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).]
Friday, May 30, 2014
Thursday, May 29, 2014
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.
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.
Monday, May 26, 2014
Crib sheet link.
https://www.writelatex.com/
This will be everyone's crib sheet for the final. I think you can edit it (add stuff).
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?)
*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?)
Saturday, May 24, 2014
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?)
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?)
Friday, May 23, 2014
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?
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?
Thursday, May 22, 2014
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.
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.
Wednesday, May 21, 2014
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.
(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/
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/
Monday, May 19, 2014
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.)
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.)
Sunday, May 18, 2014
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.
Wednesday, May 14, 2014
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...
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...
Friday, May 9, 2014
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
* (problem 4 corrected, Saturday 8:40 PM)
https://drive.google.com/file/d/0B_GIlXrjJVn4aGJ6X25CaWowSDg/edit?usp=sharing
Wednesday, May 7, 2014
Saturday, May 3, 2014
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).
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.
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.
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