Discussion
Density of states in conduction band for electrons assumed to be essentially free in two dimensions is proportional to :
Answer: B
In two dimensions the density of states is independent of energy.
The number of states with energy below E is proportional to the area of the circle of radius \( k \) in k-space, \( \pi k^{2} \), and \( E = \hbar^{2}k^{2}/2m \) makes \( k^{2} \propto E \). So N(E) grows linearly with E and its derivative is constant:
\[ g(E) = \frac{mA}{\pi\hbar^{2}} \]
for area A including spin. \( E^{1/2} \) is the three-dimensional result and \( E^{-1/2} \) the one-dimensional one.
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