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)