electronic heat conduction
Why are the best heat conductors — copper, silver, aluminum — also the best electrical conductors? Because in a metal the same cloud of free electrons that carries electric current also carries heat. These loose electrons roam the whole crystal, and a fast-moving electron from the hot end delivers its extra energy wherever it collides down at the cold end.
A metal's valence electrons are not tied to individual atoms; they form a sea that drifts freely through the lattice. Heat one end and those electrons pick up kinetic energy and shuttle it rapidly to the cold end, far faster than sluggish lattice vibrations can. Because there are so many mobile electrons, electronic conduction dwarfs the phonon contribution in most metals — copper's k of about 400 W/m-K is overwhelmingly electronic. The very same free electrons carrying charge are why good electrical conductors are also good thermal conductors.
This shared-carrier picture is captured quantitatively by the Wiedemann-Franz law. It also explains why alloying, which scatters electrons and cuts electrical conductivity, cuts thermal conductivity too: stainless steel, a heavily alloyed metal, conducts heat about 8 times worse than pure copper. Honest caveat: in ceramics and polymers there are almost no free electrons, so this mechanism is absent and phonons do all the work of carrying heat.
Pure copper conducts heat at about 400 W/m-K, but adding nickel and chromium to make stainless steel drops it to about 15 — the alloying atoms scatter the very electrons that carried both the current and the heat.
Whatever scatters conduction electrons cuts a metal's thermal conductivity along with its electrical conductivity.
Electronic conduction is why metals are the top heat conductors, but it is absent in insulators — do not assume a good insulator against electricity is a good insulator against heat, since phonons can still carry heat well (diamond).