Problem 1
(a) Draw the Brillouin zone of a two-dimensional metal where the charge carriers at the Fermi surface are holes.

(b) What experiments could you do to show that the charge carriers are holes?

(c) Sketch the electron density of states that corresponds to the Fermi surface you drew in (a).

(d) How could you measure the density of states experimentally?

Problem 2
(a) What can Raman spectroscopy tell you about the phonon dispersion relation of some material?

(b) What are surface plasmons? How can yo measure them?

(c) What are polarons? What kind of materials have polarons? How do you measure them?

Problem 3
(a) How can you calculate the dielectric function of an insulator from the electronic band structure?

(b) How can you measure the dielectric function of an insulator at zero frequency?

(c) How can you measure the dielectric function at optical frequencies?

(d) How does the band gap of the insulator manifest itself in the dielectric function?

(e) What are the limiting values for the real and imaginary parts of the dielectric function at high frequency? Why?

Problem 4
The electrical conductivity of a material decreases rapidly at a certain temperature. This could either be a Mott transition or a structural phase transition.

(a) Explain what happens at a Mott transition to cause a sudden decrease of electrical conductivity.

(b) Explain what happens at a structural phase transition to cause a sudden decrease of electrical conductivity.

(c) What experiments would you use to determine the type of phase transition? Describe also the experiments you would have to do to determine if it is a first order transition or a second order transition.