In recent years, capacitively coupled plasmas driven by ultra-low frequency source have received increasing attention, because they are beneficial to generating ions with high energy and small scattering angle, which aligns well with the current trend in high aspect ratio etching. Since the sheath becomes thicker when an ultra-low frequency source is applied, the secondary electron emission becomes significant. Indeed, these energetic secondary electrons can enhance the ionization process and even affect the discharge mode. In this work, a two-dimensional fluid model is employed to study the influence of secondary electrons on the dual-frequency capacitively coupled plasma under different ultra-low frequency voltages, secondary electron emission coefficients and inter-electrode gaps. The high frequency is fixed at 13.6 MHz, and the ultra-low frequency is fixed at 400 kHz. First, by using the ion energy dependent secondary electron emission coefficient, it is shown that the electron density first decreases and then increases with ultra-low frequency voltage rising. This is because, on the one hand, the higher ultra-low frequency voltage leads to thicker sheath, and therefore, the effective discharge volume is compressed. On the other hand, secondary electrons emitted from electrodes can obtain more energy, thus enhancing the ionization process. By comparing with the results obtained with a fixed secondary electron emission coefficient, it is found that in the low voltage range, the evolution of the electron density is similar to that with a fixed coefficient of 0.1. While, in the high voltage range, the growth of the electron density is even more pronounced than that with a fixed coefficient of 0.2, indicating that the enhancement of the secondary electron effect by ultra-low frequency voltage is non-linear. Finally, the influence of discharge gap on the plasma properties is also discussed. It is shown that with the inter-electrode gap increasing from 2 to 4 cm, the maximum ionization rate becomes lower, but the electron density rises significantly, and the plasma radial uniformity is improved. When inter-electrode gap is large, secondary electrons can completely collide with neutral species, so their influence on the electron density at high ultra-low frequency voltage is more significant. The results obtained in this study contribute to understanding the influence of ultra-low frequency source on the secondary electron effect, and provide some guidance for optimizing plasma processing.
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