Does the shifting of the Fermi energy level of an intrinsic semiconductor mean that $n \neq p$?

Question

It has been stressed out in the books that I've consulted that, for an intrinsic semiconductor, $n=p$.

However, with this in mind, they also derivate the following equation: $$E_{F_i}=\frac{E_c+E_v}{2}+\frac{3}{4}k_BT\ln\left(\frac{m^*_h}{m^*_e}\right) \quad\quad\quad\quad (1)$$ Which would be the Fermi energy level of an intrinsic semiconductor, depending on temperature. Meaning that for an intrinsic semiconductor, $E_F$ would be a little bit shifted from the center if the masses of the holes and electrons are different (in general they are).

This has implications if we want to calculate $n$ and $p$, which wouldn't be equal, because they have a dependance on this energy level. I guess that this is a contradiction, because you start with the assumption of $n=p$ but if you want to calculate them using (1), you end up with them being $n \neq p$. Why is that? Which one is correct?

Skip the following derivation if you already know the dependance of $n$ and $p$ on $E_F$.

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$$n=2\int^{\infty}_{E_c} \frac{g_c(E)}{1+e^{\frac{E-E_F}{k_BT}}} \ \mathrm{d}E= 2\int^{\infty}_{E_c} \frac{g_c(E)}{1+e^{\frac{E-E_c+E_c-E_F}{k_BT}}} \ \mathrm{d}E$$ Change of variables: $x=\frac{E-E_c}{k_BT}$ and $\xi_n =\frac{E_c-E_F}{k_BT}$; and supposing that for a 2D semiconductor $g_{2D}$ is independent of E: $$n=2g_{2D}k_BT\int^{\infty}_{0} \frac{1}{1+e^{x}e^{\xi_n}} \ \mathrm{d}x$$ Same goes for p, using the same arguments, and with $\xi_p =\frac{E_F-E_v}{k_BT}$: $$p=2\int^{E_v}_{-\infty} \frac{g_c(E)}{1+e^{\frac{E_F-E}{k_BT}}} \ \mathrm{d}E =2g_{2D}k_BT\int^{\infty}_{0} \frac{1}{1+e^{x}e^{\xi_p}} \ \mathrm{d}x$$ So in the end we have $$n=F_0(\xi_n) \quad \mathrm{and} \quad p=F_0(\xi_p), \quad \xi_n \neq \xi_p$$ where $F_j(-\xi)$ is the Complete Fermi–Dirac integral

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Answer 1

Your equation 1 was derived with an approximation for the Fermi-Dirac integral and was derived for 3D. That is, they used $F_{\frac{1}{2}}\left(\eta_c\right) \approx \frac{\sqrt{\pi}}{2} e^{\eta_c}$, which is appropriate for many situations of interest. You're not getting the answer you expect because you are working in 2D (and not making the same approximation). For reference, see Robert Pierret's Semiconductor Device Fundamentals section 2.5.1 and 2.5.6.

FWIW, in 2D, you can do the relevant Fermi-Dirac integral exactly, so I don't think there's any need for approximation. That said, I don't know the 2D equivalent of your first equation off hand. However, it should be simple to derive by following the steps in Pierret. I'm guessing it's somewhere in John Davies' The Physics of Low-dimensional Semiconductors, but I don't have a copy of that handy.