**Problem 1**
Consider a one dimensional lattice structure consisting of two different atoms A and B that alternate at equally spaced intervals of width a, as sketched below.

Assume that each atom contributes one valence electron. For atoms centered at the origin, the atomic orbital of an A atom valence electron is φ_{A}(**r**) and the atomic orbital of a B atom valence electron is φ_{B}(**r**). The atomic energy levels of these orbitals are E_{A} and E_{B} respectively. You may assume that φ_{A} and φ_{B} are real and spherically symmetric.

Using the tight binding method, compute the band structure energies and eigenstates for this lattice. Assume that only nearest neighbor overlap integrals are significant.

Plot the band energies vs k in the first Brillouin Zone.

**Problem 2**
Using the tight binding band structure for the π and π* bands of graphene as computed in lecture, show:

a) For **k** near **k**=0, the surfaces of constant energy are circles centered about the orign.

b) For **k** near **k**_{K} at a vertex of the surface of the first Brillouin Zone, show that the band energies are ε_{±}(**k**) ~ |**k**−**k**_{K}|.

c) For **k** near **k**_{K} at a vertex of the surface of the first Brillouin Zone, show that the surfaces of constant energy are circles centered about **k**_{K}.

d) Find an energy ε for which the surface of constant energy ε is strongly nested. Draw this surface in the first Brillouin Zone.

e) Show that the density of states g(ε) vanishes at the Fermi energy ε_{F}.