Problem 1
Consider an ionic crystal.

(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) Sketch the polariton dispersion relation for the ionic crystal. Use the same scale for the frequency axis and in part (a).

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

(d) What are polaron's? How could you measure the polarons in this material?

(e) What electronic or structural phase transitions might be observed in an ionic crystal?

Problem 2
The piezoresistive effect describes how the electrical resistivity changes when strain is applied to a crystal. Silicon has a relatively large piezoresistive effect.

(a) Applying pressure to silicon changes the bandgap. Describe how you could calculate the piezoresistive effect of silicon starting from the Schrödinger equation.

(b) What rank tensor is the piezoresistive tensor?

(c) Why would it be useful to know silicon's point group when calculating the elements of the piezoresistive tensor?

Problem 3
The quantum Hall effect can be observed in a MOSFET.

(a) What is the relationship between the cyclotron frequency and the magnetic field? (The frequency can be calculated with classical physics).

(b) The quantum Hall effect will only be observed if the energy splitting between the Landau levels is larger than thermal fluctuations. Derive a relation of the form $\frac{B}{T} > \cdots$ that tell us under which conditions the quantum Hall effect can be observed.

(c) The gate of the MOSFET can be used to change the two-dimensional electron density. What would happen to the Landau levels if the electron density were reduced?

Problem 4
Consider electrons moving in a one-dimensional potential. A band of the form $E= -\cos(ka)$ eV is half filled. Here $a$ is the periodicity of the potential.

(a) The expression for the current involves the group velocity, the density of states, and a function $f$ that describes the probability that the states are occupied. What is the expressionfor the current?

(b) What are the group velocity and the density of states in this case? How could $f$ be calculated?

(c) Under what conditions would this system go through a Peierls transition?

(d) Under what conditions would this system go through a Mott transition?