Problem 1
GaAs is a direct band gap semiconductor with a band gap of 1.4 eV. It forms a zincblende crystal structure.

(a) Describe how you could calculate the specific heat of GaAs.

(b) Draw the dielectric function for GaAs.

(c) Would you expect there to be polaritons in GaAs? Explain why or why not.

(d) How would you use GaAs to observe the Quantum Hall Effect?

(e) Is GaAs piezoelectric? Explain why or why not.

Problem 2
The discussions of the quantum Hall effect and superconductivity both started with the same Hamiltonian,

(a) The concept of field momentum was introduced to describe a charged particle in a magnetic field. What is field momentum? Does field momentum exist in superconductors?

(b) What is the essential difference between the quantum Hall effect and superconductivity? (Hint: it is related to the entropy in the superconducting state).

(c) How would you detect a superconducting transition with a magnetometer?

(d) What happens to the thermal conductivity of a metal when it becomes superconducting?

(e) Why does the resistance go to zero in the quantum Hall effect?

Problem 3
The dielectric constant of materials used in a supercapacitor can be high ($\epsilon_r$~1000). Explain why the dielectric contant 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?

Problem 4
To plot the frequency response of the magnetic susceptibility of a paramagnet, consider how a paramagnet would respond if an magnetic field were pulsed on and off. Plot the real and imaginary parts of the magnetic susceptibility of a paramagnet. Explain why the susceptibility has the form that you have drawn.

Make a similar plot of the frequency dependence of the magnetic susceptibility of a diamagnet.