1- What are the key reasons of band folding from one symmetry point to another?
2-How can we determine/examine the band folding phenomena in band structure?
3- What are the key reasons of Dirac point shifting from K to G high symmetry point or vice versa? (in general shifting of Dirac points from one symmetry point to other)
4- Why does usually Dirac point appears at K point? Why not at other symmetrical points?
Dirac points and their positions in k-space are consequences of the topology (symmetry) of the material. The reason that Dirac points "usually" end up at the K point is just that people have "usually" worked in hexagonal symmetry. I think a good starting reference for these questions could be this review:
http://www.tandfonline.com/doi/abs/10.1080/00018732.2014.927109
1. The proprieties of a materials depends on all the points of the BZ. Since this is a 3D surface, for matters of use, people pick some specific line where the symmetry, and the degeneracy (thus the amount of information) is higher.
The BZ depends on the symmetry, if related structures (for example a cell and a supercell) are taken in account, it is necessary to reconduct the BZ (and the high symmetry points/direction) of one to the other.
2. The band folding is not a phenomena, it is an human representational of reality. If a defect for example change the size of a cell, then to compare the cell with and without defect you need to fold back the supercell results in the old one.
3. This is a system dependent point. In my system we use mostly the R and M point of spacegroup 221.
4. Again depends on the topology of the system. Gor istance, in perovskite the softmode appears at M point because there is a tilting with a 2 cell lattice parameter periodicity.
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