In lecture 13 “basic principles of semi-conductors” question 2.4 asks you to compute the DoS with approximation. I am unsure how to approach this. In the beginning of the lecture notes it says the density of states can be computed by a sum of (dn/dk)*(dk/dE). I was able to calculate dk/dE which looks a lot like the answer but I am not sure if this is the correct approach and how to compute (dn/dk). Does anyone have a tip to help me further?
The scheme for computing the density of states is as follows:
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Write the number of particles as a sum over k states (varies depending on the dimension D):
N=\frac{V}{(2π)^D}\int d\boldsymbol{k} -
Expand d\boldsymbol{k} taking into account dimensionality:
1D: 2dk
2D : 2πkdk
3D: 4 πk^2dk -
Make a change of variables from k to E using your dispersion relation.
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Extract the density of states from the expresion:
N =\frac{V}{(2π)^D}\int g(E) dE
I believe this is how you learnt it in the first chapters and it is of course still applicable for semiconductors.
Hope thay helps!
Thank you very much! I was a bit confused as to what kind of dimensionality a semiconductor would take but I understand now. Just to be sure 3D is 4\pi k^{2} dk right? Without the “3/” as it is the surface of the sphere times dk.
Indeed, your are correct 
To be honest, I still don’t quite get the answer,
N=\frac{V}{(2\pi)^1}\int d\mathbf{k} \\ N=\frac{V}{(2\pi)^1}\int 2dk \\ N=\frac{V}{\pi}\int dk \\ E_c_b = E_G + t_c_b(ka)^2 \\ E_c_b - E_G= t_c_b(ka)^2 \\ \frac{(E_c_b - E_G)}{t_c_b}= k^2a^2 \\ \frac{(E_c_b - E_G)}{a^2t_c_b}= k^2 \\ \sqrt{\frac{(E_c_b - E_G)}{a^2t_c_b}} = k \\ \frac{1}{\sqrt{a^2t_c_b}} \sqrt{(E_c_b - E_G)} = k \\ \frac{dk}{dE_c_b} = \frac{1}{\sqrt{a^2t_c_b}} \frac{1}{2\sqrt{(E_c_b - E_G)}} \\ \frac{dk}{dE_c_b} = \frac{1}{2a\sqrt{t_c_b(E_c_b - E_G)}} \\ dk = \frac{dE_c_b}{2a\sqrt{t_c_b(E_c_b - E_G)}} \\ N=\frac{V}{\pi}\int dk \rightarrow \frac{V}{\pi}\int \frac{1}{2a\sqrt{t_c_b(E_c_b - E_G)}}dE_c_b \\\int \frac{V}{2{\pi}a\sqrt{t_c_b(E_c_b - E_G)}}dE_c_b \rightarrow \int g(E_c_b) dE_c_b
This is my derivation, but this implies that V = 2, I don’t quite get why that is.
If I take out the V/2Pi, then I am missing a pi in the answer, so it’s still wrong then, I really don’t get what I am missing here.
Edit: Shouldn’t 2D be pi * 2k?
I also thought it should be 2πk for 2D. At least that’s what the answers of the exam of 16 april 2019 question 3a seem to indicate. Is this correct?
The way I see it:
3D → area of a sphere with radius k (This is a 2D value which measures the outer 3D thing) = 4 x pi x k^2
2D → length of the circumference of a circle with radius k (This is a 1D value which measures the outer 2D thing) = 2 x pi x k
1D → (I can’t think of the geometrical meaning because we get to 0D) = 2
Yeps sorry. I wrote that in a random moment and made some mistakes. I think now it’s okay 
The V/(2π)^D doesn’t go into the density of states. I’ll check about the π at some point during the day.
Also don’t find a \pi in the answer. Probably a typo.
In the lecture Debye model, the section about Density of States, they do include (L/2pi)^3 in the integral.
Looking at the Sommerfeld (Free electron) model, once again I see that the density of states does include the volume.
Why are we omitting the V/(2pi)^D factor when we look at the semiconductors?
Is it because we look at the density of states inside k-space, and the first lectures looked at the density of states in real space?
I’m going to tell you things the way I learnt them, but perhaps @anton-akhmerov can clarify. Let’s focus on a 1D problem. The density of electrons is
n=\frac{1}{2\pi}\int g(k)dk=\frac{1}{2\pi}\int 2dk,
where you can already recognize the density of states as a function of k. The size of the system does not play a role at all. Then it is just a matter of making a change of variables to go from g(k) to g(E), with the particularity that you need to remember that the interval dk->dE also matters, such that
n=\frac{1}{2\pi}\int g(k)dk=\frac{1}{2\pi}\int g(E)dE
which for a parabolic potential E=ck^2 reads
n=\frac{1}{2\pi}\int 2 dk=\frac{1}{2\pi}\int \frac{1}{\sqrt{cE}}dE.
In the same way that the prefactor is not included in g(k), I do not include it in g(E). I am not certain whether this is a definition issue or some conceptual misunderstanding because this contradicts indeed what you saw in the Debye model. In case of doubt, I would stick to that in which you include the prefactor.
The question asks for the density of states per unit length. If you’ll keep the prefactor inside the density of states as usual, but divide by L, you’ll end up with the 1/pi factor, right? I guess that would explain why there is a factor 1/pi and not L.