Kronig-Penny Model: Effective Mass

Here I go cover energy/momentum diagrams in more depth, and I explain how we can use them to determine how electrons and holes move in the semiconductor through the concept of effective mass.
/ edmundsj
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.
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.

[@Jacob-ge1py]

I actually almost cried with relief when I found this. Your explanation is so much clearer than any lecture I have ever been to.

[@KeenestObserver]

Holy shit. I learned and understood SO MUCH in 13 minutes !! My lecturer's slides are completely horrible and he's bad at explaining intuitively. Thank you loads

[@JordanEdmundsEECS]

Aw thanks :)

[@rodgervinsmoke8220]

sharingan

[@JyotiDeka1989]

I completed my under-graduation 9 years ago and today, I understood the meaning of effective mass of an electron... :)... Thank you so much Jordan for these lectures...

[@bryantcombs9301]

My lecturer for Quantum Mechanics is too smart for his own good. He is horrible at explaining stuff, you are going to save my GPA and my degree, thanks Homie.

[@roonilwazlib8137]

Wow excellent explanation !! I have been trying to wrap my head around this for so long but you explained it so simply!

[@JordanEdmundsEECS]

Ye dude all of physics is basically just linear and quadratic series approximations, followed by tears and more approximations when those break down.

[@shubhamtanwar5341]

Thanks for the excellent video! Physics should never be taught without explaining what "should sound awkward" and what "should sound intuitive". Great approach.

[@JordanEdmundsEECS]

Yeah although to be honest almost everything sounds awkward at first xD. Then, after a while, everything starts to sound intuitive (which is very dangerous from a teaching perspective).

[@benwinstanleymusic]

Brilliantly explained, thank you so much!

[@KnowWithMalvika]

Just Amazing!!! Really a profound lecture.

[@alis5893]

I really enjoyed this lecture

[@jahnavilothugedda6352]

Oh Thank you so much sir.....it helped lot cuz I have an exam tmmrw.....That was literally so easy when you explained it....

[@martinsimbona6145]

Ooh man, that was so helpful. I neede to prove that formula for effective mass and found it nowhere else. Can't thank you enough, thanks for Sharing!!

[@JordanEdmundsEECS]

martin simbona Thanks! It is buried pretty deep xD I believe you can find it in the QM section of Neamen

[@anupamayadav4472]

Amazing Explanation!

[@haniefsofi]

Nicely presented. Thanks

[@user-hm1nu3qg5g]

I really appreciated it. It's very helpful for me to understand the concept of an effective mass. 😌

[@JordanEdmundsEECS]

Glad it helped you understand the concept! It can be challenging to wrap your head around.

[@trueintellect]

I finally get it! Thanks!

[@OfficialAnarchyz]

Love you so much, thanks dude

[@erikchavarin6085]

This was awesome. I so wish my prof. made electronics this exciting.

[@JordanEdmundsEECS]

I'm glad :) What about it did you find exciting?

[@erikchavarin6085]

@@JordanEdmundsEECS My prof presented energy band diagrams in a very peculiar way. We never discussed how to derive them so it was a very abstract concept with no intuitive motivation (on top of that, our textbook didn't discuss band diagrams at all) so when I saw that you used the Schrödinger equation to create the Kronig-Penney model and then used that to construct the energy band diagram I was losing my mind lol

[@venkataramanarishikesan8104]

Thank you for the wonderful explanation Jordan! I have a question. When we flatten the energy bands, does this correspond only to the CB minima and the VB maxima?

[@sharunpshaji8288]

Thank a lot, so easy to understand the concepts with your explanation

[@JordanEdmundsEECS]

You are welcome! Anything in particular that made it easy to understand?

[@sharunshaji6328]

The bandgap calculation became much more sensible after watching this video!

[@davidansi1683]

Jordan is really good Thanks

[@proffessor4203]

Thanku sir its soo help full 🖤

[@Psyduckie71]

How perfect can a video be 🥰

[@JordanEdmundsEECS]

Daww

[@eliyahomar]

Thank you so much

[@zacharythatcher7328]

This is great! I am really interested in how this relates to interactions with photons. A&M goes over why scattering happening at brillouin zone edges because there are standing waves. The fact that there are standing waves makes sense, but I can’t understand why that means photons will scatter.

[@1q1q1q1q1q1q1qw]

super nice explained

[@JordanEdmundsEECS]

Thank you 🙂

[@ArsenedeBienne]

thanks a lot!

[@lightningllama]

Is there an intuitive explanation for energy-band diagrams with multiple sub-bands? Some materials such as GaAs have different effective masses which lead to multiple overlapping E-K curves. Spin-Orbit interaction also causes band splitting. I'd love an explanation on this!

[@Saptarshi.Sarkar]

I always thought that the effective mass formula must have something to do with Taylor series. Today I understood why it is. It's just like how we find the frequency for small oscillations due to any arbitrary potential. Thanks!

[@JordanEdmundsEECS]

Yup, it's exactly the same thing. Basically all of engineering comes down to fitting parabolas and straight lines to hella complex curves :p

[@prashantupadhyay5110]

It's 1 min 46 second only,learned lot

[@nada1815]

Thank you

[@blitzkrieg5454]

thanks a lot

[@bhaveshsoni7171]

So are the conduction electrons and holes only present where value of k is less?

[@srijanisallyouneed]

From #Kolkata , India🇮🇳 Really appreciate your golden Lecture Sir But 1 question : *In solid state physics from where the Kronig penny graph(which u have drawn) come from? Is it derived by experiment or hypothetical*

[@RexGalilae]

I think that, atleast physically, the effective mass is just a way of accounting for the change in potential energy as an electron moves between potential wells. Since we stated that E = p^2/2m, we're compensating for the lack of the V term by treating mass as a non-constant. Other than that, I don't really see any reason why the mass should change in a non-relativistic scenario

[@JordanEdmundsEECS]

Yup, that's basically how I interpret it now. It's essentially a way of accounting for the "medium" the electron is embedded in (which is a periodic crystal, with a bunch of adjacent potential wells).

[@saifamu1989]

Please suggest a good book to learn solid state physics from.

[@xephyr417]

Great video! I think it would have been helpful if you'd elaborated a bit more on why the E-K diagram can be "flattened" to the band gap we see normally. To my understanding it's because for MOST cases we consider relatively small variations around k=0 (low momentums and energies) the parabola can be considered roughly "flat" to a first order approximation. Is that right?

[@JordanEdmundsEECS]

Yes sir that is correct. The Kronig-Penney model (and band structure more generally) is something I need to make more in-depth videos. Thank you!

[@alexeykulik2625]

@@JordanEdmundsEECS Please do! Your videos deserve much more views and recognition. Your channel is the first where I make sure to like every video so that the algorithm gods are merciful to you.

[@tonymagana6489]

So are k and momentum synonymous in quantum mechanics?

[@ashlynnundlall]

I thought 'p' was momentum and 'k' was the wave number so why is it called the Energy vs momentum curve when "k" wave number is the independent variable?

[@JordanEdmundsEECS]

Since p and k are just related by a constant (Planck's constant) p = hbar * k, the term "momentum" is used interchangeably to refer to either p or k. In semiconductor physics, though, it's just easier to work with 'k'.

[@siaram123]

Hi, I am sorry to bother you with my very naive question. But I want to ask you how you have calculated the effective mass of electron in silicon is 0.82 times the actual mass of an electron. Because, when I see several reports, they showed that it is 1.08 times the actual mass of an electron in the silicon lattice. Please let me know what I am missing here.

[@JordanEdmundsEECS]

Doh! I think you are correct. My mistake.

[@srchang7104]

Dear sir, the video is very helpful. Thank you very much.But I have one question. At about 3:20, you mentioned the calculation of mass by taking the vaules of E and k from the diagram. What if we take the vaules of E and k at k = 0? It gives us m = 0, which contradicts the result of calculation from the curvature, which you mentioned later.Thanks.

[@JordanEdmundsEECS]

Sure! This is actually a very good reason *not* to define mass in this way. Since the second derivative is always defined, it will always work. I just used dividing by E and k to illustrate the problems with defining a single mass. It’s not actually how we define it in practice.

[@srchang7104]

@@JordanEdmundsEECS Thanks.

[@srchang7104]

Dear sir, I am the person who asked you a question regarding calculation of effective mass from the E-k relation. I got your answer already. Thank you very much. But I would like to discuss a little bit more. The E-k relation is actually an equation containing a complicate function of E and cos k(a+b), where a is the width of the potential barrier and b is the width of the potential well. Therefore, k can be varied by its period without changing the value of E. Consequently, the value of k in the E-k curve is not uniquely defined; it can be shifted by the amount of the period. Therefore, choosing a pair of E-k values from the curve to calculate effective mass, as you mentioned at 3:20, will lead to a wrong answer. Using the curvature (calculating the 2nd derivative) can avoid the problem of k-value shift, because the curvature is not changed by the shift of k. Moreover, only the bottom of the curve (near k = 0) can be represented by a parabola function, so we use this part of the curve to conveniently ontain the 2nd derivative. Please let me know if I am right or wrong. Thanks.

[@JordanEdmundsEECS]

Sounds totally reasonable.

[@srchang7104]

@@JordanEdmundsEECS Thanks again!

[@thealiaria]

first, many thanks for your great video. second is my question that, why don't we accept that the electron effective mass is dependent on the momentum? i mean, electrons with higher momentum have higher masses than the ones with less momentum.

[@Science_is_Better]

Good video. I'm still having trouble with a related topic which is finding the effective mass when you are given multiple E and k values for a conduction band when Ec =1. I know E=Ec + (hbar^2*k^2)/2m*. But i can't just plug those numbers into this formula correct? I don't know, it seems like it should be relatively easy but the effective mass for each that I am calculating must be wrong (x10^-36). I feel like i'm missing something, or maybe i'm just too tired to think right now lol.

[@JordanEdmundsEECS]

Yeah the smallest effective mass I have ever seen is like 1/100 of the electron mass, typically between 0.1-1x the electron mass.

[@pablomejia5694]

Can u use effective mass at boxing

[@tanyasaini9257]

What is the importance of effective mass of holes in various currents in a semiconductor diode?

[@JordanEdmundsEECS]

The effective mass shows up in the expression for the reverse saturation current density. It affects pretty much everything xD

[@qplaqpla]

at 7:51 that was a partial derivative, wasnt it? sorry if this is obvious, but the derivative sign and the result acted like that's what you did. Anyway, great video! Wish I'd found it sooner!

[@hungthanh4615]

I thought electron in valenced band has negative effective mass and moves in the same direction of electric field so hole moves in the opposite direction of electric field. But why hole still moves the same direction of electric field ?

[@giorgiorossi8189]

In a lecture of a semiconductors course I am taking, it says that the Kronig Penney Model implies that a small effective mass yields a small energy gap. Intuitively the only explanation I can give myself is that steeper bands (small eff mass) would give a smaller gap compared to bands at approximately the same energy but flat (high eff mass). I don't see however how this can relate to the Kronig Penney Model. If anyone could help I would be extremely grateful.

[@JordanEdmundsEECS]

You are correct that smaller effective mass corresponds to steeper bands, but it’s also *separately* true that a smaller effective mass yields a smaller bandgap (the two phenomena are distinct). To prove this you actually have to solve for the band gap as a function of the effective mass.

[@giorgiorossi8189]

@@JordanEdmundsEECS Thank you. Do you have a physical explanation for small effective mass --> small band gap? Because I really cannot see a link, and even less why this is a result of the Kronig-Penney Model.

[@seandafny]

Why is the effective mass for a hole in the valence band positive. Is it because the function of E with respect to k for a hole is opposite for electron

[@JordanEdmundsEECS]

Yeah this is confusing - the simplest way I know of thinking about this is that when you apply a force in the form of electric field, an electron in the conduction band will move one direction (say to the right), but a hole will move in the opposite direction (to the left). One way of interpreting this is that it's just a negatively charged particle with negative mass, but this seems ass-backwards, so we just say it's a positive particle with positive mass.

[@karamveersingh1182]

I am just concerned that if the negative mass of the electron is taken as the positive mass of the hole, why are the effective masses of electrons and hole significantly different?

[@PauloConstantino167]

Your work is beautiful. I feel so sorry you only have 4k subs. Out of BILLIONS of people in our planet, only a minority understands the foundations of our society. Keep it up.

[@renatoberaldo2335]

what is effective mass ?

[@emankopeal4210]

thank you ("

[@mathsphysicsteacher]

Is k the wavenumber or momentum ?

[@mathsphysicsteacher]

Is the relation to momentum due to p=hbar k, since hbar is constant

[@JordanEdmundsEECS]

Yes!

[@MahaKaaal009]

What is the difference between the relativistic mass and effective mass, are both the same?