I have written a program that solves the Schrödinger equation numerically using the finite difference method based on the attached article. The end goal is to make a program that solves the Schrödinger and Poisson equations self-consistently for the conduction band in different heterostructures.(adsbygoogle = window.adsbygoogle || []).push({});

To do so you need to invoke different relations. For example that the occupation of the kth eigenenergy is given by:

n_k = m*/(πħ2) ∫E_k∞ 1/(1+exp((E-E_f)/kT)) dE

, where E_k is the fermi energy and m* is the effective mass in the band.

Now the problem I have come to is that my units create problems in the above equation. Using the electron mass for the effective mass (I don't know what I should use, I guess it depends on the material) and ħ and a trial potential in the same scale as these units I get eigenenergies of size ≈10-34.

But since kT≈10^(-25) for the systems I am working with the exponential in the above equation simply yields 1 for all energies.

I feel like an amateur that this is giving me problems, but I really don't know what to do at this point. Should I use a different effective mass? I guess I should but is that the main problem? Should I add some kind of conduction band offset? And also what fermi energy should I use for my system?

Hope you will take time to answer some of my questions :)

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Schrodinger equation numerical solution

**Physics Forums | Science Articles, Homework Help, Discussion**