Intrinsic carrier concentration and bandgap

Question

My understanding is that the intrinsic carrier concentration of a wide bandgap material tends to be lower than that of a narrow bandgap material.

$$ n_i = \left(N_cN_v\right)^{1/2}e^{\left(\frac{-E_g}{2kT}\right)} $$

and the open circuit voltage of a solar cell is

$$ V_{OC} = \frac{kT}{e}ln\left(\frac{J_{SC}}{J_o}\right) $$

where

$$ J_o = e\left(\frac{D}{L}\right)\left(\frac{n_i^2}{N}\right) $$ multiplied by recombination loss factors.

This makes sense to me since greater bandgap yields greater intrinsic carrier concentration, and greater intrinsic carrier concentration yields greater $ J_o $ and greater open circuit voltage.

However, I recently learned that a wide bandgap material, AlGaInP (Aluminium gallium indium phosphide), has a pretty high intrinsic carrier concentration due to its aluminum content.

How can I understand this situation where you have a wide bandgap AND high intrinsic carrier concentration??

Answer 1

The other terms in the equation for $n_i$ also matter. $N_c$ and $N_v$ depend on the effective mass. In particular, they are proportional to the effective mass to the three halves power. All things being equal, a higher band gap means a higher effective mass:**

This is something you can derive from k.p theory with a little hand waving. See for example this and this. (Sorry that I don't have a better reference handy.)

So, the equation for $n_i$ has two forces that act in different directions as $E_g$ increases: the exponential gets smaller, but $N_c N_v$ gets larger. Normally, the exponential wins. But evidently AlGaInP bucks the trend; the effective mass must be high enough that it overpowers the exponential.

Offhand, I don't know why the aluminum would have this effect, but maybe that's obvious to someone with more semiconductor knowledge than myself. Still, the takeaway is that the band gap acts in multiple ways, so it's not always the case that a larger bandgap means a smaller $n_i$.

** Of course, there are exceptions to this, but it is useful as a general rule.