Generalizing Similarity Laws for Radio-Frequency Discharge Plasmas across Nonlinear Transition Regimes

We generalize similarity theory based on the scaling and solution invariance of the Boltzmann equation, coupled with the Poisson equation, and demonstrate similarity laws for radio-frequency (rf) discharge plasmas across three nonlinear transitional regimes, namely, the alpha-gamma mode transition, the stochastic-Ohmic-heating mode transition, and the bounce-resonance-heating mode transition. Fundamental plasma parameters, e.g., the electron power absorption, under similar discharge conditions are examined via fully kinetic particle-in-cell simulations, and electron-kinetic invariance is exemplified in similar rf discharge plasmas. The results unambiguously confirm the applicability of similarity laws for rf plasmas in extended operating regimes, and strengthen the foundations and framework of similarity physics with universality.

Figure 1

(a) Schematic illustration of similar discharge systems ($S$, $S^s$) with different dimensional scales. (b) A strategy for showing that plasma properties can be predicted in an extended range of parameter regimes by combining both the scaling and the similarity laws.

Figure 2

(a) Bulk electron density versus $V_{\mathrm{rf}}$ under similar discharge conditions during the AG mode transition. In the simulations, we set $[p,d,f]=[0.3\mathrm{ Torr},6.7\mathrm{ cm},13.56\mathrm{ MHz}]$ for $S$ and $[p,d,f]=[0.6\mathrm{ Torr},3.35\mathrm{ cm},27.12\mathrm{ MHz}]$ for $S^s$ with ${\gamma }_{\mathrm{se}}=0.1$. Taking $V_{\mathrm{rf}}=500\mathrm{ V}$ (case AG5) as an example to demonstrate the dynamical similarities, (b),(c) spatiotemporal distributions of the elastic-collision rates normalized to their maxima in $S$ and $S^s$, respectively; (d) time-averaged scaled rates across the gap.

Figure 3

Similarity of discharges in rf plasmas with an electron-heating transition from the stochastic to the Ohmic regime. (a) Bulk electron density and (b) corresponding ratio between the Ohmic and stochastic heating components in similar discharge systems with $p/k$ from 0.01 to 1 Torr. Time-averaged spatial distributions of the electron heating rate at (c) $p/k=0.01$ Torr (SO1, stochastic-heating-dominated) and (d) $p/k=1$ Torr (SO7, Ohmic-heating-dominated). (e) Decomposition of the electron heating rate in $S$ and $S^s$ at $p/k=1$ (SO7). In the simulations, we have $V_{\mathrm{rf}}=300\mathrm{ V}$ and ${\gamma }_{\mathrm{se}}=0.1$ in all cases, and set $[d,f]=[6.7\mathrm{ cm},13.56\mathrm{ MHz}]$ in $S$ and $[d,f]=[3.35\mathrm{ cm},27.12\mathrm{ MHz}]$ in $S^s$.

Figure 4

Determining the BRH mode transition by tuning the gap distance using test-particle simulations. The gap distances are (a) (2.5, 1.25), (b) (4.5, 2.25) (under the BRH condition), and (c) (5 cm, 2.5 cm) in ($S$, $S^s$). (d)–(f) Spatiotemporal electron trajectories corresponding to (a)–(c). In the simulations, we have $[p,f]=[0.025\mathrm{ Torr},13.56\mathrm{ MHz}]$ in $S$ and $[p,f]=[0.05\mathrm{ Torr},27.12\mathrm{ MHz}]$ in $S^s$, with $V_{\mathrm{rf}}=40\mathrm{ V}$ and secondary electrons neglected $({\gamma }_{\mathrm{se}}=0)$ in all cases.

Figure 5

Invariance of electron kinetics under similar discharge conditions. Temporal EEPFs in full space corresponding to (a) AG1, (b) SO1, (c) BRH in $S$ and (d) AG1, (e) SO1, (f) BRH in $S^s$. (g)–(i) Time-averaged EEPFs in $S$ and $S^s$ under similar discharge conditions.