MOSFET Threshold Voltage Calculator (VT from doping, oxide thickness, gate and body bias)

Understanding the MOSFET threshold voltage calculator — concept and principle

This calculator estimates the threshold voltage of a long-channel silicon MOSFET from the substrate doping, gate-oxide thickness, gate material, fixed oxide charge and body bias. It also returns the flat-band voltage, bulk Fermi potential, oxide capacitance, depletion term, body-effect coefficient and maximum depletion width. It is used in semiconductor device courses.

Three pieces add up: the flat-band offset, twice the Fermi potential, and the charge of the depletion layer.

How it is calculated

NMOS: V_T = V_FB + 2|φ_F| + √(2 ε_Si q N_A × 2|φ_F|) ÷ C_ox

|φ_F| = (kT/q) ln(N_A ÷ nᵢ); C_ox = ε_ox ÷ t_ox

V_FB = φ_substrate − φ_gate − Q_f ÷ C_ox

Body effect: ΔV_T = γ (√(2|φ_F| + |V_SB|) − √(2|φ_F|)), γ = √(2 ε_Si q N) ÷ C_ox

Constants: nᵢ = 1.0 × 10¹⁰ cm⁻³, ε_Si = 11.7 ε₀, ε_ox = 3.9 ε₀, T = 300 K, poly gates ±0.55 V relative to intrinsic silicon. For PMOS the signs flip.

The principle

To turn an NMOS transistor on, the gate must pull enough electrons to the surface of the p-type substrate to make it as n-type as the bulk is p-type. That condition, strong inversion, occurs when the surface potential bends by 2|φ_F|. The gate voltage needed is the flat-band voltage (to cancel work-function differences and oxide charge), plus that band bending, plus the voltage across the oxide to support the depletion charge. Reverse body bias widens the depletion region, adding charge and raising V_T.

Worked example

Default inputs: NMOS, N_A = 10¹⁷ cm⁻³, t_ox = 10 nm, n+ poly gate, N_f = 1 × 10¹⁰ cm⁻², V_SB = 1 V.

|φ_F| = 0.025852 × ln(10⁷) = 0.417 V; C_ox = 3.9 ε₀ ÷ 10 nm = 345.3 fF/µm².
V_FB = −0.417 − 0.55 − 0.005 = −0.971 V.
Depletion term = 0.482 V; V_T = −0.971 + 0.833 + 0.482 = 0.344 V.
γ = 0.528 V^½; with 1 V body bias V_T = 0.576 V.
Maximum depletion width = 103.8 nm; kT/q = 25.85 mV.

Each point is the calculator re-run with only “Gate oxide thickness tox (nm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Gate oxide thickness tox (nm)	Threshold voltage VT (V)
2.5	-0.014
5	0.105
7.5	0.224
10	0.344
12.5	0.463
15	0.582
20	0.821

How to read the result

Higher doping or a thicker oxide raises V_T; the gate material shifts it through the flat-band voltage. The body effect matters in stacked transistors whose source is above ground. Modern short-channel devices behave differently, so treat these values as long-channel estimates.

Common mistakes

Using the wrong sign convention for PMOS.
Mixing cm and m units in the constants.
Forgetting the oxide charge term.
Applying long-channel formulas to nanometre-scale devices.

Keep in mind: Short-channel effects, quantum confinement, poly depletion and interface traps are not included. Real device thresholds come from measurement or calibrated simulation.

Frequently asked questions

Why does doping raise the threshold?

More dopants mean more depletion charge to support and a larger 2|φ_F|.

What is the body effect?

An increase in threshold when the source is biased above the substrate.

Why does the gate material matter?

Its work function sets the flat-band voltage.

What is strong inversion?

The surface electron density equals the bulk hole density, defined by a band bending of 2|φ_F|.

Sources: MIT OpenCourseWare 6.012 Microelectronic Devices and Circuits (Spring 2009), Lecture 7

Checked 9 Nov 2026. Threshold and flat-band relations V_T = V_FB − 2φ_p + √(2ε_sqN_a(−2φ_p))/C_ox, V_FB = −(φ_N+ − φ_p): MIT OCW 6.012 Microelectronic Devices and Circuits (Spring 2009), Lecture 7. Material constants (n_i, ε_Si, ε_ox, ±0.55 V poly), oxide-charge shift and body-effect formula: standard textbook values; k and q exact (2019 SI).

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