MOSFET Operation

MOSFET operation is governed by the relationship between gate voltage, channel formation, and drain current. The two most important parameters in this analysis are the threshold voltage and the concept of pinch-off, both of which define the boundaries between operating regions and appear heavily in GATE problems.

Threshold Voltage and Channel Formation

The threshold voltage V_T is the minimum gate-to-source voltage required to form an inversion layer strong enough to conduct significant current between drain and source. Below V_T, the MOSFET is in the cut-off region: the channel does not exist and drain current is essentially zero. The value of V_T depends on the doping concentration of the substrate, the oxide thickness, and the flat-band voltage, which accounts for fixed oxide charges and work-function differences between gate and semiconductor.

For an NMOS device, V_T is positive, meaning a positive gate voltage is needed to attract electrons and form the channel. For a depletion-mode MOSFET, the channel already exists at zero gate voltage (V_T is negative), and a negative gate voltage is needed to turn it off. This distinction is important in GATE questions that ask you to identify device type from a V_T value.

The threshold voltage expression including body effect is: V_T = V_T0 + gamma * (sqrt(2*phi_F + V_SB) - sqrt(2*phi_F)), where V_T0 is the threshold at zero source-body bias, gamma is the body-effect coefficient (V^0.5), phi_F is the Fermi potential, and V_SB is the source-to-body reverse bias. When V_SB increases, V_T increases, and for a given VGS the overdrive voltage VGS - VT decreases, reducing drain current.

Operating Regions of the MOSFET

Once VGS exceeds V_T, the MOSFET can be in one of two active regions depending on the drain-to-source voltage VDS. In the linear (triode) region, VDS is small and the channel exists uniformly from source to drain. The device behaves like a voltage-controlled resistor. Current increases approximately linearly with VDS for small VDS.

As VDS increases toward the value VGS - VT (called the saturation voltage V_DS,sat), the channel near the drain becomes thinner because the local gate-to-channel voltage drops. When VDS = VGS - VT exactly, the channel just pinches off at the drain end. Beyond this point, the channel is pinched off near the drain, but current continues to flow by carrier injection across the narrow depletion region. The drain current becomes approximately constant (independent of VDS), and the device enters the saturation region.

In saturation, the MOSFET functions as a voltage-controlled current source, which is the basis of its use in amplifier circuits. The boundary between linear and saturation is defined by VDS = VGS - VT. This line is called the pinch-off locus on the output characteristics.

Mathematical Expression

The drain current in the linear region is ID = kn * [(VGS - VT)*VDS - VDS^2/2], and in saturation it is ID = (kn/2) * (VGS - VT)^2, where kn = mu_n * Cox * (W/L). The saturation current depends only on VGS - VT (the overdrive voltage), not on VDS. This is why the curves are flat in saturation on the ID-VDS plot.

Practical Understanding

In digital logic, MOSFETs operate as switches. When VGS < V_T, the transistor is off (cut-off). When VGS is well above V_T and VDS is small, the transistor is on and in the linear region, conducting like a small resistor (on-resistance). Analog amplifiers operate in saturation where the quadratic relationship between VGS and ID enables gain through the transconductance gm = dID/dVGS = kn*(VGS - VT).

Channel-length modulation is a second-order effect where the effective channel length decreases slightly as VDS increases beyond pinch-off, causing a small increase in drain current with VDS in saturation. This is modeled by a factor (1 + lambda*VDS) multiplied to the saturation current. The parameter lambda (1/V) is the channel-length modulation coefficient, and its inverse 1/lambda = VA is the Early voltage.

Given:
kn' (mu_n * Cox) = 100 uA/V^2
W/L = 10
VT = 1 V
VGS = 3 V
VDS = 4 V

Why this formula applies:
VDS (4V) > VGS - VT (3-1 = 2V), so device is in saturation.

Formula:
ID = (kn/2) * (VGS - VT)^2
kn = kn' * (W/L)

Substitution:
kn = 100 uA/V^2 * 10 = 1000 uA/V^2 = 1 mA/V^2
ID = (1 mA/V^2 / 2) * (3 - 1)^2

Calculation:
ID = 0.5 mA/V^2 * 4 V^2
ID = 2 mA

Final Answer:
ID = 2 mA (MOSFET in saturation)

Exam Tip: Always check the region first. Compute VGS - VT and compare with VDS. If VDS >= VGS - VT, use the saturation formula. If VDS < VGS - VT, use the linear formula. Mixing up regions is the most common GATE mistake in MOSFET problems.

• Cut-off: VGS < VT, no inversion layer, ID = 0 (ideal), device is off.
• Linear region: VGS > VT and VDS < VGS - VT, channel uniform, device acts as a resistance.
• Saturation: VDS >= VGS - VT, channel pinched off near drain, ID controlled only by VGS.
• Transconductance gm = kn*(VGS - VT) in saturation; determines voltage gain of amplifier.
• Channel-length modulation (lambda) causes slight increase in ID with VDS in saturation.

Quick Revision

• VT is the threshold: NMOS VT > 0, depletion MOSFET VT < 0.
• Three regions: cut-off (VGS < VT), linear (VDS < VGS-VT), saturation (VDS >= VGS-VT).
• Saturation current: ID = (kn/2)*(VGS-VT)^2, independent of VDS (ignoring lambda).
• Linear current: ID = kn*[(VGS-VT)*VDS - VDS^2/2].
• Pinch-off locus: VDS = VGS - VT separates the two active regions.
• Body effect increases VT: more VSB means higher VT and lower drain current.
• Exam trap: when VDS equals exactly VGS-VT, device is at boundary of saturation, use saturation formula.