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?

Answer 1

The letters in the electronic band structure diagrams represent points of high symmetry in the reciprocal space, derived from the material's lattice structure. For example, K points usually correspond to corners of the Brillouin zone in graphene, while Г (Gamma) is the center point.

Step-by-step explanation:

You are trying to understand the letters like Г, K, M, T, etc., in the electronic band structure diagrams that are often used in physics to describe the electronic properties of materials like graphene. In these diagrams, different points such as K, Г, and M represent points of high symmetry in the material's reciprocal space. For instance, in graphene, which has a hexagonal lattice structure, the reciprocal space also forms a hexagon, known as the first Brillouin zone. The corners of the hexagonal Brillouin zone are notated with K, representing positions where the lattice shows six-fold rotational symmetry. On the other hand, Г is at the center of the zone and represents a point where the wavevector is zero, and the lattice symmetry is highest.

The letters themselves trace their origins to crystallography and group theory, often reflecting historical and conventional choices. For example, the letter Г (Gamma) is conventionally used to denote the center of the Brillouin zone in reciprocal space, whereas K points typically denote corner or edge positions associated with other symmetry directions in the crystal. Although these letters may seem arbitrary, they follow specific conventions used in solid-state physics and materials science. Unfortunately, understanding why certain symmetries are associated with specific points can be complex and often requires a deeper dive into group theory.