1. Quick Refresher: Where We Left the Physics

Just enough to make this chapter standalone. Skim if you remember section 4 of the physics chapter; otherwise re-read.

A semiconductor like silicon has a band structure with two important bands — a nearly-full valence band and a nearly-empty conduction band — separated by a small band gap of about 1.12 eV. At room temperature, thermal energy kicks a few electrons across the gap. Each electron that goes up leaves behind a hole in the valence band, and the hole behaves like a positive mobile charge. So a piece of intrinsic (pure) silicon at 300 K has equal numbers of electrons and holes — about $1.5 \times 10^{10}$ of each per cubic centimeter — and a tiny but measurable conductivity.

We can dramatically change this by doping — substituting a few of the silicon atoms with atoms from a neighboring column of the periodic table:

• Phosphorus has 5 valence electrons (silicon has 4). Drop one phosphorus atom into the silicon lattice and four of its electrons bond to neighbors; the fifth is loose. We call this an n-type semiconductor — "n" because the majority charge carriers are negative (electrons). Typical doping is a few parts per million, giving electron concentrations around $10^{16}$ to $10^{19}$ per cubic centimeter — many orders of magnitude more than intrinsic.
• Boron has 3 valence electrons. Drop a boron atom into silicon and it is one electron short of a full bond — effectively, it creates a hole. p-type — majority carriers are positive holes.

In an n-type sample, $n$ (electron concentration) is roughly equal to the donor doping concentration $N_D$, and $p$ (hole concentration) is suppressed:

$np = n_i^2 \approx 2.25 \times 10^{20} \text{ cm}^{-6}.$

This mass-action law holds at thermal equilibrium and is one of the foundational identities of semiconductor physics. The product of electron and hole densities is constant, regardless of how you dope. Increase $n$ by doping with phosphorus, and $p$ drops to keep the product fixed.

Carriers move under two driving forces:

• Drift. Under an electric field, with average velocity $v_d = \mu E$. The constant $\mu$ is mobility, and silicon's electron mobility is about 1350 cm²/V·s; hole mobility is about 480. Electrons are about three times more mobile than holes — the asymmetry that propagates throughout circuit design.
• Diffusion. From regions of high concentration to low, like ink spreading in water. The flux is $J = qD\,dn/dx$, with diffusion coefficient $D$ tied to mobility through the Einstein relation $D/\mu = kT/q \approx 26$ mV at room temperature.

We track the energetic state of electrons in the bands using the Fermi level $E_F$ — roughly, "the highest energy at which electrons are still found at appreciable density." In intrinsic silicon $E_F$ sits in the middle of the gap. In n-type, $E_F$ is pulled close to the conduction band; in p-type, close to the valence band.

That is the physics backdrop. Now, the action.