Understanding JFET Depletion Layer Width

A Junction Field-Effect Transistor (JFET) is a type of field-effect transistor that uses a p-n junction to control the current flow through a channel. The key to JFET operation is the depletion region that forms at the p-n junction between the gate and the channel.

The width of this depletion layer is crucial because it modulates the effective channel width, thereby controlling the current. The width of the depletion layer is dependent on the voltage applied across the p-n junction.

In a JFET, the voltage across the gate-channel junction varies along the length of the channel. At a distance \(x\) from the source, the potential in the channel relative to the source is denoted as \(V(x)\). The gate-to-source voltage is \(V_G\). Additionally, there is a built-in potential \(V_{bt}\) across the p-n junction even with zero applied bias.

The total potential difference across the gate-channel p-n junction at point \(x\) is the sum of the channel potential \(V(x)\), the applied gate-source voltage \(V_G\), and the built-in potential \(V_{bt}\). However, note that standard JFET biasing uses a reverse-biased gate-channel junction. The gate voltage \(V_G\) here usually refers to \(|V_{GS}|\) in magnitude for reverse bias, or it represents \(V_{GS}\) algebraically, where a negative \(V_{GS}\) increases the reverse bias.

The depletion layer width of a one-sided p-n junction (which is often a good approximation for the gate-channel junction where one side is much more heavily doped than the other, typically the gate is heavily doped p+ or n+) is given by the formula derived from Poisson's equation:

\( W \approx \sqrt{\frac{2 \epsilon_s (V_{bi} + V_R)}{q N_D}} \)

where:

• \(W\) is the depletion width.
• \(\epsilon_s\) is the permittivity of the semiconductor.
• \(V_{bi}\) is the built-in potential.
• \(V_R\) is the applied reverse bias voltage.
• \(q\) is the elementary charge.
• \(N_D\) is the doping concentration of the lightly doped side (the channel).

In the context of the JFET at a position \(x\), the "applied reverse bias voltage" is effectively the total potential difference across the gate-channel junction, which is a combination of the voltage drop along the channel \(V(x)\) and the applied gate-source voltage \(V_G\). Accounting for potential differences, the total potential across the junction at point \(x\) is related to \(V(x)\), \(V_G\), and \(V_{bt}\).

Considering the voltage polarity and referencing, the total potential difference driving the depletion width at position \(x\) is effectively \(V(x) + V_G + V_{bt}\). This sum represents the effective potential across the junction that influences the depletion width at point \(x\).

Therefore, the depletion layer width \(y(x)\) at a distance \(x\) from the source is given by the expression:

\( y(x) = \left[\frac{2 \epsilon_s (V(x) + V_G + V_{bt})}{q N_D}\right]^{1/2} \)

Let's look at the options provided and compare them to this standard form.

Based on this analysis, the formula matching the standard derivation for the depletion layer width in a JFET at a distance \(x\) from the source is the one where the total potential \(V(x) + V_G + V_{bt}\) is multiplied by \(2 \epsilon_s\) and divided by \(q N_D\), with the entire expression raised to the power of \(1/2\).

Revision Table: JFET Key Parameters

Additional Information on JFET Operation

The JFET is a voltage-controlled device. The gate-source voltage \(V_G\) controls the width of the depletion region. A more negative \(V_{GS}\) (for n-channel) or more positive \(V_{GS}\) (for p-channel) increases the reverse bias, widening the depletion region. As the depletion region widens, it constricts the channel, increasing its resistance and reducing the drain current \(I_D\).

The channel potential \(V(x)\) increases from the source (\(V(0)=0\)) towards the drain (\(V(L)=V_{DS}\)), where \(L\) is the channel length and \(V_{DS}\) is the drain-source voltage. This means the gate-channel voltage \(V_G - V(x)\) varies along the channel, leading to a non-uniform depletion width. The depletion region is widest at the drain end (where \(V(x)\) is highest) and narrowest at the source end (where \(V(x)\) is lowest).

When the depletion regions from both sides of the gate meet in the middle of the channel, the channel is said to be "pinched off" at that point. This phenomenon occurs at a certain gate-source voltage (pinch-off voltage, \(V_P\)) or when the potential difference across the gate-channel junction reaches a critical value.

Understanding the depletion layer width variation is fundamental to deriving the current-voltage characteristics of the JFET in both the triode (linear) region and the saturation region.