Homework 1 (due Feb 1)

1. At t=0, sketch the wave-function as a function of x over a few lattice spacings for Bloch states for which:
a) k=0;   b) k=\(\pi/a\);   c) k=\(\pi/2a\);   d) k=\(- \pi/2a\)
e) sketch a graph of the atomic state wave-function you used in your Bloch state construction.
Note that, with time dependence included,  a) and b) represent standing waves, while  c) and d) are traveling waves, traveling to the right and left, respectively.



2. Calculate the width of the depletion region for an n-p junction made with semiconductor for which: \(E_g = 1.2 eV, \quad kT=.025 eV\) and 
\(D_c = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = 3 eV\)
\(D_v = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_v = 3 eV\).
(Assume that within each band the density of states is independent of E.)
a) The case where the doping on each side is \(10^{17} \: cm^{-3}\) (donors and acceptors, respectively).
b) (extra credit, optional)  The case where the doping on the left is \(10^{17} \: cm^{-3}\) donors, while the doping on the right is \(3 \times 10^{17} \: cm^{-3}\) (acceptors).

3. Calculate n(x) in the context of the depletion ansatz (approximation) and use that n(x) to calculate the the diffusion current as a function of x.  At what value of x is the diffusion current maximum?  What is its value in electrons per second? (You may use actual the numbers from problem 2.) If you need a scattering time for this, you can use \(10^{-13}\) seconds.  (You can do the same for p(x) and the hole diffusion current if you like. It is pretty much analogous.)

4. Suppose a homogeneous semiconductor is illuminated by photons around the energy of the band gap such that 10^18 electrons per second are excited from the valance band to the conduction band. (For specificity, we can assume it is like the one in problem 2 in terms of density of states and band gap and that it is uniformly doped with donors at \(10^{16} \: cm^{-3}\). The photons can be around 1.2 or 1.3 eV.)
a) If the recombination time (for an electron to fall into an empty state) is say \(10^{-10}\) seconds, what will be the steady state density of electrons in the conduction band under this illumination?
b) If the photons are suddenly blocked, figure out an expression for n(t) that describes the time-dependent return to equilibrium.

5. Suppose the junction from problem 3 is biased by an external voltage source at a voltage of 0.2 volts with the positive voltage on the right side (p side).
a) Sketch the bands as a function of x. (That is, the bottom of the conduction band and the top of the valence band.) How has the voltage influenced the band bending in the space charge (depletion) region?
b) Sketch the Fermi level as a function of x. How has the voltage influenced the Fermi level? (non-trivial, subtle question)
c) Calculate the electron diffusion current at the right hand edge of the depletion region. Which way are the electrons going?  Are the electrons minority or majority carriers here? How is this current influenced by \(E_F\)? (non-trivial question)
d) What is/are the characteristic length scale(s) that emerges from this calculation? Discuss these length scales in terms of a random walk. Describe the electron diffusion outside the depletion region in terms of a random walk.
e) Sketch a picture of this junction under bias showing the rough size (length scale) of different regions and giving those regions descriptive names.