Intrinsic carrier concentration where did I go wrong?

[Kara386]

Homework Statement

I've looked up the intrinsic carrier concentration of silicon, and what I've got isn't close. The question says given there are $2\times 10^{22}$ electrons per cubic cm in silicon, and the bandgap is $1.1$eV, what is the free electron concentration at room temperature?

Homework Equations

The Attempt at a Solution

My first thought is that since
$n_i = \sqrt{N_cN_v} \exp\left(\frac{-E_g}{2kT}\right)$
Maybe as T tends to infinity all electrons are free electrons so that $\sqrt{N_cN_v}=n_i$, but then subbing that back in at room temp would mean $10^{12}$ free electrons per cubic cm, so too high, but then that would be assuming the densities of states are constant at different temperatures, and they're not.

For silicon I can get around this by just looking up $N_c$ and $N_v$, but I have to repeat the process for diamond and the values aren't available. I do get the right order of magnitude estimate for diamond's carrier concentration just by using the same $N_c$ and $N_v$ as for silicon, presumably that's because they're similar in terms of crystal structure.

However, my professor assures me that all I should need is the electron concentration and the bandgap. I'd appreciate any help, been stuck on this for a while! :)

 

[DrDu]

Maybe as T tends to infinity all electrons are free electrons so that $\sqrt{N_cN_v}=n_i$, but then subbing that back in at room temp would mean $10^{12}$ free electrons per cubic cm, so too high, but then that would be assuming the densities of states are constant at different temperatures, and they're not.

How is the situation at T=0?