One atom has sharp energy levels. Bring two atoms together and each level splits in two; bring $10^{23}$ together in a crystal and each level spreads into a nearly continuous band of allowed energies, separated by gaps with no states at all. The exclusion principle then fills the bands from the bottom, two electrons per state, and how the filling ends decides whether a solid is a metal, an insulator or a semiconductor.
🔑 Key idea
Conductor: the highest occupied band is only partly filled, so electrons can move into nearby empty states and carry current.
Insulator: the highest filled band (valence band) is full and a wide gap (several eV; diamond's is 5.5 eV) separates it from the empty conduction band.
Semiconductor: the same picture with a small gap (silicon 1.12 eV, germanium 0.67 eV), so heat or light lifts some electrons across, and conduction rises steeply with temperature.
A photon can move an electron across the gap only if $hf \ge E_g$; an LED emits light with $hf \approx E_g$.
💡 Why it works
In a full band, every state is taken, so an electric field can't shift electrons into states of different momentum: no net current. Conduction needs empty states just above the filled ones. In a metal they are right there. In a semiconductor, an electron needs about $E_g$ to reach them, and at temperature $T$ the chance of having that much thermal energy goes roughly like $e^{-E_g/2k_BT}$ for the number of electrons that make it across. That factor grows fast with $T$, which is why warming a semiconductor lowers its resistance, while a metal's resistance rises (its electron count is fixed, and hotter atoms scatter them more). Doping adds carriers on purpose: a phosphorus atom (5 outer electrons) in silicon (4) donates an electron (n-type); a boron atom (3) leaves a hole (p-type).
Bands in a semiconductor. Raise the temperature and watch electrons hop across the 1.12 eV gap; switch to a metal or an insulator to compare.
Worked examples
✏️ Worked example: What silicon absorbs
What is the longest wavelength of light that can lift an electron across silicon's 1.12 eV gap?
Answer: About 22 times more carriers across the gap (from this factor alone)
⚠️ Common mistake
Thinking semiconductors conduct worse when warmed, as metals do. In a metal the number of free electrons is fixed, and warming only makes them scatter more, so resistance rises. In a semiconductor, warming lifts many more electrons across the gap, and that exponential gain outweighs the extra scattering, so resistance falls. Thermistors in thermostats use exactly this.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.