Fermi velocity
/ FAIR-mee veh-LOSS-it-ee /
If electrons fill up energy levels like water filling a tank, the ones right at the surface are moving the fastest. The Fermi velocity is the speed of those topmost electrons — the ones sitting at the Fermi energy. Far from being lazy, they are racing along at hundreds of kilometres per second even when the metal is bitterly cold.
Why so fast at zero temperature? Because the slow states are all taken. With no two electrons allowed in the same state, the surface ones are forced into high-energy, high-speed slots — they have no choice. The Fermi velocity follows directly from the Fermi energy, and for most metals it sits at around half a percent of the speed of light.
It matters because this is the speed that counts for conduction, not the crawling drift of current. When you switch on a voltage, electrons drift forward at a snail's pace, but each one is really hurtling at the Fermi velocity, merely biased ever so slightly forward. A common confusion is mixing up this blistering Fermi velocity with the sluggish drift velocity — they differ by a factor of millions.
In copper the Fermi velocity is about 1,600 kilometres per second. Yet when current flows in a household wire, the electrons drift forward at less than a millimetre per second — slower than a strolling ant. The electricity arrives fast not because electrons fly fast, but because the push travels almost instantly.
Copper electrons race at ~1,600 km/s yet drift forward slower than a walking ant.
The Fermi velocity is the speed of the fastest, surface electrons — not an average over all of them. Most electrons buried below move slower, but they're frozen out of the action, so for conduction the Fermi velocity is the relevant one.