Understanding MOSFET Threshold Voltage

Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFETs) are fundamental components in modern electronic circuits, serving as switches and amplifiers. Among their critical parameters, the threshold voltage (V_TH) plays a pivotal role in defining when the MOSFET turns from the "off" state to the "on" state. This article explores the MOSFET threshold voltage in detail — its physical meaning, mathematical formulation, calculation methods, example problems, and practical implications.

What is MOSFET Threshold Voltage?

The threshold voltage of a MOSFET is the minimum gate-to-source voltage (V_GS) required to create a conducting channel between the drain and source terminals. Below this voltage, the MOSFET remains off, and above it, the device allows significant current flow.

In simpler terms, V_TH is the voltage level at which an inversion layer forms at the semiconductor-oxide interface, enabling current conduction.

Types of Threshold Voltage

• Enhancement Mode MOSFETs: Require a positive gate voltage (for n-channel) greater than V_TH to conduct.
• Depletion Mode MOSFETs: Have a built-in channel and can conduct at zero gate voltage; V_TH is typically negative.

Physical Origin of Threshold Voltage

The threshold voltage arises from the need to invert the semiconductor surface beneath the gate oxide. Initially, the semiconductor surface is either p-type or n-type doped. When a gate voltage is applied, it modifies the surface potential, and when it reaches a critical value called the surface potential at threshold, an inversion layer forms.

Key Physical Phenomena:

• Flat-band voltage (V_FB): The gate voltage where the energy bands of the semiconductor are flat, meaning no band bending.
• Oxide capacitance (C_ox): The capacitance per unit area of the gate oxide layer.
• Bulk charge (Q_B): The charge required to invert the surface.

Mathematical Expression of Threshold Voltage

The general formula for the threshold voltage V_TH is:

V_TH = V_FB + 2φ_F +    ( ^{√(2qε_sN_A2φ_F)} / _Cox )

Where:

Symbol	Description	Units
VFB	Flat-band voltage	Volts (V)
φF	Fermi potential (bulk potential)	Volts (V)
q	Elementary charge	1.602 × 10-19 C
εs	Semiconductor permittivity	F/cm or F/m
NA	Acceptor doping concentration	cm-3
Cox	Oxide capacitance per unit area	F/cm2

Explanation of Terms

• Flat-band Voltage (V_FB): This voltage accounts for work function differences between the gate material and the semiconductor and any fixed oxide charges.
• Fermi Potential (φ_F): Defines the energy difference between intrinsic Fermi level and Fermi level in the doped semiconductor:

φF = (kT/q) × ln(NA / ni)

Where:

• k = Boltzmann constant = 1.38 × 10^-23 J/K
• T = Absolute temperature in Kelvin
• n_i = Intrinsic carrier concentration

Calculating Oxide Capacitance (C_ox)

The oxide capacitance per unit area depends on the oxide thickness (t_ox) and oxide permittivity (ε_ox):

C_ox = ε_ox / t_ox

Where:

• ε_ox = ε₀ × κ, with ε₀ = permittivity of free space (8.854 × 10^-14 F/cm) and κ = dielectric constant of SiO₂ (≈ 3.9)
• t_ox = thickness of the oxide layer in cm

Example Calculation of MOSFET Threshold Voltage

Consider an n-channel MOSFET with the following parameters:

Oxide thickness, tox	10 nm = 1×10-6 cm
Acceptor concentration, NA	1×1017 cm-3
Intrinsic carrier concentration, ni	1.5×1010 cm-3
Temperature, T	300 K
Work function difference (gate-semiconductor), ΦMS	-0.95 V
Fixed oxide charge, Qox	0
Permittivity of Si, εs	11.7 × ε0 = 11.7 × 8.854×10-14 F/cm

Step 1: Calculate φ_F

φF = (kT/q) × ln(NA / ni)

kT/q ≈ 0.0259 V at 300 K

φF = 0.0259 × ln(1×1017 / 1.5×1010)
            = 0.0259 × ln(6.67×106)
            = 0.0259 × 15.72 ≈ 0.407 V

Step 2: Calculate oxide capacitance C_ox

ε0 = 8.854 × 10-14 F/cm
κ = 3.9 for SiO2

εox = κ × ε0 = 3.9 × 8.854×10-14 = 3.453 × 10-13 F/cm

tox = 1 × 10-6 cm

Cox = εox / tox = 3.453×10-13 / 1×10-6 = 3.453 × 10-7 F/cm²

Step 3: Calculate flat-band voltage V_FB

VFB = ΦMS - (Qox / Cox)

Given Qox = 0, so VFB = ΦMS = -0.95 V

Step 4: Calculate threshold voltage V_TH

VTH = VFB + 2φF + ( sqrt(2 × q × εs × NA × 2φF) ) / Cox

Calculate each term:

εs = 11.7 × ε0 = 11.7 × 8.854×10-14 = 1.036×10-12 F/cm

Calculate the numerator inside the sqrt:

2 × q × εs × NA × 2φF
= 2 × 1.602×10-19 × 1.036×10-12 × 1×1017 × 2 × 0.407
= 2 × 1.602×10-19 × 1.036×10-12 × 1×1017 × 0.814

Calculate stepwise:

2 × 1.602×10-19 = 3.204×10-19
3.204×10-19 × 1.036×10-12 = 3.318×10-31
3.318×10-31 × 1×1017 = 3.318×10-14
3.318×10-14 × 0.814 = 2.7×10-14

sqrt(2.7×10-14) ≈ 5.2×10-7 C/cm²

Divide by Cox:

5.2×10-7 / 3.453×10-7 = 1.51 V

Therefore,

VTH = -0.95 + 2(0.407) + 1.51
              = -0.95 + 0.814 + 1.51
              = 1.374 V

Analysis

The threshold voltage of approximately 1.37 V is typical for an n-channel MOSFET with the given parameters. This voltage must be exceeded by the gate-to-source voltage for the device to switch on effectively.

Note that the threshold voltage depends heavily on:

• Doping concentration (N_A): Higher doping increases V_TH because more charge is needed to invert the surface.
• Oxide thickness (t_ox): Thinner oxides increase C_ox, reducing the voltage needed to induce charge, thus lowering V_TH.
• Flat-band voltage (V_FB): Influenced by gate material and oxide charges.

Frequently Asked Questions (FAQs)

1. Why is threshold voltage important in MOSFET operation?

The threshold voltage determines the gate voltage level at which the MOSFET starts to conduct significantly. It defines the switching point and affects the transistor's power consumption and speed.

2. Can threshold voltage vary with temperature?

Yes, V_TH generally decreases with increasing temperature due to changes in semiconductor properties and carrier concentration.

3. What is the difference between threshold voltage and flat-band voltage?

Flat-band voltage (V_FB) is the voltage at which there is no band bending in the semiconductor surface. Threshold voltage (V_TH) is higher and includes the voltage needed to create an inversion layer and overcome bulk charge.

4. How does doping concentration affect V_TH?

Increasing doping concentration increases the bulk charge that must be inverted, thereby increasing V_TH.

5. What role does oxide thickness play in threshold voltage?

Thinner gate oxides increase gate capacitance (C_ox), allowing the channel to form at lower gate voltages, thus reducing threshold voltage.

6. Can threshold voltage be controlled during fabrication?

Yes, by adjusting doping levels, oxide thickness, and gate material work function, manufacturers tailor the threshold voltage for desired device characteristics.

7. What is body effect on threshold voltage?

The threshold voltage increases if the source-to-body voltage is not zero, due to additional depletion charge. This is called the body effect and can be modeled as:

VTH = VTH0 + γ(√|2φF + VSB| - √|2φF|)

where γ is the body effect coefficient and V_SB is the source-to-body voltage.

Conclusion

Understanding the MOSFET threshold voltage is essential for designing and analyzing electronic circuits that rely on MOS transistors. This parameter is rooted in semiconductor physics and controlled by material properties and fabrication parameters. By mastering the calculations and influences on V_TH, engineers can optimize device performance for applications ranging from microprocessors to power electronics.