Read the passage given below and answer the question The Metal Oxide Semiconductor Field Effect Transistor (MOSFET) is the most important device at the forefront of high‐ density integrated circuits such as microprocessors and semiconductor memories. It is also becoming an important power device. The principle of the surface field‐effect transistor was first proposed in the 1930s and the first MOSFET was reported in 1960 using the Si‐SiO 2system. The current in MOSFET is transported by carriers of one polarity only and hence it is usually referred to as a unipolar device. Although MOSFETs have been made with various semiconductors, such as Ge, Si, and GaAs and use various insulators such as SiO 2, Si 3 O 4 and Al 2O 3, the most important system is Si‐SiO 2.
In n‐channel MOSFET, the charge induced in the inversion layer per unit area at a distance y from the source is:
Understanding n-Channel MOSFET Inversion Layer Charge
This question asks about the charge distribution within an n-channel Metal Oxide Semiconductor Field Effect Transistor (MOSFET), specifically focusing on the charge induced in the inversion layer. The inversion layer is crucial for the operation of the MOSFET as it forms a channel for current conduction.
Components of Charge in an n-Channel MOSFET
In an n-channel MOSFET, when a positive voltage is applied to the gate (relative to the body/substrate), it attracts electrons to the surface of the p-type silicon substrate under the gate oxide. When the gate voltage is sufficiently high (above the threshold voltage), the concentration of electrons at the surface exceeds the concentration of holes in the bulk, creating an 'inversion layer' which is n-type. The total charge induced at the semiconductor surface is balanced by the gate charge.
The total charge density at the semiconductor surface ($Q_s$) consists of two main components:
- The charge in the depletion layer ($Q_B$).
- The charge in the inversion layer ($Q_n$).
So, the total surface charge density is $Q_s = Q_B + Q_n$.
Relating Gate Voltage to Surface Charge
The voltage applied to the gate ($V_G$) controls the charge in the semiconductor. The relationship between the gate voltage and the total charge at the semiconductor surface is governed by the gate oxide capacitance per unit area ($C_i$) and the surface potential ($\psi_s$). The voltage drop across the oxide is approximately $V_G - \psi_s$. Therefore, the charge induced on the semiconductor surface ($Q_s$) due to this voltage drop across the oxide is related by:
$\qquad Q_s \approx C_i (V_G - \psi_{FB} - \psi_s)$
where $\psi_{FB}$ is the flat-band voltage. For simplicity, and often in advanced treatments focusing on the potential relative to flat band, the relationship between the gate voltage overdrive and the surface potential change and induced charge is considered.
A more complete charge balance equation considering the gate voltage relative to the substrate can be written. The gate voltage ($V_G$) is related to the surface potential ($\psi_s$), the oxide capacitance ($C_i$), and the total semiconductor surface charge ($Q_s$). Ignoring flat-band voltage for this specific formula context (which often relates the gate voltage *relative* to the surface potential), the induced charge on the gate ($Q_g$) per unit area is balanced by the total charge in the semiconductor surface region ($Q_s = Q_B + Q_n$) per unit area. The voltage drop across the oxide is $(V_G - \psi_s)$. Thus, the charge on the gate is $Q_g = C_i (V_G - \psi_s)$. Since $Q_g = -Q_s$, we have:
$\qquad -Q_s = C_i (V_G - \psi_s)$
$\qquad Q_s = -C_i (V_G - \psi_s)$
Substituting $Q_s = Q_B + Q_n$:
$\qquad Q_B + Q_n = -C_i (V_G - \psi_s)$
Now, isolating the inversion layer charge $Q_n$:
$\qquad Q_n = -C_i (V_G - \psi_s) - Q_B$
This formula gives the inversion layer charge per unit area at a specific point along the channel where the surface potential is $\psi_s$. In a conducting channel (like in saturation), the surface potential $\psi_s$ varies along the length of the channel, dependent on the position $y$ from the source. Thus, the inversion charge density is $Q_n(y)$, and the surface potential is $\psi_s(y)$. The depletion charge $Q_B$ also varies along the channel, $Q_B(y)$, because the depletion region width depends on the substrate bias which changes effectively along the channel due to the potential variation $\psi_s(y)$.
So, the formula for the inversion charge density at a distance $y$ from the source in an n-channel MOSFET is:
$\qquad Q_n(y) = -C_i (V_G - \psi_s(y)) - Q_B(y)$
Note the negative sign. This indicates that the charge in the inversion layer ($Q_n$) is negative, as it consists of electrons (negative carriers) in an n-channel device. $V_G$ is the gate voltage, $C_i$ is the gate oxide capacitance per unit area, $\psi_s(y)$ is the surface potential at position $y$, and $Q_B(y)$ is the depletion region charge density at position $y$. $Q_B(y)$ itself is a negative quantity in a p-type substrate (due to ionized acceptor atoms).
Analyzing the Options for n-Channel MOSFET Charge
Let's compare the derived formula with the given options for the inversion layer charge in an n-channel MOSFET:
- Option 1: $Q_n(y) = [V_G + \psi_s(y)] C_i - Q_B(y)$ - Incorrect signs and terms.
- Option 2: $Q_n(y) = -[V_G - \psi_s(y)] C_i - Q_B(y)$ - Matches the derived formula.
- Option 3: $Q_n(y) = -[V_G - \psi_s(y)] C_i + Q_B(y)$ - Incorrect sign for $Q_B(y)$.
- Option 4: $Q_n(y) = -[V_G + \psi_s(y)] C_i + Q_B(y)$ - Incorrect signs and terms.
The correct formula for the inversion charge per unit area $Q_n(y)$ in an n-channel MOSFET at a position $y$ along the channel is $Q_n(y) = -C_i (V_G - \psi_s(y)) - Q_B(y)$. The negative sign reflects the charge of the electron carriers in the inversion layer. $V_G$ is the gate voltage, $\psi_s(y)$ is the surface potential at $y$, $C_i$ is the insulator capacitance per unit area, and $Q_B(y)$ is the magnitude of the depletion charge per unit area (a negative quantity) at $y$.
| Term | Description | Units (typical) |
|---|---|---|
| $Q_n(y)$ | Inversion layer charge density at position y | C/cm$^2$ |
| $V_G$ | Gate voltage | V |
| $\psi_s(y)$ | Surface potential at position y | V |
| $C_i$ | Insulator (Oxide) capacitance per unit area | F/cm$^2$ |
| $Q_B(y)$ | Depletion region charge density at position y | C/cm$^2$ |
Revision Table: n-Channel MOSFET Basics
| Concept | Description |
|---|---|
| MOSFET | Metal Oxide Semiconductor Field Effect Transistor - a semiconductor device used for switching and amplifying electronic signals. |
| n-Channel MOSFET | A type of MOSFET where the conduction channel is formed by electrons (n-type). Typically built on a p-type substrate. |
| Inversion Layer | A layer formed at the semiconductor surface where the carrier type is opposite to the bulk material (e.g., electrons form an n-layer on a p-substrate). |
| Gate Voltage ($V_G$) | Voltage applied to the gate terminal, controlling the channel conductivity. |
| Surface Potential ($\psi_s$) | The electric potential at the semiconductor surface, relative to the bulk semiconductor. |
| Oxide Capacitance ($C_i$) | Capacitance per unit area of the gate insulator layer. |
| Depletion Charge ($Q_B$) | Charge per unit area in the depletion region under the gate, consisting of ionized dopant atoms. |
Additional Information: MOSFET Operation Regions
The operation of a MOSFET is typically described in three regions based on the gate voltage and drain-source voltage ($V_{DS}$):
- Cut-off Region: $V_G < V_{th}$ (Threshold Voltage). The inversion layer is not formed or is very weak. No significant current flows between source and drain. $Q_n \approx 0$.
- Linear (Triode) Region: $V_G \ge V_{th}$ and $V_{DS}$ is small. A conductive inversion channel is formed. The channel resistance is relatively constant, and drain current is roughly proportional to $V_{DS}$. The inversion charge density $Q_n(y)$ varies almost linearly from source to drain.
- Saturation Region: $V_G \ge V_{th}$ and $V_{DS}$ is large (specifically, $V_{DS} \ge V_G - V_{th}$). The channel becomes "pinched off" near the drain end. The drain current saturates and becomes relatively independent of $V_{DS}$. The inversion charge density $Q_n(y)$ goes to zero at the drain end.
The formula $Q_n(y) = -C_i (V_G - \psi_s(y)) - Q_B(y)$ is a fundamental equation that applies across different operation regions, provided the channel is formed ($V_G \ge V_{th}$). The variation of $\psi_s(y)$ and $Q_B(y)$ along the channel is what leads to the different current-voltage characteristics in the linear and saturation regions.
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:
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:
