Ion energy distribution functions behind the sheaths of magnetized and non magnetized radio frequency discharges

The effect of a magnetic field on the characteristics of capacitively coupled radio frequency discharges is investigated and found to be substantial. A one-dimensional particle-in-cell simulation shows that geometrically symmetric discharges can be asymmetrized by applying a spatially inhomogeneous magnetic field. This effect is similar to the recently discovered electrical asymmetry effect. Both effects act independently, they can work in the same direction or compensate each other. Also the ion energy distribution functions at the electrodes are strongly affected by the magnetic field, although only indirectly. The field influences not the dynamics of the sheath itself but rather its operating conditions, i.e., the ion flux through it and voltage drop across it. To support this interpretation, the particle-in-cell results are compared with the outcome of the recently proposed ensemble-in-spacetime algorithm. Although that scheme resolves only the sheath and neglects magnetization, it is able to reproduce the ion energy distribution functions with very good accuracy, regardless of whether the discharge is magnetized or not.

💡 Research Summary

The paper investigates how an externally applied magnetic field influences the characteristics of capacitively coupled radio‑frequency (RF) discharges, with a particular focus on the ion energy distribution functions (IEDFs) arriving at the electrodes. Using a one‑dimensional particle‑in‑cell (PIC) simulation that resolves electron and ion trajectories, collisions, and self‑consistent electric fields, the authors examine a geometrically symmetric discharge in which a spatially non‑uniform magnetic field is superimposed. The magnetic field strongly confines electrons, reducing their gyroradius and making electron transport highly anisotropic. As a consequence, the electron conductivity becomes non‑uniform across the gap, which leads to an imbalance in the instantaneous current flowing to each electrode. Even though the applied voltage waveform remains symmetric, the magnetic field induces a DC‑like voltage asymmetry – a phenomenon analogous to the recently reported electrical asymmetry effect (EAE) but arising from magnetic confinement rather than waveform shaping.

The induced voltage asymmetry modifies the operating conditions of the sheath at each electrode. The sheath on the high‑voltage side becomes thicker and sustains a larger voltage drop, while the sheath on the low‑voltage side is thinner with a smaller drop. Because ions are much heavier than electrons, they are not directly magnetized; they are accelerated solely by the electric field within the sheath. Therefore, the IEDF at each electrode is dictated primarily by the local sheath voltage and ion flux, both of which are altered by the magnetic‑field‑induced asymmetry. The simulations show that the high‑voltage electrode receives a broad, high‑energy ion spectrum, whereas the low‑voltage electrode experiences a narrower, lower‑energy distribution.

To test whether the detailed electron dynamics and magnetic effects are essential for predicting IEDFs, the authors compare the PIC results with those obtained from the recently proposed ensemble‑in‑spacetime (EST) algorithm. EST resolves only the sheath region, neglecting magnetization and electron motion, and requires as input the ion flux entering the sheath and the sheath voltage drop. Remarkably, EST reproduces the IEDFs from the full PIC simulations with excellent accuracy for both magnetized and non‑magnetized cases. This agreement demonstrates that the IEDF is governed almost exclusively by sheath parameters; the magnetic field’s role is indirect, acting through its influence on electron transport, bulk plasma resistance, and consequently on the sheath voltage and ion flux.

The key insights of the study are: (1) a spatially inhomogeneous magnetic field can asymmetrize a geometrically symmetric RF discharge, independently of the electrical asymmetry effect; (2) this magnetic‑induced asymmetry changes sheath operating conditions, thereby reshaping the ion energy spectra at the electrodes; (3) sheath‑only models such as EST are sufficient to predict IEDFs, provided that accurate sheath voltage and ion flux values are supplied, even when the bulk plasma is strongly magnetized.

These findings have practical implications for plasma processing technologies. In semiconductor etching, surface modification, and thin‑film deposition, control over ion energy is critical for selectivity and damage mitigation. The ability to tailor IEDFs by adjusting magnetic field profiles offers a new degree of freedom that can be combined with conventional waveform engineering (EAE) to achieve precise ion‑energy control without altering electrode geometry. Moreover, the validation of EST as a fast, accurate tool for IEDF prediction simplifies process design and optimization, reducing the need for computationally intensive full‑scale PIC simulations in routine engineering work.