Electrical conductivity is electron motion, and if you have electrons to carry charge, you also get thermal conductivity. So metals (in the general sense) are g

Electrical conductivity is electron motion, and if you have electrons to carry charge, you also get thermal conductivity. So metals (in the general sense) are good electrical and thermal conductors. If you have very hard materials, with high phonon conduction, you can get thermal transport via phonons. But, these are almost all poor electrical conductors, as you need very tight binding to get hard materials with high phonon velocity. The prototypical example of the second case is diamond.