MOSFET threshold voltage (Vth) calculator
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The threshold voltage of a long-channel MOSFET is Vth = VFB + 2φF + Qdep/Cox, set by the work function difference between gate and substrate, the surface potential 2φF needed for strong inversion and the depletion charge. Vth rises with higher substrate doping and thicker oxide. Since HKMG, metals with different work functions are used to create multiple threshold voltages.
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This is the classical long-channel formula for silicon at 300 K. It does not include short-channel Vth roll-off, DIBL, quantum confinement or polysilicon gate depletion. In FinFETs and GAA transistors the channel is fully depleted, so Vth is set almost entirely by the gate work function and depends little on substrate doping.
Substrate concentration and threshold voltage
Oxide thickness and threshold voltage
| EOT | Cox | Vth | γ |
|---|---|---|---|
| 1.2 nm | 2.878 µF/cm² | 0.033 V | 0.142 |
| 2 nm | 1.727 µF/cm² | 0.123 V | 0.236 |
| 3 nm | 1.151 µF/cm² | 0.236 V | 0.354 |
| 5 nm | 0.691 µF/cm² | 0.461 V | 0.590 |
| 10 nm | 0.345 µF/cm² | 1.023 V | 1.180 |
| 20 nm | 0.173 µF/cm² | 2.148 V | 2.360 |
The four terms that set the threshold voltage
The threshold voltage is the sum of the work function difference between gate and substrate (φms), the fixed charge in the dielectric, the surface potential needed for strong inversion (2φF) and the depletion charge built up before inversion (Qdep/Cox). Raising substrate doping increases both φF and Qdep, so Vth rises; thinning the dielectric increases Cox and shrinks the Qdep/Cox term.
In the polysilicon gate era, the threshold was set by the gate doping type and a channel implant (Vth adjust implant). Since high-k/metal gates (HKMG), nMOS and pMOS use different work function metals, and their composition and thickness are varied to create multiple threshold voltages (multi-Vth).
FAQ
What is the body effect?
When reverse bias is applied between source and substrate, the depletion charge grows and the threshold voltage rises. The shift is γ(√(2φF + VSB) − √(2φF)).
Does this formula apply to FinFETs?
In FinFETs and GAA transistors the channel is fully depleted, so Vth is set almost entirely by the gate work function and depends little on substrate doping. The classical formula here suits planar long-channel devices.
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Last updated: October 23, 2026