1. Quantum particle-in-the-box model by Sommerfeld.
The very first description of electrons in a solid was suggested by Sommerfeld who treated the electrons in a metal as quantum particles - fermions - put in a box
..work in progress..
2. Nearly free electron gas. Energy gap opening.
3. Peierls transition.
4. Superconductivity.
5. Spin-orbit interactions. Rashba splitting.