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.

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 Phys. Plasmas 13 083501, 2006. The results change the understanding of 2f-CCPs considerably.

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The so-called 'electrostatic' approximation postulates that the electric field can be represented by the gradient of a scalar potential, even under dynamical conditions. This assumption reduces the set of Maxwell's equations to the much simpler Poisson equation and is often employed for modeling and simulation of radio frequency driven capacitive low pressure discharges. While it is now widely acknowledged that the neglect of induction phenomena breaks down for large-area plasma sources driven at high frequencies (such as used for VLSI processing), smaller experimental devices excited at moderate frequencies (e.g. 13.56 MHz) are generally thought to be uncritical. This paper demonstrates the opposite: even small plasma reactors of the size of the Gaseous Electronics Conference reference cell exhibit a considerable skin effect in the low pressure, high density regime and render the electrostatic approximation invalid. The point is made, however, that this phenomenon is not 'fully electromagnetic' (in the sense that its analysis requires the full set of Maxwell's equations), but can be understood by means of a simplified model which assumes quasi-neutrality and may therefore be called 'magnetostatic'.

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 electrode's 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.