Mosfet Threshold Voltage Calculator - VT0, Gamma, and Body Effect

The MOSFET threshold voltage is the minimum gate-to-source voltage that creates a conductive inversion layer between source and drain of a metal-oxide-semiconductor field-effect transistor. Below this voltage the channel is essentially off, and above it the channel conducts. The threshold voltage sets the boundary between the cutoff, triode, and saturation regions in an n-channel device.

• • Design a digital inverter: Choose oxide thickness and channel doping so VT lands between 0.4 V and 1.0 V for 5 V logic or near 0.3 V for modern CMOS.
• • Model the body effect in analog circuits: Predict how much the threshold voltage rises when the source is held above the substrate, as in cascode stages.
• • Check temperature behavior: Compare VT at 200 K, 300 K, and 400 K to see how the surface potential for inversion shifts with temperature.
• • Estimate depletion vs enhancement mode: A negative VT0 from the formula means the device is depletion mode and conducts at zero gate bias.

For an enhancement-mode n-channel MOSFET the threshold voltage is positive and the channel only opens once VGS climbs above it. For a p-channel device the same equation holds with opposite doping, so VT comes out negative and the channel opens when VGS is more negative than VT.

Two parameters control the threshold: the fabrication-driven zero-bias value VT0, and the body-effect shift that grows when the source sits above the substrate. The calculator returns both so you can compare the device behavior with and without a source-to-body bias.

When you also need the I-V curves and transconductance that come from a chosen threshold voltage, MOSFET calculator carries the rest of the device analysis.

The threshold voltage is built from three intermediate quantities: the surface potential for inversion (2*phi_f), the body-effect coefficient (gamma), and the zero-bias threshold voltage (VT0). The body-effect term then adds the impact of Vsb to give the final VT.

• C0: Oxide capacitance per unit area of the gate dielectric, in F/cm^2.
• NA: Acceptor doping of the p-type substrate, in cm^-3.
• ni: Intrinsic carrier concentration of silicon, in cm^-3.
• T: Absolute temperature in kelvin, used to compute kT/q.
• Vsb: Source-to-body voltage in volts; set to 0 when the source and substrate are tied together.

Step one computes the thermal voltage kT/q. Step two uses the ratio NA/ni to extract the surface potential 2*phi_f that marks the onset of strong inversion. Step three scales gamma with the substrate doping and oxide capacitance. Step four combines them into VT0, and step five adds the body-effect term so that the final VT tracks any source-to-body bias you apply.

The same equations appear in Sedra/Smith and Neamen, which the Omni article cites. All About Circuits uses the same body-effect form when explaining why cascode stages see a higher effective threshold than the bare device.

According to Omni Calculator, the threshold voltage of a MOSFET follows VT = VT0 + gamma * (sqrt(2*phi_f + Vsb) - sqrt(2*phi_f)), with VT0 = sqrt(2 * epsilon_Si * q * NA * (2*phi_f)) / C0 - 2*phi_f and gamma = sqrt(2 * epsilon_Si * q * NA) / C0.

According to NIST CODATA, the elementary charge used inside this formula is exactly 1.602176634e-19 C after the 2019 SI redefinition, and the exact value flows directly into both kT/q and the gamma coefficient.

If you only know the oxide thickness and permittivity instead of C0 directly, capacitance calculator converts those into the capacitance per unit area the threshold equation needs.

These four concepts are the building blocks of the threshold voltage equation. Each one maps to a term in the formula, so working through them in order keeps the calculation transparent.

Once the channel is on, the linear-region current still follows a resistive law, and Ohm's law calculator is the right tool when you want to size a load resistor around a known VGS - VT.

Enter the fabrication and operating parameters for your device, then read the intermediate results and the final threshold voltage.

1. 1 Enter the oxide capacitance C0: Use the SiO2 thickness and permittivity if you do not have a measured value. A 50 nm SiO2 layer gives roughly 6.9e-8 F/cm^2.
2. 2 Enter the substrate doping NA: Use the acceptor concentration of the p-type substrate. Heavier doping raises VT0 because it increases 2*phi_f.
3. 3 Enter ni and T: Use the textbook value 1.5e10 cm^-3 for ni at 300 K, then change T to see how temperature shifts 2*phi_f and VT.
4. 4 Set the source-body voltage Vsb: Set Vsb = 0 when source and body are tied together. Raise it to model a cascode or any circuit where the source sits above the substrate.
5. 5 Read the results: The primary result is VT. Inspect 2*phi_f, gamma, and VT0 to understand which term drives the final answer.

For a 5 V logic n-channel MOSFET with NA = 1e16 cm^-3 and an oxide that gives C0 = 3e-8 F/cm^2, the calculator reports VT0 = 0.9058 V, gamma = 1.9205 V^0.5, and 2*phi_f = 0.6934 V. With Vsb = 0 the final VT = 0.9058 V, well inside the 0.7 V to 1 V range typical of 5 V logic devices.

The tool gives you every intermediate quantity in one pass, so you can both check a textbook problem and explore how device geometry changes the threshold.

• • Fast VT0 and VT in one calculation: Skip rewriting the formula and plugging numbers into a calculator by hand.
• • See the body-effect shift clearly: Compare VT with Vsb = 0 and a non-zero Vsb to see exactly how much the source bias raises the threshold.
• • Adjust temperature to model real conditions: Change T to predict how the threshold drifts between -73 C and 127 C in automotive or aerospace environments.
• • Confirm depletion vs enhancement mode: A negative VT0 in the output tells you that the device is depletion mode and conducts with zero gate bias.
• • Cross-check textbook and SPICE results: Use the intermediate 2*phi_f and gamma values as a sanity check against simulation decks and lecture slides.

The same kT/q thermal voltage that scales 2*phi_f also sets the Boltzmann factor for carrier populations across the bandgap, and the Boltzmann factor calculator evaluates exp(-E/(kB*T)) across the same temperature range this tool sweeps.

Four device parameters dominate the calculation. Adjusting any of them shifts VT in a predictable way.

• • The model assumes a long-channel device. Short-channel effects like velocity saturation, drain-induced barrier lowering, and quantum confinement are not included, so VT predictions for sub-100 nm devices should be taken as a starting estimate.
• • The body-effect term is computed for a uniformly doped substrate. Retrograde wells, halo implants, and silicon-on-insulator devices need a more detailed treatment.
• • The intrinsic carrier concentration ni is treated as a fixed user input rather than recomputed from temperature. For wide temperature swings you may want to update ni alongside T.
• • Depletion-mode devices with channel implants that pre-shape the threshold need SPICE-grade models rather than this first-order textbook formula.

According to All About Circuits, the body effect raises the threshold voltage whenever the source-to-body voltage increases, following VT = VT0 + gamma * (sqrt(2*phi_f + Vsb) - sqrt(2*phi_f)), which is why cascode stages see a higher effective threshold than the bare device.

Because 2*phi_f is built from kT/q and the natural log of NA/ni, both the threshold voltage equation and the Nernst equation fold a concentration ratio into the thermal voltage, and the Nernst equation calculator shows that same logarithmic pattern when you change temperature or activity.