CuO2 bands.

This post deals with the nature of the CuO2 bands (within the compound \(La_2CuO_4 \) ). The 3 bands that one can construct from the \(O-p_x, O-p_y, Cu-d_{x^2-y^2}\) orbitals are somewhat interesting in their own right, and they also provide a format in which to understand how bands acquire "mixed character" and what it means to say a band is 70% Cu / 30% oxygen or something like that.

This first video derives the LCAO (linear combination of atomic orbitals) nearest-neighbor matrix. Maybe later I can do another one that looks at and interprets the eigenvectors. Feel free to do that on your own also. What eigenvectors do you get at k=0 and how would you interpret them in terms of, e.g., percent of Cu content? This is just exploratory, so feel free to make assumptions about the magnitude of t (maybe around 1 to 6 eV?) and about the diagonal terms, Ed and Ep.



Hmmm... I see now that my k=o suggestion has some problems and limitations. Why is that? What happens at k=0? Something kind of interesting, but a bit special compared to everywhere else in the BZ.

Here is another idea. There might be some specific zone-boundary spots that would be interesting to explore. Places where something is pi/a; or both things are pi/a. That might be more interesting and better. I think for this compound it is the zone-boundary regions that are most important and interesting.

Notes on the final.

A couple of notes:
On problem 1 b) an estimate just based on the location of \(E_F\) within the band is pretty accurate. There is no advantage to integration. (By the way, for 1a, if you did not have a calculator, but you got the form correct that is going to get you most of the points. I think 1a was the only problem where you needed a calculator.)  What do you get for problem 1?

In problem 5a the voltage across the resistor or junction is essentially the open circuit voltage which is 1.5 V.  The \(e^{-60} I_{ill}\) tells you that the highest voltage possible is 1.5 volts since (1.5 eV/KT) = 60. I would say this is a "large resistance" in this context. What do you think? What did/do you get for I?

In problem 5b you can ignore the exponential part of the I-V relation. The \(V^2\) term is dominant everywhere below 1.3 volts or so, I think. With just the constant negative term, \(I_{Ill}\), and the quadratic term, I think you can solve for the maximum power in closed form. (differentiate and set to zero). That is the approximation that would really help for that one. I am not sure how many people got that. What R value do you get for maximum power?

Problem 6 I believe the total energy has two off-center minima representing ferromagnetic states for alpha greater than one eV.  The x=0 point is the stable equilibrium point for alpha less than 1 eV, but becomes unstable above that value of alpha. This illustrates the nature of the instability leading to the ferromagnetic state when the bandwidth is narrow and the interaction is strong. Does anyone have an idea where those off-center minima are for alpha = 1.4?

Any comment or thoughts you have, feel free to share them here.

Final thing: 2 page email pdf.

As we discussed, please send me a roughly one or two page document in pdf or plain text format, equations are fine, but not required at all. Please focus on one or two things that you found particularly interesting --what they are, why you found it interesting including a clear and cogent explanation of your chosen topic and anything else related to that. (The length suggestion is just a rough guideline. Whatever length seems appropriate to you is fine.)
zacksc@gmail.com    (pdf format)

Thanks very much. I really enjoyed this quarter.
Best,
Zack

Last minute advice and thoughts.

There will be no problem on FETs.

There are a number of short problems that put emphasis on conceptual understanding and descriptions, as well as some longer problems where appropriate approximations are important. Sleep is good.

1. What essential aspect allows a semi-conductor to function as a semi-conductor?
--a gap and a malleable (easily move-able) chemical potential. you can dope it.

2. What is one essential key thing you need to understand to be able to understand and model inhomogeneous semi-conductor systems like junctions and FETs?
--band bending

3. What essential characteristics do metals manifest?
--a Fermi boundary (inside which essentially all states are filled). Very quantum. Pauli exclusion plays a big role.

4. Why are some metals weird?
--electron-electron interactions can make metals do strange things. Unusual collective behavior, for example ferromagnetism, anti-ferromagnetism in Mott-Hubbard insulators and superconductivity, results from electron interactions.
(Normal metal are sometimes unstable. What is the origin of instability.)
--e-e interaction 

Summary/preparation post II.

I was trying to think of something profound to say about the test. How it will test your understanding and will be the kind of test where getting a lot of sleep the night before will be really important. But I haven't been able to conjure up the right way to say that yet. Anyway, regarding Wednesday, please come prepared with your questions, and also feel free to ask more questions here.

And as you study your more detailed list of topics remember these essential things:
1. What essential aspect allows a semi-conductor to function as a semi-conductor?

2. What is one essential key thing you need to understand to be able to understand and model inhomogeneous semi-conductor systems like junctions and FETs?

3. What essential characteristics do metals manifest?

4. Why are some metals weird?
(Normal metal are sometimes unstable. What is the origin of instability.)

Polls are broken.

There is a system-wide blogspot polling problem. If you would peruse and remember the choices on the problem preferences poll and give me your feedback tomorrow in class, that would be very helpful and appreciated. Also, please comment here on your problem preferences, positive and negative.

Tuesday Class.

I am thinking that this Tuesday's class should be at least partially a question-driven review of key topics. Please come prepared with your questions! Also, you can post your questions here.

Summary/preparation post.

In addition to understanding how to do problems, I would suggest seeking an understanding of the underlying reasoning and concepts. This may be obvious, but I though I would mention it anyway. Also, I recommend that you save time by making appropriate approximations wherever you can. The ability to recognize and use approximations shows understanding and confidence (or as someone once said "if you see a fork in the road, take it").

You'll be asked to calculate n for a given mu or visa versa.  You will wish to have a good understanding of the fermi function and its approximations -- including how to use approximations to get n to very few significant figures quickly. If you don't have a calculator you may want to bring a  table of e^{-n} where n is an integer. Also, be able to sketch, understand, visualize and estimate the area of integrands. Understand how the Fermi function pertains to metals as well as semiconductors.

Understand Fermi surfaces and how to construct and visualize them in novel 2D k-space.

Understand band bending and how it pertains to junctions and FETs. Understand "everything" about a forward biased or illuminated p-n junction. Understand circuits, the relationships between current and voltage.

What else? Your input, comments and questions are welcome and encouraged!

Draft practice problems:
1. Consider a crystal for which the density of states is given by a half cycle of a cos function centered at \(E=E_o\). That is,
 \( D(E) =D_o cos ((E-E_o)/b) \, states/(cm^3-eV) \).
Outside that half cycle, D(E) is defined to be zero.
a) plot the density of states. what is the band width?
b) Suppose that the total number of states in the band is 10^22 cm^{-3} and KT=0.025 eV. Estimate, calculate or determine the carrier density to just one sig fig for \(\mu\) :
0.5 eV below the band edge
0.2 eV below the band edge
0.2 eV above the band edge
1.0 eV above the band edge
\(\mu = E_o\)
Make each calculation as simple as you can by making appropriate approximations.

2. Sketch a Fermi function with \(\mu = 0.5 \, eV\) from 0 to 1 eV for: a) KT=0.05 eV,  and, b) for KT = 0 (or .00001 eV if you prefer).
c) Sketch the integrand for a calculation of n.

3. Sketch the bands and chemical potential for an unbiased and a biased n-p junction.

4. Describe the journey of: a) an electron that is created in an illuminated n-p junction, and b) one that travels through a biased junction in "LED mode".

5. Explain the concepts associated with an illuminated n-p junction in series with a simple resistor. What does the resistor represent? What does it do? Why does its R value matter to how much power is generated? How does the resistor effect the n-p junction? How does the junction effect the resistor? What are the concepts involved in optimizing power generation?

6. Sketch the bands of a MOSFET, with an inversion layer, that is "on" due to:
a) an applied bias voltage
b) due to a work-function "mismatch" issue (normally on).
c) what is an inversion layer?

7. Illustrate and explain the origins of a ferromagnetic instability of a half-filled band. What is the possible benefit, to a metal, of becoming ferromagnetic?

8. Illustrate and explain the origins of a mott-hubbard instability of a half-filled band. What benefit can such a Mott insulator gain from having its spins arranged antiferromagnetically?

9. Sketch the structure of graphene. Why are there 2 pz derived bands and not just one? How many total filled bands are there in graphene? Explain, and put them in groups according to nature or function.

10. Here is a problem related to ferromagnetism. I am not sure how computationally difficult and time consuming it is, and it may not be worth your time, but on the other hand, it might illustrate the nature of ferromagnetism in an interesting way. Using the D(E) from problem 1 above, and assuming the band is half-filled in the normal state, with a density of electrons, n. We can divide the density of states into two identical parts, one for spin-up and the other for spin-down electrons.
a) Calculate the band energy by integration. How does the band energy depend on \(n_{\uparrow}-n_{\downarrow}\)?
b) For what range of bandwidth would a ferromagnetic state \(n_{\uparrow} \neq n_{\downarrow}\) have a lower overall energy than the normal state \(n_{\uparrow} = n_{\downarrow}\) , given that there is an iteracttion energy in addition to the band energy that can be approximated as (4 eV) n - \((U/n) (n_{\uparrow}-n_{\downarrow})^2\) where U = 2 eV.
c) Explain where such an interaction energy could come from and how that origin involves Fermi statistics.
d) extra credit: Approximately what value of bandwidth would lead to \(n_{\uparrow} = 6 n/10\). [Hint: expand the band-energy term around \(x= n_{\uparrow}-n_{\downarrow}=0\) and keep only the terms you need (probably x^2 and x^4).]

Wednesday Review. Pizza issues.

On Wednesday at 5:00 PM we will have a special meeting in ISB 235. There we will review, discuss, and ask questions about everything we have learned. This is important. Please don't miss it. There will be Pizza. Chris Kinney is in charge of getting the pizza and perhaps other things to eat.  Please post here regarding what kind of pizza you prefer, etc.

Things to think about:
1. what is the essential thing (or things) that makes a semi-conductor a semi-conductor?

2. what one key thing (or maybe more) do you need to understand to be able to model and understand most inhomogeneous semi-conductor systems (that is, semi-conductor devices?

3. what is the essential thing (or things) that makes a metal a metal?

4. Why are some metals weird?

Think about it and perhaps try to work toward really short answers (while still retaining in your mind the sense of nuance and complexity that these subjects deserve). For example, one might be able work toward two-word answers for 2, 3 and 4, more or less.

Crib sheet link.

https://www.writelatex.com/1111062ymybkn

This will be everyone's crib sheet for the final. I think you can edit it (add stuff).

Video on graphene band calculations.

I'll post a video here later today. This structure provides a perfect example of both lattice structure and band calculations*. If you understand how this structure is characterized, mathematically, then I think you can understand all crystal structures. Additionally, if you understand to some degree the band (E vs k) calculation for this structure, and why there is a 2x2 matrix, then I think you will have the potential to understand a wide range of band calculations.

*Band calculation means how you get from an atomic "orbital" to a periodic crystal eigenstate, and the energies of those crystal eigenstates, E(k), relative to that of the originating atomic orbital.



Here is part 2. This completes the calculation and also discusses the concept of Brillouin Zone, which is very important! (Why do we invoke a special zone? What is the reason?)

HW 8: 2D crystal carbon.

Hmm. I think it probably makes sense not to try to turn this in a get it graded. Just hold on to it after you have finished.

Two-dimensional crystalline carbon, know as graphene, has an interesting structure as well as very interesting and unusual electron properties. The structure is established by \(sp^2\) hybridized 2s and 2p wave-functions; the unusual electron properties are associated with a band formed from the left-over \( p_z\) wave-function.

Warm-up problems:
1. How many valence electrons does carbon have? How many (per atom) are locked up in bonding, and how many (per atom) are available to go into the non-local states of the \(p_z\) band?

2. a) Sketch the graphene structure.
b) How many nearest neighbors does each atom have?
c) If the C-C distance for nearest neighbors is a, then what is the distance to next-nearest neighbors? What is the distance to next-next-nearest neighbors?

3. a) What two vectors can be used to generate the Bravais lattice structure of graphene?
b) What is the difference between the Bravais lattice and the crystal lattice in this case?
c) what is a unit cell in this case?
============== end of warm-up====

4. For our usual 1D crystal (not graphene), the BZ can be chosen to cover any interval of width \(2 \pi/a \). The usual choice is from -pi/a to +pi/a.  The choice 0 to 2pi/a, for example, is equally valid.
a) graph E vs k for each of these interval choices.
b) Why is the range of \(k_x\) limited to one of these ranges? Why not less? Why not more?

5. For graphene the BZ extends from \( k_x = - 4 \pi /(\sqrt{27} a)\) to \( k_x = 4 \pi /(\sqrt{27} a)\) and from \( k_y = - 2 \pi /(3a)\) to \( k_y = 2 \pi /(3a)\). And it is a hexagon with a corner on the \(k_x\) axis. Draw this BZ.

6. For graphene, using the result of the nearest-neighbor band calculation we discussed in class (see video), plot E vs k along the \(k_x\) axis:
a)  from \( k_x = -4 \pi /(\sqrt{27} a)\) to \( k_x = +4 \pi /(\sqrt{27} a)\).
b)  from \( k_x = 0 \) to \( k_x = +8 \pi /(\sqrt{27} a)\).

7. Our nearest-neighbor band calculation yields two bands that arise from the \(2p_z\) "orbital".
      For one of the bands:
\( E_{2p_z,1}(k_x,k_y) = E_{2p_z} - \lvert t + 2t e^{-i (3/2) a k_y} cos ((\sqrt{3}/2)a k_x) \rvert\),
where the constant term is the atomic energy of the 2p state. The first term inside the absolute value symbols (t) comes from an atom in the same unit cell as the "starting atom" (a distance a above it). (It has no phase factor because it is chosen to be in the same unit cell.)  The other term, proportional to 2t, comes from the nearest-neighbors below-and-to-the-right and below-and-to-the-left which combine to create the 2 cos term.
      For the other \(2p_z\) -derived band:
\( E_{2p_z,2}(k_x,k_y) = E_{2p_z} + \lvert t + 2t e^{-i (3/2) a k_y} cos ((\sqrt{3}/2)a k_x) \rvert\),
a) Why are there two bands? Why not just one band?
b) If t = 1 eV, then what is the band width?
c) What is the Fermi energy?

8. interesting extra credit:  Given t=1 eV, as before, now suppose you find a way, via doping or a gate voltage of some sort, to increase the Fermi energy by .01 eV (10 meV) above its normal value. What is the size and shape of the Fermi surface and what is the density of electrons in the conduction band? For example, is the Fermi surface a circle or an ellipse?
Hint: If we assume a constant D(k) in 2D k-space, which is true, then the electron density will be proportional to the area inside the k-space Fermi surface. Also, I think the total area of the BZ times the k-space density of states should be related to the density of carbon atoms. (So hopefully you do NOT need D(E), only the much easier to work with D(k) (which is a constant and not dependent on k).
Note also that if one were to shift the choice of BZ boundary by \(\pi/3a\) in the \(k_y\) direction, then one would have two "Dirac cones" inside the BZ, instead of 6 on the corners (each shared 3 ways). Two inside might be nice for visualization, no?
b) What are some other cool things we can ask/explore here?
c) PS. If anyone can get D(E), that might be interesting too.

9. People talk about making p-n junctions with graphene, e.g., for solar energy. Based on these band theory calculations, what might be challenging about that (or at least different from Silicon)?

10. (see video, extra credit)
a) In the calculation of the 2pz-derived bands one gets a 2x2 matrix. (see video) Why is that? Why not just one band (from one atomic state)?
b) If one were to endeavor to calculate the Bloch (band) states associated with the sp2 bonding of graphene, how large a matrix would you expect? Explain. (Assume starting with ordinary orbitals and letting the hybridization emerge naturally in your calculation.)
c) In that calculation, where would you find the information expressing the nature of the hybridization?  (in the ________________. What is the missing word?)

How did the last HW go?

How did you do on problem 4? The other problems?  Please post comments on what you got, how you did it, etc, here. Your comments will be much appreciated!

Did anyone notice how the Fermi surface evolves from closed to open in problem 4?  What does open mean? What does closed mean?

Bonus question for discussion here: For the E vs k relation of problem 4, at what filling does the Fermi surface transition from closed to open?

Plan for June 3-5th.

June 3 (Tuesday): Regular class
June 4 (Wednesday): Special review section: 5-7 PM in ISB 231 (tentative plan)
June 5 (Thursday) : In class test using a collectively created reference sheet.*

Please email me or post here if that 5-7 PM time does not work for you. We can make the start time later.  Also, that is tentative at this point because I have not gotten confirmation on the room yet.

* The plan is to create a reference sheet that everyone can utilize using something like google docs that includes latex (for equations). This will be the only thing you can use for the in-class test, so please participate and help ensure that it has everything you might wish for. The test will be comprehensive so the equation sheet should be as well. Let's plan to finish this by June 2nd.

Homework 7 with solution.

You can discuss these problems here. Especially the methods that you might use. Discussion of 4, how to approach it, might help everyone. 
(These problems should not be too hard. It is okay if we have them due on Friday?)
errors corrected Thursday. question about thermal speed deleted.

1. Consider a conduction band like we derived the first week with a dispersion relation \(E = Eo - 2t cos(a k_x)\).
a) What is k_f for a half filled band (in terms of a and B, the bandwidth)?
b) What is the Fermi velocity for a half-filled band?
c) If a is 0.08 nm and B is 5 eV, what is the Fermi velocity (in cm/sec)?
d) If a is 0.08 nm and B is 5 eV, what is the (unit-less) effective mass?

2. For a semiconductor with a scattering rate (transport) of 10^-12 sec and an effective mass of 0.5.
a) what is the average speed of an electron for an applied electric field of 1 Volt/cm?
b) how does the speed from a compare with that of an electron at the Fermi surface in problem 1. 
c) what is the mobility for this semi-conductor? (cm^2/Volt-sec). Is that very good?

3. Back to the case of problem 1 with a = 0.08 nm and B = 5 eV:
a) calculate v_f for the cases of a 1/4-filled, half-filled, and 3/4-filled band. How do they compare?
b) What is E_f for each case?

4. For a 2-dimensional cubic crystal where \(E(k) = E_o - 2t(cos(a k_x) + cos(a k_y))\) sketch the Fermi surface for the following cases:
a) 1/8 filled band.
b) 1/4 filled band.
c) 1/2 filled band.
d) 3/4 filled band.
e) which is most like a circle in k_x, k_y space.
f) What is E_f in each case? (extra credit)
 Note on 4: There is no easy, magic way to do this that I know of. You might have to get your hands dirty and explore. The 3/4 case is really unusual and requires your interpretation and understanding. The shape of the 1/2 filled case surprised me.
Here is an example of the sort of thing you could try to explore with:

I think in that plot the B Z would extend from -pi to pi and be squared shaped. In k-space states are evenly distributed, so 3/4 filled means the area inside the Fermi surface is 3/4 of the total area of the B Z. (Of course, this represents a k-space plot. I used x and y cause Wolfram probably prefers that.)

Here is a page of solution notes. This are pretty short and may not be so clear, so please feel free to ask questions here (and to answer other peoples questions). Your comments will be much appreciated. This will be probably on the final.

Reading, Fermi surface, graphene...

You can see pictures of Fermi surfaces online. Gold is a common one that is pretty simple. If you can find any 2-dimensional materials that would be cool since those are easier to visualize and might be simpler.

Reading about k-space and the 1st Brillouin zone would help you understand things better. If you encounter anything about "extended zone scheme", my opinion is that it is confusing, counter-intuitive and not a good idea. Ignore it. The way we do things, starting with atomic levels and letting each one broaden into a different band, it never comes up.

Graphene: I was thinking graphene might be a good example Fermi surface because it is 2D and should be easy to visualize. Then I realized that it is not really a metal, so it doesn't really have a normal k-space Fermi surface. In a way, the Fermi surface consists of 6 dots in k-space. See if you can find those online and understand them a bit. Then, if you were to dope it with extra electrons, the Fermi energy would move upward and each dot would evolve into a circle. So then the Fermi surface would be 6 little circles each one centered at a "K" point. If you can understand what a K point is for graphene in 2D, then I think you understand k-space and the Brilloiun zone concept.

I am thinking it would be interesting to cover graphene for several reasons: Easy to visualize because it is 2D. interesting material with interesting properties. of current interest...   Also, it is an example of sp2 bonding. However, note that the band of interest is not part of the sp2 bonding, but rather it comes from the pz state of the carbon atom (which is one of the 4 orthogonal 1st-excited states of the attractive -1/r potential. pz wave-functions overlapping with pz wave-functions from neighboring atoms is what leads to the valence and conduction bands of graphene.

http://demonstrations.wolfram.com/GrapheneBrillouinZoneAndElectronicEnergyDispersion/

Conductivity in a metal. Priorities.

Tomorrow I am thinking that we could discuss conductivity in a metal. It is not as simple as you might think because of all the occupied states. Most electrons are in a state where essentially every nearby state (in energy or k) is occupied and they are pretty hemmed in by that. So it is pretty different from conductivity in a semi-conductor where almost every state is empty (in the conduction band).

So how are electrons in a metal able to move if all nearby states are occupied? What is a microscopic way to view metal conductivity?

(Also in tomorrow's class let's try to set priorities for future classes.)

What Topics are you interested in?

I am interested to get a sense of what topics you are interested in. Please post on that here and feel free to discuss whatever comes up here from other students.

Reading: Fermi surfaces, broken symmetry...

Here are some things you could read about relevant to our class tomorrow and after that:
Fermi Surface
Fermi sphere
conductivity in a metal
Fermi surface instabilities.
Ferromagnetism
Mott-Hubbard insulator
spontaneous symmetry breaking
(spontaneously broken symmetry)
superconductivity
anti-ferromagnetism
...
"more is different" (Anderson)
try online sources...

Midterm. due Tuesday, noon.

OK, here it is. Please post here regarding any questions, errors or ambiguities. This seems a little long, longer than I was hoping for. If you identify any questions that seem not so important-- that do not seem to have a point -- please post a comment on that here and maybe we can try to shorten it. Your thoughts are welcome. Please keep checking here for possible updates, corrections or clarifications.
* (problem 4 corrected, Saturday 8:40 PM)

https://drive.google.com/file/d/0B_GIlXrjJVn4aGJ6X25CaWowSDg/edit?usp=sharing

Homework solutions are posted.

In the homework posts.

N-P Junction Videos.

Illuminated junction:


Junction that emits photons:

Reading about FETs for this week.

I would encourage you to read about FETs this week. FETs all have a source, a drain, and a gate, as far as I know. Reading about any of them is fine. MOS FETs are perhaps the most important.

Also, if you wanted to read about Schottky and Ohmic contacts that would be worthwhile. Note how the position of the Fermi energy of the metal relative to mu in the semiconductor matters. Focusing on metals contacted to an n-doped semiconductor would be fine. I think a metal Fermi level in the gap yields a Schottky contact, which is a lot like a p-n junction (but with half of it compressed to almost zero volume). A metal Fermi level above the CB edge tends to lead to an Ohmic contact, I believe, which is what you want in a lot of circumstances (e.g., where a wire meets a source or drain, or at the ends of an n-p jucntion LED or solar cell).

HW 6 wi solutions.

This week's classes will focus on FETs, so you are on your own, so to speak, with problems 1-3, which involve things we have already covered (but might be pretty challenging). (i.e., don't wait thinking that we will cover them in class.)

These problems are tricky to design and to get the parameters in a range that works. I am interested to see your results for 1c and 2a, which are the easier parts. Please email me those as soon as you get something if you like. (Then we can see if the problem is going to work out okay.  The voltage range should hopefully stay below the band bending.)
1. Consider a semiconductor with an energy gap of 1.5 eV, \( D_2 = 0.8 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.8 \times 10^{22} \, states/(eV-cm^3)\), \(D_n = D_p = 100 \, cm^2/sec\) and \(\tau_r = 10^{-10} \, sec\). Suppose you make an n-p junction with an area of 1 cm^2 where \(\mu\) is 0.2 eV away from the majority carrier band on each side. [1a) and b) are warm-up problems and should be pretty easy and quick to do. c) is where it gets more difficult.]
a) What is the total band bending in this case (the difference between E_c on the left and on the right)?
b) What are the majority and minority carrier concentrations on each side? (away from the interface region) [Don't be discouraged if the minority carrier density seems too small; that might be a natural consequence of the larger gap (1.5 eV).]
c) If you connect a battery to the junction, how much bias voltage would you need to get a current of 1 amp?
d) How much voltage would you need to get a current of 0.5 amps?
e) Which way does the current go (most easily)? Sketch a picture.

2. Suppose you get rid of the battery and instead have a resistor, R, in its place. Then you illuminate the junction with \(10^{19}\) photons per second per cm^2. Each photon has an energy of 1.5 eV.
a) If half the photons are absorbed, in the junction region, then how much current would you get for R=0 Ohms?
b) Which way does the current go through the resistor? Sketch a picture.
c) How much current would you get for R = 1 Ohm?
d) Is this difficult? Why? (discuss here) Would it be easier if you were asked to get an estimate of I to just one or two sig figs?

https://drive.google.com/file/d/0B_GIlXrjJVn4djlsd2ExZElKNHc/edit?usp=sharing

3. We can think of the resistor as a stand-in for something that we would like to provide power to.

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.

Homework 1. Due Tuesday, April 1.

This is a pretty short assignment. Please start in it as soon as you can and try to do it before our first class. I think you will understand our first class on Tuesday much better if you have the experience of working with these problems.

These questions are not designed to be clear, so it is natural to want to ask questions about them and discuss them. Please read and engage with them as best you can and ask questions and join in the discussion here. Part of your learning is to figure out what is being ask here, what these things mean, and what is expected. I view posting here as a sign of engagement and I appreciate that.

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 Added Note:  
How functions work: Consider the function f(x) = x^2. What is the nature of the related function f(x-9a)?
       At x= 9a, what is f(x-9a)? zero right?  And at x=10a one would get a^2, I think. True?

It is no different with our function, v(x) above. Whenever the argument is zero, the rules of the function yield v=-10 eV;
whenever the argument is greater than L/2 then the function is zero. For the function v(x-2a) the argument is zero when x=2a. For the function v(x+2a) the argument is zero when x=-2a. Although it could be something in the way I worded the problem or defined the function, I think in this case the issue is that you have not been taught clearly what a function is. If in resolving this you acquire a clearer concept of that, that would be the best outcome.  The argument of the function is the key thing.