Schüngel are with the Department of Physics, West Virginia University, Morgantown, WV 26506 USA (e-mail: felixjulian. schulze@mail.wvu.edu; edmundschuengel@gmx.net). A. Derzsi, I. Korolov, and Z. Donkó are with Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest 1053, Hungary (e-mail: derzsi.aranka@wigner.mta.hu; ihor.korolov@gmail.com; donko.zoltan@wigner.mta.hu). T. Mussenbrock is with the Institute for Theoretical Electrical Engi- neering, Ruhr-University Bochum, Bochum 44801, Germany (e-mail: mussenbrock@gmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2014.2306265 The electrode gap is 3 cm, the pressure is 3 Pa, the secondary electron emission coefficient is γ = 0.2, and 20% of the electrons are reflected at the electrodes [1,3]. Fig. 1 shows the spatio-temporal distribution of the electron heating rate within one fundamental RF period. The heat- ing dynamics observed here are significantly more complex compared with single frequency discharges and cannot be explained by the classical picture of stochastic heating caused by the expansion of a sheath, whose motion is sinusoidal, and cooling during the phase of sheath collapse. The more complex voltage waveform induces a strong electrical asymmetry [4], that leads to a dc self bias of about -465 V. At such low pressures, this asymmetry leads to the self-excitation of strong plasma series resonance (PSR) oscillations, i.e., the current is no longer proportional to the driving voltage or its temporal derivative [5]. The PSR causes high frequency oscillations of the electron heating rate at the beginning of the fundamental RF period (0–30 ns) due to corresponding oscillations of the electric field until they are damped by collisions. During the time of high sheath voltage at the powered electrode (5–65 ns), heating of secondary electrons inside the sheath is observed [6]. During sheath collapse (65–74 ns), an electric field reversal causes additional heating [7]. Fig. 2 shows a zoomed-in-view into the dashed rectangle in Fig. 1, i.e., the time of initial sheath expansion. At a distance of about 0.7 cm above the powered electrode, the ambipolar electric field is maximum due to the steep ion density profile at this position and additional heating induced by this field as well as interference effects with the PSR oscillations are observed. Fig. 3 shows the net density, n i - n e , within the same spatio- temporal region. In the region of high ambipolar electric field, a double layer of positive and negative space charge is observed due to a change of the local gradient of the electric field around 0.7 cm. During the initial phase of sheath expansion at the powered electrode, the ensemble of electrons is compressed by the fast expanding sheath and a beam-like negative space charge propagates at a veloc- ity of about 1.5 × 10 6 m/s into the plasma bulk (dashed rectangles in Figs. 2 and 3). This negative space charge is followed by a region of positive space charge similar to a negative streamer. This moving space charge oscillates on the timescale of the inverse electron plasma frequency and relaxes quickly. Its presence affects the heating caused by the PSR oscillations. In conclusion, we observe complex electron heating mech- anisms in multi-frequency geometrically symmetric capacitive discharges operated at low pressures. The complexity of 0093-3813 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.