Homework 8:

suggested activities:  Watching the videos in the Feb 21 class notes post. Also, it may be helpful to read about moderately doped semiconductors and carrier generation and recombination processes in semiconductors. Also, p-n junctions, but just the most superficial things. I think we can understand them in a simple yet accurate manner. We will focus on p-n junction LEDs, lasers and solar cells.

8. $v= (Bbc/2\hbar c) sin(kb) = (Bbc/2\hbar c) (\pi/20)$
     = (4 eV 1 A/(2 1972 eV-A))  (pi/20) c
     = 4.8x10^6 cm/sec


9. One more problem! (the focus of this problem is to help you build your skills of actually calculating numbers.)
Consider an electron in the conduction band of a crystal for which the energy is given by:
$E_c (k) =  E_o - (B/2) cos (kb)$.
Then confirm that,
$v(k) = (Bb/2 \hbar) sin(bk)$,
and, for the case b = 1 angstrom and B=4 eV,
a) calculate v(k) at $k = \pi /(20 b)$ in Angstoms per second and in cm/sec. Get numbers!   (Soon we may try to show (numerically) that this is the electron speed associated with this state.)


Due Friday, 3 PM

1. a) If a semiconductor has a band gap of 1 eV what frequency of photon would you require in order to create an electron-hole-pair excitation? What color is that?

b) What color of photon would a semiconductor with a 1.9 eV band gap emit in a transition from the bottom of the conduction band to the top of the valence band?

2. Suppose you have a 1-dimensional semiconductor that is 2 cm long. Suppose the left half is doped with 10^4 acceptors per cm and the right half is doped with 10^4 donors per cm.

a) graph the density of electrons in the conduction band and the density of holes in the valence band (empty states) as a function of x. You can call the density of electrons in the conduction band, n, and the density of holes in the valence band, p.
(hint: don't make this too complicated. It is really simple!)

3. Consider a 1D semiconductor, our typical one, with a band gap of 0.5 eV between the top of the valence band and the bottom of the conduction band. Suppose you substitution dope the right-hand half of it with 10^4 donor atoms per cm and then apply a voltage (electron field) that pushes the mobile electrons in the conduction band (on the right) toward the left, into an region where they all fall into the valence band with each one emitting a photon. Let's assume that, in response to the applied voltage, the electrons move with a speed of 3 $10^6$ cm/sec.
a1) sketch a picture of that...

a2) Would the region on the left where the downward transition takes place need to be doped at all? why or why not? In what way would you dope it to allow downward "recombination" transitions. 

b) How many photons would be emitted per second?

c) what would be the energy of each photon (more or less)?

d) how much total power would be emitted? (in eV/sec, and Joules/sec = Watts)

4. Now consider a 3D semiconductor with a band gap of 0.5 eV and doped on the right-hand half of it with 10^17 donors/cm^3. Apply a voltage (electron field) that pushes those electrons toward the left (at a speed of 3 $10^6$ cm/sec) into a region on the left where they all fall into the valence band with each one emitting a photon.

a) would the region on the left need to be doped? In what way? Why?

b) How many photons would be emitted per second? (are the units the same or different than in #3? Why?)

c) what would be the energy of each photon (more or less)?

d) how much total power would be emitted? (what are the units of that?)

5. a) What semiconductor band gap would be best suited to creating a red LED?

b) a) What semiconductor band gap would be best suited to creating a green LED?

6. Discuss how substituting N in GaAs might possibly help you obtain a green LED (i.e., help you change a red LED material into a green one?)

7.  Consider a 1D semiconductor with an energy gap of 0.8 eV. Suppose the density of states near the bottom of the conduction band is g(E) = $2 \times 10^9$ states/(eV-cm). Further, suppose we treat g(E) as constant within the band and zero in the band gap).

a) sketch a plot of this approximate g(E) (which is either zero or $2 \times 10^9$ states/(eV-cm)).

b) Integrate the product g(E) times f(E) over the range of the conduction band,  where f(E) is the Fermi function. Put the Fermi energy $E_F$ right in the middle of the gap and set KT=0.025 eV = 1/40 eV.
(important hints:
 hint 1) What is the largest value of f(E) for E in the range of the conduction band? If it is small, then you can ignore the +1 in the denominator and the integrand becomes much easier. Perhaps if you are lucky the integrad becomes a constant, g(E), times a simple decaying exponential? (Do you feel lucky?)
  hint 2) The upper limit of integration (from the top of the conduction band) will give you a very, very, very small number. I would suggest that you totally ignore it.)
c) What are the units of your result from b)?
What is the numerical value of your result? (Do you need to have anything else given?)

d) Explain why this calculation might give you the density of electrons in the conduction band at room temperature for this semiconductor when it is un-doped (intrinsic).

8. a)  What is the relationship between the structure of diamond and the structure of Silicon?
b) Comparing the C-C bond in Diamond, and the C-C bond in Graphene: which is strongest? Why do you think that might be?

Appendix:
The Fermi function is:
$f(E) = (e^{(E-E_F)/KT} + 1)^{-1}$.
It tells you the probability, in a thermal physics/statistical physics sense, that a particular state at energy E is occupied. Note that it can't be bigger than 1 (Fermions) or less than zero.
You are given that KT= .025 eV (so you don't need to know K) and also where $E_F$ is (in the center of the gap).  Note that whenever $E-E_F$ is greater than about 3KT you can approximate f(E) by a simple exponential.