Unless otherwise specified, for the following assume that you are dealing with a 3D semiconductor* for which: \(E_g = 1 eV, \quad kT=.025 eV\) and
\(D_c = 4 \times 10^{21} states/eV*cm^3, \quad B_c = 6 eV\)
\(D_v = 8 \times 10^{21} states/eV*cm^3, \quad B_v = 3 eV\).
(You can assume that for the undated case there are exactly enough electrons to have a filled valence band and an empty conduction band.)
1. What value of \(E_f\) corresponds to \(n = 10^{15} electrons/cm^3\)? What is p for this case?
2. What value of \(E_f\) corresponds to \(n = 10^{16} electrons/cm^3\)? What is p for this case?
3. What value of \(E_f\) corresponds to \(n = 10^{17} electrons/cm^3\)? What is p for this case?
4. What value of \(E_f\) corresponds to \(n = 10^{18} electrons/cm^3\)? What is p for this case?
5. What value of \(E_f\) corresponds to \(n = 10^{22} electrons/cm^3\)? What is p for this case?
6. What is the value of \(E_f\) for this semiconductor if it is undoped. What are n and p in this case?
7. What value of \(E_f\) corresponds to \(p = 10^{15} holes/cm^3\)? What is n for this case?
8. What value of \(E_f\) corresponds to \(p = 10^{16} holes/cm^3\)? What is n for this case?
9. What value of \(E_f\) corresponds to \(p = 10^{17} holes/cm^3\)? What is n for this case?
10. What value of \(E_f\) corresponds to \(p = 10^{18} holes/cm^3\)? What is n for this case?
11. For each of the above questions one assumes that the valence and conductions bands are -- ------- -----------. What is the missing phrase (3 words)? This will turn out to be important.
12. a) Graph \(E_f\) as a function of n or \(E_f\) as a function of p or maybe n and p as a function of \(E_f\). What graph(s) best illustrates the relationship between carrier density and \(E_f\)? Discuss this here!
b) Added note: Here is an idea for a plot that I think will illustrate this relationship. For a 1000 atom crystal suppose there are 3 bands, with bandwidths of 1 eV, 3 eV and 6 eV, respectively and band gaps of 2 eV and 1eV. Plot the number of electrons in the crystal as a function of \(E_F\) starting with \(E_F\) below the lowest band and ending with \(E_F\) above the top of the 3rd band. what is the domain of this graph?
[additional note: This is a theoretical exercise, not something that one can do with an ordinary material. (Although there are unusual materials where something like this is possible via "gating".]
13, 14 and 15 were edited on Friday at 8 PM: (this version is less difficult)
13. a) What value of n do you get for \(E_f\) 0.3 eV below the conduction band edge?
b) What value of p do you get for \(E_f\) 0.3 eV above the valence band edge? Is this value of p larger or smaller than the value of n you got in part a)? Or is it the same? Explain.
c) What value of n do you get for \(E_f\) 0.3 eV above the conduction band edge? How does this compare with the value of n you got in part a)? Discuss.
14. Density of states near the bottom of a conduction band in two-dimensions (2D): Assume a conduction band dispersion relation for a 2D crystal of the form \(E_{3,k} = E_3 - \Delta T cos(ak_x)- \Delta T cos(ak_y)\), where kx and ky range from \(-\pi/a\) to \(+\pi/a\).
a) What is the bandwidth for this band?
b) Derive the approximate form of E vs kx and ky near the bottom of the band in the effective mass approximation. (That is, to quadratic accuracy.) extra credit: Over roughly what range of energy or k is this approximation reasonable? Post comments here on this? You can ask, "what is reasonable" or speculate about what might seem reasonable to you.
c) Calculate the density of states as a function of energy near the bottom of the band.
15. Density of states near the bottom of a conduction band in three-dimensions (3D): Assume a conduction band dispersion relations for a 3D crystal of the form \(E_{3,k} = E_3 - \Delta T cos(ak_x)- \Delta T cos(ak_y)-\Delta T cos(ak_z) \), where kx, ky and kz each range from \(-\pi/a\) to \(+\pi/a\).
a) What is the bandwidth for this 3D band?
b) Derive the approximate form of E vs kx and ky near the bottom of the band in the effective mass approximation (that is, to quadratic accuracy) and use that to calculate the density of states as a function of energy near the bottom of the band. Is this D(E) similar to the one you got in the previous problem or different? Plot them vs E and discuss.
c) extra credit: What effective mass corresponds to a bandwidth of 10 eV in this 3D case.
Good question. How about if we all set \(E_v =0\)? Even though these are all bound state energies (and so they are naturally negative) it would be convenient for the purposes of this problem to put the zero of energy at the top of the valence band. The zero of energy is pretty much arbitrary, so I think we can do that. It will make it much easier to express the answers to these problems.
ReplyDelete(Just so that we don't forget that these are bound state energies and that this zero is an arbitrary choice.) Does that make sense?
E_g is short for E-gap. It is the energy gap between the valence and conduction band. (Between the top of the valence band and the bottom of the conduction band.)
ReplyDeleteSo basically, yes.
That makes good sense. I thought about placing the zero at the top of the valence band as it would simply the calculation, but I was not sure I could validate my reasoning for doing so. Thanks!
ReplyDeleteI was thinking of something else. It might seem a bit random. What is in the fermi function beside \(E_F\)
ReplyDeleteAnd I guess I should mention that \(E_V\) is the top of the valence band. Or does everyone already know that?
ReplyDeleteSo I got the chance to start on the homework. I am wonder if electron and states are units. For some reason, I can't somehow to remove them in my calculations for 1-8.
ReplyDeleteI was wondering the same thing. For example, my answer for #7 is in terms of states, not electrons, which is how n was described in previous problems.
DeleteIf you think of the Fermi function as being electrons per state, then that gets you from states to electrons.
DeleteOne can also regard the Fermi function as unitless, and in that case just drop the states.
hmmmm, I wanna say that the bands will have the same temperature because they correspond to the same object and by increasing the temperature you just make the electrons in the object go to a higher energy state on average.
ReplyDeleteI thought that having the temperature constant is the assumption, but I'm less sure after reading this if we can say that the T in the n and p equations is always the same T.
I was thinking along similar lines. In my thinking, the 2nd word starts with t and the 3rd word starts with e.
ReplyDeleteHere is an idea for a plot that I think will illustrate the relationship between Ef and electron density as requested in problem 12.
ReplyDeleteFor a 1000 atom crystal suppose there are 3 bands, with bandwidths of 1 eV, 3 eV and 6 eV, respectively and band gaps of 2 eV and 1eV, for example. Plot the number of electrons in the crystal as a function of \(E_F\) starting with \(E_F\) below the lowest band and ending with \(E_F\) above the top of the 3rd band. what is the domain of this graph? (the vertical scale)
Good point/ question! Let's think about how to do that for problem 14 (2 dimensions). First, I would say that the actual number of states would be 2N in the band, as you say, and that those states occupy a square in k-space. The area of that square is 4 pi^2/a^2. Within that square there are 2N allowed values of k. Now suppose we consider the approximate form where E is proportional to k^2. There is a circle in k-space that has the same area as that square. Consider the k value that defines that circle. I think if you integrate the D(E) that you got up to the energy corresponding that k, and set that equal to 2N, then that would give you a valid normalization. Having done that, then go back to viewing only the lower energy part of that D(E) as actually accurate. I hope that makes sense.
ReplyDeleteFor problem 12 I am still unsure of how to do the plot. My first idea was to graph the n function we have but to sum three exponential terms each with a different E_c for the bottom of the three bands. This gives me an extremely sharp graph that predicts a huge amount of electrons. I’m unsure how to account for the N=1000 in my calculation to maybe reign in my prediction. Any insight would be much appreciated.
ReplyDeleteYou don't need to do any calculation. Use your understanding to inuit what the graph must look like and what the scales must be.
DeleteYou can assume T= room temperature.
DeleteOkay so its more of an intuition problem than an exact calculation. Then i would say that each band will gain a maximum on 2N electrons for the range of this graph.
Delete"So I assume that logic could hold true for the case where our domain is a cube as opposed to a square, problem 15, where there would be a sphere in K space that would have the same area as the cube?"
ReplyDeleteI agree.
Regarding 2N, I am assuming that N means the total number of atoms in the crystal, so in 1D, N=L/a; in 2D, N=(L/a)^2... Does that make sense?
So if L/a were 1000 (atoms) then in 1D there would be 1000 atoms and in 2D 10^6 atoms
Is Ec the top of the conduction band as well, or is it the bottom of the conduction band?
ReplyDeleteThe bottom. So then Ec-Ev=Egap. Does that make sense?
ReplyDeleteHmm, yea that could be difficult. Maybe approach to this way. Using that E(k) is isotropic one could write \(2 \pi g(k) d^2k = D(E) dE\) where g(k) is the k-space density of states, which does not depend on k.
ReplyDeleteoops. I should have written just \( g(k) d^2k = D(E)dE\) for that line.
ReplyDeleteThen that can be rewritten as:
ReplyDelete\( 2 \pi g(k) dk = D(E)dE\) (Jacobean in 2D i think?).
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ReplyDeleteoops. make that: \( 2 \pi k g(k) dk = D(E)dE\)
ReplyDeleteSo then \(D(E) = 2 \pi g(k) k /[dE/dk]\).
ReplyDeleteDoes that make sense?
(This takes full advantage of the isotropic nature of E vs k in this approximation.)
Yes, that was what I was thinking, but I just wanted to make sure. Thank you.
ReplyDeleteThat is exactly the problem that it solves. E is proportional to just k^2, not to two dimensions of k. It is just a simple function of one variable. That k is a scalar.
ReplyDeleteWell, it is proportional to 1, but I think it is actually something like 2N/(2 pi/L)^2 (in 2D) since you want to integrate it over kx and ky and get 2N.
ReplyDeleteAssuming ranges of -pi/a to +pi/a.
ReplyDeleteThanks Louis, I did use those formulas but I am not understanding what I am doing really, so when it comes to the tough problems like 13-15, I am lost.
ReplyDeleteYes, exactly. You can use the model from the 1st problem. Good point.
ReplyDeletewhat lecture did we get play with 2d crystal bands???
ReplyDelete