# 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 *ħ*ω *>* E_gap. The density of directly associated states is abbreviated as DDAS. The quantity DDAS(*ħ*ω)**ħd*ω is the number of directly associated states in a unit volume of the crystal with energy difference in the range ħω to *ħ*ω + *ħd*ω.

**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 < *ħ*ω

**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 (*ħ*ω - E_g)? Sketch the probability of absorption vs. *ħ*ω 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.