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Valence and conduction band structure. Larger conduction band gapping, the weaker of a conductor it will be.

Metals have a high level of overlap in their conduction band structure (bandgaps, we'll call them) This makes them a natural candidate for conductivity.

I could get into Fermi levels and photon interaction, but I'm not entirely sure this is necessary, as long as you know that conductivity is at an atomic / molecular level, and basically hard-coded into the structure of a substance. Keep in mind that nano science can greatly alter the conductivity of some substances (see carbon vs. carbon nanotubes for a very prime example)

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What makes one material better conductor than another?

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