Extra: High Symmetry Points
High symmetry points
If you have a particular structure and you want to find out which are the important k-points, then this website is a useful tool.
Finding appropriate high symmetry points and their labels for a band structure plot is beyond the scope of this course, but generally you need to find the Brillouin zone for the system you're interested in, along with the names of the high symmetry points, and how these should be represented in terms of your reciprocal lattice vectors.
These can often be looked up in a table for a given structure, while keeping in mind that even for the same structure you may come across papers where some zone boundary high symmetry points will be labelled differently. You can access these tables online at e.g. the Library of Crystallographic Prototypes by typing the name of the mineral into the search box. This will give you the space group number, which you could then use with e.g. the Bilbao Crystallographic Server to find the reciprocal space coordinates of the high symmetry points and their labels. Another good reference is TU Graz which has nice interactive visualizations of the most common types along with their labels.
For the diamond lattice example, we might do this as follows:
- The diamond lattice is FCC. We can find find the space group number from http://aflow.org/CrystalDatabase/A_cF8_227_a.html and see that it is number 227.
- We can enter this number at http://www.cryst.ehu.es/cryst/get_kvec.html
and find several high symmetry points. Often, you'll want to pick the same
path that was chosen in some previous work to ensure you can reproduce it
correctly. In
02_C_diamond_nscf.inwe have set up an input file for the pathΓ-K-X-Γ'-L-X-W-LwhereΓ'indicates the gamma point in a different Brillouin zone. Note - these labels don't exactly match points given in the table. Many points have a number of equivalent positions on the Brillouin zone surface, and often different conventions can be used for different materials with the same structure.