0
$\begingroup$

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 $T=300K$:

Distributions

Density as function of potential

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.

$\endgroup$
  • $\begingroup$ Could you please cite the paper? Maybe there are assumptions in the paper that you're missing. $\endgroup$ – Massimo Ortolano Feb 9 '17 at 19:17
  • $\begingroup$ 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 $\endgroup$ – Ronan Tarik Drevon Feb 9 '17 at 19:25
  • $\begingroup$ Your chemical potential is given relative to Ec. What is the reference level for the one in the paper? $\endgroup$ – nasu Feb 9 '17 at 20:26
  • $\begingroup$ 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. $\endgroup$ – Ronan Tarik Drevon Feb 9 '17 at 20:58
0
$\begingroup$

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 : enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.