Physics

Band Theory

One atom has levels. Bring a line of them close enough to notice each other and every level splits into a band. Which bands are full and how wide the gap is decides, on its own, whether you are holding a metal, a semiconductor or an insulator.

A chain of atoms with one s and one p level each, solved exactly by tight binding; everything on screen comes out of that one model

Verdict—

—

Atomic spacing a 2.50 Å

squeezedfar apart

Electrons per atom 2

Temperature 300 K

Applied field 0.30

nonestrong

Time rate 1.0×

Show the individual levels

Numbers

Verdict—

Band gap E_g—

Lower band width—

Fermi level E_F—

Free carriers—

Drift velocity—

Thermal energy kT—

What to observe

Squeeze the chain and each sharp atomic level spreads into a band, until the gap closes and the two bands overlap. In k each band is a cosine across the Brillouin zone, flat at its edges ±π/a just like thephonon dispersion.
With two electrons per atom the lower band is exactly full. Turn up the field and the drift stays at zero: a full band shifted is still full.
With one electron per atom the same band is half full. The sea leans under the field and the drift velocity jumps: a metal.
Give two electrons per atom a gap near 1 eV and heat it. Electrons cross the gap as e^−E_g/2kT and the free carriers climb by orders of magnitude: a semiconductor.

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