ELCT 780 Homework 3
The Bandgap
1. Kronig Penney Model
Using Equation 21a in Chapter 7, Kittel,
a. Write Matlab code to derive the left hand side, for a given energy, E.
b. From the given E and the LHS value, derive the corresponding k value
corresponding to the E.
c. Plot this dispersion for 2 values of V0.
d. Vary the magnitude of the periodic potential V0 from 0.1 to 1eV. Plot
the first bandgap as a function of this potential.
Assume a lattice constant of 4x10-10m, and a duty cycle of 50%. Note: there is
a singularity in the LHS of the equation, when E=V0, leading to funny results
in Matlab. If you add this value to all the energies, you remove the singularity
without changing the physics of the situation. Why is this? What happens to
the dispersion and bandgap?
2. Choose one of the periodic potential values you used above. For this value of
bandgap, use the central equation result we derived in class to plot the
dispersion of the electron. Remember to divide V0 by 2 to account for the
factor of 2 we get in front of the cosine by adding 2 exponentials. Compare
the 2 results i.e. square potential with perfect cosine potential. Comment on
this.
3. Choose a k-value near the Brillouin Zone boundary. Solve for the C(k) values.
Plot ||2 in Matlab. Is this a traveling wave or a standing wave?
4. Choose one of the bandgap values in problem 1. Plot the imaginary part of
the k-vector you obtain. Do the same for the one in problem 2. What does this
represent? Explain this in terms of the wave-function. This is the analog of
arcing in electromagnetics. What quantum phenomenon are we describing?