Band Structure & the Fermi Level: Why Some Solids Conduct and Others Don't

From atomic levels to bands

A lone atom has sharp, discrete energy levels. Bring roughly 10²³ of them together into a crystal and each level does not stay put — it splits into as many closely spaced sub-levels as there are atoms, because every electron now feels the periodic potential of the whole lattice rather than a single nucleus. Pack enough atoms in and those sub-levels merge into a quasi-continuous band of allowed energies. Between neighbouring bands there can be a range of energies with no allowed states at all — a band gap — and whether that gap exists, and how wide it is, is the single fact that decides if a solid is a metal, an insulator or a semiconductor.

Bloch's theorem and the E(k) dispersion

Felix Bloch showed in 1928 that an electron moving in a perfectly periodic potential has wavefunctions of a special form: a plane wave modulated by a function with the same periodicity as the lattice. Each allowed state is labelled by a wavevector k, and plotting energy against k gives the dispersion relation E(k) that defines a band.

Bloch's theorem:  ψ_k(r) = e^{ikr} u_k(r),   u_k(r + a) = u_k(r)

Kronig-Penney (1D square-well lattice of period a) gives the
implicit dispersion relation:

  cos(ka) = cos(αa)·cosh(βb) + (α² − β²)/(2αβ)·sin(αa)·sinh(βb)

Solutions exist only for ranges of energy that satisfy |RHS| <= 1 —
the forbidden ranges in between are the band gaps.

Gaps open at the edges of the Brillouin zone, where the electron's wavelength matches the lattice spacing closely enough that forward- and backward-travelling waves interfere the way X-rays do in Bragg diffraction. The two resulting standing waves pile electron density either on top of the positive ion cores or in between them; the Coulomb energy difference between those two charge distributions is exactly the width of the gap.

Metals, insulators, semiconductors

Whether a solid conducts comes down to how its highest occupied band is filled at absolute zero. A metal has a partially filled band: there are empty states an infinitesimal energy above the topmost filled ones, so any applied field can push electrons into them and current flows freely. An insulator has a completely full valence band separated from the empty conduction band by a large gap — several electron-volts — so essentially no electron has the thermal energy to cross it. A semiconductor is built exactly the same way as an insulator, just with a small gap (around 1.1 eV for silicon, 1.4 eV for gallium arsenide), small enough that a meaningful fraction of electrons are thermally excited across it at room temperature.

Doping and the Fermi level

The occupation of every state is set by the Fermi-Dirac distribution, and the Fermi level Ef is the energy at which the probability of occupation is exactly one half. In pure ("intrinsic") silicon, Ef sits almost exactly in the middle of the gap. Replace one silicon atom in a million with phosphorus — one more valence electron than silicon — and that extra electron sits in a shallow donor level just below the conduction band edge; at room temperature it ionises easily and Ef shifts up toward the conduction band (n-type). Replace a silicon atom with boron — one electron short — and the resulting acceptor level near the valence band edge pulls Ef down (p-type).

Fermi-Dirac occupation:   f(E) = 1 / ( exp((E − Ef)/kT) + 1 )

Intrinsic carrier density:   ni = sqrt(Nc·Nv) · exp(−Eg / 2kT)

Nc, Nv = effective density of states of the conduction/valence band
Eg     = band gap energy

Temperature and carrier concentration

Because ni depends exponentially on −Eg/2kT, carrier concentration is brutally sensitive to temperature. At very low temperature, donors and acceptors freeze out — there isn't enough thermal energy to ionise them, and conductivity collapses. Across the normal operating range the extrinsic carriers from doping dominate and conductivity is roughly flat. Push the temperature high enough and thermally generated electron-hole pairs from across the full gap overwhelm the doping, and the material behaves intrinsically again — which is exactly why the simulation lets you slide temperature and watch the Fermi level and carrier population respond.

Direct vs indirect gaps

Not all gaps are created equal for optics. In a direct-gap material like gallium arsenide, the conduction band minimum and the valence band maximum sit at the same k, so an electron can emit a photon and drop straight across the gap — a photon carries essentially no crystal momentum, so momentum is automatically conserved. In silicon the gap is indirect: the conduction minimum sits at a different k than the valence maximum, so radiative recombination also needs a phonon to make up the momentum difference. That three-body process is far less likely, which is why LEDs and laser diodes are built from direct-gap compounds, while silicon — superb for transistors — is a poor light emitter.

Frequently asked questions

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