That extra MOSFET problem

Thinking more about the advanced problem we spoke about in class, which had to do with charge as a function of voltage in a MOSFET, I realize it could be pretty difficult. I think we're imagining starting with a p-doped semi-conductor starting out in a flat band situation in which there is no space charge and no band bending. The initial application of positive voltage leads to the development of a depletion layer, associated with the valence band bending away from the chemical potential. The width of this depletion region will tend to grow linearly with voltage, I think, and in that way Q=CV is satisfied (with a constant C, which I would expect depends simply on the width of the insulating oxide and its dielectric coefficient). Does that make sense? I hope so. Because, really, that's the easy part.

Then, as more voltage is applied, the conduction band will begin to get too close to the chemical potential to be ignored. And at that point, a large electron concentration, associated with what we might call deep inversion, will develop close to the interface between the semiconductor and the oxide.

I guess I was thinking that if one could assume some sort of exponential dependence of the charge density on z and then one could integrate that twice to get a potential that depends also exponentially on the z, and that somehow that might be self consistent. But then I realized, the charge density, which in that region is associate with the conduction electron density (n(z)), depends exponentially on the potential so we have an exponential of an exponential which doesn't seem very self consistent or easy to work with at all. HMMM…

So anyway, strike one, I guess. If anybody has any ideas how to model the inversion region self-consistently, or how to make a convincing argument that the width of the depletion region stops increasing when strong inversion occurs, that would be really great. Then at least we would have a pretty clear idea where the charge goes as a function of voltage. I'm still wondering how one might characterize the functional form and width of the strong depletion region. Maybe its not that easy.

Final, please comment, Due Weds noon...

Lasers and LEDs

Comments, questions and discussion are encouraged.

Why n+ to p to p+ ?
Where is the p-n junction?
Where is the light emitted?

What role(s) does the (GaAl)As play?
How does it help?

Circuit elements

Here are some pictures of "circuit elements" that use FET's. Please comment, question, etc.
Beaucoup extra credit for any ones who explain what these are and how they function.

Last HW assignment

For this week's HW perhaps you could write a two page report outlining what you have learned in this class, (including what you expect to learn about lasers and photo-detectors next week). You can also post outlines of this here or comments, questions etc. related to this matter.

Also, suggested problems for the final are encouraged.

Key moments in the life of a MOSFET

Key moments for a MOSFET tend to occur as a function of z, distance from the oxide (in the semicondoctor), or V (applied gate voltage). They generally involve the relationship of the bands to $\mu$. Focusing on the phenomenology of the space-charge:
1) one obvious "key moment" is when the valence band has moved a few kT away from $\mu$ indicating the onset of depletion,
2) a second, less obvious "key moment", which generally occurs at a smaller value of z, is when the the electron concentration in the conduction band, n(z), becomes greater than $N_a$. At that point one is leaving the depletion regime, in which the space charge can be approximated as a constant ($-e N_a$), and entering a realm in which the space charge will be strongly (exponentially) dependent on z.

(Actually, n(x) always depends exponentially in $E_c (z)$, but that is "hidden" until the rubicon of "2)" is crossed as I understand it. )

(Comments welcome.)

PS. a third key moment might be when $E_c(z)$ crosses $\mu$, but we should probably focus on understand the first 2 before delving into that.

MOSFET picture clarified

Maybe this will help clarify some of the issues that came up in today's class. This focuses on the nature of a metal-oxide-semiconductor(p-doped) sandwich, where-in the chemical potentials of the metal and doped semiconductor are perfectly aligned. In the left-hand picture the voltage is zero. If you apply a positive voltage, that leads to depletion in the p-doped semiconductor (center picture). All the equations pertain to the center illustration and to the use a a depletion ansatz for that case. It is interested to see how the applied voltage, V, influences the depletion width, $z_0$, and the band bending, $\Delta \Phi$.

Looking at these relationships, which seem pretty understandable, i am thinking we should have started with this simpler zero chemical potential difference case. For extra-credit, who can figure out/ calculate how a higher chemical ptential in the metal would effect the equations for the depletion width and $\Delta \Phi$.

Homework 6, revision

I think at the heart of the MOSFET and related MISFET-type FET's, including those based on GaAS, lies the metal-insulator-semiconductor structure, which, in some ways, is essentially a capacitor. What I hadn't realized until today was the critical role that the chemical potential (or Fermi surface) of the metal plays in the whole thing.

These problems should help us understand that role. Once we understand it, maybe it won't seem so important or mysterious, but i think that until we do there may be a cloud of confusion hanging over whatever we do regarding metal-insulator-semiconductor structures.

So I would recommend doing this problem (really 3 problems) before any of the others assigned this week.

1. Consider a metal-insulator-semiconductor sandwich. Suppose the semiconductor is p-doped, and, for definiteness, suppose the band gap is 1 eV and $\mu - E_v$ = 0.2 eV far to the right (far from the metal) (moderate doping).
Sketch the band edges, the space-charge density, electric field, etc., in the semiconductor as a function of z (distance from the insulator):
a) for the case where the chemical potential of the metal is matched to that of the semiconductor.
b) for the case where the chemical potential of the metal is (or would like to be*) somewhat higher than that of the semiconductor.
c) for the case where the chemical potential of the metal is (would like to be*) somewhat lower than that of the semiconductor.

* (The problem statement is a little confusing, since the chemical potential should be independent of z, so maybe someone can clarify that here via comments. Maybe someone would like to outline their understanding and approach.)

PS. Even though there is no charge in the insulator, there is probably an electric field, in some cases, so it might be important to include that as well in lining things up.

PPS. This draws on exactly the same skills and principles we developed in studying p-n junctions.

Diamond structure post

In class on Thursday Zack and I seemed to have different opinions on the distinguishability of sites in the Si crysal structure (Zincblende, or diamond). This structure has bravais lattice vectors $a(\frac{1}{2},\frac{1}{2},0)$,$a(\frac{1}{2},0,\frac{1}{2})$,$a(0,\frac{1}{2},\frac{1}{2})$. The basis vectors for this structure are $(0,0,0)$ and $a(\frac{1}{4},\frac{1}{4},\frac{1}{4})$.

For those unfamiliar with crystal structures, the basis of a crystal is the unit that has translational symmetry, that is, the part that repeats in space. So, this structure has two types of atoms that repeat, one for each basis vector, and the two atoms are connected by the vector $a(\frac{1}{4}-0 , \frac{1}{4}-0 , \frac{1}{4}-0)$, namely the difference vector between the two basis vectors.

Now that we have our basis, we put a copy of the basis at each point generated by the bravais lattice vectors. In other words, we take the bravais lattice vectors and form all possible linear combinations with integer (...,-2,-1,0,1,2,...) coefficients. (For example, $5a(\frac{1}{2},\frac{1}{2},0)+3a(\frac{1}{2},0,\frac{1}{2})-2a(0,\frac{1}{2},\frac{1}{2}) = (4a,\frac{3a}{2},\frac{a}{2})$ is one such point generated by a linear combination of the bravais lattice vectors with integer coefficients 5,3,-2.) Then, we put a copy of our basis on each of these points. Since our basis is $\{(0,0,0),(\frac{a}{4},\frac{a}{4},\frac{a}{4})\}$, we add these vectors to our generated points to find out where all the atoms are. For our example of the point $(4a,\frac{3a}{2},\frac{a}{2})$, we have atoms at $(4a,\frac{3a}{2},\frac{a}{2})+(0,0,0)$ = $(4a,\frac{3a}{2},\frac{a}{2})$ and $(4a,\frac{3a}{2},\frac{a}{2})+(\frac{a}{4},\frac{a}{4},\frac{a}{4})$ = $(\frac{17a}{4},\frac{7a}{4},\frac{3a}{4})$.

With a picture of the structure (somewhat) firmly established, let us now come to the apparent disagreement between me and Zack.

Zack commented that the two sites in our basis ($(0,0,0)$ and $(\frac{a}{4},\frac{a}{4},\frac{a}{4})$) are identical by symmetry. From this conclusion, he mused that it was puzzling two basis vectors were required to describe the structure.

On the other hand, I argued that the problem was more subtle. The sites were identical and not at the same time. While this seems an obvious contradiction, this is not the case.

I agree with Zack in saying that the two points of the basis are indistinguishable in one sense. Let atom A be at $(0,0,0)$ and atom B at $(\frac{a}{4},\frac{a}{4},\frac{a}{4})$.

If we translate the whole crystal by $(-\frac{a}{4},-\frac{a}{4},-\frac{a}{4})$ and then rotate by 90 degrees about the z-axis, atom B now looks like it is at $(0,0,0)$. (Atom A is off in a different place than $(\frac{a}{4},\frac{a}{4},\frac{a}{4})$, but that doesn't matter--we just needed to show that both atom A and atom B could be seen as at point $(0,0,0)$.)

So, this seems to show that atom A and atom B are identical, since either can be at $(0,0,0)$. However, this is not the whole story. The difference between the two lies in their positions relative to each other. Say atom A is at $(0,0,0)$ and atom B is at $(\frac{a}{4},\frac{a}{4},\frac{a}{4})$.

There are 8 possible locations for atom A and atom B to have nearest neighbors. The eight locations relative to either atom A or atom B comprise all choices for $(\pm\frac{a}{4},\pm\frac{a}{4},\pm\frac{a}{4})$. The distinction between atom A and atom B lies in that atom A has exactly 4 of these neighbors, and atom B has the other 4--there is no overlap. No matter how you rotate or translate the crystal, atom A will have 4 neighbors in 4 different directions, and atom B has 4 neighbors in 4 different directions, but the directions never coincide.

If we choose atom A to be at $(0,0,0)$, make our origin the bottom front left, make the front axis going to the right the x-axis, the line going into the board the y-axis, and the vertical the z-axis, here are the directions of the nearest neighbors for atoms A and B:

(NOTE: the directions are actually multiplied by $\frac{a}{4}$, but this seemed cumbersome to repeat over and over.)

atom A: (+,+,+);(-,+,-);(-,-,+);(+,-,-)

atom B: (-,-,-);(+,+,-);(-,+,+);(+,-,+)

So, while either atom A or atom B can be at $(0,0,0)$, the two sites are distinct from each other. In fact, there are actually two classes of atoms here--those with neighbors in the same direction as atom A's neighbors and those with atoms in the same direction as atom B's neighbors. This is why we need to have a basis of two atoms--if atoms A and B were truly identical, then only one basis vector would be required, namely $(0,0,0)$.

This problem shows that it is critical that we look at the neighbors of an atom to judge whether it is identical to another atom, for this is often how we can tell different sites apart. Moreover, in order to find the smallest building block for a crystal (our basis), we need to know how many unique atom sites there are.

If any of this is incorrect or confusing, please comment!

Homework 6

FET's are inherently complex and there are many varieties which are similar in some ways, but differ in important details. My suggestion is: let's read chapter 8 some more and let's do some end of chapter problems from that chapter. Your suggestions are welcome. I am thinking of some problems related to MOSFET depletion and inversion region formation etc., numbers 8.6 to 8.9 in my version. I will scan some pages tomorrow. In the mean time, your suggestions are welcome.

P.S. Here is a warm-up problem:

1. Consider a structure that involves a metal gate, a (GaAl)As insulating layer, and undoped GaAs (in that order). Suppose the insulating region is doped with $10^{11}$ Si atoms per $cm^{2}$, so that with zero gate voltage there is a thin conducting region of areal carrier density $10^{11}$ electrons/cm^2 on the GaAs side of the interface.
a) Sketch this structure.
b) Calculate the influence of gate voltage on the carrier density at the interface. What voltage would be required to (turn off) reduce the carrier density to zero? (You can use an $\epsilon$ for (GaAl)As of about 12, or look up a value (for $Al_{0.3}$) and post it here.)
c) (What approach would make this problem not too difficult and use epsilon?)

Solutions: Homework # 4

Here is HW #4. Can you label the 4 currents in the graphs on page 2. Please feel free to comment and question. There is no claim of infallibility associated with this, or any other post, on this site.

Solutions: Midterm

Here are some solutions. Please do not assume perfection and feel free to comment here if you find a discrepancy with your own results, or if you are just not sure about something. For example, in #5, I am puzzled about the value used for $n_i$. I thought it was about $10^{10} cm^{-3}$ ?
Also, maybe the drift current should be negative...

Reading, May 17, chapter 8

I would recommend reading chapter 8. You can skip chapter 7, and focus your attention on chapter 8.

Homework 5

Let's try to do a lot of this via discussion here:
1. a) Describe a picture of how a solar cell could work (which you could compare and contrast to an LED if you like). What processes are important; what currents are generated? In what regions of the junction?
b) What might be the influence of different biases (+, -, magnitude) that could arise? Consider, for example, connecting the junction +- infinity regions to each other via a capacitor (or a resistor).
c) if you are familiar with photosynthesis, compare, or contrast, its process(es) with those of a solar cell.

2. a) What does MOSFET stand for?
b) sketch a MOSFET and describe how it works.
c) what is the difference between "normally on" and "normally off"?
(discuss here)

3. extra credit: discuss here. What does the C in CMOS stand for? What is an advantage of having both hole and electron FET's? (discuss here)

3. Come up with another problem to add to this assignment. (and present it here as a comment)

4. For our usual p-n junction parameters mu=5000, Nd=10^17, etc.
( z) Would someone please post our usual parameters here.)
a) what value of the recombination time (which we have been calling $\tau_p$ on the right side, where diffusing holes are the minority carriers) would make the width of the diffusion region 5 or 10 times larger than $x_0$? (width defined by 1/e..., feel encouraged to ask about that or explain it in detail here.)

5. (optional, xc) For a "Scientific American" type article, do an illustration of a p-n junction under forward bias that includes all important regions: (depletion, recombination or diffusion, and, hmm, what should we call that outer one?)
Discuss here: the best names for each region? (This is an audience with no preconceptions regarding the names of the regions.)
a) Is "depletion" an ideal choice? Can we do better?
b) recombination vs. diffusion vs other choice??? what is most accurate, descriptive or helpful.
c) what about the outer region.

PS. You can choose to focus on photon emission (LED), or phonons, i.e., whereby the energy of recombination turns into heat (or both).

What to cover for the rest of the quarter

I am working on an outline of what to cover for the rest of the quarter and would be interesting in the thoughts and ideas that you have about that. You can post those here as comments or send them by email if you prefer.
Thanks,
-Zack

Homework 4, continuation post (and summary of today's class)

all about J, under bias, continued...
Homework 4 and today's class are completely linked. I got the sense that today's class was difficult and confusing; but I think we can simplify and clarify the issues by breaking them down into parts.

From a mathematical perspective, we obtained a differential equation and solved it with boundary conditions at $x=x_0$ and $\infty$. However, the boundary condition we used at $x=x_0$ was very non-obvious and important. In fact, the entire essence of p-n junction phenomenology, including diode behavior, LED's and perhaps solar cells, hinges on understanding the origin and implications of this b.c..

There are two parts which are described in more detail in the 1st comment below:
1) the first part involved getting a solve-able differential equation for the hole current in the region $x \ge x_0$, which is discussed in the 1st comment below.
2) the second part involved establishing the boundary conditions for that solution.

Boundary conditions are usually pretty boring, but in this case the boundary condition at $x = x_0$ is not at all boring and in fact involves a bold step that takes us outside the realm of equilibrium physics in a way that most of us have not experienced before. Let's focus on that first, and let's separate explaining what we did from justifying it.
What we did was to take the chemical potential that was valid on the far left, and use it at $x = +x_0$. Pretty weird, huh? The consequence of that assumption was that we obtained an enhanced value for p(x) at $x_0$ which leads to an enhanced p(x) for $x \ge x_0$ and an enhanced and unopposed $J_p (x)$ for $x \ge x_0$. That enhancement is exponential and it leads to a hole diffusion current at $x = x_0$ of $J_p (x) \approx (e n_{i}^2/N_d) [e^{eV/kT} - 1] (D/\tau_p)^{=1/2}$. There should be a similar electron diffusion current on the other side ($x \le -x_0$. These represent charge current flow in the same direction; holes to the right, electrons to the left and may provide a model basis fro understanding current-voltage relation for a p-n junction.

One implication of the enhanced minority carrier concentration (e.g., enhanced p(x) for x gt x_0) is that the ordinary relationships between n, p, $\mu$ and KT are not applicable. I believe you might also describe the regions around $x_0$ as "hot", though that may be more confusing than helpful.

Please post discussion of this below. There are a number of aspects to focus on including:

• the math,
• the assumptions,
• the results, including the magnitude of the current, its dependence on voltage,
• and the picture that emerges from all this including where recombination occurs, etc.

Connecting math and assumptions to a physical tangible understanding is the essence of physics.

Added HW problem and discussion point.

How would you justify using $p(x_0) = (n_{i}^2/N_d) e^{eV/kT}$?
(or, equivalently, $p(x_0) = (N_a) e^{-e(\Delta \Phi-V)/kT}$? )
(did i get that right?)

Important post on pn junction current-flow, generation, recombination and more.

Last class an interesting point came up (i think from Dave M) regarding whether primarily generation or recombination (g or r) occurs in the the regions just outside the depletion region. As I recall, our discussion of that issue left it unresolved and confused. The more I think about it, the more I think that is critical and profoundly important to understanding p-n junction current flow.
To be ready for our next class, please think about and discuss that here. For example:

1. What is generation? What is recombination?
2. which is dominant for x less than $\color{white} -x_o$?
3. which is dominant for x greater than $\color{white} x_o$?
4. (Hmm. Maybe this is a poorly considered question. Thinking about majority carriers may only cause trouble. Feel free to completely ignore this part (4) .) What do you think about the assumption of enhancement of both n(x) and p(x) in these regions? Would that make things a little simpler to think about g-r issues? In what way might it fail to capture the essence of what is happening in a junction, or not?

If we imagine that each recombination event involves emission of a photon (of energy Egap), what does that imply?

Regarding the in-class midterm, if you can stay ready for that, that would be great. No date is set yet: there will definitely be some notice in advance in a blog post here and probably also in class. It would cover everything we have covered involving:
1) homogeneous semiconductors
2) p-n junction physics:
a) in static equilibrium, where counter-flowing drift and diffusion currents make Jn(x) and Jp(x) each individually zero at all x (though the drift and diffusion parts separately have some interesting x-dependence, as we have been exploring),
b) in steady state with an applied voltage. (key things to understand might include: excess diffusion current, recombination, bias (forward / reverse), hmm... what else??? ...

Homework 4: Due Tuesday (all about J)

Notes: a) V== Vext refers to an externally applied voltage which can cause current to flow across a p-n junction.
b) This is a very important assignment!
c) Please check here frequently for comments. Also, you will get a 5 point HW bonus for your comments. Also, you can use LAtex in your comments! WooWoo! (thanks to Brad)


1. For equilibrium (V=0) show that the electron drift current in a p-n junction can be expressed as a function of x using the: electric field at x=0 (negative), the potential step and depletion length scale $\color{white} e \Delta \Phi & x_0$, also kT, etc (see "Image" above. Also, correction note: Jn(x) should refers to just the electron drift current, not the total Jn(x). The label is misleading. Sorry.)

a) what is the range (x) of validity for each line of the 3 lines of the eqn for Jn(x) ? Why?

b) For $\color{white}\mu =5000 (cm^2/V sec) , N_d = N_a = 10^{17} (cm^{-3})$ and $\Delta \Phi =0.8 eV$ and $E_g = 1.0 eV$ (no tricks, as you see), find the values of E(0) and $\color{white} x_0$ and use them to obtain largest value (max or min) of the drift current. (Is it positive or negative?)

c) graph this drift current vs x (for x in the range of validity of lines 2 and 3 of the eqn for Jn(x) which is...?)

2. a) Graph all 4 currents for this system. Why do you not really need to do any more calculations to do this and, for example, why do you not really Need to calculate Jn(x) drift for x less than zero in order to do an okay graph. (Answer here. That will help people.)

b) Why do we not worry much about generation and recombination (or dn/dt) for the unbiased p-n junction?
------

For a biased p-n junction:

3.
a) explain why and when we can neglect the time derivative term in the continuity equation?

b) for x greater than x_0, explain how we can simplify the expression for hole current (and why).
Then combine that with the continuity equation to obtain a solve-able diff eq for p(x). (You may post your result here, if you like.) (latex is enabled.)

c) Assume $\color{white} p^0 (x) = (n_i)^2/N_d$ . Justify and discuss that here. (PS. What is $\color{white} p^0 (x)$. Elucidate that here for an extra bonus.)

d) Assume $\color{white} p (x_0) = ((n_i)^2/N_d) e^{eV/kT}$.
What is the significance and meaning of this assumption or ansatz? This is worth a lot of thought! (and a major bonus: what is $\color{white} p (x_0)$?

4. With the above assumptions, solve for the hole current for x greater than x_0.

5. a) For V=.05 Volts, how big is this hole diffusion current at x= x_0? How about for V= 0.1 or 0.2 Volts?
b) Just out of curiosity, how does it compare in magnitude to the (individual) currents you calculated in problem 1 (and 2)?

6. Discuss the x-dependence of this hole diffusion current for x gt x_0? Describe and explain what is going on?

7. Think about the dependence of J on x in the depletion region (and outside). Discuss any thoughts or speculations you have about the dependence of J on x in the depletion region (or anywhere else) (comment here).

8. Think about the dependence of n(x) and p(x) on x, both in and outside the depletion region. Discuss and comment here.

Homework 3, more solutions

Notes:

• 1 and 2 soln's were posted earlier.
• Regarding 3, in response to this sort of question you are expected to express n(x) as $\color{white} N_c e^{-(E_c(x) - \mu)/(k_B T)}$ . It is also helpful to note that: $\color{white} N_c e^{-(E_c(\infty) - \mu)/(k_B T)} = N_d$. Then what remains is to evaluate a function defined in sections (piecewise) using a band-edge energy you have calculated from an assumed charge density in the depletion region $\color{white} E_c(x) = E_c(\infty) -e(\Phi(x)-\Phi(\infty))}$, where $\color{white} \Phi(x)}$ is related to the charge density. You do an analagous calculation to get p(x). n(x) and p(x) are important because they help us understand J(x).
• In #4 you may notice that i tend to calculate values in a quirky way that avoids mass in kg by using c^2 factors. There are many other ways to get numbers. Also, I am wondering if my result for part b) is off by 100 or so. Extra credit if you find this error (or any error in HW solutions.)
• For #5, I was thinking that, unlike a metal where there is "fixed" Fermi velocity (usually about 5x10^7 cm/sec), in semiconductors the carrier speed tends to depend on electric field. Then for a given scattering time*, the mean-free-path would be field-strength dependent (v=u E = e tau E /M*) and, since it depends of E, not that useful.
• In 6 the sign of the terms is important. I think there was perhaps a sign error in the problem originally? Is that true? One has to think carefully about gradients and flow and the sign of charge carried to get these signs correct.
  

Graph of current vs x (and more)

Here is a plot which i think is proportional to J_drift (x) . What does the 16 reflect? Where iss the 1-x associated with? What is x_0.

Also, can you tell which drift current this could be? why?

When i see these graphs I think about what's next. I hope maybe you do too.
With that in mind, please consider the following semi-optional problem(s)/fun thing(s) to do and think about for Thursday:

What does this curve suggest about dn(x)/dt (=dJn(x)/dx) ?
(Like if this were the total Jn (x).....

Starting with this curve, plot all 4 J(x) curves.
How do they effect your thoughts about dn(x)/dt?

In general, what is missing from the equation:
dn(x)/dt (=dJ(x)/dx) ? (what else could effect n(x,t) and what form might it have?

PS. Any comment here (in the next 26 hours) is worth 5 bonus points on your midterm!

HW #3 problems 1 and 2 solutions: Please comment and critique.

HW #3 (along with HW #2b) contains a lot of what we understand about p-n junction equilibrium. This includes the depletion ansatz, the charge density, electric field and electric potential near the junction, how the depletion width is established through its relationship to the potential step need to allow mu to remain constant.

In actually doing these problems i realized the importance of starting on the left, x<-x_0, and systematically integrating to get phi(x) in all regions. Because of the way integrals are defined and used, that seemed the best way. (Setting phi(x) =0 for x less than -x_0 got things started. then one can integrate from -x_0 to zero and then use the value of phi(x) at zero to set the integration constant for the 0 to +x_0 integration.

In problem 1, you can see where i initially forgot the 11.8 (epsilon) and then added that in. What do you think of the values of 76 nm (1), and 145 nm and 3 nm (problem 2). Do those seem correct?

HW #2b Solutions (more or less)

The solution to problem 2 refers to a HW#3 solution for the calculation of electric potential as a function of x. (That is because I did the HW #3 solutions first.) At the end of problem 3 you will see the table I am working on. Clearly it needs more entries. Hopefully you get the idea. If not, if anything is unclear, feel free to ask. This works best as a dialogue.
The last upload is of a graph made with "grapher" (Mac OS10....), which refers to the calculation on page 3 (and represents a graph of that for x=0 to x_0). The 16 comes from the ratio of the band displacement energy (deltaE) to kT, and we treat x_0 as 1 (horizontal axis) and think of 1 as 10^17 (cm-3) for the vertical axis.

Latex math on blogspot

http://watchmath.com/vlog/?p=438

Check out the url above. It looks like one can "install" latex in blogspot. Can someone help me with that?, maybe Tuesday after class or by phone this weekend?

email me (or comment here)

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."
------------------------------

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)

Band calc, and HW#1, part I: start date: Mar 31

The enclosed 5 pages of jpeg notes contain a calculation on the origin of bands from quantum atoms using Bloch eigenstates. The key thing is how a band of states emerges from a single quantized state, and how the width of that band depends on the overlap integral t.

For homework (see below*) you may rederive this result (optional) and also, most importantly, graph Eq as a function of q (from -pi/a to pi/a) (not optional). Before you graph you should be sure of the sign of every term in your equation, and also have some sense of what their relative magnitudes might be. (Also units.)

PS: Please note the page numbers. I am not sure how to control the order of multiple page uploads here, but the pages are numbered...

PPS. Please feel free to ask and answer questions here!

* Homework summary as of 3-31-10 (HW#1):

1. a) Sketch the two lowest energy states of a potential which consists of two equal finite square wells.

b) Sketch the four lowest energy states of a potential which consists of four equally spaced, equal strength finite square wells.

c) Sketch the six lowest energy states of a potential which consists of six equally spaced, equal strength finite square wells.

d) (extra credit/optional) Sketch the three lowest energy states of a potential which consists of three equally spaced, equal strength finite square wells.

e) (optional) Sketch the fifth lowest energy state of a potential which consists of four equally spaced, equal strength finite square wells.

2. What are the units of t and I (see notes). Graph Eq vs q from -pi/a to pi/a. Which is larger, 2t or E0 ?

Braket notation review

the braket notation we used in class yesterday is essentially a way of writing inner products. The accompanying jeg is a review of braket notation, which was prepared by a helpful student. Some review of linear algebra, especially eigenvalue / eigenvector equations / formalism, may also be helpful. Also, some review of basic 1D quantum mechanics.

Welcome to the blog

Welcome to the physics 156 blog.

Reading: chapters 2 and 3

HW: The problem we discussed in class: finding the energy eigenstates of an electron in a spatially periodic potential. (see notes in later post)

I would recommend becoming a follower, and posting questions, comments, etc., as well as answering other peoples questions when you can.