Density of States Derivation Part 2

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In this video, I complete the derivation for the density of states, and I show how finding the most convoluted way possible to count apples helps us do this.

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.

[蔡船船船]

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.

[srdhar]

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??

[stephan]

These videos are great. Love the top-down approach. Looking forward to, as Neel said, f(E) derivation

[menghanxu]

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

[kirakirievsk]

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

[ShanFardeen]

can someone tell me why is the volume taken as ( pi/ L)^3 in one step and L^3 in another

[ipinlojunureni]

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

[nl]

In which video do you find f(E) : probablity that the states are occupied

[T_tintin]

Damn I was suffering with for so long ..finally i get it!!! Thank you so much!😭😭😭😭

[顧峻愷]

Why do we consider the 1/8 spherical shell instead of the 1/8 sphere?

[yewedmund]

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?

[deepvision]

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.

[arduinoguy]

Sir, when the value of “h” is (6.625*10^-34)^3 the result is 0. May I know how to fix this error ?

[omercohen]

Shouldn't the spacing between k points be 2 pi over L?

[_saqibjaveed]

Why are we dividing by L^3 ..while the volume under consideration is 4πk^2 delta k

[oraange]

a state (solution of Schrödinger Equation ) occupies a volume ?

[swsaber]

can we do these things in real space instead of k space because volume in k space little confusing .. :(

[nitismishra]

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?

[rafiamurtaza]

why we used momentum space (k space) when we were doing fine in real space (r space) ?

[SiddharthRanjan]

A single state can have only one electron. So why did you multiply by 2 ??