I would like to ask how to use PHONOPY to define which band is LO, TO or LA, TA, or ZO or ZA. I attach a picture from https://iopscience.iop.org/article/10.1088/0953-8984/28/10/103005
There are two major techniques used to model phonon interactions - The frozen phonon method and density functional perturbation theory (DFPT). Phonopy is used to carry out calculations in the frozen phonon scheme. ProfM does a fantastic job delineating the differences between the two methods here.
The first step in such a finite displacement scheme is to generate supercells with small 'q' shifts in different directions. Next, the forces between atoms in the 'displaced' crystals are calculated to construct a force constant matrix. Diagonalizing this matrix leads us to the normal modes at any particular wavevector 'q'. First, we need to generate supercells using phonopy - This is the simplest step and there are different ways to do this, whether you work with QE (Quantum ESPRESSO) or VASP. (Check phonopy documentation here: https://phonopy.github.io/phonopy/vasp.html). Once phonopy generates the supercells, your first task (the 'pre-processing' step in phonopy) is complete.
Typically, the scheme to calculate the vibrational modes would be to perform a charge density calculation for all these supercells. Once this is done, the outputs of these files can be used to construct the force constant matrix (this is the 'calculation of forces' section in phonopy).
The information regarding eigenvectors of the modes can be found by generating the mesh.yaml file, which is part of the post-processing process listed on the phonopy webpage. Again, this can be generated with a simple one-line command. I was able to scrap one of the mesh.yaml files for a system (which has 6 atoms in the cell) that I worked on a long time back and here's a snippet:
frequency: 2.5939095837 eigenvector: - # atom 1 - [ 0.14423900354589, 0.00000000000000 ] - [ -0.00000000000000, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - # atom 2 - [ -0.14423900354589, 0.00000000000000 ] - [ -0.00000000000000, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - # atom 3 - [ -0.32762271902471, 0.00000000000000 ] - [ -0.00000000000000, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - # atom 4 - [ 0.32762271902471, 0.00000000000000 ] - [ -0.00000000000000, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - # atom 5 - [ -0.60980198739832, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - # atom 6 - [ 0.60980198739832, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ] - [ 0.00000000000000, 0.00000000000000 ]
The first line lists the frequency of the mode in TerraHertz. What follows next is the eigenvector of the modes - The real and imaginary components are listed along all three cartesian directions. From this snippet, one can infer that all atoms move along 'x'. Additionally, the imaginary parts of all eigenvectors are zero - This has nothing to do with my particular system, but due to the fact that first-order Raman spectroscopy can only be used to probe near the Gamma point (because momentum of incident photon is much less compared to size of the Brillouin zone).
Now that I have inferred that all atoms in this vibrational mode move along 'x', I would have to take a look at the supercell and observe the direction of the 'q' shift. If the 'q' shift is also along 'x', then this mode would be a longitudinal mode and if the 'q' shift was along 'y', this mode would be transverse. Whether the modes are optical or acoustic, can usually be logically inferred because the three lowest frequency modes are acoustic.
I have not performed a phonon calculation for graphene. But, the modes reported in the paper referenced in the question, can be inferred in the way that I've described above - whether it be TA, LA, LO, TO. ZO and ZA are not much different - they just indicate bending out-of-plane (along 'z') and this can also be checked by looking at the 'z' components of the eigenvectors of modes in question.