EE 505 – Nanostructures: Fundamentals and Applications
Spring 2009
Prof. Carmen S. Menoni
Homework #2
Due Date: March 11, 2009
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1. Explain which are characteristics of the electron wavefunction in a crystal. Provide a
comparison for Schroedinger equation for a crystal compared with that for a
hydrogen atom. Contrast also the eigenfunctions.
2. Explain band formation in a solid and provide main differences between a metal,
insulator and semiconductor in terms of the band picture.
3. What is a hole?
4. Calculate the density of states in the conduction band of Ge for an excess energy
E=60 meV about the conduction band minimum. Calculate the total number of
electrons at T=100K.
5. Explain how you would design a quantum well with a bandgap around 1.8 eV using
III-V semiconductor materials. Calculate the number of confined eigenstates.
6. Explain the variation of the bandgap energy in a quantum dot material with dot size.
Then explain main features of the figure in page 74 of the notes.
7. How is the interaction between an electron in a crystal and a photon modeled? How
is Schroedinger equation modified?
8. Describe the differences between absorption and emission processes. What are
conservation laws. Explain this in the context of the band diagram for a direct
bandgap semiconductor material.
9. What is an exciton? What are the main signatures of exciton formation? What are
the main differences in the absorption in a semiconductor quantum well with and
without excitonic contributions?
10. Which type of materials do you expect have the largest excitonic binding energy?
11. Why is it important to know (E) in a low dimensional material system?
12. Explain what type of recombination processes are possible in quantum confined
semiconductor systems.
Questions related to the Paper by R.J. Ellingson et al. “Highly Efficient Multiple Exciton
Generation in Colloidal PbSe and PbS Quantum Dots”.
13. Explain what is the growth mechanism of the colloidal Quantum dots. How are the
different size materials realize?
14. Which are the main optical transitions observed in Figure 1 and how are their energy
values calculated?
15. Which are the dominant recombination mechanisms?
16. Which are typical exciton formation and decay time? Do these characteristic times
depend on dot size?
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17. The ionization energy of the hydrogen atom in its ground state is Eion=13.6 eV.
Calculate the frequency and wavelength of the electromagnetic radiation that just
ionize the atom.
18. The magnetic dipole moment μ of a current loop is defined by   I A , where I is the
current and A is the area of the loop. A current loop may be represented by a charge e
rotating at constant speed in a small circular orbit.
19. Use classical reasoning to show that the magnetic dipole moment of the loop is
related to L the orbital angular momentum of the particle by
e

L
2m
where m is the mass of the particle.
If the magnitude of L is
h
2
, calculate the magnitude of μ for (i) and electron
and (ii) a proton
20. Solve Schroedinger’s equation for a rectangular finite potential (page 29 of notes).
Calculate the probability density for the n=1 to n=3 confined states. Plot the
probability density versus x. Please use MATLAB, MATHCAD or similar programs
to do this.
21. What is the meaning of the bandgap energy for electrons?
22. Explain using the concept of confinement elaborated in Q.2, what is a quantum well,
a quantum wire and a quantum dot.
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