Question

For a two-dimensional free electron gas, the electron density \(n\) and Fermi energy \(E_F\) are related as:

• A. \( n = \frac{(2mE_F)^{3/2}}{3\pi^2\hbar^3} \)
• B. \( n = \frac{mE_F}{2\pi \hbar^2} \)
• C. \( n = \frac{(2mE_F)^{3/2}}{\pi \hbar} \)
• D. \( n = \frac{(2mE_F)^{3/2}}{\pi \hbar} \)

Hint:

The electron density in a 2D electron gas is linearly dependent on the Fermi energy.

Answer 1

The Correct Option is B

For a two-dimensional electron gas, the density of states is proportional to \( m / \hbar^2 \), and the relation between the electron density \(n\) and Fermi energy \(E_F\) is given by:
\[ n = \frac{mE_F}{2\pi \hbar^2} \]