Homework #6

This homework will test your knowledge and skills in band structure concepts and calculations.

Send the report as a single MS Word or pdf document to the instructor by email.

1. (3 pt) Use the orthogonal tight-binding code skband to calculate the electronic band structure of a linear carbon chain.

• Start by copying the input file C-dia-bas.skin onto skin
• Align the chain along the x-direction in space
• Use 1.3 Å as the interatomic distance.
• Keep two atoms in the unit cell. The unit cell must be twice as long as the interatomic distance, which is 2.6 Å
• Use the following 6 k-points in the input file skin:
  

    0.000  0.000  0.000  G   F                       Gamma
    0.100  0.000  0.000
    0.200  0.000  0.000
    0.300  0.000  0.000
    0.400  0.000  0.000
    0.500  0.000  0.000  X
  

• Plot the band structure.
• Is there a band gap?
• If there is a band gap, at which k-point is it narrowest?
• Where is the Fermi level?
• Is the system metallic, semimetallic, or insulating?

2. (2 pt) Use the orthogonal tight-binding code skband to calculate the electronic band structure of a dimerized linear carbon chain.

• Start by modifying the input file of problem 1.
• Keep the chain aligned along the x-direction in space.
• There will still be 2 atoms per unit cell. The new unit cell should be as long as in problem 1, 2.6 Å.
• Use 1.2 Å and 1.4 Å as alternating interatomic distances.
• Use the same k-point sampling as in problem 1.
• Plot the band structure.
• Where is the Fermi level?
• Is the system metallic or insulating?

3. (6 pt) Use the orthogonal tight-binding code skband to calculate the electronic band structure of graphyne, shown in the middle panel of the image below. In comparison to graphene with two atoms per unit cell, graphyne has 12 atoms per unit cell. Still, the shape of the unit cell and the symmetry of the Bravais lattice are the same, so graphene and graphyne have the same set of k-points.

• Make a graphyne.xyz file with the coordinates of the 12 atoms per unit cell. All interatomic distances should be 1.3 Å. Display the molecular structure with Discovery Studio Visualizer. Label the atoms by their number in your solution.
• How long are the lattice constants? (Get the inspiration from the C-gra-skin file).
• Generate the skin file for graphyne.
• Calculate and plot the band structure in the energy range between -5 eV and +5 eV (you will need to modify the y-range called yr in skbnd.gpl).
• How many electrons are in the unit cell? How many bands are occupied?
• Where is the Fermi level?
• Is the system metallic, semimetallic, or insulating?
• Where (at which k-point) is the top of the valence band?
• What is the character of the state at the top of the valence band?

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by David Tomanek - tomanek@pa.msu.edu