Collisionless Heating in Capacitive Discharges Enhanced by Dual-Frequency Excitation

We discuss collisionless electron heating in capacitive discharges excited by a combination of two disparate frequencies. By developing an analytical model, we find, contrary to expectation, that the net heating in this case is much larger than the sum of the effects occurring when the two frequencies act separately. This prediction is substantiated by kinetic simulations, which are also in excellent general quantitative agreement with the model for discharge parameters that are typical of recent experiments.

Figure 1

Collisionless heating power as a function of space and time, calculated by the particle simulation, with ${\tilde{J}}_l=10\mathrm{A}{\mathrm{m}}^{-2}$, ${\omega }_l=2\pi \times 2\mathrm{MHz}$, ${\tilde{J}}_h=20\mathrm{A}{\mathrm{m}}^{-2}$, and ${\omega }_l=2\pi \times 26\mathrm{MHz}$, and the plasma parameters discussed in the text. The ion sheath edge is at $x=0$ and the electrode is at $x=s_{\max }$.

Figure 2

Temperature of electrons in the sheath computed from the analytic theory of the text (solid line) and from the particle-in-cell simulation (dashed line), for the conditions noted in Fig. 1.

Figure 3

Normalized collisionless heating power as a function of $\beta \equiv {\tilde{J}}_h/{\tilde{J}}_l$, with ${\tilde{J}}_h$ as the parameter varied (so that $H_l$ is constant) and with $\alpha ={\omega }_h/{\omega }_l=13$.

Figure 4

Time averaged electron heating power resolved in space for three cases corresponding to the two frequencies acting separately and both acting together. Note that the ion sheath edge in each case is approximately located at the transition to negative power absorption, and that the sheath width is appreciably influenced by the application of the high frequency [4], relative to the low-frequency acting alone. The parameter values used are ${\tilde{J}}_l=10\mathrm{A}{\mathrm{m}}^{-2}$, ${\omega }_l=2\pi \times 2\mathrm{MHz}$, ${\tilde{J}}_h=36\mathrm{A}{\mathrm{m}}^{-2}$, and ${\omega }_l=2\pi \times 26\mathrm{MHz}$.

Figure 5

Time averaged electron heating power as a function of $1/\beta \equiv J_l/J_h$, where the parameter varied is the low-frequency current density $J_l$. Other parameters held constant are the high frequency current density $J_h=36\mathrm{A}{\mathrm{m}}^{-2}$, and the two frequencies ${\omega }_l=2\pi \times 2\mathrm{MHz}$ and ${\omega }_h=2\pi \times 26\mathrm{MHz}$. The points are simulation data. The short dashed line is the model of Eq. (15), the long dashed line the model generalized using the ansatz of the text.