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How do free electrons give metals electrical and thermal conductivity?

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electron hole: movement — materials science Movement of an electron hole in a crystal lattice.

In materials science, the high electrical and thermal conductivity of metals is primarily attributed to the presence of non-localized valence electrons, often referred to as "free electrons".123

Electrical Conductivity

In a metallic solid, atoms are closely packed, causing their outermost electron shells to overlap. Consequently, valence electrons are not bound to any specific atom but are free to wander throughout the entire crystal lattice. When an electrical field is applied, these mobile electrons accelerate, creating an electric current. The efficiency of this process depends on both the number of available free electrons and their mobility—the ease with which they move through the solid without scattering off defects or vibrating ions.124

Thermal Conductivity

These same free electrons are responsible for the efficient transport of thermal energy. Because they are not localized, they can move rapidly through the crystal structure, effectively carrying kinetic energy from warmer regions to cooler regions of the metal.35

While the free-electron model provides a foundational understanding for simple metals, the band theory is used to explain more complex materials, such as transition metals, where electron behavior involves interactions with d-states.435