If N is the doping level of substrate and V is the effective voltage across the junction in a MOSFET, then the depletion region width for the junction is given by:
Understanding MOSFET Depletion Region Width
The question asks for the formula that describes the width of the depletion region in a MOSFET junction. This depletion region forms between the semiconductor substrate and the channel region (or under the gate oxide) when a voltage is applied. Its width depends on several factors, including the doping level of the substrate and the effective voltage across the junction.
Factors Affecting Depletion Width
The key factors mentioned are:
- \( N \): Doping level (concentration of impurity atoms) of the substrate.
- \( V \): Effective voltage across the junction.
- \( \epsilon_{st} \): Relative permittivity (dielectric constant) of the semiconductor material.
- \( \epsilon_0 \): Permittivity of free space.
- \( q \): Magnitude of the electronic charge.
Deriving the Depletion Region Width Formula
The depletion region is formed due to the diffusion of carriers across the junction, leaving behind fixed charged ions. In a MOSFET, this typically refers to the depletion region formed in the substrate under the gate when a voltage is applied, or the depletion region of the body-source/drain PN junctions.
The width of the depletion region (W) in a semiconductor junction is generally derived from solving Poisson's equation, considering the charge distribution in the depletion region and the applied voltage. For a single-sided junction (which is a reasonable approximation for the depletion under the gate in many MOSFET cases, especially when the substrate is much lighter doped than the channel or source/drain regions), the depletion width is related to the voltage and doping concentration.
The formula for the depletion width \( W \) in a region with doping concentration \( N \), under an effective voltage \( V \), and made of a semiconductor with permittivity \( \epsilon_s = \epsilon_{st} \epsilon_0 \) is given by:
\( W = \sqrt{\frac{2 \epsilon_s V}{q N}} \)
Substituting \( \epsilon_s = \epsilon_{st} \epsilon_0 \), the formula becomes:
\( W = \sqrt{\frac{2 \epsilon_{st} \epsilon_0 V}{q N}} \)
Analyzing the Given Options
Let's compare this derived formula with the given options:
- Option 1: \( \sqrt{\frac{2 \epsilon_{st} \epsilon_0 V}{q N}} \) - This matches our derived formula exactly.
- Option 2: \( \sqrt{\frac{2 \epsilon_{st} \epsilon_{0}qv}{N}} \) - This includes 'qv' in the numerator instead of just 'V', which is incorrect dimensionally and physically.
- Option 3: \( \sqrt{\frac{2 \epsilon_{st} N}{qV}} \) - Here, N and V are in the numerator and denominator respectively within the square root, which is the inverse of the correct relationship for V and N.
- Option 4: \( \frac{2 \epsilon_{st} \epsilon_0 V}{q N} \) - This formula is not under a square root, which is incorrect for the depletion width formula.
Based on the derivation and comparison, Option 1 correctly represents the depletion region width.
Revision Table: Key Parameters
| Parameter | Symbol | Description | Typical Units |
|---|---|---|---|
| Depletion Width | \( W \) | Width of the region depleted of mobile carriers | meters (m) |
| Substrate Doping Level | \( N \) | Concentration of impurity atoms in the substrate | cm\(^{-3}\) or m\(^{-3}\) |
| Effective Voltage | \( V \) | Voltage across the junction (e.g., band bending potential) | Volts (V) |
| Relative Permittivity of Semiconductor | \( \epsilon_{st} \) | Dielectric constant of the semiconductor material (e.g., Silicon) | Dimensionless |
| Permittivity of Free Space | \( \epsilon_0 \) | Permittivity of vacuum | F/m |
| Electronic Charge | \( q \) | Magnitude of the charge of an electron | Coulombs (C) |
Additional Information on MOSFET Depletion
The depletion region width is a critical parameter in MOSFET operation. In the context of a MOS capacitor or the gate region of a MOSFET, the depletion width in the substrate varies with the gate voltage. As the gate voltage increases (for an n-channel MOSFET with a p-type substrate), the depletion region extends further into the substrate until the onset of inversion, where a channel of mobile electrons forms at the interface. Beyond the threshold voltage, the depletion width ideally saturates at its maximum value, and the additional charge induced by increasing gate voltage is accommodated by the carriers in the inversion layer.
Understanding the depletion width is essential for analyzing several MOSFET characteristics, including:
- Threshold voltage calculation.
- MOS capacitor capacitance (which varies with depletion width in depletion mode).
- Body effect (variation of threshold voltage with substrate bias).
The formula \( W = \sqrt{\frac{2 \epsilon_{st} \epsilon_0 V}{q N}} \) applies when the substrate is uniformly doped and provides a good approximation for the relationship between voltage, doping, and depletion extent.
Important Questions from MOSFET - Teaching
In enhancement type of MOSFET, at zero gate bias:
A. Channel conductance is very low
B. Channel conductance is very high
C. Channel resistance is very low
D. Channel resistance is very high
Choose the correct answer from the options given below:
The maximum operating frequency of a MOSFET is
A. \(\rm \frac{\omega_m}{2\pi}\)
B. \(\rm \frac{\mu_nV_D}{2\pi L^2}\)
C. \(\rm \frac{\mu_nL^2}{2\pi}\)
D. \(\rm \frac{\mu_nV_D}{2\pi L}\)
Choose the correct answer from the options given below:
In a MOSFET, the transconductance in linear region ean be expressed as:
For a thermally grown silicon dioxide layer, in a MOSFET, the leakage current between the gate and channel is:
For n MOSFET fabrication the substrate required is:
