Semiconductor doped with acceptors

[Kara386]

Homework Statement

A semiconductor has an acceptor concentration of $6.7 \times 10^{16}$cm$^{-3}$, with no donors. Its intrinsic carrier concentration is $4.2\times 10^{17}$ at 650K.

What's the concentration of free electrons at 290K? Estimate the temperature of the crossover between intrinsic and extrinsic behaviour.

Homework Equations

The Attempt at a Solution

For the first part I've calculated $N_c, N_v$ so now I just need what the concentration of holes is, because
$np = N_c N_v e^{-\frac{E_g}{kT}}$

Where $N_c = N_v = AT^{\frac{3}{2}}$, and I used that equation at $T=650K$ to calculate A, which is the same at all temperatures then. So then I can use it to calculate $N_c,N_v$ at 290K. I'm not sure it's true that they're equal actually but I assumed so because the intrinsic carrier concentration is the same for electrons and holes. Now I think about it, maybe those two aren't equal, but I guess that's essentially the same problem - I don't know what the concentration of holes is! Is it the intrinsic concentration plus the number of acceptors?

As to the second part, no idea about the method for that.

Thanks for any help! :)

 

[anathema]

Hello,

Thank you for your post! It seems like you have a good understanding of the equations and concepts involved in this problem. To answer your first question, the concentration of holes in this semiconductor would indeed be equal to the concentration of acceptors, since there are no donors present. So the total number of holes would be 6.7 x 10^16 cm^-3 at 290K.

For the second part, the crossover between intrinsic and extrinsic behavior occurs when the concentration of free carriers (electrons and holes) is equal to the concentration of dopants (acceptors and donors). So at this temperature, the concentration of free electrons would be equal to 6.7 x 10^16 cm^-3, and the concentration of free holes would be equal to 4.2 x 10^17 cm^-3 (since the intrinsic concentration of both electrons and holes is the same at this temperature). To estimate the temperature at which this crossover occurs, you can use the equation you mentioned, np = NcNve^(-Eg/kT), and solve for T. This would give you an estimate for the temperature of the crossover.

I hope this helps! Let me know if you have any further questions.