Plasma immersion ion implantation is an effective surface modification technique. Unlike conventional beam-line ion implantation, it features ion acceleration/implantation through a plasma sheath in a pulsed mode and non-line-of-sight operation. Consequently, the shape of the sample voltage pulse, especially the finite rise time due to capacitance effects of the hardware, has a large influence on the energy spectra of the incident ions. In this article, we present a simple and effective analytical model to predict and calculate the energy distribution of the incident ions. The validity of the model is corroborated experimentally. Our results indicate that the ion energy distribution is determined by the ratio of the total pulse duration to the sample voltage rise time but independent of the plasma composition, ion species, and implantation voltage, subsequently leading to the simple analytical expressions. The ion energy spectrum has basically two superimposed components, a high-energy one for the majority of the ions implanted during the plateau region of the voltage pulse as well as a low-energy one encompassing ions implanted during the finite rise time of the voltage pulses. The lowest-energy component is attributed to a small initial expanding sheath obeying the Child-Langmuir law. Our model can also deal with broadening of the energy spectra due to molecular ions such as N2+ or O2+, in which case each implanted atom only carries a fraction (in this case, half) of the total acceleration energy.
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