What is the Threshold for Interband Optical Transitions in a 2D Square Lattice?

[CAF123]

Homework Statement

A two dimensional solid has two electrons per unit cell and has a Bravais lattice with primitive vectors $\vec a = \ell \hat x$ and $\vec b = \ell \hat y$. The crystal potential is weak and the solid behaves like a free electron metal.

a)In a reciprocal space diagram, sketch the first Brillouin zone and Fermi contour.
b) Draw an energy band diagram for this solid
c) Estimate the threshold for interband optical transitions in terms of $\epsilon_f$, the Fermi energy.

Homework Equations

2 electrons per unit cell imply first B.Z fully occupied.

3. The Attempt at a Solution
a) and b) are done I think, I am just wondering really how to begin with c). By interband transition, I think it means we are considering a transition or promotion of an electron in one band to another. It does so by some external means giving it enough energy to overcome the $\epsilon_f$ energy gap. However such a means is not specified.

The diagram in b, given that the crystal potential is weak, shows the fermi energy cutting two bands, so each band is partially filled and conduction can take place. I am just now sure how to begin. If the crystal potential was stronger, the band gap opens and the solid tends toward an insulator.

Many thanks for any help!

 

[M Quack]

Hmm, can you show us a picture of your answer to b)?

As to the means, it says "optical". Implying energies in the eV range and negligible momentum transfer.