Energy Bands and Carrier Statistics

Syllabus: Energy bands in intrinsic and extrinsic semiconductors, equilibrium carrier concentration, direct and indirect band-gap semiconductors.

4 hand-picked visual resources on this topic, 2 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Equilibrium carrier concentrations

At equilibrium the product of electron and hole densities is fixed by the material and temperature alone — doping shifts the split, never the product.

$np=n_i^{2}$
mass action law, equilibrium only
$n\approx N_D,\qquad p\approx\frac{n_i^{2}}{N_D}$
$n_i^{2}=N_CN_Ve^{-E_g/kT}$

Where marks are lost

$np=n_i^{2}$ fails under illumination or bias — those are not equilibrium.

See it

Fermi level position

Where $E_F$ sits encodes both the doping type and how heavy it is. Inside a band means degenerate — Boltzmann statistics no longer apply.

$n=n_ie^{(E_F-E_i)/kT}$

Direct vs indirect gap

In a direct-gap material the conduction minimum sits directly above the valence maximum, so recombination emits a photon. Indirect materials need a phonon too, which makes them poor light emitters.

$E_g=\frac{hc}{\lambda}\approx\frac{1.24}{\lambda(\mu m)}\ \text{eV}$

Carrier Transport →

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