Finding the potential inside a semiconductor

[Christoffelsymbol100]

Homework Statement

My question is more about understanding the task itself, not about calculation.

I am supposed to use the poisson equation, to derive the potential inside a semiconductor for a barrier with potential height $\phi_B$ and a donator doping with $N1 > N2$. Then I should use the schroedinger equation to derive the probability density for electrons and assume that m1 = m2. I have drawn the situation below.

https://imgur.com/a/JXAlLf5

Homework Equations

Poisson Equation: $\frac{d^2V}{dx^2} = \frac{\rho}{\epsilon_0\cdot \epsilon_r}$

Time-Independent Schroedinger Equation: $-\frac{\hbar^2}{2m}\frac{d}{dx}\psi + V\psi = E\psi$[/B]

The Attempt at a Solution

As I said, it is more about understanding the question. I already talked to my teacher but didn't understand.

First, I have to use poissons equation to calculate the potential. The charge density is given by the donator density N1 and N2 in the specific regions and the free electrons densities. I can plug this in and solve the poisson equation. On ther other hand, in the drawing, isn't the potential already given as this barrier?

Then I should use schrödingers equation to get the probability densities of the electrons. I thought about plugging in the potential from the poisson equation and if I am lucky, I can solve this analitically to get the wave function. The probability density then is the amplitude squared. However the presence of the barrier suggest, that this is just a simple textbook tunneling problem. If that is the case however, I am just not sure how this task is then connected to the one above.

 

[Henryk]

Then I should use the schroedinger equation

No you don't. The dimensions are much larger than atomic and semi-classical approximation will do just fine.
You need to use statistical physics to get probability distribution. In non-degenerate case, the Boltzmann statistics will do.
Next simplification comes from the fact that at ambient temperature, kT ~ 26 meV, that is much smaller than typical barrier heights in semiconductor junctions.
Therefore, you can say that within the barrier, there are no free carriers and charge density is equal to the concentration of donors.
The result is a quadratic potential barrier.