Problem 1
The first Brillouin zone of a two-dimensional metal is a hexagon. This metal has a valence of 2. Draw the Fermi surface in the nearly free electron approximation and indicate the regions in k-space where electrons are found and where holes are found.

Solution

Problem 2
In Kittel's book it says, "It came as a surprise that optical spectroscopy developed as an important experimental tool for the determination of band structure."

(a) What can optical measurements tell you about the electronic bandstructure of a material?

(b) How can causality be used to check optical measurements?

Causality: a bell rings after you strike it, not before you strike it.

Problem 3
Because of the mathematical difficulties of solving the Schrödinger equation including the electron-electron interactions, we often consider systems of non-interacting electrons. Landau's theory of a Fermi liquid shows that we should really consider non-interacting quasipartcles instead of non-interacting electrons. Describe these non-interacting quasiparticles. What properties do they have?

Problem 4

(a) Explain the Peierls transition.

(b) In some materials the Peierls transition is a second order phase transition. In this case, how would the gap grow as a function of temperature?

(c) Would there be a Peierls transition in a one-dimensional semiconductor? Explain your reasoning.