Fermi Function Derivation
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- Опубліковано 15 вер 2024
- / edmundsj
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I derive the fermi function using the Gibbs factor, a central idea in thermodynamics (I cheat a little bit to make it easier to write down in saying it's the exponential of the kinetic energy. While this is technically true for this system this is not the case for more complex ones).
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.
In my career as a physicist I was more involved with photonics than electronics, and I have forgotten about the Pauli exclusion principle through the years working in the industry. Electrons are fermions and photons are bosons - what a delight to review these concepts again. Thanks to your series.
Welcome back :)
This is such a great series. Thank you so much for making it!
Thank you:)
Woooooooow. This is so much better than whatever nonsense is happening inside of Ashcoft & Mermin. Also, for those of you who are not familiar with the Gibbs factor, Leonard Susskind derives it in lecture 3 of his series on statistical mechanics here on youtube. But do not watch that video without also watching or reading an explanation of lagrange multipliers. Kahn academy had some great videos and reading on lagrange multipliers. Total it will take probably around 2 hours to get up to speed from knowing nothing to understanding the statistics.
Lmao
I am actually just going through the same section in Ashcoft & Mermin.
actually susskind gives overview of lagrange multiplier too in the previous lecture near the end, that series for stat mech should get one started!
Thanks for making this series! I'll try to become a patron at some point in the near future...
Thanks a lot for these videos. I have been utilizing them to get started in my research project.
I am from INDIA.I have watched your video before one night of my examination.
Masterpiece
Why E(N) isnt the total energy but kinetic energy? Its more confusing for me
Hello Jordan, I started this playlist a few days ago. It is very helpful and nice of you to do this!!!
Now I will ask a question:
I am still a bit confused. One particular energy level can have many states. Each state can then possess an electron. It might have it or not. That makes sense.. When we did the g(E) derivation, we talked about each state having a spin up or down and thus multiplied the equation by 2. Isn't it the same thing here as well. Each energy level can have many buckets with a space for one electron each, but that electron can have two spins? Does it make sense what I mean?
Here the “buckets” are states, and states account for spin. So you will have two buckets at each energy.
Hello sir, so the last replacement between E and KE. Does it mean in the gibbs factor we have to do the replacement as well? Because that was the very base of this derivation.
A very consistent question? waiting for the response??
yes
saying that Ef is the potential energy is a bit confusing. this would imply a negative kinetic energy if E < Ef. i guess Ef is related to the potential energy of an ensemble. i might be wrong as i left university 25 years ago.
Very nice video .I really loved ,but still I have a question.The Gibbs measure is derived or its concluded via experiments?Fhank you very much for your time!
Look up Grand Canonical Ensemble in statistical mechanics. Your doubt will be clarified.
Very clear and precise waw
Why in the last part, the energy term in the equation changed from the total energy to just kinetic energy? If someone understood that part can you please tell?
you r my hero :)
Really Helpfull:-)
Glad you think so :D and thanks for reminding me that this was missing
Thank you for the simple explanations. You have mentioned a state being occupied by 0 or 1 electron. What about P(N=2)?
Just saw the same question below! Got your answer. Thanks, Jordan!
Pauli exclusion principle say that one state can be occupied by one electron.
So just to be clear , the probability of the the electron being in the "bucket" include the 2 electrons with the up spin and down spin. If so ,what about when an orbital which only contains 1 electron like in say the outer shell of a group 1 metal such as sodium; is their a probability for an orbital containing 1 or 2 electrons?
Excellent question! It cuts to the heart of what the Fermi function actually describes, which is not atomic orbitals of individual atoms, but the states that result from the overlap of many adjacent atomic orbitals. Strictly speaking it is only applicable for “bulk” crystals, and describes only the behavior of the outermost electrons. Spin is also not dealt with in the Fermi function, but is contained in the density of states. Atomic orbitals of metals like sodium are an entirely different beast than what we deal with in device physics.
@@JordanEdmundsEECS Thank you. BTW, your videos are really helpful for an intuitive understanding of such concepts. You are way better at explaining topics from the basics than my university which focuses on rote learning. Sadly, even learning institutions have become a place of business and internal politics instead of focusing on the fun of learning these awesome concepts. They present topics in such a boring way that you forget that these are the tools that humanity has used to build itself and not just for some grades on an exam.
Thanks :D that’s what I’m aiming for, EE can be a beautiful story
I had the same question going on my mind while watching , Thanks for the awesome answer and the awesome videos. @Jordan Edmunds
Great video thank you very much!
question: I thought every energy state could hold up to 2 electrons (of different spin)? So can't we have 3 possibilities? 0, 1, or 2 electrons per state?
I had the same doubt but I guess it's because for the density of state's function we multiplied by two because of the two electrons in a state. Then afterwards we work with one electron per state. EDIT: sorry haha, it was already answered just two comments below haha.
how could you determined the constant is reciprocal (1/z), instead of z?
You will get same final expression it doesn't matter
U r great at doing this! thanks
Jordan,
Why U=NE(f)? in another words , Why the potential energy of the the electrons is equal to number of electrons times the fermi energy?
Good question, that’s just poor notation on my part. Here U is the total energy of the system of N electrons.
@@JordanEdmundsEECS I thought that's what E represented (in E=K+U) ?
@@rvlli2377 In this video, U is denoting potential energy, insteead of the total energy.
1:00 why is the probability distribution proportional to an exponential distribution?
Great question. This is what's called a "Boltzmann factor", and the fundamental reason is because entropy involves a logarithm, and when you invert that you get an exponential. To really understand it you'll want to take a class on statistical mechanics / thermal physics. It's an awesome course.
Thanks for such a great video, but I'm wondering should we consider P(2)? In which there are two electrons with different spin in a single state.
we cant fit two electrons in a single state
i really got confused at the last part. u were writing about f(E) that has exp^(E) but later that exp has kinetic energy instead of total energy.
Same doubt! Did you get an answer by any chance?
7:14 HAHAHAHAHAHAHAHAHA ITS NOT EVEN FUNNY