- #1
Christoffelsymbol100
- 19
- 1
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.