Describe the development of the metal theory, from the Drude model, to the Sommerfeld model, to the band theory. You don't need to provide very detailed description, but need to show the important assumptions, problems, changes (as compared to the previous model, such as how what changes that Sommerfeld model made as compared to the Drude model, etc). Use the electronic band theory to explain why metal can conduct current but insulator cannot. See the temperature dependence of resistivity below, which one is for a typical metal and which is for a typical insulator? For each of the metals and insulators, what is (are) the most important factor(s) that determines the temperature dependence of resistivity? Resistivity (p) Resistivity (p) Temperature T(K)y Temperature 7(K) Fermi surface for ideal free electron gas is a sphere. However, in real materials, the Fermi surface shows distortion, especially when approaching the Brillouin zone boundary. Explain why. Quantum oscillations due to Landau quantization is an established method to study Fermi surface and electronic state. What important information can quantum oscillation provide? Explain the acoustic branch and optical branch for lattice vibration (like what are the general properties, in what system they can appear, etc). Provide simple description of the physics picture