Plasma parameter calculator (Debye length and sheath)
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KEY POINTS
In a typical etch plasma (electron density 10¹⁰ to 10¹² cm⁻³, electron temperature 2 to 5 eV) the Debye length is a few tens of µm to about 0.1 mm, and a sheath tens of times thicker forms on the wafer. The number of ions reaching the wafer is set by plasma density and Bohm velocity, and their energy by the sheath voltage; ICP tools control the two with separate power supplies.
Inputs
An estimate assuming a collisionless sheath and Maxwellian electrons. Above a few tens of mTorr, ions collide inside the sheath and the ion energy distribution spreads toward lower energies. When the ion transit time is well above the RF period (ratio greater than 1), ions are accelerated by the average sheath voltage.
Values by electron density (Te = 3 eV, Ar⁺, sheath voltage 300 V)
| ne (cm⁻³) | Debye length | fpe | Ion current density | Sheath thickness |
|---|---|---|---|---|
| 1×10⁹ | 407 µm | 284 MHz | 0.0263 mA/cm² | 1.02 cm |
| 1×10¹⁰ | 129 µm | 898 MHz | 0.263 mA/cm² | 3.23 mm |
| 1×10¹¹ | 40.7 µm | 2.84 GHz | 2.63 mA/cm² | 1.02 mm |
| 1×10¹² | 12.9 µm | 8.98 GHz | 26.3 mA/cm² | 323 µm |
Typical plasma scales
A plasma is electrically neutral overall, but charge imbalance is allowed over distances shorter than the Debye length. In a typical etch plasma (electron density 10¹⁰ to 10¹² cm⁻³, electron temperature 2 to 5 eV) the Debye length is a few tens of µm to about 0.1 mm, and a sheath (a charged layer) tens of Debye lengths thick forms at the wafer and electrode surfaces.
The number of ions reaching the wafer is set by plasma density and Bohm velocity, and their energy by the sheath voltage. Tools that separate the power source that raises density, such as ICP or microwave sources, from the wafer bias supply can control ion flux and ion energy independently. The ratio of ion transit time to RF period decides whether the ion energy distribution has one peak or splits into two, which is the basis for choosing the bias frequency.
FAQ
Why is the sheath much thicker than the Debye length?
The Child law gives a sheath thickness that grows with (2V/Te)^(3/4). With sheath voltages of hundreds of volts and Te of a few eV, the sheath is tens of Debye lengths thick.
Why does bias frequency matter?
If ions take longer than one RF period to cross the sheath, they respond to the average sheath voltage and the energy distribution has one peak. At lower frequencies they follow the oscillation and the distribution splits into two peaks.
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Last updated: October 23, 2026