Our sample was specifically designed to have finite quantum wells within it's structure. This arises from the different badgap energies of the materials used. The differing thickness of the layers give rise to different "lengths" in the quantum wells. See schematic below.

Stimulating our sample with a Helium Neon laser excites some electrons in these quantum wells. These electrons can only dwell at certain energy levels dictated by quantum mechanics. The depth of the well depends on the band gap of the heterostructure materials. The confined electron energy levels should lie between the well and barrier bandgap energies. Collecting the luminescence emitted from the material should yield a spectrum that gives us an experimental reading for the ground state energies of these finite square wells.

Theoretically, we can find the energy levels by treating the quantum wells as either finite or infinite. The infinite case works best for wider wells, but we expect a large error associated with the thinner layers. The energy levels of an infinite square well can be found from the expression:

E = n^{2}*h^{2}/8*m*a^{2},

where "a" is the thickness of the well, h is Plank's constant, and m is the effective mass of the electron. Research tells us that the effective mass of the electron in our experiment will be 0.067 times the rest mass of the electron.

The finite square well model is much more precise for the thinner wells, but the energy levels are harder to calculate. We used a technique by which you find the intersection of two equations, both describing the odd eigenstates. These equations were as follows:

x*(Cot x) = -y and x^2 +y^2 = p^2 , where p^2 = 8(pi)^2 ma^2 V/ h^2

Solving for the x coordinate of the intersection, and knowing that x = ka we could solve for k at the intersection. Finally, we could get the energy level from the relationship:

h^2 k^2/ 8(pi)^2 m = V - E

Although rather roundabout, this method yielded excellent results, as you can see from our Data.

References:

1. Liboff, Richard L. *Introductory Quantum
Mechanics*. Holden-Day, Inc. Oakland, CA. 1980.

2. Yu, Peter Y. and Manuel Cardona. *Fundamentals
of Semiconductors: Physics and Materials Properties*. Springer-Verlag
Berlin Heidelberg. Germany 1996.

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