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
NaCl is an ionic crystal and an electrical insulator.

(a) Sketch the dielectric function for this material. How could the Kramers-Kronig relation be used to check the form of the dielectric function?

(b) Polaritons are observed in NaCl. Sketch the polariton dispersion relation for the ionic crystal. Use the same scale for the frequency axis as in part (a).

(c) Sketch the reflectance for the ionic crystal for normally incident light. Use the same scale for the frequency axis as in part (a).

(d) What are polarons? How could you measure the the properties of polarons in NaCl?

Problem 3

The Seebeck Effect relates the gradient of the electrochemical potential to the gradient of the temperature.

(a) What is the electrochemical potential?

(b) What rank tensor is the Seebeck coefficient?

(c) How can you calculate the Seebeck coefficient?

(d) If you wanted to calculate the Seebeck coefficient for a free electron gas, what dispersion relation would you use?

Problem 4
The dielectric constant of materials used in a supercapacitor can be high ($\epsilon_r$~1000). Explain why the dielectric constant is temperature dependent. The index of refraction $n=\mathcal{Re}\left[\sqrt{\epsilon_r} \right]$ is not so large for these materials ($n$~2). Why is this so?