MOSFET Threshold Voltage Calculator — V_th from Device Parameters
Compute the long-channel MOSFET threshold voltage V_th from fundamental device parameters: substrate doping N_A, gate oxide thickness t_ox and flat-band voltage V_fb. The step-by-step breakdown shows the Fermi potential, oxide capacitance and depletion charge contributions.
nm
V
Gate voltage at which the inversion channel forms (long-channel, 300 K)
Fermi potential φ_F
Oxide capacitance C_ox
Max depletion charge |Q_dep|
Threshold voltage V_th
- 1
Fermi potential φ_F
(kT/q) × ln(N_A ÷ n_i) = 0.02585 × ln(10^17 ÷ 1.5×10¹⁰) = 0.4062 - 2
Oxide capacitance C_ox
ε_ox ÷ t_ox = 3.9 × 8.854×10⁻¹⁴ ÷ (10 × 10⁻⁷ cm) = 0Gate oxide capacitance per unit area (F/cm²). - 3
Depletion charge |Q_dep|
√(2 × ε_Si × q × N_A × 2φ_F) = 0 - 4
Threshold voltage V_th
V_fb + 2φ_F + Q_dep ÷ C_ox = -0.9 + 0.8123 + 0.4755 = 0.3879
How does this calculator work?
MOSFET threshold voltage: V_th = V_fb + 2φ_F + Q_dep/C_ox. Fermi potential φ_F = 25.85 mV × ln(N_A / 1.5×10¹⁰). Oxide cap C_ox = 3.9ε₀/t_ox. Depletion charge Q_dep = √(2×11.7ε₀×q×N_A×2φ_F). Valid for long-channel, uniformly doped p-substrate at 300 K.
Formula
How this is calculated
The threshold voltage marks the gate bias at which a strong inversion layer (the channel) forms at the silicon surface. It has three additive components. The flat-band voltage V_fb accounts for the work-function difference between the gate material and the silicon, plus any fixed oxide charge — it is typically around −0.9 V for an n⁺ polysilicon gate on p-type silicon.
The surface potential term 2φ_F is twice the Fermi potential, the energy level that separates filled from empty electron states in the bulk. φ_F = (kT/q) × ln(N_A / n_i), where n_i ≈ 1.5×10¹⁰ cm⁻³ for silicon at 300 K. The body-effect term Q_dep/C_ox is the voltage required to sweep the depletion charge out of the channel region: Q_dep = √(2ε_Si·q·N_A·2φ_F) is the maximum depletion charge, and C_ox = ε_ox/t_ox is the gate-oxide capacitance per unit area.
This formula assumes a long-channel, uniformly doped p-substrate at 300 K. Short-channel effects (threshold roll-off, drain-induced barrier lowering), non-uniform retrograde doping, polysilicon depletion and high-κ dielectrics all modify the result. The constants used are: n_i = 1.5×10¹⁰ cm⁻³, ε_Si = 11.7ε₀, ε_SiO₂ = 3.9ε₀, q = 1.602×10⁻¹⁹ C, kT/q = 25.85 mV.
Frequently asked questions
V_fb is the gate voltage at which there is no band bending at the silicon surface — the semiconductor is electrically flat. It depends on the gate-to-silicon work-function difference and fixed oxide charge. For an n⁺ polysilicon gate on p-type silicon, V_fb ≈ −0.9 V is a common starting estimate; for metal gates it varies widely by metal species.
Higher substrate doping N_A raises both φ_F and |Q_dep|, increasing V_th. This is the basis of threshold-voltage adjustment implants: ion-implanting the channel with acceptors raises V_th; donor implants lower it. The dependence is roughly logarithmic in N_A for φ_F and as √N_A for the body-effect term.
The body effect describes how V_th increases when a reverse bias V_SB is applied between the source and the body. The modified formula is V_th(V_SB) = V_th0 + γ(√(2φ_F + V_SB) − √(2φ_F)), where γ = √(2ε_Si·q·N_A) / C_ox is the body-effect coefficient. This calculator computes V_th0 (V_SB = 0).
Also known as
TG we-Calculate Editorial Team. (2026). MOSFET Threshold Voltage Calculator — V_th from Device Parameters [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/mosfet-threshold-voltage-calculator
TG we-Calculate Editorial Team. "MOSFET Threshold Voltage Calculator — V_th from Device Parameters." TG we-Calculate. 2026. https://we-calculate.com/calculator/mosfet-threshold-voltage-calculator.
TG we-Calculate Editorial Team, "MOSFET Threshold Voltage Calculator — V_th from Device Parameters," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/mosfet-threshold-voltage-calculator
@misc{wecalculate_mosfet_threshold_voltage_calculator, title = {MOSFET Threshold Voltage Calculator — V_th from Device Parameters}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/mosfet-threshold-voltage-calculator}}, year = {2026}, note = {TG we-Calculate} }
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