Physics

Band Theory

One atom has levels. Bring a line of them close enough to notice each other and every level splits into as many as there are atoms, packed 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 a rock.

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

Verdict—

Band gap—

Lower band width—

Fermi level—

Free carriers—

Drift velocity—

Thermal energy kT—

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.

What to observe

Start with the atoms far apart. Each band collapses to a single sharp level, the level an isolated atom has. Now bring them together. Every level splits into as many levels as there are atoms, because no two electrons may share a state, and the split widens the closer they get. That fan is where a band comes from.
Watch the two fans as you squeeze. First they widen, then the gap between them narrows, and at close enough spacing they overlap and the gap is gone. Nothing about the atoms changed. Being an insulator or a metal can be a matter of how far apart you hold the same atoms.
Set two electrons per atom and look at the occupied states: the lower band is exactly full. Now turn the field up. The whole sea of occupied states tries to shift, and nothing happens, because a full band shifted is still a full band. For every electron moving one way there is one moving the other. That is what an insulator is, and it is a counting argument, not a strength argument.
Change to one electron per atom. The same band is now half full, there is a Fermi surface, and the sea can lean sideways. Watch the drift velocity jump and the electrons in the crystal start moving. Odd electrons per atom and you have a metal, almost for free.
Go back to two per atom and set a spacing where the gap is about one electron volt. Now raise the temperature. A few electrons make it across the gap, leaving holes behind, and the conduction climbs steeply because the number of carriers goes as e^−E_g/2kT. This is a semiconductor, and this is why heating one makes it conduct better while heating a metal makes it worse.
Compare kT against the gap in the readout. At room temperature kT is about 0.026 eV, so a 5 eV gap is untouchable and a 1 eV gap is merely very unlikely. The whole difference between a diamond and a chip is that ratio.
Look at the density of states. It piles up at the band edges and thins in the middle, and the Fermi-Dirac curve beside it is a soft step whose softness is a few kT wide. The states are what the crystal offers, the curve is what temperature takes up, and everything electrical is the product of the two.

Shortcuts: space run/pause · s step · r reset · f fullscreen