Take home midterm: Due Tuesday (high noon)

Extra HW problem/ MT practice problem...

One more HW problem:
Consider a semiconductor in which mu=5,000 cm^2/Volt-sec, and kT = room temp (25 meV) and n is independent of y and z but depends on x such that dn(x)/dx = 10^16 cm-4.

Calculate the diffusion current density for the diffusion current that will flow in the x direction in this problem. (or is it the -x direction? ) Also, what are the units of the diffusion current density?

P.S. In your triage process, i would recommend giving this problem priority over asymmetric junction issues.

What did we learn today? (and more)

Well, several things, depending on how you look at it:

1) How to start problems in which the proximity of mu to the band edges is "given" (specified), instead of Na and Nd.

2) How Ec(x) behaves and looks for an asymmetric p-n junction (i.e., a junction in which Nd =/ Na). (And also rho(x) and E(x) (electric field).)

3) That things can become a little grungy, and definitely more difficult computationally, when we have less symmetry.

4) Anything else?? (your comments are welcome)

(Oh, i thought of one other thing. With regard to our process and the failure of our original method. What we (I) originally thought was that we could divide the "step" change in Ec(x) into two equal parts. That would have made the problem easier, but it was wrong and so we had to retool when we realized that.
In physics we often try to use intuition and symmetry to simplify; in this case we went too far and had to reconsider our simplifying assumptions. I think we learned the meaning of the phrase: "things should be made as simple as possible, but not simpler".)

Perhaps even more important than these specific things, we got a chance to spend more time in the world of band edges (Ec(x) is a "band edge"). This experience will serve us well when we take on the surprisingly daunting task of understanding exponential current flow in a biased p-n junction (that is, the exponential dependence of I on V).

At this point Ec(x), Ev(x) and mu should be your 3 closest companions (your new bff's, if you will). Otherwise the I-V analysis of a p-n junction will seem hopelessly complicated, when really it is merely hopelessly subtle and mysterious.

Cool stuff from Kelsey + sample questions:

Please post all comments and questions in the previous post.

Here are some sample questions you could ask:

I am confused about Ec-mu. What about Nd and Na? I don't understand how to work with this?

I don't understand question X?

What are the units of Y?

How can we graph XX when we don't know YY ?

Why would XXX effect YYY at all?

Question X seems impossible. How would one even start to think about that?

Reminder, sometimes questions and comments lead to cookies. Discussion is good!

HW #3, begin immediately, due Tuesday if possible

As we have discussed, one can start with a depletion approximation, in which we "pretend" n(x) and p(x) are both zero in the "depletion region", then, starting with that, on can calculate V(x) and hence Ec(x) and Ev(x) and then use them to recalculate n(x) from n(x) = Nc exp{-(Ec(x)-mu)/kt}, and p(x) from an analogous relation.

1) Suppose Ec-mu = mu-Ev = 0.1 eV, and Egap=1. eV (and kT=.025 eV)
a) Graph Ec(x). What is total variation in Ec(x) from left to right in your graph.
b) where are the boundaries of the depletion region?

2) Suppose Ec-mu =0.15 ; mu-Ev = 0.05 eV, and Egap=1. eV (and kT=.025 eV)
(How does that change things?)
a) Graph Ec(x). What is total variation in Ec(x) from left to right in your graph.
b) where are the boundaries of the depletion region?

3) (re)Calculate n(x) for each case.

4) For a homogeneous doped semiconductor, given n=4x10^17, m*=.2me, and hbar/tau=3 meV:
a) what is tau in sec?
b) what is sigma in (Ohm-cm)-1 ?
c) what is mu in cm^2/V-sec ?
d) What is rho? and what is R (resistivity) in Ohms for a "strip" 1 mm wide, 100 nm thick and 2 cm long?
e) What is I for that strip for a voltage of 10 microvolts end to end? How much heat will that generate?

5) how come nobody uses the concept of mean-free-path for semiconductors? (but they do for metals)

6) In a p-n junction, one can write J=n(x) e mu Ex(x) - e D dn(x)/dx (eqn 1) (c.f. eq 5.3 or so in Streetman), where n(x) is the density of electrons in the CB, mu is the mobility, Ex(x) is the electric field in the x direction and D is the diffusion coefficient. Turns out D = mu KT/e (Einstein relation).
a) Show that we can rewrite eqn. 1 (above) as:
J=n(x) e mu Ex(x) - n(x) mu KT d(ln{n(x)}/dx

b) take the derivative using the expression way above for n(x) = ...

c) evaluate at a value of x such as x=0 for a nice symmetric junction (as in problem #1). Show that this is, or is not, zero. [side points: What is Ec(x)? How is it different from and related to Ex(x) in terms of definition, units, physical relationship, etc. ]

7) Consider two different junctions, one in which Ec-mu =mu-Ev = 0.1 eV, and another in which
Ec-mu =mu-Ev = 0.15 eV. What is the diffusion current across the junction interface (x=0) in each case? How does the diffusion current depend on "delta mu"? It is a weak or strong dependence?

8) Suppose you apply an external voltage of 0,3 Volts. You may represent that by a linear dependence of mu through the depletion region (but mu is independent of x outside of that range). Do it in a way the reduces the offset of the band edges (so that Ec and Ev have less x dependence.)

a) How does this effect the extent (length scale, x_0) of the depletion region?

b) Sketch the band edges and mu

c) (major extra credit) Starting with the expression for J (in #6, above) consider, analyze and discuss how this effects the current through the p-n junction. Big change, small change...???

Questions and discussion are welcome and encouraged.

Fun things to do

Fun things to do between now and Thursday:

1) Read about "transport". J = sigma E, sigma =n e mu, mu =ne^2 tau/m* (whew)

2) read about the other kind of transport: diffusion... related to dn(x)/dx (i.e. density gradients)

3) reexamine the n(x), rho(x) calculations. Does our approximation get better or worse as T gets lower?

4) find the depletion approx equilibrium x_0's for an p-n junction with 10^16 cm-3 acceptor doping on the left and 10^17 donor doping on the right? Graph Ec(x) for that...

5) if this is the 4th week, then you might want to be think about preparing for your upcoming take home and in-class midterms. (10 week qtr, "mid term"=5th week right?)

6) other suggestions of things to do or study that relate are welcome

Possible picture and coord system

The comments people have been posting look excellent, and if you are already on your way to solving and understanding n(x) feel free to ignore this. I just thought I would make the mistake of posting a possible picture and coordinate system (setting 0 coincident with Ev on the far right), etc. Using deltaE separates the problem into two distinct parts and subsumes everything you have done with rho, Electric field and potential into the parameter, deltaE, which is a coefficient in the quadratic dependence of the band edge energies ...

HW 2b, due Tuesday (p-n junction calcs)

The problems below, for Tuesday, 4-20, I think will warrant some discussion. I don't think you will just look at them and say, "oh, no problem, I can get drunk and do those Monday night". So feel free to say, "wow, I am stuck. what ...?" It is okay to ask any questions. I think other students will appreciate your attempt to articulate your stuckness; I believe I will too.

Let's try posting all questions and thoughts as comments to this post since that other one has dropped down quite a bit.

The accompanying notes summarize some of what we need for our "symmetric" p-n junction problem. If you complete the 2nd half of V(x) (the x greater than zero part), then you have a change in potential across the depletion region, which is associated with the space charge regions on either side of the interface. You can use that to create a model picture in which the chemical potential is constant (i.e., does not depend on x), and the band edges are continuous, as a function of x, have the "correct" proximity to the chemical potential in the large x regions (large and negative, large and positive).

Please try the following:
(Assume Nv=Nc=5x10^18 cm-3 and Eg = 1.0 eV)

1. Guess (use your intuition to consider) which has a wider depletion region: the above model with a doping of 10^16 cm-3 on each side or with a doping of 10^17 cm-3 on each side. Explain your reasoning.

2. Calculate and compare the widths of the "depletion region" (2x_0) for dopings of 10^16 and 10^17 cm-3, respectively.

3. For the 10^17 cm-3 case calculate n(x), that is, the density of carriers in the conduction band. On what does that depend? (I imagine this is probably best done numerically since it requires exponentiating a more-or-less quadratic function. You can choose about 9 equally-spaced points covering the region from -x_0 to x_0, if you like, (an odd number allows you to "hit" x=0), and make a table of x, Ec(x)-mu, and n(x).

4. Graph n(x) on a linear scale. How does it look?

5. Do the same for p(x).

6. Here is an interesting conundrum. This is supposed to be equilibrium, right? But at x=0 there is a big, fat electric field, and the carrier densities are not exactly zero right? So would there be current flow? In what direction? If so, what kind of equilibrium is this? If not, why not???

Do not hesitate to ask clarifying questions. Are these problems not vague?

Temporary office hour modifications: (in the interest of reducing the clutter of "too many different posts", i'll put this here.)
Today and next Friday I will have office hours at 3:10 PM. Also, next week my Thursday office hours are canceled we will only have office hours on Friday at 3:10 PM.

HW 2a Soln notes

Notice how in problem 1 there is a sot of symmetry in that the energy from the energy to the band edge is the same for holes and electrons for the same doped carrier concentration. ("Carrier" is a general term that refers to either holes or electrons.)
However, in problem 4 that symmetry is gone. Because of the higher state density in the valence band, one achieves the same carrier concentration with a mu that is further from the relevant band edge.

In this "asymmetric" model, one also finds that in the undoped (intrinsic) circumstance, mu is not in the exact center of the gap (and the n_i and p_i are a little higher because the state density is higher.

(It is important to carefully distinguish between state density and carrier density in your thinking and your language. The later depends on mu, the former does not.)

HW 1 Soln notes

For the problems in which you are sketching states for potentials with several identical wells, the approach is to: 1) work with the ground states for the single wells, and 2) combine them, using the +- freedom, in such a way as to increase the number of nodes by one each time. The ground state has zero nodes, 1st-excited state has one node, 2nd excited state has 2, etc. That principle can guide you in creating solutions to all the problems except 1e, which goes one state beyond what you can create with the ground state. There you have to use the single-well 1st-excited state for each well (you can't mix ground and excited states; that is related to momentum "conservation"). In that case you get 4 nodes in the well centers (none between the wells). There is a pretty big jump in energy between state 4 and 5 in that case (see picture and graph). Conceptually, that state will belong to a different band than the four below it.

Problem 2 involves a graph of state energy vs "q", which has units of inverse length. The E vs q function has a minimum at q=0 (long wavelength and is biggest at the largest values of q (+-pi/a). For this to make sense, 2t should be pretty small compared to E_0.

Problem 9 gives us a rough intuitive sense of the nature of the bound state that forms around an impurity, including the effect of dielectric screening, and carrier effective mass, in making this state only very weakly bound.

We will discuss q and effective mass more when we consider transport properties, e.g., the movement of electrons in the conduction band.

For Thursday homework reminder

As we discussed in class, for Thursday your HW is to calculate x_0 so that the following constraints are all satisfied:
1) a constant chemical potential (independent of x),
2) an appropriate relationship between mu and the CB and VB edges based on the doping should be maintained in the "bulk" regions far from the interface, and
3) no discontinuities in either the CB or VB (edges) as a function of x.

This is all in the context of an all or nothing ansatz: i.e., at a given value a x, either all the CB electrons escape to the VB on the other side, or none do...

Steps probably involve graphs of, and relationships between, charge, electric field, potential and energy...

Advanced problem #1 (optional)

Since the HW assignment due on Tuesday may be fairly short and perhaps insufficiently challenging, you are invited to work on the following extended problem.

Consider a semiconductor model characterized by: Eg=1.eV, Nv=Nc=5x10^18 cm-3. For x less than 0 it is doped with 10^16 acceptors/cm-3. For x greater than zero, it is doped with 10^16 donors/cm^3. (pause and draw a picture here)
Let us assume --and this is a big leap-- that all the excess electrons in the "slice" between x=0 and x=d (d is some distance, as yet undetermined), sneak over to the x less than zero side and reside in the region between x=-d and x=0.
Graph the (net) charge as a function of x. Graph the electric field associated with that charge as a function of x.

Feel free to ask questions or discuss here via comments. All comments should be in your own words; do not site outside authorities.
You are encouraged to work on this, but it is optional. Work you hand in on this should be well-presented with appropriate size graphs (i.e., small, e.g., 3"x3" more or less) embedded along with text.

Reading, April 8

I know there are a lot of posts here. I hope and expect that you will read all of them, and that you will take responsibility for understanding and responding to their content as needed. If you are accepting of this approach (and not longing for a single syllabus in which every reading assignment and homework is pre-established) I think you will find it beneficial.

For the near future, please read chapter 5, or other material related to equilibrium conditions for p-n junctions. Later we will come back to chapter 4, which discusses "excess carriers", e.g., photon induced carriers in semiconductors. That is really interesting and we expect to cover it in some depth later in the quarter.

Currently our focus is on equilibrium carrier concentration. We are moving from exploring equilibrium carrier concentration in homogeneous semiconductors (what we did last class) toward equilibrium carrier concentration in an inhomogeneous semiconductor. That means a semiconductor where different parts are doped differently and includes the iconic p-n junction.

Homework 2a: Due Tuesday.

preface: For the problem we worked on in class, we characterized our model (which represented a "homogeneous" semiconductor) by an energy gap (1 eV) and by an intrinsic carrier density (at room T) of 10^10 cm-3. An alternative approach to modeling (which is not really much different) is to specify Eg, Nc and Nv, where Nc and Nv are related to the state density at the edge of the CB and VB, respectively (as defined in our discussions in class).

rationale/due date: I got to thinking that if there was no HW due until next Thursday, some people might not work on these things until possibly next Wednesday. I believe, however, that it may be valuable for students to work problems frequently, so that these concepts (like the relationship between mu and carrier concentration) sink in and become "second nature". So with that in mind I decided to creat this short assignment due next Tuesday.

Homework 2a (due Tuesday, April 13):

1. Suppose our semiconductor "model A" has:
an energy gap of 1.0 eV
Nc=5.0 x 10^18 cm-3
Nv=Nc

a) Calculate mu for a doping level of 10^16 donors/cm^3 (e.g., phosphorous atoms).
b) Calculate mu for a doping level of 10^17 donors/cm^3 (e.g., phosphorous atoms).
c) Calculate mu for a doping level of 10^16 acceptors/cm^3 (e.g., boron atoms).
d) Calculate mu for a doping level of 10^17 acceptors/cm^3 (e.g., boron atoms).

2. What doping level would correspond to mu=0.7 eV for this model?

3. What is n_i for the above model?

4. For a model, with Eg=1 eV, but with Nc=5.0 x 10^18 cm-3 and
Nv=2Nc:
a) Calculate mu for a doping level of 10^17 donors/cm^3 (e.g., phosphorous atoms).
b) Calculate mu for a doping level of 10^17 acceptors/cm^3 (e.g., boron atoms).
c) Calculate n_i.

5. Discuss the differences between the 2 models.

6. (Extra credit/optional) Consider a semiconductor (model A) in which the left half (x<0)>0) is doped with 10^17 cm-3 donors. What is it like?

Student questions, comments, HW related questions, etc

Let's try focusing all questions and discussion*, on HW and everything else, as comments to this one post. That way if you want to see what everyone has posted, all you have to do is look here. (This will be the active "comment post" for one week starting now and going until next Tuesday's class.)

* The one exception at this time would be your quiz response. Post that as a comment to the quiz post itself rather than here. Everything else, here!

Thoughts on exploring mu

There are no new hw problems that i have to assign today, but here are some thoughts on what you might do "for fun" to follow up on our exploration of doping and mu today.

Try doing our problem today, i.e. calculating chemical potential, mu, for a few different values of doping (i.e., phosphorous substitution concentration). Like maybe: 10^17 and 10^18. See where our approximations break down... (like the apprroximation of f(E). was that okay today? when would it not be okay?
(hint: if one finds that mu is venturing into, or really close to the CB, that would not be good, right?) What if we started out saying that n_i was 10^11 (cm-3) instead of 10^10? How would they effect the breakdon pt and why?...

What about lower dopings like 10^14 or so? Does anything get weird there?...

So anyway, one could continue to explore. Try making Nc and Nv different, see what effect that has. Try specifying Nc and Nv explicitly, instead of implicitly through n_i or ...

PS. What would you think of a surprise pop quiz on this on Tuesday in class?

April 5: Reading & preparation; HW and Quiz 1 due date

To prepare for this weeks classes, reading about chemical potential (or Fermi energy) and its relationship to carrier density and "doping" will be very useful. This weeks classes will focus on exploring and understanding the relationship between chemical potential and the number of occupied states in the conduction band (or unoccupied states in the valence band), i.e.,the relationship between chemical potential and carrier density in semiconductors. This will prepare us to understand a p-n junction, the iconic structure in which hole-doped and electron-doped regions meet and say, "this semiconductor ain't big enough for both of us."
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Homework 1 (parts I and II) and quiz 1 are due Thursday.

-In general, homework may be assigned on Tuesday, Thursday and Saturday (on the web) and will be due on Tuesday and Thursday. Generally you will have a week, or sometimes more if the problems are especially difficult, to work on an assignment. (Assignments will overlap.) The problems will range in difficulty from not-that-difficult to very-very-challenging. Feel free to seek help from classmates in person or by posting questions on the web site. If you are confused other people probably are also; asking/formulating questions is a great way to learn, and the dialogue that follows questions from one student's question or comment can help everyone.
Please initiate and contribute to dialogue.

Office hours, grade breakdown

My office hours this quarter will be Thursday, 2:00 - 3:45 PM.

homework and quizzes: 25%
midterm: 35%
final: 40%
all are important. keeping up all along the way will be essential.

Due Dates for HW etc

For each post I will put a start date and then you can start it, see how hard it is, and then we can discuss/negotiate the due date.
Speaking of which, what do you think are reasonable due dates for HW1 ? for quiz 1?
(The sooner the better as far as I am concerned; the faster we go the more you will learn (up to a point).)
RSVP
-Zack

PS. If anyone has not found this web page, they would be getting behind pretty fast in this class. So since you are reading this, that would be not you. But if you have any friends (or aquaintances) who may not have found it, perhaps you could: help them. cajole them, guide them...

Quiz #1: start date April 2

Here is our first quiz. As soon as you feel ready, please take this quiz online by posting your responses as a comment to this post (and writing your name in your comment so i know who you are). The sooner you do this the better, moreover, it is okay to update your responses (answers) with a 2nd comment later (and i can even delete your first comment if you request that). I hope that makes sense.

Anyway, here is pop quiz #1: (PS. Let me know if you would rather have gotten this at the beginning of class? (just wondering) I thougth ths might be less stressful and offer more oppoptunities for learning.

1. What 3 essential "components" do you need in order to make a minimal model of a semiconductor?

2. What does the center energy of a band of states in a crystalline solid primarily depend on?
(what is the most important thing?)

3. What does the bandwidth of a band of states in a crystalline solid primarily depend on?
(what is the most important thing?)

4. Suppose a model solid is constructed from a periodic array of identical finite square wells. Each well is 10 eV deep, has a width d less then a, the periodic spacing, and, in isolation, each well would have a ground state at -9 eV and a 1st excited state at -5 eV.
Sketch what the bands for this model solid might tend to look like. Where are they roughly centered? How wide are they? Which one is wider?
(You can get extra credit if you notice and explain the subtle things associated with the difference in sign of the overlap integrals that play a role in the respective bandwidths for these bands and how that effects the graph of E vs q.)

5. What is exp{E/KT} for T = room temperature and:
a) E= 1 eV
b) E= 2 eV

Homework #1, part II, start date April 1

I am putting the 2nd part of HW#1 in a separate post because it involves a different topic. These problems, on energy scales, are not difficult I think. If you start on them right away it will get you thinking about things that will help you get more out of our upcoming classes on the nature of semiconductors. Energy scales play a big role in that.

#5. a) Roughly, what is "room temperature" in Kelvin? b) What is KT, in eV, where K is Boltmann's constant and T is room temperature?

#6. What is the band gap of Si, Ge and of diamond in eV?

#7. What is exp{E/KT} at room temperature if E is 1 eV?, 0.5 eV?

#8. What is the wavelength and energy, in eV, of a photon of red light and of green or blue light?

#9. a) If you substitute one phosphorus atom for a Si atom in silicon, what is the energy of the localized bound state that forms in the neighbor of the P atom?
b) What about for Au doped into Ge?(optional)