Paschen's law (breakdown voltage) calculator
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KEY POINTS
Breakdown voltage is a function of pressure × electrode gap (pd) and follows a valley-shaped curve, Vb = B·pd / (ln(A·pd) − ln(ln(1+1/γ))). Plasma tools put the process chamber at easy-breakdown conditions and design gas lines and the space behind the wafer, where discharges are unwanted, at pd values far from the valley.
Inputs
On the left side of the Paschen curve, lowering the pressure (or narrowing the gap) makes breakdown harder. This is the design rule used to prevent unwanted discharges behind the wafer and in narrow gaps in plasma tools.
A and B are experimental constants fitted over a specific range of E/p, so accuracy drops at very small or very large pd. Breakdown voltages for RF discharges (such as 13.56 MHz) are lower than for DC because electrons oscillate between the electrodes and are lost less easily.
Paschen curves
Gas constants and minimum breakdown voltage (γ = 0.01)
| Gas | A (1/cm·Torr) | B (V/cm·Torr) | pd_min | Vb,min |
|---|---|---|---|---|
| Air | 15 | 365 | 0.836 Torr·cm | 305 V |
| Nitrogen (N₂) | 12 | 342 | 1.045 Torr·cm | 358 V |
| Argon (Ar) | 12 | 180 | 1.045 Torr·cm | 188 V |
| Helium (He) | 3 | 34 | 4.182 Torr·cm | 142 V |
| Hydrogen (H₂) | 5 | 130 | 2.509 Torr·cm | 326 V |
| Carbon dioxide (CO₂) | 20 | 466 | 0.627 Torr·cm | 292 V |
Why breakdown has a pressure valley
For a discharge to start, electrons leaving the cathode must collide with gas molecules often enough on the way to the anode to multiply into enough electrons. At too high a pressure, collisions are so frequent that electrons cannot gain energy; at too low a pressure, there are too few collisions for multiplication. The breakdown voltage therefore forms a valley-shaped curve against pressure × distance (pd), and breakdown is easiest at the pd at the bottom of the valley.
Plasma tools choose easy-breakdown conditions in the process chamber and keep pd far from the valley where discharges are unwanted: gas lines, the helium cooling grooves of electrostatic chucks and the back of the wafer. The same reason explains why high-voltage parts in vacuum systems arc most easily not at atmospheric pressure or in high vacuum but somewhere in between, around a few hundred Pa.
FAQ
Why is breakdown harder at both low and high pressure?
At high pressure, electrons collide too often to gain enough energy; at low pressure, they collide too rarely to multiply. Breakdown is easiest at the pd at the bottom of the curve.
Is RF breakdown the same as DC?
No. In RF discharges such as 13.56 MHz, electrons oscillate between the electrodes and are lost less easily, so the breakdown voltage is lower than the DC value from Paschen's law.
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Last updated: October 23, 2026