How to Determine Fermi Energy in a Quantum Well Without Given Temperature?

[Martin89]

Homework Statement

See below

Relevant Equations

2D density of states $g\left(E \right)=\frac{m^{*}}{\pi\hbar^2}$

Fermi energy in a quantum cascade laser $E_{F}=E_{i}+k_{B}Tln\left[exp\left(\frac{\pi\hbar^2n^{2D}}{k_{B}Tm^{*}} \right)-1\right]$

Problem Statement: See below
Relevant Equations: 2D density of states $g\left(E \right)=\frac{m^{*}}{\pi\hbar^2}$

Fermi energy in a quantum cascade laser $E_{F}=E_{i}+k_{B}Tln\left[exp\left(\frac{\pi\hbar^2n^{2D}}{k_{B}Tm^{*}} \right)-1\right]$

I've been stuck on this problem for a few days now, I really can't see how to proceed. I believe I have successfully completed the first part of the question but can't do the second part. I have found the equation for the Fermi energy in a quantum cascade laser but it depends on temperature and I am not given a temperature to work with. Any help would be really appreciated.

 

[berkeman]

Your attachment is unreadable for me. Can you please make a better scan, or better yet, type your work into the forum window using LaTeX? There is a LaTeX tutorial under INFO, Help at the top of the page. Thank you.