Homework 3. Not due. Skip this and go to HW 4.

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 undoped 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 might 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. 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.

Extra Credit:
(note: Delta T is related to bandwidth and has nothing to do with temperature.)
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.