Fermi temperature
/ FAIR-mee TEM-per-uh-chur /
The Fermi energy is an energy, but it's handy to ask: how hot would you have to make an ordinary gas for its particles to be that energetic? That equivalent temperature is the Fermi temperature. For a typical metal it comes out staggeringly large — tens of thousands of degrees, hotter than the surface of the Sun.
It's a translation of the Fermi energy from energy units into temperature units, dividing by Boltzmann's constant. It is not the actual temperature of the metal — your copper wire is cool to the touch. It simply tells you the energy scale the electrons already carry, just from being packed in and forbidden to share states.
It matters because comparing room temperature to this huge Fermi temperature reveals why metals behave so strangely: room temperature is only well under one percent of the Fermi temperature, so the electron sea is effectively "ice cold" and almost frozen. That's why only a tiny sliver of electrons can respond to heat, and why their heat capacity is so small.
Copper's Fermi temperature is about 80,000 kelvin. Compared to that, a room at 300 kelvin is almost absolute zero for the electrons — which is why heating the metal stirs up only the thin top layer of the electron sea, not the whole thing.
Copper's Fermi temperature (~80,000 K) dwarfs room temperature, leaving the electron sea nearly frozen.
Don't take the Fermi temperature literally as how hot the metal is — the metal can be sitting in your hand at room temperature. It's a yardstick for the electrons' energy, not a thermometer reading.