Semiconductor Band Structure: The Gap That Runs Electronics

Bands, not levels

In an isolated atom, electrons occupy sharp, discrete energy levels. Pack billions of atoms into a crystal lattice and those levels smear out into continuous bands of allowed energy, separated by gaps of forbidden energy where no electron state exists at all. This band structure is a direct consequence of quantum mechanics applied to a periodic potential: Bloch theorem shows that electron wavefunctions in a periodic crystal take the form of a plane wave modulated by a function with the crystal's periodicity, and solving for the allowed energies at each crystal momentum produces the band diagram.

Conductors, insulators, semiconductors — the same physics, one number

What separates a metal from an insulator from a semiconductor is simply the size of the gap between the highest band that's (mostly) full of electrons — the valence band — and the next band up, which is empty at absolute zero — the conduction band. A metal has no gap at all, or bands that overlap, so electrons sit right at the edge of mobility. An insulator like diamond has a huge gap (~5.5 eV) that thermal energy at room temperature (~0.025 eV) can't realistically bridge. A semiconductor sits in between — silicon's gap is 1.12 eV, germanium's is 0.67 eV, gallium arsenide's is 1.42 eV — small enough that a useful population of electrons gets thermally excited across it at room temperature, but large enough that the material isn't simply a metal.

material       bandgap E_g (eV, ~300K)
diamond        5.47   — insulator
silicon        1.12   — semiconductor
GaAs           1.42   — semiconductor
germanium      0.67   — semiconductor
(metal: E_g ≈ 0, bands overlap or touch)

The Fermi level and Fermi-Dirac statistics

At any temperature, the probability that a state of energy E is occupied by an electron follows the Fermi-Dirac distribution:

f(E) = 1 / ( exp((E − E_F)/k_B T) + 1 )

E_F  — Fermi level (chemical potential for electrons)
k_B T at 300K ≈ 0.026 eV

The Fermi level E_F is the energy at which occupation probability is exactly 50%. In an intrinsic (undoped) semiconductor, E_F sits almost exactly in the middle of the gap, and raising the temperature thermally promotes more electrons across the gap into the conduction band, leaving behind holes — the absence of an electron in the valence band, which behaves like a mobile positive charge carrier in its own right.

Doping: engineering the Fermi level on purpose

The entire semiconductor industry runs on deliberately shifting E_F by doping — replacing a tiny fraction of the host atoms with impurities that have one more or one fewer valence electron. Adding phosphorus (5 valence electrons) to silicon (4) contributes a spare electron that sits in a shallow donor level just below the conduction band; this n-type doping pushes E_F up toward the conduction band and floods it with mobile electrons. Adding boron (3 valence electrons) instead creates a shallow acceptor level just above the valence band that readily grabs an electron, generating a mobile hole; this p-type doping pushes E_F down toward the valence band. Every diode, transistor, and integrated circuit works by joining regions of different doping (a p-n junction is the basic building block) and controlling how their Fermi levels line up and how carriers flow across the junction.

Direct vs indirect gaps

One more detail matters for optoelectronics: whether the conduction-band minimum and valence-band maximum sit at the same crystal momentum. GaAs is a direct-gap semiconductor — an electron can drop straight from the conduction band to the valence band emitting a photon, with no extra momentum needed, which is why GaAs and similar materials are used in LEDs and laser diodes. Silicon and germanium are indirect-gap semiconductors — the transition requires a simultaneous change in crystal momentum, which needs a phonon to participate, making photon emission far less efficient. That's why silicon dominates logic chips (it's cheap, abundant and its oxide is an excellent insulator) but essentially all commercial LEDs and laser diodes are made from direct-gap III-V materials like GaAs and its relatives.

Frequently asked questions

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