Quantum Well Density of States

/ edmundsj
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Perhaps almost as important as the density of states of a 3D semiconductor, the density of states of a 2D semiconductor (also known as a quantum well) can be found in a very similar way. By counting the number of states in 'k-space', you can determine a variety of properties of a quantum well - in particular the electron concentration.
This is part of my graduate series on optoelectronics / photonics, and is based primarily on Coldren's book on Lasers as well as graduate-level coursework I have taken in the EECS department at UC Berkeley.
Hope you found this video helpful, please post in the comments below anything I can do to improve future videos, or suggestions you have for future videos.

[@darioquintero5008]

Thank you so much for doing these videos!

[@onurunal9948]

That was a great explanation!

[@TheGoldennach]

Really nice and helpful video!!! I only have one question. Why is it a quarter of a circle in k-space? Can't we have negative kx and ky?

[@JordanEdmundsEECS]

This has to do with how you want to solve the Schrodinger equation. You can solve it in terms of complex exponentials (in which case you can have negative kx and ky, this corresponds to traveling waves going -x and -y). *Or* you can solve it taking only the sinusoidal solutions, in which case kx can only be negative (or positive and negative kx mean the same thing). If you use the sinusoidal solution, you use 1/8 of k-space, if you use the exponential, then all of k-space. It's just a little harder to do with complex exponentials.

[@TheGoldennach]

@@JordanEdmundsEECS Really great explanation! Thank you!!

[@tzahigeller6266]

In the beginning of the video you briefly went over how to calculate the energy density of a bulk material, introducing concepts like Boltsman approximation Fermi Dirac equation and more. However they were not explained in this playlist, can you refer to videos in which you elaborate on these concepts?

[@tzahigeller6266]

Maybe it's a silly question but why can K be a negative number? I am associating it with position vector x that can be with flipped sign. If what I am writing is correct then the energy density should be larger by a factor of 4, since 2*pi*k*dk will be a full square Thanks

[@sosavelazquez2914]

GOLD, but can you talk about parabolic potentials?

[@JordanEdmundsEECS]

Mmm harmonic oscillators. Might have to make a few videos on those.

[@sushilsharma691]

Does this expression work for massless spin 1/2 fermions??

[@keerthanaskumar96]

How is this different if it is a finite quantum well?

[@JordanEdmundsEECS]

If the quantum well is small, it’s usually better to treat the individual states separately rather than as a density. The approach in this video works well in practice as long as the dimension is large (>> 10nm or so)