Problem 1
The Quantum Hall effect is measured in samples where electrons are restricted to a two-dimensional plane. Often the electrons are confined to the interface between two materials.

(a) In a quantum Hall experiment, the magnetic field is held constant and a gate electrode is used to change the electron density. While the gate is changed, the conductivity is measured as in the Shubnikov-de Haas oscillations. Sketch the conductivity vs. electron density and explain why it has the form you have drawn?

(b) What other properties of a two dimensional electron gas will depend on the electron density?

Problem 2
(a) Sketch the dielectric function of an insulator.

(b) Draw the corresponding reflectance of light striking the normal to the surface from vacuum.

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

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

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

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

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

Problem 3

(a) Explain why electron-electron interactions are difficult to describe.

(b) A simple model for electron-electron interactions is electron screening. Explain what screening is and how it can be used to explain a Mott transition.

(c) Single electron effects are another simple way to include electron-electron interactions. Explain the single-electron charging effect.

(d) How can the density of excitons be related to a metal-insulator transition?

Problem 4
The pyroelectric coefficient describes how the polarization of a crystal changes as the temperature changes.

(a) Express the pyroelectric coefficient as a derivative of the Gibbs free energy.

(b) Assume Landau's theory of a second order phase transition can be used to describe polarization of the crystal. Sketch the pyroelectric coefficient as a function of temperature.

(c) If you measured the pryoelectric coefficient as a function of temperature and the electric susceptibilty as a function of temperature you would be able to predict the specific heat as a function of temperature. Explain how this is possible.