N-n or p-p junctions?

1. Mar 23, 2014

Petr Mugver

I have some difficulty applying the depletion region approximation to calculate all the quantities of interest (carrier density, charge density, eletric field, potential) in a n-n or p-p junction.

Here is what I've got. In a 1-dimensional problem, let $N(x)=N_{d}(x)-N_a(x)$ be the difference of the donor and acceptor dopants densities, $n(x)$ the unbound electron density and $p(x)$ the hole density. Let's consider a junction of the type $N(x)=N_1$ for $x<0$ and $N(x)=N_2$ for $x>0$. Far away from the junction, I have $n(x)=N(x)$ and $p(x)\approx0$ if N is positive, or $p(x)=N(x)$ and $n(x)\approx0$ if N is negative.
In the depletion approximation, I suppose that both n(x) and p(x) drop abrubtly to zero in a certain region $(-x_1,x_2)$ around the origin. From this, knowing that the charge density is $\rho(x)/e=N(x)+p(x)-n(x)$, I can calculate the electric field, the potential etc. I can also calculate $x_1$ and $x_2$ imposing zero total charge and appropriate boundary conditions.

But what if $N_1$ and $N_2$ are both positive or both negative? In this case it makes no sense to me that the carrier densities drop to zero around the origin. Maybe the minority carriers drop to zero and the majority drop to the smallest of $N_1$ and $N_2$ (in absolute value)? Maybe the depletion region approximation is not applicable in this case? If so, why?