The carriers of current, revisited
We can now tell the conduction story properly. Connect a battery to a metal, and the electric field gives every electron a gentle push. As we saw with Drude, this turns the wild random motion of the electron gas into a slow collective drift, and that drift is the current. The electrical conductivity measures how much current you get for a given push — how easily charge flows.
The quantum picture adds a refinement to Drude. It is not the whole electron sea that drifts usefully — it is the thin crowd at the Fermi surface, the only electrons free to shift into new states. Conductivity is high when those surface electrons can coast a long way between collisions. The typical coasting distance is the mean free path, and the typical coasting time is the relaxation time. Fewer obstacles, longer coast, better conductor.
What gets in an electron's way
Here is a beautiful surprise. A perfectly regular crystal — atoms in a flawless, repeating grid — does not scatter electrons at all. In the quantum world, an electron can glide through a perfect lattice as if it were empty space. So what causes resistance? Two things only: the atoms wobbling away from their proper spots because of heat, and defects — stray impurity atoms, missing atoms, cracks in the order.
This explains two everyday facts at a stroke. First, warm a metal and its resistance rises: more heat means more atomic wobble, more scattering, a shorter relaxation time. Second, a dirty or strained metal conducts worse than a pure, well-ordered one — the defects act as permanent obstacles that no amount of cooling can remove. Cool a very pure metal toward absolute zero and the wobble nearly vanishes, leaving only the defects to limit the current.
The same electrons carry the heat
Now notice something lovely. The mobile electrons do not only carry charge — they carry energy too. Heat one end of a metal bar and the electrons there speed up; as they rush about and collide, they pass that extra energy along, ferrying warmth toward the cold end. This is electronic thermal conductivity, and it is why metals feel cold to the touch: they whisk heat away from your skin far faster than wood or plastic can.
It helps to picture the electrons as a courier service that happens to deliver two kinds of parcel at once. Each electron rushing past carries a packet of charge and, if it has been warmed, a packet of energy too. The busier and more unobstructed the couriers, the more of both they deliver. So a metal that is good at moving charge is, almost by necessity, also good at moving heat.
A law that ties heat to electricity
Here comes the elegant payoff. The very same electrons carry both charge and heat, and they are slowed by the very same collisions. So their two conducting abilities ought to rise and fall together. They do — and astonishingly, the relaxation time that limits both simply cancels out of the ratio. The result is the Wiedemann-Franz law: across nearly every metal, the ratio of heat conductivity to electrical conductivity is the same fixed number times the temperature.
(heat conductivity) ÷ (electrical conductivity) = (a universal constant) × temperature the scattering details cancel — only fundamental constants remain
Pause to feel how remarkable this is. Take gold, copper, silver, tin — wildly different metals — and this ratio comes out almost identical for all of them. It is one of the clearest signs that a single, simple idea, the free electron, really is doing the heavy lifting inside every metal. A crude cartoon, refined by one quantum rule, predicts a law that nature obeys with quiet precision.
Coaxing electrons out, and where to go next
The free electrons roam happily inside the metal, but they are not free to leave it — there is a wall at the surface. To pull a single electron out into empty space costs a definite amount of energy, called the work function. Shine bright enough light on a metal, or heat it fiercely, and you can give electrons enough energy to leap over that wall. This is the engine behind solar light-meters, old vacuum tubes, and the electron beam in a classic television.
So the free-electron story carries us a long way: conduction, heat flow, the Wiedemann-Franz law, even how electrons escape a surface. Yet it still cannot tell us the deepest question of all — why some materials are metals at all, while others are insulators or semiconductors. For that, we must finally stop ignoring the regular grid of atoms and let it shape the electrons' allowed energies. That is the doorway to energy bands, and the next track waiting just ahead.