Final answer:
The free electron behavior in crystalline solids is altered to form energy bands through the overlapping of orbitals as atoms come closer, resulting in bands and gaps determined by quantum mechanics and the Pauli exclusion principle. Electrical conductivity varies based on the band gap size; conductors have a small or nonexistent gap, insulators have a large gap, and semiconductors are in between. Superconductivity, with zero resistance, emerges at low temperatures due to electron pairing and condensation into the same energy level.
Step-by-step explanation:
The free electron behavior in crystalline solids is modified to form energy bands due to the overlapping of atomic orbitals as atoms in a crystal come closer together. Two main approaches used to determine the energy levels of electrons in a crystal are solving Schrödinger's equation for a periodic potential and studying the electron energy structure as the atoms are grouped. The energy levels form bands and gaps because of the quantum nature of electrons and the Pauli exclusion principle.
Materials differ in their electrical conductivity due to the size of the band gap between the valence and conduction bands. In conductors, the band gap is small or non-existent, allowing electrons to move freely and conduct electricity. Insulators have a large band gap that prevents electron movement, making them poor conductors. Semiconductors have a band gap that is intermediate, which can be modified by doping to alter their conductive properties.
Superconductors exhibit a completely different behavior, where the material conducts electricity with no resistance below a critical temperature. This phenomenon is explained by the formation of Cooper pairs, in which electrons pair up and condense into the same energy level, as described by the BCS theory.