HW 5 with Solutions.

1. Using the equation we obtained for n(x) last Thursday (April 24) for an n-p junction with an applied voltage that pushes electrons to the right:
a) Calculate the diffusion current in the region \(x \ge 2d\).
b) How does the size of this current depend on the doping level of the p-type material? How does it depend on the applied voltage?
c) How does it depend on the recombination time?
d) Does our result from problem 1 for the overall magnitude (prefactor) of the current seem correct? I haven't really checked any books regarding this. Is there anything wrong with this result for the current prefactor? Does our result agree with what other people get?

There are more solutions, as well as the original problems, below the break. I think for 5 it should say "electric field". Please post any comments below.

HW 4 and HW 4 solutions

Here is a solutions link for HW 4. Please feel free to comment.

https://drive.google.com/file/d/0B_GIlXrjJVn4eGRGLWNJR1g3TXc/edit?usp=sharing


I was thinking that, now that we have the motivation of wanting to understand a p-n junction, maybe we should back up a little and review the different kinds of current (drift and diffusion). How does that sound? Also, perhaps you would like to read about the two kinds of current:
1) drift current (current directly associated with the force that an Electric field exerts on an electron.
2) diffusion current (current associated with a density gradient!).
Also, you can read about the continuity equation for electrons in a semiconductor, and the recombination time.  I think that is related to the life-time of an excited electron (and there is one for holes as well). Here is a guiding question for that: suppose you excited some electrons into the conduction band (with photons) and thus created a state not in (thermal) equilibrium. What would happen over time after that? What mathematical description might be suitable to describe the return to equilibrium? What would the characteristic time scale be? What is it called?

For the following problems let's use: \( D_2 = 0.6 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.3 \times 10^{22} \, states/(eV-cm^3)\). (Or would you rather do them with D2 and D3 equal? comment here if you have an opinion on that. (simplicity vs generality))

Warm-up problems related to \mu.
1. Calculate the intrinsic carrier concentrations (\(n_i\) and \(p_i\)) for a semiconductor:
a) with a gap of 1 eV at room temperature.
b) with a gap of 0.25 eV at room temperature.
c) with a gap of 0.25 eV at 60 Kelvin.

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?

Density of states and band structure for Si.

Integration

You probably already know this, but anyway, here is and approach to integration that you may find useful. In this class we want to get results and not get bogged down in integration methods any more than necessary. In this class, most of our integrals will be definite and thus simpler than indefinite integrals.

Band Theory video lecture

Below are videos on:
1) Energy bands in crystals
2) Effective mass
Let's discuss these here and, then also have a live discussion on Friday at 5:30 PM.

This is a video on bands and bandwidth. An isolated atom has bound states. When an infinite number of atoms are arranged to form a spatially periodic crystal, we find that there are an infinite number of electron (energy) eigenstates, for the crystal, coming from each single atom eigenstate. Please post any questions, thoughts comments, associations, etc. here.




Here is a video on effective mass:

About Physics 156.

My office is ISB 243.
The best way to contact me is by email at:
zacksc at gmail.com
Please use just this email (not any other emails you may have for me).

Our final, assuming I am reading the UCSC schedule correctly, is on Wednesday, June 11, 2014. That is scheduled by UCSC and set in stone. Other things are not completely certain at this time, but I expect to have a midterm close to the middle of the quarter.

Our midterm will take one full class.  We'll see if we want to have quizzes or not.

Homework will be due more often on Tuesday than Thursday, but both are possible. Expect a lot of homework. Please keep all your returned HW in a homework portfolio to be turned in at the final.

This Blog will play a key role in the class. Please check it frequently and please participate in the discussions here. This is the place to ask questions.

Book: For people who like to have a book, Streetman, 6th or any edition, is probably pretty good. I think it is available for about $25 on ABE books and other sites, but it is expensive on amazon for some reason.

We will start with and examination of energy bands in crystals. A crystal is a system made of atoms in a perfect periodic arrangement. Energy bands refer to the crystal energy eigenstates (wave-functions) that arise when we solve the problem of a single electron in a spatially periodic potential. This is analogous to the manner in which the periodic table provides a starting point for understanding the electron configuration of atoms. These approaches emphasize symmetry.  That is spherical symmetry for the case of an atom, hence the s, p, d, f nomenclature, which comes from grouping energy eigenstates according to their symmetry (technically the representation of the rotation group to which they belong, but never mind that now). For the crystal it is a sort of remnant spherical symmetry, hence the s, p, d and f origins tend to survive, along with a partial translational symmetry (e.g., invariance under translation by "a"). (The translational symmetries can get more complex in 2 and 3 dimensions. Graphene is an interesting example of a 2D crystal structure if you are curious.) Crystals can be either semiconductors (Si, GaAs) or metals (Au, Fe, Pb). (An intrinsic semiconductor is essentially an insulator with a small energy separation between bands.) Bands are the natural starting point to understanding semiconductors and semiconductor devices, metallic behavior, magnetism, superconductivity and pretty much anything else that is based on the quantum behavior of electrons in a periodic potential.

So my plan is to start with bands. (The approach we are taking is called "tight-binding" (bad name) or, more descriptively, linear combination of atomic orbitals (LCAO). Reading about "free electron bands" may confuse you; that is a different approach that is not helpful for us and less reflective of the nature of anything real.) Then after "bands" I am thinking that we will transition quickly to understanding doped semi-conductors and then some semi-conductor devices such as p-n junctions and FETs (MOSFET for one).  Maybe also how electrons can be confined to the surface of GaAs and how a 2D metal forms there, isolating single electrons and other stuff. I am curious to learn about your level of interest in various topics...

If you want to read about something I would recommend reading about doped semiconductors, chemical potential, p-n junctions and FETs and other semi-conductor physics and device stuff.

Homework 2 (some solution notes added)

Please point out any errors, and ask questions and share thoughts here freely. Problem 7 is now finished (updated) following a discussion of that in the comments section (and it includes a special extra-credit part!).
     The assignment had two distinct parts. Problems 1-4 relate to finding the bands (of electron states) for our microscopic 1D model. If you get stuck on 4, which is pretty difficult, I would recommend skipping it and going to 5. Also, you can ask questions (about 4) here, and that is encouraged.
Problem 5 is a transitional problem. It introduces the Fermi function. It is important at this stage to be able to visualize the Fermi function as a function of energy! The energy dependence is the important part.
Problems 6-8 use a somewhat simples DOS and involve calculations of what states are occupied, using the Fermi function. While this DOS is simpler that the one you would get in problem 4, it is still not trivial and understanding the juxtaposition of bands (regions of non-zero DOS) and gaps (regions of zero DOS) is very important. In fact, that, along with visualizing the Fermi function vs energy, is probably the most important thing.
PS. I think 8 is a pretty difficult problem, really 2 hard problems molded into one. Leave yourself enough time for that. Get actual numbers; accurate numbers. The part where you calculate the number of empty states in the valence band involves an approximation for the Fermi function that is somewhat more difficult than the approximation we use for the Fermi function in the conduction band.

1. Our key result from Tuesday's class was:
 \( E(k) = E_1 - 2t \, cos(a k) \)
where
\( t = - \int^\infty_{-\infty} \phi_1(x) v(x-a) \phi_1(x-a) dx   \)
 a) What does \(E(k) \) represent?
b) What does \(\phi_1(x) \) represent?
c) What does \(v(x) \) represent?
d) What does \(E_1\) represent? Thinking about its origins, would you think that \(E_1\) would be positive or negative? Why?
e)  What does \(\psi(k) \) represent?

2. Assume that t is positive and that the magnitude of \(E_1\) is about 10x larger that the magnitude of 2t.
a) With just your understanding of the cos function and the relative magnitude of \(E_1\) and t, sketch a thoughtful plot of \(E(k)\). Graph only from \(k= - \pi/a\) to \(+\pi/a\). Take your time on this. Do it thoughtfully and look at it.
b) Where is \(E(k)\) largest?  Where is it smallest?
c) What is the difference, in energy, between those highest and lowest points?
d) Looking at this graph, what might you call the bandwidth? (That is, how would you define bandwidth?)
e) extra credit. How would you define the effective mass associated with this band? To what part of the band does effective mass refer? How are effective mass and bandwidth related?

3. Extra credit: Suppose that \(\phi_1(x) = \frac{1}{\sqrt{b}} e^{-|x|/b} \) (This is a reasonable atomic wave-function in the sense that it decays exponentially away from a cusped center (like an H-atom ground state). Additionally, suppose that  \(v(x) = -\alpha \delta (x) \) (a delta function).
a) Do you know how to do an integral involving a delta function? If no, ask someone about it or pst a question here. (A lot of people don't seem to know how to do this.) Computationally, it is really easy to do the integrals once you know what to do. Conceptually, think of the delta function potential as a very, very narrow square well. So narrow that \(\phi_1(x)\) is pretty much the same everywhere inside the well, which is all you need to know to integrate since \(v(x)\) is zero outside the well.
b)  Calculate t. (The overlap integral)
c) Use t to obtain an expression for \( E(k)\) that is based on the approximation of keeping just the n=j=1 and n=j=-1 terms from "the sum".
d) Plot \(E(k)\) from \(k= - \pi/a\) to \(+\pi/a\). What is the bandwidth?  Assuming \(E_1\) is -16 eV, what is the lowest energy state in the band and what is the highest?
e) What are the largest terms form the sum involving \(v(x\) and also from the other sum (over just n). Explore whether they are actually smaller than the terms we kept (e.g., n=j=1) and how including them might alter the band, \(E(k)\).

4. Starting with the 1D band-structure:
\( E(k) = E_1 - 2t \, cos(a k) \)
where  \(E_1\) is the atom state energy of the atom state from which the band is derived,
t is not time but is an overlap integral, as we discussed in class on Tuesday (except that we changed the sign of t so that it would be positive for a band made from a symmetric state. a is the lattice parameter, and k is the key variable.
a) What are the units of density of states as a function of energy for our 1-dimensional crystal?
b) What is the density of states as a function of energy? To calculate this you can start by assuming that states are uniformly distributed along the k axis. Also, you can assume that the total number of the states in the band, per cm, is equal to the number of atoms per cm.
[thoughts on problem 4: One can find the energy dependence by differentiating followed by substitution to eliminate k (and get things just in terms of E). I think you will get a density of states (as a function of energy) with "integrable singularities" (at the band edges). The integral of D(E) over the band should be related to the number of atoms in the crystal, since there are two states per atom if you include spin (one state per atom if not). Please post questions, thoughts or comments here.] There are some solutions here and in the link.

https://drive.google.com/file/d/0B_GIlXrjJVn4eXg0NlNzV21JMk0/edit?usp=sharing
There are more problems below this break.