Problem 1

(a) How could you determine the Fermi surface of titanium theoretically?

(b) How could you determine the Fermi surface of titanium experimentally?

(c) How could use a description of the Fermi surface to calculate the electronic contribution to the specific heat of titanium?

(d) How could use a description of the Fermi surface to calculate the electrical conductivity of titanium?

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
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 4
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?