Homework 3 & solutions link.

Here is a link to solutions:
https://drive.google.com/file/d/0B_GIlXrjJVn4YnByRnF3bDJaQVE/edit?usp=sharing

Please comment on any errors you find or any questions or thoughts you have. I recommend starting soon. I doubt this set this set can be done in one sitting. 2, for example, is a bit difficult and time consuming I think, but you may learn a lot from it. 5 and 6, which involve diffusion current, are way harder, and also interesting imo. Note the break in this post, and that there are problems below the break!! As always, a lively ongoing discussion is the best way to tackle these problems.


1. Consider a semiconductor with a 1 eV gap and with: \( D_2 = 0.6 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.3 \times 10^{22} \, states/(eV-cm^3)\).
a) Suppose it is doped to a level of \(n = 10^{17} \,electrons/cm^3\). What is \(\mu\)?
b) Suppose it is doped to a level of \(p = 10^{17}\, holes/cm^3\). What is \(\mu\)?
c) Which \(\mu\) is larger? By how much? Are they both in the gap? What are the units of \(\mu\)?
d) Which  \(\mu\) is closer to "its band edge"? why?


2. When two things are in intimate contact,  \(\mu\) has to be the same in both (in equilibrium). In this sense \(\mu\) is like temperature. ( \(\mu\) is to carrier density as T is to entropy.)      Suppose you put the material from 1a) into intimate contact with the material from 1b). (Say there is an interface at x=0 and for \(x \le 0\) you have the material from 1a) and for \(x \ge 0\) you have the material from 1b).
a) What could happen near the interface to enable \(\mu\) to be the same on both sides?
b) Okay, that is pretty difficult. Maybe we can break that difficult problem into parts? Maybe a bunch of electrons from the left side sneak over to the right side? If that happened, would the crystal still be neutral (uncharged) everywhere? If not, qualitatively describe and sketch the net charge density as a function of x.
c) Hmm, Let's make a radical approximation. Let's assume that, within a distance d of the interface, all the electrons that would normally be in the conduction band on the left go to the other side. Let's assume that they fill any empty states in the valence band. With that assumption, plot the net charge density as a function of x. What is its maximum value?
d) Calculate and plot the electric field (associated with that charge) as a function of x. [hint: Gauss's law (GL) can be used to get the value of the E field at x=0 and also that E is zero outside the charged region.] Where is the maximum of E and what is the form of the dependence of E on x? (Feel free to discuss this here.)
e) What is the maximum value of E? Does that depend on d?
f) What is the difference in the potential energy of an electron from one side to the other due to the charge layers? (in eV) Does that depend on d?

3. When you have found the difference in potential energy (2f) that equals the difference in mu (1c), then you have found the equilibrium state of the junction.
a) do that and thereby determine d.
b) Would d be different if the doping levels were different? Discuss.
c) (x cred) email me your cogent discussion of the influences of doping on d.

4. Did we leave something out that you need to solve problem 2? something unit-less? something related to E&M? (It is no big deal if you solved problem 2 without that. It is a pretty easy correction to put in after.)

5. a) Why does a "barrier" form around the interface anyway? What role does it play?
[hint: I think that the reason it forms is to inhibit, stop and/or cancel out what is called "diffusion current". What is diffusion current? Diffusion current refers to flow from regions of higher density to lower density.  In this case it is an electrical current associated with an electron density that depends on x. Focusing on the CB, \( J_{diff}(x) = q D_n \frac{dn(x)}{dx}\), where q is the charge of an electron, \(D_n\) is the diffusion coefficient, n(x) is the density of electrons in the CB and J(x) is current density (coul/(sec-cm^2)) in the x direction.]
a') What are the units of \(D_n\)?
b) Using the electric field, E(x), that you calculated in problem 4, calculate the associated potential V(x) for the left hand side of the junction. To make this mathematically less messy and easier, let's do this only for the left side, and let's (for this problem) move our zero of x to the left hand side of the junction region. That way V(x) is proportional to \(x^2\), I believe.
c) Use this to calculate n(x). [mu is constant now and the band moves away from mu as x increases.] Your range of x for this is 0 to d, i.e., from the left edge of the "junction region" to the center. n(x) is proportional to an exponential involving the energy difference between the band edge and mu.)
d) graph n(x) vs x.
e) calculate the diffusion current and, f) graph it as a function of x.
g) which way does this current flow? which way do the electrons flow?

6. a) For the same range of x, calculate the "regular current" (proportional to the electric field, E(x)), i.e.,  \( J_{E}(x) = q^2\, \tau \, n(x) E(x)/m^*\).
b) graph this current vs x.
c) What is the direction of electron flow for this current? How does it compare with the electron flow from diffusion?
d) Is there a relationship between \(q \, \tau/m^*\) and \(D_n\) that would make these currents equal and opposite at every value of x so that they perfectly cancel?

optional:
7. How would you describe the exponential growth of current in an appropriately biased n-p junction? That is, what happens to our perfect equilibrium? Does one current becomes dominant or do both change or ...?

8. Looking at the actual density of states for Si, posted below, one can see why we would just use a constant density of states approximation. However, there is another possible approach that might be equally reasonable. That is, one could assume that near a band edge there is a single quadratic minimum in E vs k that dominates the DOS (near the edge).
a) Calculate the density of states in 3D in terms of m* using that approach. [I think it has a simple dependence on E, perhaps proportional to \((E-E_c)^{1/2}\) for the CB, for example?]
b) Calculate the density of electrons in the conduction band as a function of mu, for mu in the gap and not too close to the CB, using that DOS.
c) which approach (constant DOS or this one) do you prefer? (email me if you like)