(b) At what temperature would the Boltzmann approximation become invalid for an intrinsic semiconductor with a bandgap of $E_g$?

(c) Explain the relationship between the effective masses of the electrons and holes and the shape of the electronic density of states near the conduction band edge $E_c$ and the valence band edge $E_v$.

(d) Why do the electron and hole mobilities depend on strain?

(a) Do electron like-states and hole-like states appear in the empty lattice approximation? Why or why not?

(a) How do you calculate the number of electron-like states and hole-like states for some metal?

(c) If a metal had both electron-like states and hole-like states, what consequences would this have for the electrical conductivity and the Seebeck effect?

(d) Cyclotron resonance is sometimes used to measure the effective masses of electrons and holes. Explain how this works.

Problem 3
In nonlinear optics, the Faraday effect describes how the electric susceptibility changes as the DC magnetic field is changed.

(a) How could you calculate the Faraday effect from the band structure of a material? Would we need to consider the Landau Levels?

(b) What rank tensor would be used to describe the Faraday effect?

(c) The tensor for the Faraday effect must obey the symmetries of the point group for a crystal but it must also obey an intrisic symmetry that is not associated with the crystal. What is this intrinsic symmetry?

Problem 4

(a) Coulomb interactions are responsible for ferromagnetism and antiferromagnetism. Explain why this is so.

(b) Magnons are quasiparticles that describe excitations from the ferromagnetic ground state. Explain how you could calculate the magnon dispersion relation.

(c) How could you measure the magnon dispersion relation?

(d) Sketch the pyromagnetic coefficient $\frac{d\vec{M}}{dT}$ as a function of temperature.