Problem 1 (a) Describe the optical properties of a low electron density metal. (b) Describe the thermal and electrical properties of a low electron density metal. (c) Describe the magnetic properties of a low electron density metal. Consider the regime $k_BT < < \hbar\omega_c$. (d) There is an electric field at the surface of a metal. Explain where this electric field comes from. Problem 2 (a) How does the resistance of this semiconductor change near room temperature? (b) Explain what the Boltzmann approximation is and if it can be used for this semiconductor at room temperature. (c) How would you calculate the specific heat of this semiconductor? (d) Consider the response of this semiconductor if a $\delta$-function electric field pulse is applied at room temperature. What would the dielectric function look like at room temperature? Problem 3 (b) The piezoresistive effect describes how the electrical resistivity changes when strain is applied to a crystal. Describe how you could calculate the piezoresistive effect of silicon starting from the Schrödinger equation. (c) How do frequency doubling crystals work in nonlinear optics? Problem 4 (a) What does the entropy of a superconductor look like as a function of temperture? Explain why you have drawn the entropy the way you have. (b) How is the entropy of a superconductor related to it's thermal conductivity? (c) Why do the phonons in superconductors have long quasiparticle lifetimes?
A semiconductor has a bandgap of 25 meV.
(a) If the point group of a crystal contains inversion, all odd rank tensor quantities vanish. Provide an example of a rank 1, a rank 3, and a rank 5 tensor.
Superconductivity is a second order phase transition.