2026 MATSE 413 Homework 11

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 silicon, which has a band gap at room temperature of 1.12 eV and an electron effective mass m*=0.98me.

Calculate the density of electrons in the conduction band of intrinsic silicon at room temperature in units of m-3. (2 points)

Calculate the conductivity for silicon using your results from part a, an electron mobility of μe=1000 cm2/Vs, and a hole mobility of μh=450 cm2/Vs. Use units of S/m. (3 points)

Now imagine silicon is n-doped at a lower level of 1×1018 atoms/m3. Calculate the conductivity of this doped silicon at 300C and 350C. Does the conductivity increase or decrease? Assume the mobilities of holes and electrons are the same as in part b. Explain your answer. (6 points)

Consider the band structure shown below.

Is this a direct gap semiconductor or an indirect gap semiconductor? (2 points)

At what energy do you expect to begin to see absorption? (2 points)

At what energy do you expect an inflection point in absorption? (2 points)

Do you expect to see high or low optical absorption at the band gap energy? (2 points)

Consider the absorption plot below. The different colored curves are for AlGaInP alloys with increasing amounts of aluminum.

Is the material represented by the black curve a direct or indirect gap semiconductor? How do you know? (3 points)

Is the material represented by the gray curve a direct or indirect gap semiconductor? How do you know? (3 points)

Estimate the band gap in eV of the material represented by the black curve, material represented by the red curve, and material represented by the green curve. (3 points)

Shown below is a plot of carrier density versus temperature for an intrinsic semiconductor.

Calculate the band gap at 400 K and 200 K assuming that the electron effective mass is 1. Give your answer in eV. (3 points)

Calculate the band gap assuming that the electron effective mass is 0.01. Give your answer in eV. (3 points)

BONUS: Why are your answers different for different temperatures? (3 points)

The following question is extra credit. You are not obligated to do it. If you do the problem correctly, you get extra credit points. If you do the problem incorrectly or do not do the problem at all, you are not penalized.

The binding energy of a donor electron in a semiconductor can be calculated by assuming that the extra electron moves in a hydrogen-like orbit. Estimate the donor binding energy of an n-type impurity in a semiconductor by using the ground state energy of the hydrogen atom from quantum mechanics: E1=-m2ℏ2e24πε02. Give your answer in eV. Use the effective mass m*=0.8m0 in place of the bare electron mass and use the dielectric constant of the semiconductor εr=16ε0 in place of the permittivity of free space ε0. This binding energy tells you how much energy must be added to remove the free electron from its donor ion. Will this dopant be fully ionized at room temperature? For reference, the thermal energy in a system at room temperature is ~26 meV=0.026 eV. (3 points)