Consistent simulation of capacitive radio-frequency discharges and external matching networks

Abstract

External matching networks are crucial and necessary for operating capacitively coupled plasmas in order to maximize the absorbed power. Experiments show that external circuits in general heavily interact with the plasma in a nonlinear way. This interaction has to be taken into account in order to be able to design suitable networks, e.g., for plasma processing systems. For a complete understanding of the underlying physics of this coupling, a nonlinear simulation approach which considers both the plasma and the circuit dynamics can provide useful insights. In this work, the coupling of an equivalent circuit plasma model and an electric external circuit composed of lumped elements is discussed. The plasma model itself is self-consistent in the sense that the plasma density and the electron temperature is calculated from the absorbed power based on a global plasma chemistry model. The approach encompasses all elements present in real plasma systems, i.e., the discharge itself, the matching network, the power generator as well as stray loss elements. While the main results of this work is the conceptual approach itself, at the example of a single-frequency capacitively coupled discharge its applicability is demonstrated. It is shown that it provides an effective and efficient way to analyze and understand the nonlinear dynamics of real plasma systems and, furthermore, may be applied to synthesize optimal matching networks.

A generic method for equipping arbitrary rf discharge simulation frameworks with external lumped element circuits

“…The current resulting from the disturbed excitation is denoted by Î. To now account for the variances in the current due to the specific difference in the excitation at the m-th entry in the voltage, equation (11) can be reformulated to…”

“…The global plasma model utilized in this work is based on considerations introduced and discussed in [4,6,8,17], while its interaction with an external electric circuit has been studied using ngSPICE [18] in previous works [11,12]. In the following, the model is only briefly reviewed.…”

“…, respectively, resulting from a Matrix sheath model [1]. Again, a detailed discussion about this model coupled to an external electrical circuit can be found elsewhere [11]. The resulting equivalent circuit is depicted in Figure 3 on the far right side attached to a generator, a matching network and stray elements.…”

“…The values chosen for the plasma parameters and the network elements are the same as published in [11] and are listed in Table I.…”

“…Many plasma simulation techniques focus on the plasma dynamics itself and external circuits are often neglected or drastically simplified, e.g., to a simple bias capacitance. Electrical equivalent circuit models allow to comprisingly incorporate the plasma and complex external circuits in the solution algorithms [11,12]. In contrast, Particle-in-Cell (PIC) simulations have been coupled to external series circuits consisting of a resistance, an inductance and a capacitance via conservation of charge by Verboncoeur et al [13].…”