# Electron density in Semi Conductors

I want to determine the relationship between the carrier density $N$ and the chemical potential $\mu_e$ in the conduction band of an intrinsic semi-conductor (GaN for example).

The band structure of the conduction band is assumed parabolic with effective mass $m_e^*=0.2m_e$. Using a $3D$ free electron carrier model the relationship is then given by : \begin{equation} N = \int_{E_c}^{\infty}g(E)f(E)dE ~~~~\mbox{with}~~~~ f(E) = \frac{1}{1+e^{\frac{E-\mu_e}{k_BT}}} ~~~~\mbox{And}~~~~ g(E)=\frac{m_e^*\sqrt 2}{\pi^2 \hbar^3}\sqrt E \end{equation}

Since the semi conductor is meant to work at very high carrier concentrations ($N\approx 10^{19} cm^{-3}$), we won't take advantage of the approximation $E_c-\mu_e\gg k_BT$. Using Matlab to compute these integrals, I obtain the following curves for the distributions $f(E)g(E)$ and $N(\mu_e)$ at $T=300K$:  We can clearly see that $N=2.10^{19}cm^{-3}$ is achieved with $\mu_e-E_c=11.87meV$ confirming that the approximation $E_c-\mu_e \gg k_B T$ can not be used. However, in a research paper, I have found that using the exact same values they end up with $\mu_e=130meV$. Paper is :

Modal Gain in a Semiconductor Nanowire LaserWith Anisotropic Bandstructure A. V. Maslov, Member, IEEE, and C. Z. Ning, Senior Member, IEEE

I really do not see where the discrepancy could come from after spending an afternoon on it.

• Could you please cite the paper? Maybe there are assumptions in the paper that you're missing. – Massimo Ortolano Feb 9 '17 at 19:17
• I jut edited the post including the paper. The model is rather complex for the valence band, but the conduction but should have been straightforward. Details in page 5 of the paper – Ronan Tarik Drevon Feb 9 '17 at 19:25
• Your chemical potential is given relative to Ec. What is the reference level for the one in the paper? – nasu Feb 9 '17 at 20:26
• That is a good question. I have not been able to tell exactly. I have the feeling looking at their band structure graph that the origine is taken at $E_c$ for the conduction band. Looking at the paper they get their dispersion relation from, the origin of energy is taken such that $Ec=3.53eV$. However, we can clearly see from my plos that the chemical potential should be above $Ec$ to get $N=2.10^{19}/cm^3$ so $\mu_e=130meV$ cannot be relative to this origin. – Ronan Tarik Drevon Feb 9 '17 at 20:58

The error was due to the density of states function that was starting that should have been $\approx \sqrt{E-Ec}$ instead of $\sqrt{E}$. Indeed, at the bottom of the conduction band $E_c$, the density should be equal to $0$ hence the graphs for $f(E)g(E)$ should more look like the standard bell-like shape : 