Problem 1
Many equilibrium properties of a free electron gas can be determined from the electron density.
(a) To calculate the optical properties and transport properties of a free electron gas, a second parameter besides the electron density is needed. What is this second parameter and why is it needed?
(b) Sketch dielectric function of a metal. Indicate the relevant frequency range for microwave engineering.
(c) In an EELS experiment of a metal, how would you distinguish between phonons, magnons, and plasmons?
(d) Why does the conductivity of a metal oscillate when the magnetic field is changed?
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 band gap of GaAs is 1.42 ev and is direct. The band gap of AlAs is 2.16 eV and is indirect. The semiconducting alloy AlxGa1-xAs has a direct band gap for x < 0.44 and an indirect band gap for x > 0.44. Consider alternating layers of GaAs and Al0.6Ga0.4As stacked on top of each other in a periodic pattern.
(a) Draw the band diagram (conduction band, valence band, chemical potential) along a line perpendicular to the layers.
(b) Draw the carrier concentrations (electrons and holes) along a line perpendicular to the layers.
(c) If no voltage is applied, the total current will be zero. Nevertheless, a diffusion current can be deduced from the minority carrier concentrations. This implies that electric fields exist in this structure. Describe the diffusion currents and the electric fields.
(d) How could we calculate the electrical conductivity perpendicular to the layers? Keep in mind that the electrons in the conduction band are concentrated around different $k$ values in the two materials.
Problem 4
Metal - insulator transitions can have several causes.
(a) Explain how a structural phase transition can cause a metal-insulator transition.
(b) Explain a Mott transition.
(c) Explain a Peierls transition.
(d) What experiments could you use to determine which type of metal-insulator transition occurs in some material?