Maybe you are missing a step here? At around 6:14 you said that g(E)dE = (something-something)*dE . Where did the dE in the Left-hand side of the equation come from??
@JordanEdmundsEECS5 жыл бұрын
Excellent catch! When I wrote g(k), I should have written g(k)*dk. This quantity refers to the total number of states within a shell, so it needs to be written density*length.
@-NikoLee3 жыл бұрын
@@JordanEdmundsEECS had the same question with Rishi....Jordan thank you so much this is really helpfull with my physics courses! :-)
@RoyMustang0273 жыл бұрын
@@JordanEdmundsEECS First of all, thanks for the excellent videos! Regarding this question, I don't understand why you should have written g(k)*dk, if you replaced N/L^3 by g(k), which is indeed a density of states.
@suruchiverma483 жыл бұрын
@@JordanEdmundsEECS i had the same question, wherefrom did g(E) appear on the left hand side? Can you pl elaborate at what point we should add dk? Am not fully clear.
@suruchiverma483 жыл бұрын
@@RoyMustang027 exactly, pl explain
@user-lr3ui2nk3x Жыл бұрын
As mentioned in comments below. The expression for g(k) should not include dk. We can add further view on the derivation on g(k). g(k) = N(k)/dk, where N(k) represents the number of states within the shell, and dk represents the thickness of the shell.
@stephan63345 жыл бұрын
These videos are great. Love the top-down approach. Looking forward to, as Neel said, f(E) derivation
@JordanEdmundsEECS5 жыл бұрын
Stephan Flumm Thanks :D I’m looking forward to making it
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@aatt32094 жыл бұрын
thank you for a fantastic lecture.
@menghanxu80143 жыл бұрын
Hi Jordan, I am not quite sure why you mentioned below one of the comment that you should have written g(k)*dk. You wrote about g(k), the density of states, from N/L^3 which is also density of state and which has already had a delta K term in it. I am not sure why still need to incorporate the dk
@kirakirievsk40355 жыл бұрын
Great video. Thanks a lot! I didn't catch why do we have 1/8 before. You took infinitesimal resiprocal volume 4PiK^2dK and decided it by volume of of one state in KSpace Pi/L. why do we need 1/8
@JordanEdmundsEECS5 жыл бұрын
Thanks :D The 1/8 comes from the fact we are only considering one-eighth of the volume of a sphere (one octant of the 3D state space).
@Odiskis14 жыл бұрын
why are we considering only 1/8 of the sphere? Is that a choice or a necessity?
@Odiskis14 жыл бұрын
Think I got it! Since n's are only positive, you only look at 1/8 of the 3D coordinate system (where x,y and z axis are positive). That makes sense to me at least
@nitismishra5514 жыл бұрын
@@Odiskis1 you cleared my doubt as well. Thanks man
@deepvision88772 жыл бұрын
The introduction of calculus here assumes the density of states is continuous. This doesn't make sense as amount of states should only be a multiple of pi/l. To my understanding, as electrons are restricted to a wavenumber of npi/l, then the smallest possible wave number is pi/l. Therefore in K space, the number of states is : N= 2(1/K)^3*Vk where Vk is the volume in K space and 2(1/k)^3 is the density of states in k space including spin. For any volume, N can be a non-integer. But the number of states would just be the integer part of N. Now for energy, a similar argument should be used. As only integer values of pi/l for wave number are allowed, then there is only an integer value if Es is allowed for energy. Here Es is the energy corresponding to the wavenumber pi/l. Therefore the number of states at a given energy E should be: N= 2E/Es Where E = (Es+Es+Es+Es+...)=nEs (the factor of 2 accounts for spin) Which in 3D is made of three components (assuming Ex=Ey=Ez for a cube well): E= Sqrt( (uEs)^2+(wEs)^2 + (bEs)^2) = Es(sqrt(u^2+w^2+b^2))= nEs Therefore the density of states for energy is 2/Es= 2 ( 1/(hbar^2k^2/2m))= 4mL^2/hbar^2pi^2. This makes sense as the density of states is not a function of energy, but rather of the quantum well width L. I'm not sure how the density of states (the number of states per unit energy) would increase with energy. That would only be true if L increased. For example, If I had a nanoscale silicon transistor, the density of states should be much smaller and more discreet than that for a block of silicon with a large L value. I'm not sure what's wrong with my understanding, but the calculus approach should be an approximation (that only works for large L values). Meaning there should not be a state that exists between K and K+dK, but rather a state that exists between K and K+ pi/l.
@JordanEdmundsEECS2 жыл бұрын
Yes! That's exactly it. This implicitly assumes that L is large enough so that the number of states is huge, and you can even use differential calculus at all. Its typical application is in the "bulk" - crystals larger than ~100nm or so. For nanoscale devices smaller than this, this is not the most accurate approach, but often those devices are only small along 1 dimension (i.e. MQW lasers), in which case you can use this approach with the other 2 dimensions.
@T_tintin2 жыл бұрын
Damn I was suffering with for so long ..finally i get it!!! Thank you so much!😭😭😭😭
@JordanEdmundsEECS2 жыл бұрын
Yeah the DOS is a pretty frustrating and challenging subject for most people (myself included). Glad you get it now! :)
@ipinlojunureni87809 ай бұрын
Much appreciation for this video. I have been struggling with how the g(E) is obtained in optical properties of solids textbook by mark fox
@ninjanothing83434 жыл бұрын
Thank you! Very understandable video
@tashwinsj38372 жыл бұрын
Thank you very much sir .
@n33l855 жыл бұрын
In which video do you find f(E) : probablity that the states are occupied
@JordanEdmundsEECS5 жыл бұрын
I actually haven’t made that video yet D: definitely in the next month or two
@n33l855 жыл бұрын
Jordan edmunds : Thanks this videos are really helpful.........👍👍👍
@stephan63345 жыл бұрын
yay :)
@user-lw5xh4jl6h5 жыл бұрын
Why do we consider the 1/8 spherical shell instead of the 1/8 sphere?
@JordanEdmundsEECS5 жыл бұрын
Because we are interested only in the *differential volume* so we can use this derived equation in an integral, not the *total volume*.
@yewedmund3 жыл бұрын
Hi Jordan, ultimately we want to find the number of electrons in the entire semiconductor cube so if we integral P(E)g(E)dE, does that only give you total number of electrons per volume in the semiconductor?
@sageofsixpack226 Жыл бұрын
Yes, you are correct. And this is often called concentration of charge carriers, which is key to finding current and voltage relationships
@ummatilawan5797 Жыл бұрын
Very helpful to me. Thanks a lot
@Dr.kcMishra5 жыл бұрын
why do we considere only one-eighth of sphere?. why not complete sphere?
@JordanEdmundsEECS5 жыл бұрын
Great question . Each state we are counting represents a different solution the the Schrodinger equation, and a different wavenumber k. We only want to count the positive wave numbers since the negative ones do not give unique solutions, and so the 1/8 comes from the fact that only in one octant of the sphere are all the wave numbers positive.
@christopherhallett78615 жыл бұрын
Super helpful thanks!
@040_saqibjaveed44 жыл бұрын
Why are we dividing by L^3 ..while the volume under consideration is 4πk^2 delta k
@mihirgupta38243 жыл бұрын
L^3 is the "unit" volume in the actual space, which is exactly what we want. Number of electron states per "unit" volume. I think this is mostly right, if I am wrong, please correct me!
@SiddharthRanjan61975 жыл бұрын
A single state can have only one electron. So why did you multiply by 2 ??
@JordanEdmundsEECS5 жыл бұрын
Great question. A single *state* can have only one electron. But at a single *energy* we can actually have two states, or two spins (these states are called degenerate). This is we need to multiply by two.
@SiddharthRanjan61975 жыл бұрын
@@JordanEdmundsEECS Thanks 🙌
@jimitsoni184 жыл бұрын
Sorry for asking a very stupid question but what actually is a state in density of states?
@JordanEdmundsEECS4 жыл бұрын
Not at all! A ‘state’, technically speaking, is a solution to the time-independent Schrodinger equation for a 3D quantum well. Intuitively speaking, it’s a particular speed and direction the electron is moving which is allowed by quantum mechanics (and these are discrete).
@haozheyang30174 жыл бұрын
@@JordanEdmundsEECS This is really great explanation .
@vikaspandey28505 жыл бұрын
how g(k)*dk is equal to number of states within a shell. g(k) is no. of states per unit volume to get number of states within a shell you have to multiply by volume and dk is not volume
@JordanEdmundsEECS5 жыл бұрын
You are absolutely correct, g(k)dk*L^3 is the number of states within a shell. Perhaps I misspoke.
@omercohen58335 жыл бұрын
Shouldn't the spacing between k points be 2 pi over L?
@JordanEdmundsEECS5 жыл бұрын
Ah, yes, that is extremely subtle. The answer is that it depends on how you want to solve the Schrödinger equation. If you assume the solution is a standing wave (a sinewave with nodes at the boundaries), then your spacing is pi/L and you only sum over 1/8th of k-space. If instead you assume the solutions are complex exponentials (which is true, but only a specific combination of them satisfy the boundary conditions), then your spacing is 2pi/L and you count all of k-space.
@justanon58124 жыл бұрын
@@JordanEdmundsEECS could you plz plz explain why it works that way? My book has assumed those conditions and now I am extremely confused and torn between the two proofs
@vikaspandey28505 жыл бұрын
i did not understand from where dE is come because N/L3 is states density so it should be equal to g(E) only
@JordanEdmundsEECS5 жыл бұрын
g(E) is the density of states per unit volume per unit energy, and g(E)dE is the density of states per unit volume for a small ‘slice’ of energy. I agree the units are confusing.
@nitismishra5514 жыл бұрын
sir are we taking a cube instead of a line because while calculating k, we assumed only one axis. But we can do the same for the other two axis as well. Is it correct?
@JordanEdmundsEECS4 жыл бұрын
Yessir, that is correct.
@oraange3 жыл бұрын
a state (solution of Schrödinger Equation ) occupies a volume ?
@JordanEdmundsEECS3 жыл бұрын
In a sense, yeah, because each state has a different momentum (k), and these are some distance (pi/L) apart from each other. So a single state is (pi/L) away from all the neighboring states. In this sense it occupies a volume of “k-space”. Not a literal volume.
@rafiamurtaza3074 жыл бұрын
why we used momentum space (k space) when we were doing fine in real space (r space) ?
@JordanEdmundsEECS4 жыл бұрын
Because solving the Schrodinger equation (and Maxwell’s equations, and virtually all differential equations in circuits) which is nearly impossible in real space becomes trivial in k-space. Also known as frequency space (but now the frequency is in space not time).
@rafiamurtaza3074 жыл бұрын
@@JordanEdmundsEECS alright! Thankyou :)
@drscott110 ай бұрын
👍🏼
@arduinoguy7413 Жыл бұрын
Sir, when the value of “h” is (6.625*10^-34)^3 the result is 0. May I know how to fix this error ?
@kiprotichaviator39352 жыл бұрын
Watching in Muranga university of technology
@prithwieboedhoe25089 ай бұрын
i think there is a factor of 2 missing in the last equation g(E)dE
@swsaber8199 Жыл бұрын
can we do these things in real space instead of k space because volume in k space little confusing .. :(
@nostradamus9132 Жыл бұрын
No because we use the dispersion relation that connects q and E space to get D(E). So we have to start in q space.
@MrWeblapfoto2 ай бұрын
Where is part 3????? 😰😰😰
@joxa61194 жыл бұрын
why 1/8 ?
@JordanEdmundsEECS4 жыл бұрын
Because we only want to work with positive k in x, y, and z
@aselim20.4 ай бұрын
I watched it once and I didn't take notes.
@suruchiverma483 жыл бұрын
Pl pl explain why it should be g(k)dk. Am really not following.
@deeyakaur2 жыл бұрын
It is probably easier to explain with g(E) first. g(E) is the volume density of electron states per energy interval. N/L^3 is only the volume density of electron states, not per energy interval (dE). We have to account for this missing dE so we write g(E)dE. More intuitively what this means is that we can determine g(E) when we have some specific value for dE (i.e. what energy interval we are looking at). It is the same principle for g(k)dk except we have just substituted our E with k. Instead of requiring some energy interval we need to know what wavenumber interval we want to look at.
@librealgerien2 жыл бұрын
Math murdered with this g(k)=k^2dk. There is an easier derivation that doesn’t commit such an atrocity. Just work your way from E and relate it to the radius of the sphere sqrt(n_x^2+n_y^2+n_z^2).