Detailed Derivation of Channel Charge and Drain Current in MOSFETs

Overview

To understand MOSFET behavior—especially for NMOS transistors—it is essential to analyze how channel charge is generated and how it contributes to current flow. This article walks through the key equations and physical insights behind the MOSFET I-V characteristics, covering the linear (triode), saturation, and subthreshold regions.

---

1. Channel Charge Formation

Uniform Channel Charge

Once the gate voltage $V_{GS}$ exceeds the threshold voltage $V_T$, an inversion layer forms in the channel, accumulating electrons (for NMOS).

The channel charge density $Q_c$ can be expressed as:

$$
Q_c = -W C_{ox} (V_{GS} - V_T)
$$

• $W$: Channel width
• $C_{ox}$: Gate oxide capacitance per unit area
• $V_{GS}$: Gate-to-source voltage
• $V_T$: Threshold voltage

Note: The negative sign indicates that the charge is due to electrons (negative carriers).

---

2. Local Charge Density (When Drain Voltage is Non-zero)

If a drain voltage $V_{DS}$ is applied, the channel potential $V(x)$ varies along the channel length from source (0) to drain ($V_{DS}$).

The local charge density at position $x$ becomes:

$$
Q(x) = -W C_{ox} (V_{GS} - V(x) - V_T)
$$

This spatial variation in charge density drives the current along the channel.

---

3. Drain Current Derivation

Basic Equation

Drain current is the product of charge and carrier velocity:

$$
I_D = -Q(x) \cdot v(x) = W C_{ox} (V_{GS} - V(x) - V_T) \cdot v(x)
$$

Carrier velocity is proportional to the local electric field:

$$
v(x) = \mu_n \frac{dV(x)}{dx}
$$

Substituting and integrating along the channel length $L$:

$$
I_D = \mu_n C_{ox} W \int_0^L (V_{GS} - V(x) - V_T) \frac{dV(x)}{dx} dx
$$

---

Triode (Linear) Region

Assuming $0 < V_{DS} < V_{GS} - V_T$, solving the integral gives:

$$
I_D = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_T)V_{DS} - \frac{1}{2}V_{DS}^2 \right]
$$

In this region, the MOSFET acts like a voltage-controlled resistor.

---

Saturation Region

If $V_{DS} \geq V_{GS} - V_T$, the channel pinches off near the drain, and current saturates:

$$
I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_T)^2
$$

Current becomes independent of $V_{DS}$ and only depends on $V_{GS}$.

---

4. Subthreshold Region (Weak Inversion)

Even when $V_{GS} < V_T$, a small current flows due to thermionic emission:

$$
I_D = I_0 \exp\left( \frac{V_{GS} - V_T}{n V_T} \right) \left(1 - \exp\left(-\frac{V_{DS}}{V_T} \right) \right)
$$

• $I_0$: Pre-exponential factor depending on device geometry
• $n$: Subthreshold slope factor (typically 1.1–1.5)
• $V_T$: Thermal voltage (approx. 25 mV at room temp)

This is important for low-power and leakage-sensitive designs.

---

5. Channel Length Modulation (CLM)

In reality, drain current increases slightly even in saturation due to a shortening of the effective channel length as $V_{DS}$ increases:

$$
I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_T)^2 (1 + \lambda V_{DS})
$$

• $\lambda$: Channel length modulation parameter (analogous to Early effect in BJTs)

---

Summary Table of I–V Regions

Region	Condition	Current Equation
Cutoff	$V_{GS} < V_T$	Exponential (Subthreshold)
Triode	$V_{GS} > V_T,\ V_{DS} < V_{GS} - V_T$	Linear w.r.t. $V_{DS}$
Saturation	$V_{DS} \geq V_{GS} - V_T$	Quadratic in $V_{GS} - V_T$

---

Notes

• For plotting I-V curves, use $I_D$ vs. $V_{DS}$ for fixed $V_{GS}$, and log($I_D$) vs. $V_{GS}$ to visualize subthreshold behavior.
• These models are foundational for both analog and digital CMOS design.