Homework on Quantum Wells (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material and be in your own words. Students who turn in identical (or close to identical) homework assignments will be asked to explain their answers orally to the TA or prof. A student who cannot explain how he or she arrived at a given answer will be charged with academic dishonesty.

You should justify all of your answers for full credit.

Quantum Wells

You want to design a device that will emit ultraviolet laser light of 15.5 nm. You decide to build a quantum well, in which the transition between the n=2 and n=3 states provides the desired light.

	First, assume you can design an infinitely-deep well, so the probability of an electron escaping your well is zero.
1.	To what energy (in eV) does this wavelength of light correspond? What is the energy in Joules? E = (1240 eV nm) / (15.5 nm) = 80 eV. or 1.28x10^-17 J
2.	The energies in an infinite well depend on the well width, a. Find an expression for the energy emitted when an electron moves from the n=3 state to the n=2 state as a function of a (i.e., plug in the rest of the numbers in the expression for the energy so you get a number times a to some power). E = (h²n²) / (8ma²); h = 6.63x10^-34 Js, m = 9.11x10^-31 kg, gives E = (6.03x10^-38 J m²) (n² / a²). Plugging in for n and subtracting gives E = (3.02 x 10^-37 J m²) / a², or E = (1.88 eV nm²) / a².
3.	Set this expression equal to your desired energy (in Joules) from the earlier question, and find the desired width a of your well. *0.153 nm*
	In the real world, we create finite wells rather than infinite wells, and the expression for the allowed energies does not have a closed form. Use the simulation at http://phys.educ.ksu.edu/vqm/html/eband.html that you used in Activity 21 to answer the next few questions. (If you missed this activity, you should contact your professor, TA, or classmate for an explanation of the energy scale (messed up) and how to use the simulation.)
4.	Set the well depth to its maximum value of 400 eV, Find Energies, and record the values of the energies allowed in the well. What is the energy difference between the n=2 and n=3 energies (in eV)? Roughly, n=1 is -377 eV, n=2 is -299 eV, n=3 is -177 eV,* and n=4 is -5 eV. The difference is 122 eV.*
5.	Do you need to make the well wider or narrower to produce your desired transition energy? How do you know this? Wider. This allows the energy levels to be more closely spaced.
6.	Experiment with the simulation until you find a value of a that produces your desired transition energy, and record it. Also record the energies for n=3 and n=2 when you achieve the desired transition. 0.133 nm appears to produce a transition energy of 80 eV. n=3 is roughly -258 eV and n=2 is roughly -338 eV.