Finding the Electron Concentration in a Semiconductor

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If you want to see more of these videos, or would like to say thanks for this one, the best way you can do that is by becoming a patron - see the link above :). And a huge thank you to all my existing patrons - you make these videos possible.
After much trial and tribulation, I am finally able to describe how we calculate the electron concentration of an intrinsic semiconductor under equilibrium conditions using the density of states, effective mass and E/k diagrams, fermi-dirac statistics, and some basic calculus (and yes, *technically* it should be a sum because quantum states are discrete, but the error incurred is so small as to be negligible).
This is part of my series on semiconductor physics (often called Electronics 1 at university). This is based on the book Semiconductor Physics and Devices by Donald Neamen, as well as the EECS 170A/174 courses taught at UC Irvine.
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.

[@emmanuelnyamekeh7313]

Your videos are very nice I've never seen anyone who breaks down solid states like you do kudos my boss

[@ninjanothing8343]

I understood more from these videos than from my whole semester class

[@markpascual100]

:o could it be THE strawberryhacker?

[@RaphiTheOne]

Those short video are far more understandable than the book that was recommended for my exam. They are very much appreciated and will probably watch them all if some in accelerated.

[@neydora2912]

I am just studying for my defence and wanted to repeat all that stuff again since my university time is now more than 4 years ago - your videos are just amazing! Watching the Semiconductor Physics list :) Thanks a lot!

[@xturtleparadex]

Bless your soul. Sincerely, A current UCI EECS 170A student lol.

[@JordanEdmundsEECS]

:DDD A fellow anteater! Good luck on finals (if you aren't already done with them).

[@kaylo1680]

I'm currently being thrust into a course in Semiconductor physics at my university now (to which I and my peers lack some of the required previous knowledge for, because for some reason our program hasn't sorted that blunder out yet) and your series is absolutely saving my ass.

[@NicolasSchmidMusic]

You videos are good and you probably use the same book as in my course (semiconductor physics and devices?), so all the playlist is exactly what I have to learn. Thanks!

[@xandersafrunek2151]

In your previous video you define DOS as (4*pi*(2m)^(3/2)/h^3)*sqrt(E), but in this video use (4*pi*(m)^(3/2)/h^3)*sqrt(E). I think maybe this is just a typo? Also, I think the final equation should have an e^-(E-E_F)/(k*T), not e^+(E-E_F)/(k*T)

[@shubhamtanwar5341]

These are some high quality explanation skills! Kudos to you, Sir. I will donate as soon as I start step out of student life

[@JordanEdmundsEECS]

I'll hold you to that :)p Good luck with your studies!

[@nostradamus9132]

​@@JordanEdmundsEECSin the video you use the fermi energy in the fermi dirac statistic. But I think this is wrong, you should use the chemical potential. They are only the same at 0 K. Can you comment on that so that I know if this assumption of me is correct?

[@BillDemos]

1:41 The hand text: "We've Anally done it!". Wow. That must have hurt ahahahahah.... Besides the lough, man already subscribed, TOP NOTCH CONTENT

[@user-oq9ym7lp8s]

Thanks for your video. It's helpful for me to organize my knowledge related to semiconductors. Let me point out a mistake in this video: At 6:15 f(E) should be approximated by exp(-(E-E_F)/kT) instead of exp((E-E_F)/kT)

[@dannchan00]

Hi Jordan, I'm very grateful that u made such videos and for people like me that study biology and has no basis in physic, this is a really good study material for me. Keep the good effort going! However, I found that the formula of DOS , g(E) function 6:07 in this video is not similar to the DOS part 2 video 7:52, ( 8pi m* and the other one is only 4pi m* regardless of the dE)? Why is that so? I'm sorry if I ask a stupid question because my differentiation and integration knowledge is limited and need to be sharpened in the future.

[@xephyr417]

Great video again! Question, is the density of states function we developed under the assumption of the 3-D infinite potential well still accurate? It is what gave us the n*pi/L that we used to derive that equation. I had assumed that we would need to recalculate that function using the kronig-penney model, but we didn't. Doesn't the spacing in k-space technically change? Or are we assuming that (again) for relatively small variations around k=0 the infinite well model is accurate enough as the KP E-K diagram is approximated by the parabola of the infinite well model?

[@JordanEdmundsEECS]

So the density of states function does indeed originate from the infinite potential well model. The parabolic model is actually the model for a free electron, not that of a quantum well, and that is what allows us to treat the electron dynamics semi-classically. The spacing in k-space will change, and I actually don’t currently know qualitatively how it will change. However, I *believe* that you can use Bloch’s theorem to separate out the effects of the atoms and the effect of the larger quantum well structure and my guess is that it does change it, but doesn’t much change the overall result. Your question goes even deeper than that, though, because I think what you are really asking is *how do we know when to use which physical model*? From what I have learned so far, the answer seems to be this: the simplest model that will give the required accuracy. My guess is that this model happens to do that, but if you figure out exactly why, let me know. I’ll see if I can figure it out.

[@sazzadhossain3014]

please include this video in the playlist

[@JordanEdmundsEECS]

Done! Thanks for letting me know it wasn't in there.

[@hchen854]

INSPIRING!!!

[@okropiri142]

Where is density of states g(E) derivation? Should it not be before this video in this playlist? @Jordan Edmunds

[@colosolizer2184]

What is the value of the k constant

[@xAmiSarahx]

thank you.......

[@Biologiehilfe]

Hello there, why exactly did you set f(E) = exp((E-Ef)/kT)? I get the fact, that you approximated f(E) for E>Ef as exp(-(E-Ef)/kT) but why did you then put exp((E-Ef)/kT) as f(E) into n? Great videos by the way :)

[@NinjaTommyd]

Think he just forgot the minus.

[@nellvincervantes6233]

Question sir. Why m* = 2m ? How to derive this equation? From your vid about density of state, g(E) = 4(pi)(2m)^(3/2)(E)^(1/2)/h^3 Then in here, g(E) = 4(pi)(m*)^(3/2)(E-Ec)^(1/2)/h^3 So m* = 2m

[@UsmanKhan-nb4pd]

According to my knowledge, the derivation of f(E), utilized in this lecture, has not been discussed in previous. If it is true and suitable, please include in the playlist.

[@JordanEdmundsEECS]

You are correct, I haven't actually made that video yet, I will add it to this playlist when I make it :)

[@codewithlax]

The DOS funtion value is missing 2me as you have written only me

[@sourabhpatil2965]

I'm wondering how can Fermi function give a finite probability for an electron to exist in forbidden energy area?

[@JordanEdmundsEECS]

That’s an excellent question, I think the best way to interpret the Fermi function is the *percentage of existing states that will be occupied at that energy*. So your fermi function might be 0.3 (30% of states occupied) within the forbidden region, but the number of states you have is 0. This will be given by the density of states function. 30% of 0 is 0, so you still don’t have any occupied states.

[@mitchellmckay5448]

@@JordanEdmundsEECS Oh I have been waiting for this answer

[@saidteacher3331]

Also our professor use a very weird strategy , he calculated the reciprocal lattice volume and devised on it...i find your method much more forward and logical

[@JordanEdmundsEECS]

Yup that becomes necessary to derive the "density of states" (which I have a couple of videos on), but it's a terrible starting point xD

[@saidteacher3331]

First thankyou so much! . Then, can you please tell more what the effective mass is? oir professor just dropt it from the skies and siad (dont worry u find it in tables) . Thanks!!

[@JordanEdmundsEECS]

Yeah! I actually have a whole video on the subject: kzbin.info/www/bejne/p569Xn2OZr-fpNk