While defining a graphene structure

[collpitt]

Hello!
I am a student and have just started studying about graphene. However I am having quite a lot of problems understanding the crystal structure, specifically, I am unable to place certain terms. These being :
1. Basis vectors.(I do understand what a basis vector is but am having difficulties in realizing them in the hexagonal structure.)
2. Reciprocal lattice, brillouin zone.
I would be very thankful should someone help me out or suggest references.

 

[lww]

You can find the answers in any fundamental textbook concerning SSP, e.g. book by Ashcroft.

 

[tejas777]

I have a hunch that you are having difficulties in understanding the crystal structure of graphene, i.e. a honeycomb lattice, because it’s not a Bravais lattice. The honeycomb lattice is a "lattice with a basis." When you put a collection of atoms, which are arranged in a particular way with respect to each other, on every lattice site of a Bravais lattice then you get a lattice with a basis. In two dimensions there are five Bravais lattices. The Bravais lattice corresponding to the honeycomb lattice is the hexagonal (or triangular) lattice. You can find a very good description of this in sections 1.2.1-1.2.3 in this book:

https://www.amazon.com/dp/0471177792/?tag=pfamazon01-20

Take a look at figure 1.4. In (A) you will see a hexagonal lattice where the black dots represent the lattice sites. Now, (B) represents the same hexagonal lattice except now you have two atoms at every lattice site; these atoms are horizontally separated (check the math in the book to see the amount of separation). The solid lines in (A) and (B) represent the edges of the Wigner-Seitz cell (this is also defined in the book). When you remove the solid lines the honeycomb lattice is easily apparent in (C).

You can obtain the basis vectors for the lattice with a basis by simple vector addition (see the book for details). Given the basis vectors for the lattice with a basis, you can find the basis vectors of its reciprocal lattice using equation (5.3) of Ashcroft and Mermin. Using these new basis vectors you can construct the reciprocal lattice and define the Wigner-Seitz cell in the same way as in the real space. This new Wigner-Seitz cell is the (first) Brillouin zone. It’s a good exercise to do these calculations and check your results against existing literature.

Here’s a nice link where you can play around with the applet to get a better feel for this:

http://demonstrations.wolfram.com/GrapheneBrillouinZoneAndElectronicEnergyDispersion/