Solid State Physics, Physics 551Homework Assignment 2, due February 25, 2013

Electronic Band Structure

Problem 1. Problem 9-1.

Problem 2. Problem 10-1.

Problem 3. Problem 10-2.

Problem 4. Consider a square lattice in two dimensions with a crystal potentialU(x, y) = U0 cos(2πx/a) cos(2πy/a). Apply the approximation of a weak periodic potentialto evaluate the energy gap at the corner point (π/a, π/a) of the first Brillouin zone.

Problem 5. Consider a ring of a large number N � 1 of atoms in 1D chain withinteratomic spacing a. In isolation each atom has a single bound atomic orbital with energyE0 and wavefunction ψ0(r) with r = 0 coincides with the center of an atom. (a) Find thevalues of wavenumber k that satisfy the periodic condition on the ring ψ(r) = ψ(r +Na).Show that the number of states ∆Γ in an interval [k, k + ∆k] of this 1D electron system isgiven by ∆Γ = (2L/2π)∆k. Define the first Brillouin zone for eectrons in the limit N →∞.(b) Apply the tight-binding model, Eq. (10.15), for electrons on this ring and obtain thedispersion relation E(k). (c) Apply the result of part (b) to determine the correspondingdensity of states in energy,

g(ε) =dΓ

dε=L

π

dk

dε

and plot g(ε) qualitatively as a function of ε. Comment on any singularities.

updated February 12, 2013