Semiconductor basics: bands, doping, and the PN junction

Materials have two key energy bands for valence electrons: the valence band (where electrons live when bound to atoms) and the conduction band (where they're free to carry current). The gap between them, $E_g$, decides everything.

• Metals: bands overlap. Electrons are always free. Always conduct.
• Insulators: gap is huge (~5+ eV for diamond). Nothing crosses. Don't conduct.
• Semiconductors: gap is narrow (~1.1 eV for silicon, ~0.7 eV for germanium). At room temperature, some electrons get enough thermal energy to jump.

That "some" is what makes the material engineerable. With doping, light, voltage, or temperature you can turn conductivity on, off, or anywhere in between. Insulators are too closed; metals are too open. Silicon is the Goldilocks zone.

When an electron jumps from the valence to the conduction band, it leaves an empty spot behind — a hole. Holes act like positive charges: when a nearby valence electron drops into the hole, the hole effectively moves to where the electron came from.

So intrinsic (pure) silicon at room temperature has two carriers:

• Electrons in the conduction band — negative charge, mobile.
• Holes in the valence band — positive charge, mobile (in the bookkeeping sense).

The concentration of each is tiny: $n_i \approx 1.5 \times 10^{10}$ per cm³ for Si at 300 K. Compare to $\sim 10^{22}$ atoms per cm³ — fewer than one atom in a trillion contributes a carrier. Pure silicon is a lousy conductor. The trick to making it useful is doping.

Silicon has four valence electrons; each atom sits in a tetrahedral bond with four neighbors. Replace a tiny fraction of Si atoms with elements that don't fit cleanly:

• Phosphorus has 5 valence electrons — one left over. That electron drifts to the conduction band; the crystal stays neutral overall but now has a population of free electrons. → n-type.
• Boron has 3 valence electrons — one short. It pulls one from a neighbor, leaving a hole behind. → p-type.

Typical doping: 1 phosphorus per million silicon atoms multiplies carrier concentration by $10^5$ to $10^7$. A tiny chemical change turns insulating silicon into a controllable conductor. Every modern chip is doped silicon — billions of carefully placed phosphorus and boron atoms patterned at the nanometer scale.

Where p-type and n-type meet, carriers recombine and leave behind charged dopant ions — the depletion region.

Slap p-type silicon next to n-type. At the boundary, electrons from the n-side diffuse into the p-side; holes from the p-side diffuse into the n-side. Both kinds of carrier hit the other side and recombine — they annihilate.

What's left in a thin region around the boundary: no free carriers, just the fixed ionized dopants (positive phosphorus on the n-side, negative boron on the p-side). This is the depletion region, and it carries a built-in voltage $V_{bi}$ (typically 0.6–0.7 V for silicon) pointing from n to p.

That built-in voltage is the barrier. It exactly opposes further diffusion at equilibrium: as many carriers diffuse across as are pushed back by the field. Net current is zero.

Connect a battery so the p-side is more positive than the n-side. The applied voltage cancels part of the built-in voltage. The depletion region shrinks, the barrier drops, and carriers from each side flood across.

The current grows exponentially with applied voltage:

$I = I_s \left(e^{V/V_T} - 1\right)$

where $V_T \approx 26$ mV at room temperature and $I_s$ is the saturation current (tiny, in nA to pA). The exponential is why a 100 mV voltage swing changes the current by orders of magnitude. It's also why every diode IV curve looks like a hockey stick around 0.6–0.7 V — the famous "knee voltage" of silicon.

Flip the battery: p-side negative, n-side positive. The applied voltage adds to the built-in voltage. The depletion region grows, the barrier rises, and almost no current flows — just the saturation current $I_s$ leaking through.

The PN junction is now a one-way valve: easy current forward, almost none in reverse. That asymmetry is the diode, the simplest semiconductor device.

The valve isn't perfect:

• Reverse leakage — $I_s$ rises with temperature (every 10 °C roughly doubles it).
• Breakdown — at a high enough reverse voltage, the field strips electrons off atoms and current floods. Avalanche breakdown. Engineered junctions (Zener diodes) use this on purpose; in regular diodes it's a failure mode.

Forward bias is a knee. Reverse bias is a wall — until it cracks.

Three takeaways the rest of semiconductor engineering builds on:

• Band gap decides whether a material can be a switch. Silicon is the historical winner; SiC and GaN are taking specific niches (high voltage, RF).
• Doping is how you steer carriers. Every transistor, every memory cell, every diode is doping patterned at nanometer scale.
• Junctions are where the action is. A PN junction by itself is a diode. Two back-to-back junctions are a bipolar transistor. A junction modulated by a gate is a MOSFET. Three back-to-back junctions are a thyristor.

The rest of this cursus walks up that ladder: diode → transistor → CMOS logic → chip → fab → scaling. Every step is a cleverer rearrangement of the same doped silicon.