Foundations — Formation of a PN junction

This page is the toolbox. Before you can follow the parent note Formation of a PN junction, every letter, arrow, and word it uses must first mean something and look like something. We build them one at a time, each resting on the one before.

0. The atom picture we start from

Everything below lives inside a crystal — atoms locked in a neat repeating grid, like oranges stacked in a crate. In a pure (intrinsic) semiconductor like silicon, every atom shares its outer electrons with its neighbours, so almost no charge is free to move.

To make anything interesting happen we add tiny amounts of a different atom. That is doping, and it gives us the two building blocks below.

1. The symbol $q$ — the elementary charge

Why the topic needs it. Every current and field is built from charges moving. $q$ is the conversion factor between "how many carriers" and "how much charge / current".

2. p-type and n-type — and their two families of characters

Doping comes in two flavours (see Intrinsic vs Extrinsic Semiconductors).

The word fixed is the whole plot: these ions are bolted into the crystal lattice and cannot move. Only electrons and holes move.

3. The concentration symbols $n$, $p$, $n_i$, $N_A$, $N_D$

These are all counts per unit volume — "how many of this thing packed into one cubic centimetre." Think of them as crowd densities.

Worked reading of this law. On the n-side we know $n_n\approx N_D$ (nearly all donors gave an electron). The minority holes there are $p_n=n_i^2/N_D$. On the p-side $p_p\approx N_A$ and the minority electrons are $n_p=n_i^2/N_A$. These four numbers are exactly what the built-in-potential formula compares.

4. Two ways charge moves: diffusion and drift

The whole junction story is a tug-of-war between two currents (deep-dived in Diffusion and Drift Currents).

Why the topic needs both. Diffusion starts the junction (carriers spread across). The field it uncovers then causes drift back. Equilibrium is the moment these two exactly cancel.

5. Electric field $\vec{E}$, potential $V$, and their link

Here $\frac{dV}{dx}$ is a derivative: "how much $V$ changes for a tiny step $dx$ in position." We need the derivative (not just $V$) because the push depends on how steeply the height changes, not on the height itself — a steep cliff pushes hard, a gentle ramp pushes softly, even at the same altitude.

6. Thermal voltage $V_T$ and the Einstein relation

7. Current density $J$

The equilibrium condition of the whole topic is simply $J_{\text{net}}=0$: as much charge drifts back as diffuses forward, so the net stream through any window is zero.

How it all feeds the topic

Read it top-down: charge → carriers → doping → fixed ions → field, while concentrations → diffusion. The two currents meet at the balance, and out pops $V_{bi}$.

Equipment checklist

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