Formation of a PN junction

WHAT is a PN junction?

Key players before contact:

HOW it forms — the story in 4 steps

Step 1 — Diffusion (concentration wants to even out). There are way more electrons on the n-side and way more holes on the p-side. A concentration gradient exists.

Step 2 — Uncovering fixed ions (the depletion region). When an electron leaves the n-side, it leaves behind a fixed positive donor ion $N_D^+$. When a hole leaves the p-side, it leaves behind a fixed negative acceptor ion $N_A^-$. Near the junction the mobile carriers are swept out.

Step 3 — Built-in electric field. The exposed ions form a dipole: $+$ ions on the n-side, $-$ ions on the p-side. This creates an internal electric field $\vec E$ pointing from n→p (from + charge to − charge).

Step 4 — Equilibrium (drift cancels diffusion). Diffusion keeps trying to push carriers across; the growing field pushes them back (drift). The region grows until: $J_{\text{diffusion}} = J_{\text{drift}} \quad\Rightarrow\quad J_{\text{net}} = 0$ At this balance a stable built-in potential $V_{bi}$ exists across the depletion region and no external current flows.

Deriving the built-in potential $V_{bi}$ from first principles

Why this step? We use $E = -\dfrac{dV}{dx}$ (field is the negative slope of potential) to swap field for potential: $0 = -q\,\mu_n\, n\,\frac{dV}{dx} + q\, D_n \frac{dn}{dx}$

Why this step? Divide by $q\,n$ and rearrange to separate variables: $\frac{dV}{dx} = \frac{D_n}{\mu_n}\,\frac{1}{n}\frac{dn}{dx}$

Why this step? Use the Einstein relation $\dfrac{D_n}{\mu_n} = \dfrac{kT}{q}$ (thermal voltage $V_T = kT/q$), which links diffusion to thermal energy: $dV = \frac{kT}{q}\,\frac{dn}{n}$

Why this step? Integrate from the p-side ($V_p$, electron conc. $n_p$) to the n-side ($V_n$, electron conc. $n_n$): $V_n - V_p = \frac{kT}{q}\ln\!\frac{n_n}{n_p}$

So:

Depletion width intuition (Forecast-then-Verify)

Flashcards

Connections

• Doping of Semiconductors — makes the p and n regions in the first place.
• Intrinsic vs Extrinsic Semiconductors — source of $n_i$ and mass-action law $np=n_i^2$.
• Diffusion and Drift Currents — the two currents that balance at equilibrium.
• PN Junction under Bias — what happens when you apply external voltage.
• Diode I-V Characteristics — the exponential curve built on this barrier.
• Einstein Relation — the $D/\mu = kT/q$ bridge.

Concept Map

Hinglish (regional understanding)