Dennis Ziegler

A nonlinear global model of a dual frequency capacitive discharge

The behavior of dual frequency capacitively coupled plasmas is investigated. Assuming a realistic reactor configuration represented by effective geometry factors and taking into account two separate sinusoidal voltage sources operating at different frequencies, an ordinary differential equation is derived which describes the nonlinear dynamics of such discharges. An exact analytical solution of the equation is presented and employed for a parameter study of the discharge current characteristics. Simulation results for various gas pressures (=various electron-neutral collision rates), various amplitude ratios of the two independent rf sources, and various integer frequency ratios are shown. When the two frequencies are comparable, surprising nonlinear effects are observed. Particular under study is the heating at the plasma series resonance, either by direct excitation or via the nonlinear electron resonance heating mechanism.

Nonlinear dynamics of dual frequency capacitive discharges: a global model matched to an experiment

The dynamics of dual frequency capacitively coupled plasmas (2f-CCPs) is investigated using an approach that integrates theoretical insight and experimental data. Basis of the analysis is a recently published model which casts the high-frequency behavior of asymmetric 2f-CCPs in terms of a nonlinear second-order differential equation, or equivalently, a lumped element equivalent circuit (Mussenbrock et al 2006 Phys. Plasmas 13 083501). The model comprises a nonlinear capacitor (the electrode boundary sheath), a lossy inductance (electron inertia and Ohmic losses in the bulk), a blocking capacitor and two ideal voltage sources in series (the 2f excitation). In contrast to Mussenbrock et al (2006 Phys. Plasmas 13 083501) which conducted a general parameter study, the current work bases the choice of its model parameters on the data obtained by an actual 2f-CCP experiment conducted by Semmler et al (2007 Plasma Sources Sci. Technol. 16 839, and 2008 private communication). A good quantitative correspondence is obtained. The analysis shows that the system is governed by a nonlinear interaction of the applied RF with the inner dynamics of the discharge, particularly with the collective oscillation mode known as the plasma series resonance (PSR). With respect to the power dissipation, two distinct paths can be identified which contribute in approximately equal parts. The first path is non-resonant and corresponds to the traditional picture of 2f-CCPs; the second path is resonant and identical with the mechanism of nonlinear electron resonance heating (NERH) proposed by Mussenbrock et al (2006 Phys. Plasmas 13 083501, 2006 Appl. Phys. Lett. 88 151503). The results change the understanding of 2f-CCPs considerably.

The influence of the relative phase between the driving voltages on electron heating in asymmetric dual frequency capacitive discharges

The influence of the relative phase between the driving voltages on electron heating in asymmetric phase-locked dual frequency capacitively coupled radio frequency plasmas operated at 2 and 14 MHz is investigated. The basis of the analysis is a nonlinear global model with the option to implement a relative phase between the two driving voltages. In recent publications it has been reported that nonlinear electron resonance heating can drastically enhance the power dissipation to electrons at moments of sheath collapse due to the self-excitation of nonlinear plasma series resonance (PSR) oscillations of the radio frequency current. This work shows that depending on the relative phase of the driving voltages, the total number and exact moments of sheath collapse can be influenced. In the case of two consecutive sheath collapses a substantial increase in dissipated power compared with the known increase due to a single PSR excitation event per period is observed. Phase resolved optical emission spectroscopy (PROES) provides access to the excitation dynamics in front of the driven electrode. Via PROES the propagation of beam-like energetic electrons immediately after the sheath collapse is observed. In this work we demonstrate that there is a close relation between moments of sheath collapse, and thus excitation of the PSR, and beam-like electron propagation. A comparison of simulation results to experiments in a single and dual frequency discharge shows good agreement. In particular the observed influence of the relative phase on the dynamics of a dual frequency discharge is described by means of the presented model. Additionally, the analysis demonstrates that the observed gain in dissipation is not accompanied by an increase in the electrodes dc-bias voltage which directly addresses the issue of separate control of ion flux and ion energy in dual frequency capacitively coupled radio frequency plasmas.

Temporal structure of electron heating in asymmetric single-frequency and dual-frequency capacitive discharges

The power dissipation in asymmetric capacitively coupled low-pressure plasmas can be drastically enhanced by the self-excitation of collective resonances. This applies to both Ohmic heating, which is related to collisions of electrons with neutrals, and stochastic heating which reflects the energy transfer from the oscillating plasma boundary sheath to the electrons. This work studies the phase-resolved structure of the power dissipation in asymmetric single-frequency and dual-frequency discharges on the basis of a self-consistent global model. A resonance-free global model (which represents the “traditional” scenario of the heating process) is consulted for comparison. It is shown that the energy dissipation in asymmetric capacitive discharges is enhanced around the moment of the sheath collapse but assumes the nonresonant value when the sheath extension is large. In the dual-frequency case, the effect is synchronized with the low-frequency cycle but strongly dependent on the value of the high-frequency ...

Nonlinear dynamics in dual frequency capacitive discharges

This paper proposes an effective nonlinear model which derives from a mathematically demanding spatially resolved model by means of concentrating on the fundamental mode. Based on the obtained mode the interaction of the linear bulk and the nonlinear boundary sheath is analyzed. Comparing the results to traditional linear theory shows that nonlinear effects highly complicate the interaction of bulk and sheath. The low driving frequency as well as the high driving frequency contributes to electron heating within the discharge. It can be shown that this effect of nonlinear electron resonance heating considerably contributes to the total power budget of the discharge. Comparing calculated rf currents with experimentally obtained data provides qualitatively and quantitatively good results.

Nonlinear resonance effects on electron heating in capacitive discharges

In capacitive radio frequency discharges operated at gas pressures below 20 mTorr two mechanisms of electron heating play a major role: i) ohmic heating due to collisions of electrons with neutrals of the background gas and ii) stochastic heating - often referred to as Fermi heating - due to momentum transfer from the oscillating boundary sheath. In this contribution we show that the plasma series resonance due to interaction of the plasma bulk and the nonlinear sheath significantly effects the electron heating. The series resonance can enhance both the ohmic and stochastic heating by factors of 2-5. We conclude that the nonlinear plasma dynamics has to be taken into account in order to describe quantitatively correct electron heating in low-pressure capacitive radio frequency discharges.