Hello,
I wanted to know if this way to calculate the DOS for the Free-electron model and nearly-free electron model is correct:
The energy for the free electron model can be expressed as:
\( E_{\vec{k}} = \frac{\hbar^{2}||\vec{k}||^{2}}{2m}
\)
And for the nearly-free electron model, we express:
\( E_{\vec{k}} = \Delta + \frac{\hbar^{2}}{2m_{eff}}||\vec{k}||^{2} \)
So, if we want to have the number of quantum states below a given energy \( E_{k} \):
\( \Phi(E_{k}) \times (\frac{2\pi}{L})^{D} = 2 \times V_{D} \)
where D is the dimension, and \( V_{D} \) is the distance in 1D \( ( V_{D} = k ) \), the area of a circle in 2D \( ( V_{D} = \pi k^{2} ) \) and the volume of a sphere in 3D \( ( V_{D} = \frac{4}{3} \pi k^{3} ) \).
From here, we express \( \Phi(E_{k}) \) in function of the energy, and we make the derivation to obtain the DOS.
Thank you in advance for your answer