# Why are Brillouin zones for graphene and monolayers of transition metal dichalcogenides the same?

The geometrical model of graphene is the flat honeycomb lattice, so the Brillouin zone is also flat honeycomb lattice. However, monolayer of transition metal dichalcogenides is not flat as it consists of three atomic layers. So why is Brilloin zone for TMDs still flat?

I'm not expert in the solid state physics, but, in my understanding, the first Brillouin zone is primarily constructed from a unit (primitive) cell of a crystal. Mathematically it's done if we take a Fourier transform of the unit cell of the crystall in real space (meaning cartesian coordinate system, xyz). So, Fourier transform of the 2D flat graphene geometry in real space produces flat geometry in reciprocal space. However, a monolayer of TMDs is depicted in literature as three layers of atoms, and from geometrical point of view it's not in the plane anymore. So unit cell is no longer flat, but some sort of parallelepiped, I guess (or maybe some other 3D shape). So Fourier transform of a 3D object should also be 3D. That's where I'm confused.

• May I suggest Chemistry SE? – N.S.JOHN Jan 23 '16 at 3:41
• Yes, any input would be OK. – Capo Pavel Mestre Jan 23 '16 at 4:04
• The transition metal dichalcogenides is being modeled as a 2D structure as opposed to a 3D structure. Even mono-planar graphite would be 3D in absolute reality. – MaxW Jan 23 '16 at 4:08
• @MaxW, Could you please give a reference? – Capo Pavel Mestre Jan 23 '16 at 4:31

To construct a crystal you need a lattice and a basis. The lattice represents the translational symmetry of the system. Namely, graphene has a hexagonal lattice, meaning the two lattice vectors are 60 degress apart. Since the brillouin zone is constructed by inverting the lattice vectors, the brillouin zone is shaped based upon the lattice, but not the basis.

If you consider a single point and repeated apply the lattice vectors of graphene, what you will get is a hexagonal crystal which is different than graphene's honeycomb structure. This is because graphene has 2 atoms in its basis.

In short, TMDs have the same hexagonal lattice type as graphene, but have more atoms in their basis than graphene.

TMDs have a 3 atom basis with one atom at each z value, whereas graphene has 2 atoms at the same z value in its basis.

Edit: Since you mentioned Fourier Transforms I thought a little picture would help.

If you take the fourier transform of these three structures, you will find the same frequency components in each:

A  A  A  A  A

AB AB AB AB AB

ABBABBABBABBABB