1. Discuss briefly the origin of the differing electronic properties of metals, semiconductors and insulators. Under what circumstances will the nearly-free electron model be useful for describing the band structure of a solid?

A weak periodic potential

W(z) = W_0 cos (2*pi*z)/a

with W_0 > 0, is imposed on a one-dimensional free-electron gas lying along the z direction. Discuss the form of the resulting wavefunctions at the first Brillouin zone boundary. Explain why a gap arises.

A simple model of a two-dimensional band structure can be obtained by summing together two one-dimentional band structures. Consider the energy band

E(k_x, k_y) = U cos (k_x * a) - U cos (k_y * a)

with U > 0, corresponding to a particular two-dimensional crystal with square lattice and period a. Give the wavevectors and energies of the highest and lowest energy states in teh band. Draw labelled constant energy contours in the first Brillouin zone near these particular states. Comment on the form of the constant-energy contours near the centre fo teh first Brillouin zone.