2026 MATSE 413 Homework 10

Please clearly indicate your answer to aid in grading. You are welcome to use any math software you like (or none at all). If you use software and do not show your work, you cannot receive partial credit. If you show your work either by writing down your equations or by taking a screenshot of your program, you will be eligible to receive partial credit. If you use Python or similar, please comment your code if you wish to receive partial credit. Remember: if your answer does not have SI-based units, you will receive zero points! Note: in this class, I generally expect that you can look up constants like the mass of the electron or the speed of light.

Consider copper, which has a Fermi energy of 7eV.

Calculate the velocity of electrons at the Fermi energy (i.e. the Fermi velocity) in units of m/s. (2 points)

Calculate the electron density in electrons/cubic meter using the Fermi energy and assuming all electrons have the bare electron mass (the effective electron mass in copper is the bare electron mass). (2 points)

Calculate the electron density in electrons/cubic meter using the density of copper. Hint: you will need the density of copper, Avogadro’s number, and the atomic weight of copper. Make sure to check your units. (2 points)

Compare the values from parts b and c and discuss the reason for any discrepancies. (4 points)

At what temperature can we expect a 10% probability that electrons in copper have an energy which is 1% above the Fermi energy? (4 points)

If the relaxation time for copper is 10-14 s, calculate the mean free path for copper. Give the mean free path in units of microns. (4 points).

Using the Drude model and your results from part f, calculate the conductivity for copper. Give your answer in units of S/m (Siemens per meter) where 1S=1Amp/Volt. To make the units work correctly, you need to use the electron density rather than the number of electrons. Use the value you calculated from part b. (4 points)

Consider the band structure of germanium, shown below.

What is the size of the band gap at the Γ point in eV? Do you expect this material to be a good conductor? Why or why not? (3 points)

Consider the bands shown in the dotted red box. Does the higher energy band (which is really two degenerate bands) have a larger or smaller effective mass compared to the lower energy band? (2 points)

The red arrow is pointing to a specific valence band near the Δ point. Calculate the group velocity of an electron in this band near this point. Give your answer in m/s. Note: the distance in reciprocal space from the Γ point to the X point is 2π5.66 Å. (4 points)

What is the group velocity for an electron in the Γ7 valence band (in the dotted red box) at the Γ point? Give your answer in m/s. (4 points)