Notations for high symmetry points in the 1st Brillouin zone

Question

I am trying to understand how I should interpret the letters like Г,K,M,T etc., that are usually there in the electronic band structure diagrams.

So, let's assume we have graphene with its hexagonal structure. Then we make a Fourier transform of real space and obtain another hexagonal structure in reciprocal space, with a single hexagonal cell representing the 1st Brillouin zone. And there are letters "K" in the corners, "Г" in the center, M between two corners and other letters. And they are called the points of high symmetry.

So how does it work? For example, we have 6-fold rotational symmetry. How does this kind of symmetry convert into one point in the Brillouin zone? And how do we know where exactly the point is: in the center or in the corner or somewhere else? And how are letters chosen initially? Why is it the letter "K" is in the corner and "Г" is in the center? I assume these came from the group theory... So can I find some easy explanation of the notation without dipping into the theory itself?

Thanks!

Answer 1

For any crystal, the First Brillouin Zone is found using the Wigner-Seitz construction for the reciprocal lattice. The high-symmetry points are labeled by certain letters mainly as a convention--like you said Gamma for (0,0,0) etc.

The important thing to realize as far as the group theory, is that the group of the wavevector at the Gamma point has the full point group symmetry of the real space lattice. However, certain high symmetry wavevectors, labelled by the different Greek letters, are subgroups of this group. That is, only a certain number of the symmetry operations of the point group at the Gamma point (rotations, mirrors, etc.) will leave the new high symmetry point invariant. Thus the benefit of symmetry is that you only have to consider an even smaller region of the BZ to get all of the reciprocal space information about the crystal. This is called the Irreducible Brillouin Zone, and paths along the high symmetry points of the IBZ are used as the x-axis in band structure diagrams.

Use this website to explore different Brillouin zones: http://www.cryst.ehu.es/

This paper gives a thorough treatment of many Brillouin zones: Setyawan, Wahyu, and Stefano Curtarolo. "High-throughput electronic band structure calculations: Challenges and tools." Computational Materials Science 49.2 (2010): 299-312.

The following Book is a great resource which has tables of the high symmetry points in the FBZ and their point groups: Dresselhaus, Mildred S., Gene Dresselhaus, and Ado Jorio. Group theory: application to the physics of condensed matter. Springer Science & Business Media, 2007.