Final discussion post.

Not sure is many people are still looking at this blog, but, just in case, here is a place to discuss the final if you like. Some people had trouble with #5. (Illuminated n-p junction and resistor..) Does anyone want to explain how to handle that? #6 as well.

Overall, I was impressed by how much people learned. Nice work!

Physics 156 Final. Wednesday, 12:00 noon

The final should provide an opportunity for you to show your understanding of semiconductors and metals. E vs k relationships play a big role. Where do they come from? Can you utilize them to calculate fermi velocity, fermi boundaries,…? Do you understand the significance of the Brillouin zone? Can you distinguish occupied and unoccupied states?

For semiconductors, the fermi energy is in an energy gap. Do you understand what an energy gap is and what the density of states as a function of energy typically looks like for a semiconductor? Special approximations can be used to estimate carrier density for cases where Ef is in a gap. Other approximations can often be used when Ef is not in a gap. The ability to recognize and distinguish those cases is important. Understanding the nature of those approximations, and what they rely on, is also important.

What is the essential nature and phenomenology of an n-p junction?

What role does shifting of the fermi boundary play in the conductivity of a metal?

Do you understand the origin of ferromagnetism?

I’ll post more here later. I just wanted to get the ball rolling and provide a place for questions and discussion here.
Added  notes: Perhaps it would be good to have a problem on illuminated n-p junctions. How do you feel about that? What about a ferromagnetism problem.
    It might be a good idea to test yourself with Fermi boundaries. You may wish to test your ability to identify where a Fermi boundary crosses the kx axis, realizing that the ky term is not zero when ky=0. (Same thing for the ky axis.)

Equations are show here: (What else do we need? I can't think of very many basic equations from the second part of the class.)

Conductivity of a Metal

Here is a video on conductivity of a metal that talks about the shift of the Fermi boundary and the electron speed. Please feel free to discuss and ask questions here.

Homework 7 solutions.

Here is a link. When you show a Fermi boundary, it is important to also show the extent of the (1st) Brillouin zone. The (1st) Brillouin zone is a domain in k-space in which each state can be found exactly once. Going beyond that would lead to encompassing some states more than once (over counting). The point of the Fermi boundary is to be able to specify which states are occupied (and which states are not occupied).

https://drive.google.com/file/d/0B_GIlXrjJVn4R09ISFh6Nl92NU0/view?usp=sharing

Superconductivity

A number of people have indicated they would like to learn about superconductivity. Perhaps on Monday we can discuss that.  In superconductor there electrons near the fermi surface pair with each other. This is pretty surprising.  (Electrons typically repel each other.  Why would electrons pair?)    

Anyway, these pairs of electrons are bosons and a superfluid bose condensate can form that is able to transport  current (charge) with no resistance. Superconductivity is an unusual and completely unexpected phenomenon. The idea that "more is different" in physics are partly motivated by the emergence of the superconductivity and the difficulty of explaining it in a reductionist manner.

Ferromagnetism and anti-ferromagnetism.

    The relationship between ferromagnetism and what is called "anti-ferromagmetism" is not simple.   Ferromagnetism occurs in metals and is driven by the tendency of electrons to, in some circumstances, use spin alignment in order to avoid each other. In this way the overall coulomb energy associated with electron-electron repulsion can be diminished.
     Antiferromagnetism, on the other hand, usually involves electrons that are confined (that is, electrons in non-metallic systems). Confinement has increased the kinetic energy of these electrons. The electrons seek a way to lower their confinement-related kinetic energy.   Communicating through their spins, the electrons find a way to lower their KE through an inherently collective, cooperative behavior involving spin alignment. In an antiferromagnet, each spin tends to point in the opposite way of its nearest neighbors. Antiferromagnetism is not just the alternating of classical spins.  It is a sophisticated many-electron quantum collective state.
    The names, FM and AFM,  tend to be rather misleading. The magnetism part makes sense, however, ferro is just from an ancient greek name for a material in which magnetism was observed long ago. It tells us nothing about the underlying origin. One might instead call it magnetism from spontaneous spin alignment, or just "spontaneous magnetism".
     More importantly, anti-ferromagnetism is not really the opposite of ferromagnetism. It is its own collective phenomenon with its own origins. These are very different and probably even more intriguing and subtle than those of ferromagnetism. Anti-ferromagetism involves a collective state with broken symmetry which arises to enable electrons to lower their quantum kinetic energy.

Ferromagnetism and HW 8. With Solutions.

This assignment is optional. You can get extra-credit for it or you can not turn it in.

...we will look at the origins of magnetism, particularly ferromagnetism in metals. We will look at the role of the Fermi energy, Pauli exclusion principle, the fermion nature of electrons. Electron-electron repulsion plays a key role. Why is that?  How do electron spins get involved? What causes spins to align?

HW 8.
1. Band energy: The total band energy can be defined as the sum of the energies of all electrons in single-electron-states. That is, it is the sum of the energies of the occupied single-electron states.  Considering a band of bandwidth B for which we make the simplifying assumption that the density of states, \(D_c\), does not depend on energy.
a) What is the total band energy in terms of E_f and E_c?
b) What is the total band energy of we set \(E_c = 0\) ?
c) What is the total band energy in terms of n?
d) What is the relationship between n and Ef?
e) How many total states are there on the band

Suppose that just, for the fun of it, we divide the density of states into two parts: one associated with spin-up single-electron states and the other associated with spin-down single-electron states.

2. On what basis does one choose the spatial direction (axis) with respect to which spin up and spin down are defined?

3. a) What is the density of states in the spin up band? (in terms of \(D_c\))
b) What is the relationship between \(n_\uparrow\) and \(E_f^{\uparrow}\)?
c) What is the relationship between \(n_\downarrow\) and \(E_f^{\downarrow}\)?
d) Suppose for a fixed value of n, we allow \(n_\uparrow\) and  \(n_\downarrow\) to vary. What is the band energy of all n electrons as a function of  \(n_\downarrow\) and  \(n_\downarrow\)?
e) Express this band energy as a function of  \(n_\uparrow - n_\downarrow\).
f) Plot this band energy as a function of  \(n_\uparrow - n_\downarrow\). What is the domain of this graph? What configuration(s) of the system  have the lowest energy?

4. For most of this quarter we have been using single-electron states (filling them with many electrons), and we have been ignoring the coulomb repulsion between electrons. Sometimes the electron-electron (e-e) doesn't make much difference, but sometimes it makes a lot of difference. A fairly simple model from the e-e interaction if to write:  \(V(n_\uparrow, n_\downarrow) = U n_\uparrow n_\downarrow /n\).
a) Plot this as a function of  \(n_\uparrow - n_\downarrow\).
b) What is its value at  \(n_\uparrow - n_\downarrow = 0 \)?
c) What is its value at  \(n_\uparrow - n_\downarrow = n \)?

d) What simple symmetry does this function have?

e) What configuration of the system would be favored by this coulomb interaction?

5. In the Stoner model one combines the coulomb interaction energy associated with electron-electron repulsion with the band energy associated with the filled one-electron states.
a) Do this. For a given value of U, for what values of bandwidth the system will have a ground state with spin alignment (a ferromagnetic ground state).
b) For the case where the system is spin-aligned (magnetic), explain the nature of the origin of the magnetism? What is the driving mechanism that leads to the ferromagnetic state?

6. On can add a entropy related term to this model and explore why magnetism disappears at higher temperature. Generally, \( S= k ln(\Omega)\). For small B, perhaps one can approximate \(ln(\Omega) \) by \(ln(\Omega) \approx \frac{2}{3} n - \frac{1}{2n} (n_\uparrow- n_\downarrow)^2 - \frac{1}{8n^3} (n_\uparrow- n_\downarrow)^4 \).
a) For U=2 eV and B = 1 eV, find the ground state of the system as a function of temperature, T. You can assume the the ground state is the state with the lowest free energy, that is, the lowest value of F= E-TS.
b) Plot the ground state value of \((n_\uparrow- n_\downarrow) \) as a function of kT.
c) extra credit: basically one can think of \(\Omega\) as the number of possible states of the system with a given value of n and \(n_\uparrow\). What is that? That is, what is the value of \(Omega\) in terms of \(n_\uparrow\) and \(n_\downarrow\).

7. another optional special project: Calculate the magnetic susceptibility of the system from problem 6 as a function of temperature. Do this only in the temperature range above where it becomes magnetic,  and show that it diverges as you approach the transition temperature from above. What is the form of that divergence? What sort of "power law"?

8. Optional special project: This project I think will show the how electrons with align spins tend to avoid each other.  Consider an infinite square well with two electrons in it. Let's try making two electron states from products of the one electron states. Let's call the one electrons states \(\psi_n\) where \(\psi_n(x) = \sqrt{2} sin (n \pi x)\) where x is in nm and the well extends from x = 0 to x= 1 nm.
a) Consider a two-electron state in which the 11th and 12th states are occupied. Is \(\psi_{11}(x_1) \psi_{12}(x_2) \) an appropriate two electron state? Why or why not?
b) Suppose both electrons have the same spin (\(\uparrow \uparrow\) ). What is the appropriate two electron (spatial) state for this case?
c) On the other hand suppose that the 11th and 12th states are occupied, but the spin state is \(\frac{1}{\sqrt{2}} [ \uparrow \downarrow - \downarrow \uparrow ] \), then what is the two electron (spatial) state?
d) How are these (spatial and spin) states different?  Why? Discuss here if you like. Don't wait too long. Do it now.
The expectation value of \( |x_1 - x_2| \) can tell us how close the electrons tend to be in these two-electron wave states.
e) for the state from b), calculate the expectation value \( |x_1 - x_2| \).
f) for the state from c), calculate the expectation value \( |x_1 - x_2| \).
g) are they different? by how much? what do you infer from this?
========
Solution notes link:

https://drive.google.com/file/d/0B_GIlXrjJVn4V0g5V0NERFVlU3M/view?usp=sharing

Illuminated Junction With A Capacitor

 Daniel and I have been messing around with the differential equation (HW6 P.5) Zack gave us in class. Instead of attempting to solve it analytically we plugged it into Mathematica and got a solution and some cool graphs.

This is the equation Zack turned into a DE

\(I = I_{ill} + I_o(1-e^\frac{-eVa}{kT})\)

where \(I = \frac{dQ}{dt}\) and \(Va = \frac{Q}{C}\)  our \(Va\) is negative and \(C = 10 Farads\)

So this becomes

\(\frac{dQ}{dt} = I_{ill} + I_o(1-e^\frac{eQ(t)}{CkT})\) and the exponential becomes \(e^{4Q(t)}\)

This is the code for Wolfram Mathematica:

DSolve[{Q'[t] == 12.8*10^-3 + 3*10^-11*(1 - Exp[4 Q[t]]), Q[0] == 0}, Q, {t}]

Then it gives you this solution:

\(Q(t) = -0.25 Ln[2.34375*10^{-9} + 7.8125*10^{-10} * e^{20.9701 - 0.0512 t}]\)

 Here is the solution graphed

(Time is in seconds and Charge is in coulombs)

It takes around 400 seconds to charge

 Here we used a value of \(I_o = 3x10^{-7}A\)

Notice the cap doesn't fully charge and steady state is achieved faster.

Here is also a link to some work Jimmy Layne did on this. It looks like he got an analytic solution for the time dependence.

https://drive.google.com/file/d/0B_GIlXrjJVn4aHJFaDhvd1o5X3M/view?usp=sharing

Extra Special Bonus Pictures (not related to things above) 

 This is a three dimensional graph of E vs K where the energy surface is displayed

\(E(k_x,k_y)\)

\(E(k_x,k_y,k_z)\) Contour plot (Surfaces of constant energy)

  This picture is interesting anyone have some cool thoughts about it?

Homework 6 solution notes.

Homework 6 included some problems that illustrate some of the issues involved in converting the energy from electron excitation due to absorbed photons to useable energy. This includes a highly non-linear I-V relationship for the n-p junction and the circuit it is part of.

In problem 7 the current is pretty much constant up to R=16 Ohms and, because of that, the power generated in the resistor is essentially linear in R in that range. At R=32 Ohm there is just a hint of non-linearity.  (We also know already from our work with the capacitance that the current drops to zero just before Va = -.5 volts.)  At R= 64 Ohm, I think that the current will have dropped a lot. Picturing the graph of I vs V is very helpful here I believe.

Can some people comment on whether my numbers in the table agree with what you got. Please discuss what you learned here. Also, what did you find for the R, I, V values that give you the highest power generated?

https://drive.google.com/file/d/0B_GIlXrjJVn4MVV2VmVnaVI0eFU/view?usp=sharing

HW 7.

Some people asked for extra time and I think it would be okay to just turn this in Wednesday in class.

For any of these problems, assume a scattering rate of \(\tau = 10^{-13} s\) unless otherwise specified.

1. a) Show that you can combine two 1D Bloch states to obtain a standing wave.
b) How are their k values related?

2. Suppose the E vs k relation for the partially filled band of a metal is:
\(E(k) = E_o - (B/2)  cos(ak)\) where k ranges from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a) What is the relationship between the density of atoms, N, and the integral of g(k) dk from \(-\pi/a\) to \(+\pi/a\) ?
b) If the density of electrons is N, what is \(k_f\)?
c) If the density of electrons is N/2, what is \(k_f\)?

3. (Same system) When an electric field is applied, the Fermi boundaries (in k space) shift.
a) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N? Illustrate.
b) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N/2?

4. When the Fermi boundary is shifted, electrons can be viewed as belonging to two classes: those that have an occupied partner state with which they can form a standing wave state and those that do not. What percentage electrons fall into each class for:
a) n=N
b) n=N/2

5. Some people say that the current density of metal is proportional to: the density of electrons in states that have no partner, multiplied by the fermi velocity.  (Let's use a bandwidth of 6 eV for this problem.)
a) Argue for or against that as a good approximation. What other factor do you need?
b) What is the fermi velocity for a half-filled band (n=N)?
c) What is the fermi velocity for a quarter-filled band (n=N/2)?
d) How does that compare with: the thermal velocity of electron in a semiconductor with an effective mass of 1? the average speed of an electron in a semiconductor conduction band with an electric field of 10 V/cm applied?

Fermi Boundaries:
A Fermi boundary in k-space is the set of points that separate a regions of occupied states from a region of unoccupied states. In 1D it is just two points -- one at kf, the other at -kf. In order to explore the concept of a Fermi boundary we therefore need to go up to at least 2D.

6. Consider a two-dimensional (2D) metal for which the E vs k relationship for the conduction band  is: \(E(k) = E_o -  b cos(ak_x) - b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm. This range of k includes all the states of the band (2N states), however, they are not all filled.
a) What is the bandwidth of this band?
b-e) Figure out and illustrate the Fermi boundary for the cases where the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.*
f) Which one of these is simplest? Which one of this is not like the others? Describe and discuss your results and illustrations.
g) To what 2D lattice structure do you think this E vs k relationship corresponds? How come?
* If doing this for a specified filling fraction (1/8, 1/4...) is too difficult, then maybe we could use a sequence of fermi energy values instead. Would that be more manageable? What do you think? What might be a reasonable selection of values of Ef that meet our goal of exploring?

7. One can make this anisotropic in the following way:
 \(E(k) = E_o -  b cos(a_xk_x) - \frac{b}{4} cos(a_yk_y)\)
a) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/8\).
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/4\).
c) Figure out and illustrate the Fermi boundary for the case where the fermi energy is ... your choice.
d) What would you guess is a lattice structure to which that this might correspond??

6.2 (problem added on Wednesday)  I think it might be interesting to also examine the fermi boundaries for an "upside down" E vs k relationship. That is, suppose:
\(E(k) = E_o + b cos(ak_x) + b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a-d) Figure out and illustrate the Fermi boundary for the cases where the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.
e) Discuss

8. The previous problems are for a simple structure, but it is not a very realistic or stable structure  (because the atoms are not very efficiently (closely) packed). A more realistic 2D metal could have an E vs k relationship for the conduction band of:
\(E(k) = E_o -  b cos(ak_x) - b cos(\frac{1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) - b cos(\frac{-1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) \).
a) Explore the nature of the Fermi boundaries for this band for a some particular examples of filling (corresponding to different values of Ef). What structure would you guess that this could be the and structure for? (educated guess --based on symmetry considerations perhaps)

Hi everyone,

These are some of the responses that I gave for the midterm. There may be some mistakes or misunderstandings but it would be helpful and interesting to see how other students may have solved these problems. Also, could please correct me on any mistakes I may have made, this way if other students made similar mistakes we can all benefit.

Homework 6

If you like solar cells, try to make sure you are able to spend a lot of time on problems 6 and 7. Those are the problems that really help you understand solar cells. Especially problem 7.

Extra credit: Can some of you solve problem 1 soon and post what you get here? That might be helpful in contextualizing some of the later problems of this homework.

For these problems, assume a symmetric n-p junction doped to \(10^{17} cm^{-3}\) on either side.  [and to a semiconductor for which  \(E_g = 1 eV, \quad kT=.025 eV\) and \(D_c = D_V = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = B_V = 3 eV\)].

1. In the ideal junction approximation, I believe that we found that for a junction connected to a battery of voltage V_a, the current density will have the form:
\(J(V_a) = J_o (1 - e^{-eV_a/kT})\)
Based on the calculation we did of the diffusion current at x_d, what is a reasonable estimate for the value of J_o? What are the units of J_o?

2. (LED question) With a battery connected to the junction (in series),
a) how much current do you get through the junction for an area of 1 cm^2 and \(V_a = -0.4 V\)?
b) If 50% of the electrons involved in that current flow emit a photon, how much power is that? In what units would you like that power to be expressed?
c) how much radiated power (in photons) do you get from the junction if \(V_a = -0.5 V\)?

Midterm answers, solutions and notes.

Who would be willing to post answers and explanations to midterm problems here? Go for it. If you got something right, or see how to do it now, and would be willing to share your results, thoughts and questions here, please do. Thanks.   -Zack

PS. Here are my solutions & notes.

https://drive.google.com/file/d/0B_GIlXrjJVn4ZHV3N0J5eTlfRnM/view?usp=sharing

Midterm on Friday. see added notes.

Our midterm is this Friday, May 8. (This is a repost/top post.)

In this class you are allowed, even encouraged, to bring a calculator.  (Phones, however, are not allowed as per general department standards and practice.)

You will be asked to share your understanding of np junctions, fermi energy, carrier density and the origin of bands in crystals. Also drift current and diffusion current, mobility, effective mass and its relationship to bandwidth...

Some things that will help you:
Understand some basic definitions: n, p, bandwidth, bandgap, density of states...

A natural intuitive feel for and understanding how n and p are influenced by the relationship between the fermi energy and the band edge.

An understanding of the approximations we use (sometimes) to evaluate n and p. Good judgement regarding when some approximations are appropriate and when they are not. Confidence to decide what approximations to use.  (Failure to make approximations can hurt you. It is often not okay to use the exact expression for something when an approximation can be made.)

A sense of energy scales and their importance and relationships. e.g. kT, bandwidth, E_gap, etc

An understanding of diffusion current.

An understanding of drift current.

Understand what a band represents. How many states are in a band?... the relationship between bandwidth, density of states and effective mass.

Added notes: The ability to actually calculate numbers, for current as well as n and p, will be tested. 
Also, a deep understanding of the nature of an unbiased and biased np junction (negative on the n side) and the assumptions of the ideal junction approximations. Understand the concepts behind the assumptions, they are ideal, so not perfect, but in what way are they sort of reasonable? what thinking lies behind them? What are their consequences??

How do these equations look?

Video on enhanced value of \(n(x_d\)).

The enhancement of n(x) at \(x_d\) in a biased n-p junction is the key thing that leads to a exponential current-voltage (I-V) relationship.  This video illustrates and discusses the origin and nature of that enhancement. (Questions and comments are welcome.)

Video on np junction physics

     This video looks at the nature of np junctions. It includes both the static case (no bias, no applied voltage, no steady-state current flow), and the case of a bias that can produce flow of electrons from the n-side to the p-side of the junction. (An applied voltage that is negative on the n-side will tend much electrons over toward the p-side.)
      Interestingly, the key thing aspect of the current flow is diffusion of electrons from the n-side to the p-side, and, at the same time, diffusion of hole from the p side to the n side.
      The video is long, but it can be divided into parts.
The first 12 minutes or so (00:00 to 12:00)  has a discussion of the nature of the static junction (no applied voltage, no current flowing).
The rest deals with a biased junction with electrons flowing from the n  to p side.
12:00-17:30  Big picture discussion.
17:30-24:00  The enhanced value of n(x) at x= x_d. The ideal junction approximation and its consequences.
23:40 Minority carried enhancement...
24:00 - 44:00 Calculation of enhanced n(x) for x greater than x_d. Assuming a specific enhancement at x =x_d, what does that imply about n(x) for x greater than x_d?
44:00-50:30   Discussion of diffusion region (x>x_d) length scale. How rapidly does n(x) regress to its equilibrium value?
Please post lots of comments and questions here!

Carrier Density Enhancement Factor for Biased NP Junctions

When Chris and I were working on homework set 5, we noticed a discrepancy between the enhancement factor that we both agreed upon and that which was discussed online and in class. It was established online that the enhancement factor was about 54, or e^4 for the problem in which the applied voltage was -.1v. However, we believe that the exponent here is off by a factor of 2, and instead should be e^8. You can see our work for this in the picture provided, but note that the sign convention is that which Zack provided in the comments on HW 5.

HW 5 solution notes

Here are two links. The first is to a student's HW. The second is to some additional notes I prepared to clarify some things. The units for current is a bit trick ( e, eV, Volts related issues). I think what I posted is correct on that.

https://drive.google.com/file/d/0B_GIlXrjJVn4WHRYdkpaNlBVbm8/view?usp=sharing

https://drive.google.com/file/d/0B_GIlXrjJVn4VHQyMUNFNEZyWG8/view?usp=sharing

HW 5. Due Wednesday 10:30 AM at my mailbox in the physics mailroom.

I would suggest making two copies, ideally by hand, and keeping one so you have something to study from for the midterm on Friday.

For all these problems, please feel free to ask questions and postulate answers here. It is expected that you may wonder what is being asked. Figuring that out is part of the goal of the assignment and an opportunity to deepen your understanding of the physics. Problem 5 is especially important and difficult I think.

Note on scattering times, May 1st. The scattering times I originally put in this HW assignment seem a bit too short. Let's use instead \(\tau_t = 10^{-12}\) sec and \(\tau_r = 10^{-9}\) sec. I think that this will give us more realistic values for the length scales and mobilities that depend on these times, as well as other quantities.

Extra Credit. May 2. For x greater than x_d, use the n(x) that you get below to calculate the upper Ef as a function of x. Sketch a graph of Ef as a function of x for the entire junction.

1. Write a paragraph describing the equilibrium state of an n-p junction. Include a 3-part figure showing charge density, electric field, electric potential as a function of x. Explain the origin of depletion.

PN junction: some interesting plots.

Here some plots related to the pn junction we are exploring (for which the total band bending is 0.6 eV...). You can explore this kind of thing as well with Wolfram Alpha. One of the tricks is to strip equations down to their bare essentials to make it easier for W-A to give you what you want. For example, for the 10^17 symmetric junction the band shifts by 0.3 eV on each side, hence the equation 1+.3*(x+1)^2 for E_c(x) (the conduction band edge energy). (In all the plots x_d is scaled to 1, so they all need to be rescaled to correct that.)  In the 2nd plot, for n(x), for the vertical scale, 1 corresponds to 10^17 and the plot shows how n decreases with x. The 12 in the exponential represents 0.3 eV divided by .025 eV (which is unit-less). The third plot shows the charge density vs x (with a simplified vertical scale, i.e., multiply by 1.6*10^-2 to get coulombs). In each plot a 1 on the x axis corresponds to x_d.

This last one shows, I think, the x dependence of the drift current. With x-d set to 1,  (x+1) is proportional to the electric field. The exponential term is proportional to the density of electrons in the conduction band, n(x).

Extra credit problem & added special bonus.

I though of an interesting extra credit problem to add to the HW today. That problem is really not as hard as it looks. There is also a special bonus problem added that involves calculating/estimating a current in amperes.


Here is a hint for that problem:

Additional though:
At the end of all your work on the junction, I think that when the initial doping implies a 0.6 eV offset in Ef from the n to the p sides, then you will ultimately end up with something of the form:
\(E_c (x) = 1 eV + 0.3 eV (x+x_d)^2/x_d^2\)
for x less than zero and greater than -x_d. Does that make correspond with what you got? (of course you have to calculate the value of \(x_d\) that creates that and allows the bands to line up.)

Then for x greater than zero and less than x_d perhaps you would get:
\(E_c (x) = 1.6 eV - 0.3 eV (x-x_d)^2/x_d^2\) ?
Does that line up properly with the other side, produce the required amount of offset and have the correct functional form?

HW 3 solutions

Your comments, thoughts, questions and feedback on these solutions is greatly appreciated 

This homework assignment  encompasses a number of concepts that will be important going forward. Understanding the fermi energy, \(E_f\), the fermi function, f(E) and the density of states, D(E), as well as how they go together to establish state occupation in semiconductors and metals, is critical. It is very helpful and valuable to be able to visualize the integrand!

Here is a link to solutions to 1-13.
https://drive.google.com/file/d/0B_GIlXrjJVn4RFNZMDdwRGlsYzA/view?usp=sharing

Here is a solution to 14, the density of states in 2D. In 2D the density of states turns out to be independent of energy for the E vs k approximation we used in this problem.

Problem 15 can be done via the same method. For 3D one ends up with a k in the numerator and thus D(E) proportional to \((E-E_c)^{1/2}\).

HW 4. n-p junctions. -with solutions.

(See added special bonus)
For the following problems, consider a semiconductor for which  \(E_g = 1 eV, \quad kT=.025 eV\) and  
\(D_c = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = 3 eV\)
\(D_v = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_v = 3 eV\).
(Assume that within each band the density of states is independent of E.)
Let's set our zero of energy at the top of the valence band.
Please mention any possible typos, confusing things etc. I noticed a number of autocorrect issues and mistakes in the last HW.

It will probably help to discuss a lot of this in the comments here. For example, how do the units work going from charge to electric field to potential? How does one end up with a potential in eV? (You may have to convert from cm to meters when you use surface charge to calculate electric field if epsilon_o is in Farads/meter.)

Solutions link:
https://drive.google.com/file/d/0B_GIlXrjJVn4V3AwV1lpR0M2Zms/view?usp=sharing

1. a) If this semiconductor is undoped, then what is the value of n at room temperature?  What is the relationship between n and p? What is \(E_f\) for this un-doped case?
b) Suppose that this semiconductor is doped with 10^17 donors/cm^3. In that case we like to assume that each donor contributes one electron to the conduction band. What is \(E_f\) in this case?
c) extra credit: What do you think might be the rationale behind choosing \(12 \times 10^{21} \frac{states}{eV*cm^3}\) for the density of states? Why that value? (post here)

2. a) Plot the density of states as a function of energy from E from E = -4 eV to +4 eV.
b) Calculate n and p for \(E_f = 0.5 eV\).
c) Calculate n and p for \(E_f = 0.2 eV\).
d) Calculate n and p for \(E_f = 0.8 eV\).
e) Do a semi-log graph of n and p as a function of \(E_f\) for \(E_f\) in the range 0.1 to 0.9 eV. Can you use the approximate form of the Fermi function for this calculation?

3. For the same semiconductor, suppose you dope it with 10^17 donor atoms per cm^3. Then you get 10^17 electrons/cm^3 in the conduction band and 10^17 positively charged donor atoms/cm^3 embedded in the lattice.
a) What is the charge density associated with just the electrons in the conduction band (in coulombs/cm^3)?
b) What is the charge density associated with the positive ions (in coulombs/cm^3)?
c) What is the net charge density?
d) What would the net charge density be if all the electrons magically disappeared?

4. Consider a semiconductor as above. Suppose that it is doped with 10^17 donors for the half to the left of the plane x=0, and doped with 10^17 acceptors/cm^3 to the right of x=0. (The plane x=0 defines an interface between to two differently doped regions.) Suppose that within a distance \(x_d\) of the interface all the electrons in the conduction band from the left side cross over to the other side and fill up previously empty valance band states (holes) there.
a) For a particular value given of \(x_d\), (100 nm for example), plot the charge density as a function of x?
b) For the same given value of \(x_d\), what is the electric field at x= 0? (discuss/post here) What is the electric field as a function of x? (discuss here)
c) Using the relationship between electrical potential and electric field (or charge density) calculate the potential, V(x), as a function of x from x= -infinity to x=0 for the same given value of \(x_d\) (discuss here).
d) Plot the charge, electric field and potential as a function of x over a suitable range.
e) For \(x_d\) = 50 nm what is the change in electric potential from V(x= -infinity) to V(x=0)? (discuss)
f) For \(x_d\) = 100 nm what is the change in electric potential from V(x= -infinity) to V(x=0)? (discuss)

5. a) When \(E_f\) is independent of x, that represents an equilibrium state. What value of \(x_d\) would enable to bands to bend and shift by just the right amount to enable \(E_f\) to be independent of x?
b) Plot the conduction band edge and valence band edge as a function of x for this case.

6. Suppose the doping on each side is 10^18 cm^-3 instead of 10^17 cm^-3.
a) why might one guess that in that case the equilibrium value of \(x_d\) might be 10 times shorter than for the 10^17 case?
b) What is the actual equilibrium value of \(x_d\) for this case?  Why is it not exactly 10 times smaller? (What additional factor influences x_d?)

7. Suppose the doping on each side is 10^16 cm^-3.
What is the equilibrium value of \(x_d\) for this case?

Note added: 8 and 9 are basically just sketches with a scale on the vertical axis (energy). No need to read them as anything more than that.
8. Now let's consider applying a voltage across the junction.
a) Plot Ef as a function of x for applied voltages of 0 and 0.2 V. What is it like? Discuss here? (You may assume the voltage drop takes place just over the junction region (the depleted region).
b) Plot the band edge energies as a function of x for applied voltages of 0 and 0.2 V.

9. How about applying a voltage across the junction in the other direction.
a) Plot Ef as a function of x for applied voltages of 0 and -0.2 V. What is it like? Discuss here? (You may assume the voltage drop takes place just over the junction region (the depleted region).
b) Plot the band edge energies as a function of x for applied voltages of 0 and -0.2 V.
c) Discuss these 3 cases? (-2, 0 and +2 V)  How do they differ?

10. extra credit: a) For one of the n-p junctions you solved above, use your calculated value for \(E_c (x)\) for x less than zero to calculate n(x) as a function of x for x less than zero. Can you use the equation \(n(x) = KT D_c e^{-(E_c (x) - E_f)/kT}\)? Why or why not? (post here)
b) Graph n(x) as a function of x. Where/what is its maximum value?
c) Graph the product of the conduction electron density times the electric fields a function of x. Why would this be of possible interest? To what is it relevant? Where does its maximum value occur?

11. Special Bonus: \(n(x) e^2 \tau/m^*m\) multiplied time electric field has units of current per unit area. I think that if you use your expression for n(x) (from the previous problem) and multiply it time the electric field in the junction, which varies linearly with x, that correspond to a current associated with electrons accelerated by the electric field. Special bonus points to anyone who can calculate the peak value of that current in Coulombs/second (amperes) for a specific junction with an area of 1 cm^2, using m*=0.2 and \(\tau = 10^{-12}\) seconds. (Use the specific 10^17 doping case above. We want an actual number. Is it big, small, negligible, 10^-15 amps, 2 amps or what?)
   Hints: One can separate out the term \( \mu= e \tau/m = e \tau c^2/(m^*m c^2)\). With mc^2 in eV and c in cm/s that can have units of \(cm^2/(Volt*seconds\). (The e turns eV into Volts...).
      Does the maximum in this product occur about 1/5 of the way from -x_d to zero? That is, sort of near the edge where n(x) is not too small, but not right at the edge because that electric field is zero there?

Midterm date.

I am thinking that May 4 (Monday) would be a good date for our in class midterm. How does that sound?

Quiz on Monday.

Let's have a short quiz at the beginning of class Monday. If you can't make it for some reason, you'll be excused. The quiz will involve a graph and be based on one of this weeks HW problems.  For this graph, and graphs in general, please make the size of the graph big enough so that one can easily read the scales and connect them with the plot. 3 or 4 inches on a side is usually pretty good. Please put clear labels and scales on both axes.

Two materials

Suppose you start with  a chunk of metal, say gold, at 0 Kelvin. Then you add 1000 J of energy to the gold in isolation. Suppose then that you bring that gold into intimate contact with another material, aluminum or silicon for example, of similar size, which has had only 200 J of energy added to it (after starting at T=0 K). How would you describe, qualitatively, what might happen when the two materials are brought into contact with each other. Please comment here. Feel free to ask questions, speculate or whatever.

Finding the number of holes in the valence band for a semiconductor

Given that: $$p=D_{v}\int_{VB} (1-F(E))dE=D_{v}\int_{VB}(1-\frac{1}{e^{(E-E_{f})/KT}+1})dE$$ We can make the approximation $(1+x)^{-1}=1-x$ for $x<<1$ in this case(for problem 1). $$D_{v}\int_{VB}[1-(\frac{1}{e^{(E-E_{f})/KT}}+1)]dE=D_{v}\int_{VB}[e^{-(E-E_{f})/KT}]dE$$ $$=D_{v}\int_{E_{v}}^{E_{v}+B_{v}} [e^{-(E-E_{f})/KT}]dE$$ Approximating $E_{v}+B_{v}\rightarrow\infty$: $$p=KTD_{v}e^{-(E_{v}-E_{f})/KT}$$ I hope this is helpful! Please point out any errors or bad approximations, if you see any.

HW 2 Solutions

Please comment here regarding if any aspects of these solutions are not clear or may be incorrect or different from what you got. I thought the \(k = \pi/2a\) and the  \(k = -\pi/2a\)  states turned out to be pretty interesting. There is a real part and an imaginary part. The two states differ in terms of the sign of the imaginary part of the wave-function; the real parts are exactly the same. Although they have the same energy, they are very different states. One propagates to the right; the other propagates to the left. Note also that every state in the band has the same \(|\psi|^2\) and yet they each different and have different energies, etc.

For the question about dividing the energies the band into three equal parts (2.1c), was the answer 9, 2, 9 or maybe 8, 4, 8? or something else? What did you get?

HW 3. Due Monday.

Unless otherwise specified, for the following assume that you are dealing with a 3D semiconductor* for which: \(E_g = 1 eV, \quad kT=.025 eV\) and    
\(D_c = 4 \times 10^{21} states/eV*cm^3, \quad B_c = 6 eV\)
\(D_v = 8 \times 10^{21} states/eV*cm^3, \quad B_v = 3 eV\).
(You can assume that for the undated case there are exactly enough electrons to have a filled valence band and an empty conduction band.)

1. What value of \(E_f\) corresponds to \(n = 10^{15} electrons/cm^3\)? What is p for this case?

2. What value of \(E_f\) corresponds to \(n = 10^{16} electrons/cm^3\)? What is p for this case?

3. What value of \(E_f\) corresponds to \(n = 10^{17} electrons/cm^3\)? What is p for this case?

4. What value of \(E_f\) corresponds to \(n = 10^{18} electrons/cm^3\)? What is p for this case?

5. What value of \(E_f\) corresponds to \(n = 10^{22} electrons/cm^3\)? What is p for this case?

6. What is the value of \(E_f\) for this semiconductor if it is undoped. What are n and p in this case?

7. What value of \(E_f\) corresponds to \(p = 10^{15} holes/cm^3\)? What is n for this case?

8. What value of \(E_f\) corresponds to \(p = 10^{16} holes/cm^3\)? What is n for this case?

9. What value of \(E_f\) corresponds to \(p = 10^{17} holes/cm^3\)? What is n for this case?

10. What value of \(E_f\) corresponds to \(p = 10^{18} holes/cm^3\)? What is n for this case?

11. For each of the above questions one assumes that the valence and conductions bands are  -- ------- -----------. What is the missing phrase (3 words)? This will turn out to be important.

12. a) Graph \(E_f\) as a function of n or \(E_f\) as a function of p or maybe n and p as a function of \(E_f\). What graph(s) best illustrates the relationship between carrier density and \(E_f\)? Discuss this here!
b) Added note: Here is an idea for a plot that  I think will illustrate this relationship. For a 1000 atom crystal suppose there are 3 bands, with bandwidths of 1 eV, 3 eV and 6 eV, respectively and band gaps of 2 eV and 1eV. Plot the number of electrons in the crystal as a function of \(E_F\) starting with \(E_F\) below the lowest band and ending with \(E_F\) above the top of the 3rd band. what is the domain of this graph?
[additional note: This is a theoretical exercise, not something that one can do with an ordinary material. (Although there are unusual materials where something like this is possible via "gating".]

13, 14 and 15 were edited on Friday at 8 PM: (this version is less difficult)
13. a) What value of n do you get for \(E_f\) 0.3 eV below the conduction band edge?
b) What value of p do you get for \(E_f\) 0.3 eV above  the valence band edge? Is this value of p larger or smaller than the value of n you got in part a)? Or is it the same? Explain.
c) What value of n do you get for \(E_f\) 0.3 eV above the conduction band edge? How does this compare with the value of n you got in part a)?  Discuss.

14. Density of states near the bottom of a conduction band in two-dimensions (2D): Assume a conduction band dispersion relation for a 2D crystal of the form \(E_{3,k} = E_3 - \Delta T cos(ak_x)- \Delta T cos(ak_y)\), where kx and ky range from \(-\pi/a\) to \(+\pi/a\).
a) What is the bandwidth for this band?
b) Derive the approximate form of E vs kx and ky near the bottom of the band in the effective mass approximation. (That is, to quadratic accuracy.) extra credit: Over roughly what range of energy or k is this approximation reasonable? Post comments here on this? You can ask, "what is reasonable" or speculate about what might seem reasonable to you.
c) Calculate the density of states as a function of energy near the bottom of the band.

15. Density of states near the bottom of a conduction band in three-dimensions (3D): Assume a conduction band dispersion relations for a 3D crystal of the form \(E_{3,k} = E_3 - \Delta T cos(ak_x)- \Delta T cos(ak_y)-\Delta T cos(ak_z) \), where kx, ky and kz each range from \(-\pi/a\) to \(+\pi/a\).
a) What is the bandwidth for this 3D band?
b) Derive the approximate form of E vs kx and ky near the bottom of the band in the effective mass approximation (that is, to quadratic accuracy) and use that to calculate the density of states as a function of energy near the bottom of the band. Is this D(E) similar to the one you got in the previous problem or different? Plot them vs E and discuss.
c) extra credit: What effective mass corresponds to a bandwidth of 10 eV in this 3D case.

3.X Square Well - Negative Kinetic Energy

Consider a 1D finite square well that is 1 eV deep and L= 0.613 nm wide. For the ground state: $k=.595π/L,a = .27 nm, B/A=.594$ (a) From normalization calculate A and B in units of $1/(nm)^{1/2}$ Evanescent Left: $\psi(x)=Be^{-(x+L/2)/a}$ Evanescent Right: $\psi(x)=Be^{-(x-L/2)/a}$ Inside well: $\psi(x)=Acos(kx)$ For the ground state, we will assume that the left and right evanescent wave functions will be the same, therefore the normalization will look like: $$\int \mid \psi(x) \mid^2dx = 1 = \int_{L/2}^{\infty} 2B^2e^{-2(x-L/2)/a}dx + \int_{-L/2}^{L/2} A^2cos^2(kx)dx$$ $$=A^2[\int_{L/2}^{\infty} 2(0.594)^2e^{-2(x-L/2)/a}dx + \int_{-L/2}^{L/2} cos^2(kx)dx] $$ $$=A^2[\int_{0.613/2}^{\infty} 2(0.594)^2e^{-2(x-0.613nm/2)/0.27nm}dx + \int_{-0.613/2}^{0.613/2} cos^2(0.595\pi/0.613x)dx] $$ $$=A^2(0.0952657nm + 0.463221nm) =1$$ $$A=1.338115nm^{-1/2}$$ $$B/A=0.594$$ $$B=0.794841 nm^{-1/2}$$
 (b) What fraction of the normalization integral comes from outside of the well? $$(0.2699999nm)B^2 + (0.463221nm)A^2 = 1$$ $$(0.2699999nm)B^2 = (0.2699999nm)(0.794841 nm^{-1/2})^2=0.17$$  About $17$ percent of the normalization integral comes from the evanescent region outside the well. 
(c) Calculate the expectation value of the Kinetic Energy: $$\bar{T} = \int\psi(x)[-\hbar^2/(2m)\frac{d^2}{dx^2}]\psi(x)dx$$ $$= \int_{L/2}^{\infty} 2B^2e^{-(x-L/2)/a}[-\hbar^2/(2m)\frac{d^2}{dx^2}]e^{-(x-L/2)/a}dx$$ $$+ \int_{-L/2}^{L/2} A^2cos(kx)[-\hbar^2/(2m)\frac{d^2}{dx^2}]cos(kx)dx$$ Plugging in constants and using Wolfram Alpha to save time: $$\bar{T}=-0.177832eV + 0.293069eV = 0.115237eV=0.12eV$$ (d)What is the KE contribution from the region of space outside the well?  The kinetic energy contribution from outside of the well is $-0.177832eV$. 

HW2. Due Wednesday.

This may be a pretty difficult and long homework. Going from Bloch states to density of states to occupation, and then finally calculating numbers of thermal electrons in the conduction band will probably take some time to sort out. I would suggest starting as soon as you can and spending a couple of sessions working on this and thinking about it over the next few days. Comments and questions are very welcome here. If you are stuck ask questions here, don't wait until class. Also, please reply to other students questions, e.g., David S, below.

PS. All email parts are rescinded. You can just include that in your written work.

part 1. Bloch states.

1.1 Using \(\psi_{atom} (x) = \frac{1}{\pi^{1/4}\sqrt{b}}e^{-x^2/2b^2}\) as your atom state with b= .05 nm:

a) Plot  \(\psi_{atom} (x) \) as a function of x.
For each of the next plots, let the range of your plot cover 5 cells, that is, n=-2 to n=+2, and let the crystal lattice parameter, that is, the center to center distance between atoms, be a=.1 nm (= 2b). You can use a computer to learn about these, but your plots must be hand drawn.

b) Plot the Bloch state made from this state for \(k=\pi/a\).

c) Plot the Bloch state made from this state for \(k=\pi/2a\).
d) Plot the Bloch state made from this state for \(k=2\pi/a\).
e) Plot the Bloch state made from this state for \(k= - \pi/2a\).
f) Plot the Bloch state made from this state for \(k=-\pi/a\).
g) Plot the "probability density" for each of the Bloch states above.
h) which of these plots actually require 2 plots and which can be done with only one plot?  What is the difference between c) and e)? What is the difference between b) and f)?

1.2 Can you figure out a general relationship between the Bloch state \(\psi_{n,k}\) and the Bloch state \(\psi_{n,k+2\pi}\)?

1.3 Post a comment on the post "3X..." that addresses some of the issues raised by that problem or ask a question.

Part 2. Density of states
2.1 For a crystal, a total of N energy eigenstates form from each atom energy eigenstate. These states can be kept track of using the crystal quantum number k. Allowed values of k are pretty much \(j 2 \pi/L\)  where j is an integer ranging from -N/2 to +N/2 where N is the number of atoms in our 1D crystal and L is the overall length of the crystal (L=Na).
a) Sketch a k axis and draw a small circle each allowed value of k for N=10.
Suppose the energy eigenvalues for each of the states at these discrete values of k are given by the equation \(E_{n,k} = E_n -(B/2) cos(ak)\)
b) Divide the k-axis into 5 equal sections: How many allowed k states are in each of these 5 sections for N=20? (this part is easy once you understand it.)
c) Divide the energy axis into 3 equal sections, one from the bottom of the band to B/3 above the bottom;  etc. How many states are in each of these 3 sections for N=20?
d) Calculate the density of states, D(E), for the of large N.
e) Do a graph of D(E) with the vertical scale set for 100. Use units of eV and assume that B=2 eV.

2.2 Sketch the density of states for \(E_2 = -10 eV\) and \(B_2 = 4 eV\),  \(E_3 = -4 eV\) and \(B_3 = 6 eV\). Suppose that the E2 band is filled and the E3 band is empty. What is the band gap? How much energy does it take to excite an electron from the top of the filled band to the bottom of the empty band? (email me this number)

2.3 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\):  \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many states are occupied if T=0 and \(E_f = -4 eV\).
b) Plot the density of states vs E and show which parts are occupied and which are not.

2.4 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\):  \(E_3 = -4 eV\) and \(B_3 = 6 eV\). 

a) How many states are occupied if T=0 and \(E_f = -7.5 eV\).

b) Plot the density of states vs E and show which parts are occupied and which are not.

c) What is notable about -7.5 eV ?

(email me your a very short description of the situation of 2.3 and 2.4, respectively (ideally just a few words) and your thoughts on the essence of the difference between them.)

2.5 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\):  \(E_3 = -4 eV\) and \(B_3 = 6 eV\). 

a) How many conduction band states are occupied if kT = 25 meV (which corresponds to T= about 295 K or so) and \(E_f = -7.5 eV\).

Actually, that is not a very good way to ask the question. let's ask instead how many thermal electrons there are, on average, in conduction band states.

2.6 Is there an easier way to calculate how many thermal electrons there are, on average, in the conduction band? An approximation you can use in problem 2.5? If so, what is it? How accurate is it in this case? 

Quiz postponed.

We can discuss in class Friday a rescheduled time for the quiz. Sorry. I have computer issues...

Books and reading.

There are a number of good books for this class. Two I recommend are:
1) Pierret's Semiconductor Device Fundamentals,
2) Streetman and Banerjee's Solid State Electronic Devices (which is the one we used last year). 

I think that reading from the Harris book that you used for 102 would be confusing and not a good idea. That book uses a very different approach which does not mesh at all with the way we are describing bands and states for this class.

We are using a band theory approach. (Sometimes called tight binding or LCAO.) In this approach, which meshes well with the actual nature of semiconductors and metals, bands are associated with atomic states and atomic energy levels. Beware of reading about free electron approaches. They are useful and relevant in some situations, but can be confusing when you are first learning about crystal states, energies conductivity, etc from a band theory-based point of view.

Office hours

I'll have office hours on both Monday and Friday after class in ISB 243 unless otherwise noted.

With regard to my office hours, I would like to ask a special consideration: because my immune system is somewhat less than ideal, we should avoid proximity and contact if you might be sick or feel like you may be coming down with something. Thanks very much. Your consideration with this is really appreciated!

HW 1a solutions & gifs showing time dependence.

These videos contain some solutions to HW 1a problems.


This second one deals with time dependence (#2.1) and might also help with problem 5.3e.

The gif below shows the time dependent wave function for problem 2.1. It is a wave-function of an electron in a 1D HO potential in a state (one state) that is a (normalized) superposition of two energy eigenstates. What is the red graph? What is the blue graph?  Comment and discuss here. (PS. If you want to make these yourself, it is a gif with a bunch of frames (maybe 20 or 30) each calculated for a different time ( as part of a time sequence).) 

Square well: negative kinetic energy.

    Here is one last problem. This one, I think, is better (more relevant to what we would wish to understand about quantum) than 3.2e.
     About the title, kinetic energy cannot be negative. However, contributions to the KE from some regions of space can be negative. (This came up in class I think when we discussed the H atom problem a bit at the beginning and some people were getting a negative KE.) These negative contributions to the KE turn out to be important to understanding band states and solid state physics, so I thought I would add one problem related to that.
     This problem elucidates where negative KE contributions occur. The bottom line is that they occur where the curvature of the wave-function is upward. For example: that is no where for the infinite square well (the curvature is always downward); outside the well for the finite square well; and in the region where \(E_n \gt \frac{1}{2} k x^2\) for the 1D HO. Basically where the electron wave-function extends into a so called "forbidden region".
      3.X ) Consider a 1D finite square well that is 1 eV deep and L= 0.613 nm wide. For this well I believe that for the ground state: \(k= .595 \pi/L\), a = .27 nm, B/A=.594.
a) From normalization calculate B and A in units of nm^-1/2.
b) What fraction of the normalization integral comes from outside the well?
c) Calculate the expectation value of the KE. (You can assume that outside left is the same as outside right and do just two integrals, one for inside, one for outside.)
d) What is the KE contribution from the region of space outside the well? What is the total KE?
e) Show that the PE is the same as the fraction of the normalization integral inside the well times -1 eV.
f) What is the total energy?
g) Discuss the kinetic energy integrand and mathematically why it is positive inside the well and negative outside.

1.3e - The size of the hydrogen atom.

The expectation value for the potential energy (U) was calculated to be,

$$
<V> = \frac{1}{ \pi a^3} \int_{V} \Psi^*(x) V(x) \Psi(x) dV
$$
where \(V(r) = \frac{-e^2}{4 \pi \epsilon_0} \frac{1}{r}\) and \(dV=r^2 sin(\phi) drd\theta d\phi\).

$$
<V> = \frac{1}{ \pi a^3} \frac{-e^2}{4 \pi \epsilon_0} \int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{\infty} \frac{ e^{\frac{-2r}{a}}}{r} r^2 sin(\phi) drd\theta d\phi
$$

$$
<V> = \frac{4 \pi}{ \pi a^3} \frac{-e^2}{4 \pi \epsilon_0} \int_{0}^{\infty} r e^{\frac{-2r}{a}} dr = \frac{4 \pi}{ \pi a^3} \frac{-e^2}{4 \pi \epsilon_0} \left(\frac{a^2}{4} \right)
$$

$$
<V> = \frac{-e^2}{4 \pi \epsilon_0} \frac{1}{a}
$$

and similarly, the expectation value for the kinetic energy (T) was calculated to be,

Quick and accurate integration

Wolfram alpha gets slowed down and confused by parameters such as "a". 
This video shows how to an approach integration that uses a unitless zero parameter approach to get accurate results quickly.

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HW 1b. Quantum review part 2. Itinerant states. Solutions included.

In the other HW 1, we reviewed and examined the nature of quantum bound states (e.g., bound state wave-functions, size and energies). To understand electrons and electron wave-functions (quantum states) in crystals it is helpful to also understand some things about states that are not bound. That is states that extend over "all space". In general these can be called itinerant states. The simplest examples are free electron states. Some itinerant states may also be propagating states as we explore below.

5.1 For a free particle one can set U(x)=0 eV for all x.  Consider the state \(e^{ikx}\).
a) Is this a momentum eigenstate? If so, what is its momentum? (its momentum eigenvalue)
b) Is it an energy eigenstate? What is its energy?
c) What is the momentum operator? What is the energy operator in this case? (You can post those here.)

5.2 Consider the wave-function \(\Psi(x,t) = e^{ikx} e^{-i E t/\hbar}\). Suppose k=1nm and E=0.038 eV.
a) Are those consistent with this being the wave-function of an electron? Why or why not? (Please post your thoughts on that here.)
a) Plot this wave-function vs x at t=0. What is its wavelength?
Comment here: In what way is this slightly difficult? How can you resolve that difficulty?
b) Comment on the gif that will appear here pretty soon.
c) Extra credit: Please make a gif that shows plots of this wave function vs x (range about 10 wavelengths).  Includes a bunch of frames so that we can see how this wave-function evolves as a function of time. You can post that here or send it to me and I'll post it.

5.3 Consider a wave function which is of the form: \(\psi(x) = A e^{ik_o x} e^{-x^2/2a^2}\) at t=0.
a) I think that the normalization factor is \(A= \frac{1}{\pi^{1/4}a^{1/2}}\).  Is that right? Plot \(||psi(x)|^2\).
extra credit: Plot \(\psi(x)\). To do that plot the real part and the imaginary plot as two plots on one graph. Assume k is about 10/a or so. That way there will be lots of wiggles. (added 4-7).
b) Use the KE operator to calculate the expectation value of the kinetic energy at this moment (t=0). (extra credit for posting the operator and the integral here.)
c) Show that you get two terms, and that the cross term is zero. These two non-zero terms should have a fairly simple form. Discuss their interpretation here.
e) extra credit: Assume U(x) =0 and use Fourier analysis to write a time dependent version of this wave-function.  (That is, a time dependent version of \(\psi(x) = A e^{ik_o x} e^{-x^2/2a^2}\).)

[Hint: \(\Psi_k (x,t) = e^{ikx} e^{-i E_k t/\hbar}\) is an energy eigenstate for the right value of \(E_k\).] For k= 1 nm and a = 100 nm, calculate \(\Psi(x,t)\) and make a gif showing the evolution of the wave-function as a function of time, and also the evolution of  \(|\Psi(x,t)|^2\).

PS. I put a comment below about the two terms of the KE in 5.3 (c).

Office hours.

I'll have office hours on Friday after class in ISB 243.  With regard to my office hours, I would like to ask a special consideration: because my immune system is somewhat less than ideal, we should avoid proximity and contact if you might be sick or feel like you may be coming down with something. Thanks very much. Your consideration with this is really appreciated!

About Physics 156

My office is ISB 243. Office hours are Monday and Friday 3:20-4:20 PM (or later if people are still around.)
email: The best way to contact me is by email at:
zacksc at gmail.com    (Please use this email and not any other emails you may have for me.)

Our final, assuming I am reading the UCSC schedule correctly, is on Wednesday, June 10 from noon to 3 PM.  That is scheduled by UCSC and set in stone. Other things are not completely certain at this time, but I expect that we'll have a midterm close to the middle of the quarter.

Our midterm will take one full class. I am thinking that May 11 would be a good date for that.

Quizzes: There may be online or in-class quizzes

Homework will be due mostly on Monday and Wednesday. 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: No book is required but Streetman (Solid State Electronic Devices, any edition) is probably pretty good. There are many similar books.  Pierret (Semiconductor Device Fundamentals) is also pretty good I think.

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 are the bands of eigenvalues of crystal energy eigenstates (wave-functions) that arise when we solve the problem of a single electron in a spatially periodic potential. Crystals can be either semiconductors (Si, GaAs) or metals (Au, Fe, Pb). (An intrinsic semiconductor is essentially an insulator with a small energy gap.) Bands are the natural starting point to understanding semiconductors and semiconductor devices, as well as 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).

Outline:
1. Review of quantum physics.
--wave functions and energies
    i) bound states. atoms. understanding the theoretical basis for the periodic table.
    ii) propagating (free particle) states. wave-packets.

2.Crystal quantum physics  
-----Constructing crystal wave functions from atomic wave functions. (Bloch states)
-----Crystal wave functions and their energies. Bands.

3. Bands and density of states. 
----What is a band? E vs k. How density of states is related to E vs k.
----The difference between a metal and a semiconductor.
----Valence and conduction bands.  What is an energy gap?
----Fermi function and DOS. How to use them together.
----Effective mass. Dispersion (E vs k) in 3D.

4. Doped semiconductors. Understanding \(E_f\) (also known as the chemical potential).

5. Conductivity:
  a) scattering rate
  b) effective mass

5. Understanding n-p junctions. What happens when you put dissimilar semiconductors together? How is equilibrium reached? etc. (This is the natural starting point for semiconductor device physics.)

6. a) Current flow in n-p junctions. Non-equilibrium thermodynamics. (This is not easy.)
b) Light emitting diodes and lasers.
c) solar cells.

7. Physics of Metals
a) Fermi surface
a) conductivity
b) color

8. Basic physics of superconductors
a) pairing
b) phase coherence
c) superfluidity

9. Magnetism
a) Pauli exclusion principle and electron-electron interaction.
b) why spins align (Ferromagnetism)

Here are some other things we might want to fit in somewhere:
The structure of silicon.
The structure of graphene.
The bands of graphene.
MOSFETS
Quantum computing. What other topics are you interested in? Please post here.

Latex testing post

Here is a place where we can test any latex (mathjax) type things.

\(\int_{-\infty}^{+\infty} A^2 e^{-x^2/a^2} x^2 dx \)

HW 1a: quantum bound states.

(Quantum review homework)
*For problems 1.4 and 2.1, let's all use the harmonic oscillator wave-functions posted by Nicolas Blanc in the comments below.
For this homework assignment our main focus is on results and relationships. Integrals can be done via wolfram alpha (see video above); there is no point in doing them by hand or showing work. Please comment and ask questions below. (Extra credit for finding errors, ambiguities, etc..)

Section 1: quantum length scales
1.1 Consider an electron in a 1D quantum potential \(U(x) =\frac{1}{2} k x^2\) where k= 100 \(eV/nm^2\).  Suppose that the electron is in a state of the form \(\psi(x) = A e^{-x^2/(2a^2)}\).
a) Find A. Calculate the expectation value of \(x\). Calculate the expectation value of \(x^2\).
b) What is the expectation value of the U ?
c) Calculate the expectation value of the kinetic energy, T . What is the relationship between T and the expectation value of \(p^2\) ?
d) Graph the expectation values of T and U as a function of a. What is each one for a = 1 nm? What about for a = 2 nm?
e) What value of a gives you the lowest value for the total energy, T + U ? How does this value of a depend on each member of the "quantum triumvirate" m, k and \(\hbar\)?

1.2 For the results for problem 1.1, calculate the product of the expectation value of \(p^2\) times the expectation value of \(x^2\). On what does this depend? Discuss.

1.3 a) For an electron in the ground state of a hydrogen atom, express the quantum length scale known as the Bohr radius in terms of m, e and \(\hbar\).
b) For an electron in the ground state of a 1D harmonic oscillator, express the quantum length scale, "a", in terms of m, k and \(\hbar\).
c) discuss.
 1.3e (see below)

1.4  a) For the 3 lowest energy states of a 1D harmonic oscillator, write the normalized energy eigenstate wave-functions. (Please use the HO quantum length scale to simplify both the normalization factor and the exponential.)
[Please post your answer here. That way we can all converge on the same thing.]*
(see comments!)
b) Plot each of these wave functions as a function of x.
c) Using wolfram alpha to save time, calculate the expectation value of \(x^2\) for each of these states. I am thinking that your answer will be in the form of a number times \(a^2\).
d) Take the square root of that to get something with units of length. Plot that (3 points) as a function of the quantum number. What might this tell you about the nature of these states?