2026 MATSE 413 Homework 8

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.

a3a3a2a2a1a1Consider the material SrTiO3 (mp-5229) on the Materials Project. The primitive cell is shown below where the gray atom is Ti, the red atoms are O, and the green atoms are Sr. The lattice constants in the x, y, and z directions are all 3.91 Å. The basis vectors are indicated by the black arrows.

a3

a3

a2

a2

a1

a1

Which Bravais lattice is this? (1 point)

How many atoms and of which type comprise the basis? Your answer should be in the following format: X total atoms: L of atom A, M of atom B, N of atom C where the X, L, M, N are numbers and the A, B, C are the specific atoms. (2 points)

Describe the basis by giving the type of atom in the basis (Sr, Ti, O) and the vector to reach it in terms of the primitive vectors indicated on the primitive cell. Your answer should be in the following format: Atom A at location la1+ma2+na3 or Atom A at (l, m, n). Because you are describing the location of the basis atoms, the l, m, n do not have to be integers. (3 points)

Draw the reciprocal lattice for this crystal. Indicate the relevant distances. Which of the 14 Bravais lattices is this? (3 points)

Draw the first Brillouin zone for this material. What is its shape? (2 points)

How many phonon modes do you expect for this material? How many of those will be LA, how many TA, how many LO, and how many TO? (3 points)

The calculated phonon spectrum for SrTiO3 is shown below (from Phys. Rev. B 90, 241204(R) (2014)). The negative modes at the M and R points are called “soft phonon modes” and are beyond the scope of this class. Between which two high symmetry points do you see a number of nondegenerate modes equal to the number you predicted? Give all sets of points that meet this criterion. Choose from G-M, M-X, X-G, G-R. (1 point)

At the G point on the graph, indicate which modes are acoustic modes and which are optical modes. Please do this at both G points. (3 points)

Indicate at one point on the phonon dispersion plot at which a phonon mode has a large group velocity and another point at which the mode has a zero group velocity. (2 points)

Calculate the group velocity of the slowest acoustic mode near the G point. Note that the M point is in the center of one of the Brillouin zone edges, the X point is in the center of one of the Brillouin zone faces, and the R point is at the corner of the Brillouin zone. Assume that the x-axis of the plot is linear. Hint: you know the lattice parameter of the material, so you can calculate the distance from G to any of the surrounding points in reciprocal space. Be careful of your units! (4 points)

Si (mp-149) crystallizes in the diamond structure and GaP (mp-2490) crystallizes in the zinc blende structure.

Sketch the crystal structures or look them up on the Materials Project and paste their structures below. What is the underlying Bravais lattice for each crystal? (2 points)

How many atoms are in the primitive cell for each crystal? Describe the basis. (2 points)

The phonon dispersions of Si and GaP are shown below. How many total vibrational modes do you expect in each material? How many of those will be LA, TA, LO, and TO? (4 points)

Material 1 Material 2

Label the modes LA, TA, LO, and TO on the dispersion curves for Materials 1 and 2 near the G point. (4 points)

Which phonon dispersion corresponds to Si and which corresponds to GaP? Explain how the dispersion relation of these two crystals can be distinguished. Hint: notice that the two materials have similar phonon dispersion curves since they have similar structures. The main difference in the two plots is near the X point, where one plot has a gap and the other does not. Think about why such a gap would form or not form and how this can help you determine which plot goes with which material. Alternatively, you can look at the phonon frequencies and think about which material you would expect to have a higher phonon frequency. (4 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.

Consider a diatomic chain of masses M1 and M2. Model the bonding between M1 and M2 within a unit cell as a spring with constant κ1 and model the bonding between M2 and M1 between unit cells as a spring with constant κ2. Derive the dispersion relationship ω2(k). (8 points)