Homework #2

Due 5pm Friday of second week of class.

Band-to-band absorption (15 pts)

Based on Pankove p36

The directly associated states in the valance and conduction band have the same k-vector and are separated by an energy difference ħω > E_gap. The density of directly associated states is abbreviated as DDAS. The quantity DDAS(ħω)*ħdω is the number of directly associated states in a unit volume of the crystal with energy difference in the range ħω to ħω + ħdω.

a) Within the parabolic dispersion approximation, show that the density of directly associated states, DDAS, is

,

where the reduced mass is given by .

and η is the number of paired states with transition energy < ħω

b) In a semiconducting carbon nanotube, electron wavenumber k can only vary in one dimension. The parabolic dispersion approximation is still valid. In this case, how does DDAS scale with (ħω - E_g)? Sketch the probability of absorption vs. ħω for photons passing through a semiconducting CNT of bandgap E_g.

Note: Please neglect excitonic effects when answering part b. If you don't know what I mean by excitonic effects, you'll be fine.

Transition matrix element (10 pts)

Calculate the transition matrix element for 1s-2pz transition in atomic hydrogen. Assume the incident light is polarized in the z direction. Give you answer in the form:

(numerical constant)(a_Bohr)(e)(E_0)

i.e. I want to know the numerical constant.