Nonlinear Series Resonance Research Articles

Nonlinear series resonance and standing waves in dual-frequency capacitive discharges

It is well-known that the nonlinear series resonance in a high frequency capacitive discharge enhances the electron power deposition and also creates standing waves which produce radially center-high rf voltage profiles. In this work, the dynamics of series resonance and wave effects are examined in a dual-frequency driven discharge, using an asymmetric radial transmission line model incorporating a Child law sheath. We consider a cylindrical argon discharge with a conducting electrode radius of 15 cm, gap length of 3 cm, with a base case having a 60 MHz high frequency voltage of 250 V and a 10 MHz low frequency voltage of 1000 V, with a high frequency phase shift between the two frequencies. For this phase shift there is only one sheath collapse, and the time-averaged spectral peaks of the normalized current density at the center are mainly centered on harmonic numbers 30 and 50 of the low frequency, corresponding to the first standing wave resonance frequency and the series resonance frequency, respectively. The effects of the waves on the series resonance dynamics near the discharge center give rise to significant enhancements in the electron power deposition, compared to that near the discharge edge. Adjusting the phase shift from π to 0, or decreasing the low frequency from 10 to 2 MHz, results in two or more sheath collapses, respectively, making the dynamics more complex. The sudden excitation of the perturbed series resonance current after the sheath collapse results in a current oscillation amplitude that is estimated from analytical and numerical calculations. Self-consistently determining the dc bias and including the conduction current is found to be important. The subsequent slow time variation of the high frequency oscillation is analyzed using an adiabatic theory.

Nonlinear standing wave excitation by series resonance-enhanced harmonics in low pressure capacitive discharges

It is well-known that standing waves having radially center-high rf voltage profiles exist in high frequency capacitive discharges. It is also known that in radially uniform discharges, the capacitive sheath nonlinearities excite strong nonlinear series resonance harmonics that enhance the electron power deposition. In this work, we consider the coupling of the series resonance-enhanced harmonics to the standing waves. A one-dimensional, asymmetric radial transmission line model is developed incorporating the wave and nonlinear sheath physics and a self-consistent dc potential, for both conducting and insulating electrode surfaces. The resulting coupled pde equation set is solved numerically to determine the discharge voltages and currents. A 10 mTorr argon plasma is chosen with density m−3, gap width 2 cm and conducting electrode radius 15 cm, driven by a 500 V rf source with resistance 0.5 . We examine a set of frequencies from near 30 MHz up to frequencies more than three times as high. For most frequencies, no harmonics correspond exactly with the series or spatial resonances, which is the generic situation. Nevertheless, nearby resonances lead to a significantly enhanced ratio of the electron power per unit area on axis, compared to the average. Nearly similar results are found for insulating electrodes. Strong effects are seen for varying source resistance: high (50 ) resistance damps out most of the harmonic activity, while zero source resistance leads to a non-steady discharge with bias voltage relaxation oscillations. Stronger harmonic effects are seen for an increased radius of 30 cm, as lower harmonics become spatially resonant at lower frequencies. The radial dependence of electron power with frequency showed significant variations, with the central enhancement and sharpness of the spatial resonances depending in a complicated way on the amplitudes of the nearby series resonance current harmonics and the phase relations among the voltage harmonics driving these current harmonics. Significant center/average electron power per unit area enhancement is found even at the lowest frequencies for both high and low densities: 4.5 at 30 MHz and m−3, and 2.2 at m−3.

Electron heating via self-excited plasma series resonance in geometrically symmetric multi-frequency capacitive plasmas

The self-excitation of plasma series resonance (PSR) oscillations plays an important role in the electron heating dynamics in capacitively coupled radio-frequency (CCRF) plasmas. In a combined approach of PIC/MCC simulations and a theoretical model based on an equivalent circuit, we investigate the self-excitation of PSR oscillations and their effect on the electron heating in geometrically symmetric CCRF plasmas driven by multiple consecutive harmonics. The discharge symmetry is controlled via the electrical asymmetry effect (EAE), i.e. by varying the total number of harmonics and tuning the phase shifts between them. It is demonstrated that PSR oscillations will be self-excited under both symmetric and asymmetric conditions, if (i) the charge–voltage relation of the plasma sheaths deviates from a simple quadratic behavior and (ii) the inductance of the plasma bulk exhibits a temporal modulation. These two effects have been neglected up to now, but we show that they must be included in the model in order to properly describe the nonlinear series resonance circuit and reproduce the self-excitation of PSR oscillations, which are observed in the electron current density resulting from simulations of geometrically symmetric CCRF plasmas. Furthermore, the effect of PSR self-excitation on the discharge current and the plasma properties, such as the potential profile, is illustrated by applying Fourier analysis. High-frequency oscillations in the entire spectrum between the applied frequencies and the local electron plasma frequency are observed. As a consequence, the electron heating is strongly enhanced by the presence of PSR oscillations. A complex electron heating dynamics is found during the expansion phase of the sheath, which is fully collapsed, when the PSR is initially self-excited. The nonlinear electron resonance heating (NERH) associated with the PSR oscillations causes a spatial asymmetry in the electron heating. By discussing the resulting ionization profile in the nonlocal regime of low-pressure CCRF plasmas, we examine why the ion flux at both electrodes remains approximately constant, independently of the phase shifts.

Nonlinear Plasma Dynamics in Electron Heating of Asymmetric Capacitive Discharges with a Fluid Sheath Model

A fluid radio-frequency (rf) sheath model coupled to an equivalent circuit method is adopted to describe the nonlinear series resonance effects due to nonlinear interaction of plasma bulk and sheath in asymmetric capacitive discharges. With the fluid sheath model, we can determine self-consistently the relationship between the instantaneous potential drop across the rf sheath and the instantaneous sheath thickness. The numerical results demonstrate that the self-excitation of the plasma series resonance significantly enhances both ohmic heating and stochastic heating. Also, we observe that the effects of nonlinear series resonance increases the total power dissipation by factors of 2–5 for low pressure capacitive plasmas. Furthermore, we find that the largest harmonic is about 13 for the plasma current.

The effects of nonlinear series resonance on Ohmic and stochastic heating in capacitive discharges

The flow of electron and ion conduction currents across a nonlinear capacitive sheath to the electrode surface self-consistently sets the dc bias voltage across the sheath. We incorporate these currents into a model of a homogeneous capacitive sheath in order to determine the enhancement of the Ohmic and stochastic heating due to self-excitation of the nonlinear series resonance in an asymmetric capacitive discharge. At lower pressures, the series resonance can enhance both the Ohmic and stochastic heating by factors of 2–4, with the Ohmic heating tending to zero as the pressure decreases. The model was checked, for a particular set of parameters, by a particle-in-cell (PIC) simulation using the homogeneous sheath approximation, giving good agreement. With a self-consistent Child-law sheath, the PIC simulation showed increased heating, as expected, whether the series resonance is important or not.

NONLINEAR AMPLIFICATION EFFECTS IN A PERIOD-DOUBLING SERIES RESONANCE CIRCUIT

The influence of small near-resonant perturbations on the period-doubling bifurcations in a nonlinear series resonance circuit has been investigated in detail. The series resonance circuit contains ferroelectric triglycine sulfate (TGS) as nonlinear capacitor. If the temperature of the TGS-crystal is below the Curie-temperature this dynamical system is a good experimental realization of a Duffing-like oscillator with double-well potential. The depth of the double-well potential may be changed easily by temperature variation of the TGS-crystal. Variation of the amplitude of the driving voltage in the order of magnitude of the coercive field strength and the frequency of the driving voltage in the range of domain relaxation can be used to tune the system towards period-doubling bifurcations. An additional near-resonant perturbation has been introduced into the system by amplitude modulation of the driving voltage. The observed effects (small signal amplification, shift of the bifurcation point, multiple-side peaks around odd harmonics of half the driving frequency) are compared to theoretically derived results. They are well understood on the basis of the center manifold analysis taking into account nonlinear effects in the normal form of the period-doubling bifurcation.

Amplification of small signals caused by nonlinear properties of ferroelectrics

Abstract Polarization reversal in ferroelectrics causes strong dielectric nonlinearities. These nonlinearities are used to realize nonlinear dynamical systems, e. g. a nonlinear series resonance circuit. This circuit consists of a linear inductance and a ferroelectric capacitance. Depending on external parameters (e. g. amplitude and frequency of the driving voltage or temperature of the ferroelectric material) it is possible to tune the system to dynamical instabilities, such as period-doubling or symmetry-breaking bifurcations up to chaos. Near these instabilities the system can act to amplify small signals of suitably chosen frequencies. This work presents experimental results of a series resonance circuit containing TGS as ferroelectric. The effect of small signal amplification is investigated for different kinds of dynamical instabilities. The presented results are in good agreement with theoretical predictions.

Repolarization and chaotic phenomena in real ferroelectric crystals

Abstract The chaotic behaviour of a nonlinear series resonance circuit with a ferroelectric crystal (TGS, KDP, CDP) is investigated in dependence on temperature, as well amplitude and frequency of repolarizing a.c. electric field. Role of the domain processes in chaos emergence is discussed.