Primitive cell of graphene

Question

I have spent hours on finding the primitive cells of honeycomb lattice of graphene. Based on the definition of graphene from most of solid state physics books, as I quoted from Wikipedia, and in which P.K. Misra, Physics of Condensed Matter book agreed,

a primitive cell is a minimum volume cell (a unit cell) corresponding to a single lattice point of a structure with discrete translational symmetry.

What I understand is that in the primitive cell we define, must contain single lattice point. But, what I found from all articles explaining graphene, the cell they defined is a unit cell which contain two lattice point, as I attached below (taken from here),

Isn't that a non-primitive unit cell? Some articles just mentioned it as unit cell, but some others claiming that it is the primitive cell of graphene. And me myself, try to find the primitive cells of graphene for hours, and still no result.

So, is my understanding of primitive cells wrong? Or, I just hadn't found the primitive cell yet?

Thanks for the help.

Answer 1

I believe the confusion arrises from the confusion of the terms lattice points, lattice basis, and crystal structure.

Lattice points make a (bravais) lattice. A bravais lattice is an infinite group of lattice points that are geometrically indistinguishable from one another. That is, if I was sitting at any lattice point, my 'viewpoint' would be the exact same. My nearest neighbors would be the same, the particles around me would be the same distance away in the same orientation no matter what lattice point you are at.

However, lattice points are geometric objects. You have not yet conceptualized the crystal. All you have done is define the mathematical pattern that underlies the crystal. To make a crystal structure, you have to attach a basis to your lattice points. A basis is the atom, molecule, or collection of molecules and atoms that get tiled or repeated through your crystal. For example, sodium chloride has a two-atom basis at each lattice point: one sodium and one chlorine. You would not call the sodium one lattice point and the chlorine another lattice point.

Another example of this is a diamond crystal structure. If you called every carbon in a diamond structure a 'lattice point', it would break the rules of what a lattice point is (namely that each lattice point's view must be indistinguishable from any other lattice point)! Instead, one uses a cubic lattice structure, and then glues a basis of two carbon atoms. The first carbon atom goes directly on top of the lattice point, and the second goes $ \vec{v} = \frac{a}{4}(\hat x + \hat y + \hat z)$ away.

Finally, to your graphene example, while there are two basis atoms in the primitive cell, there is only one lattice point. The lattice is a triangle lattice with a 2-atom basis.

In the photo above, try to mentally draw triangles between the green atoms. These may be considered your lattice points. At each lattice point, you glue a two-atom basis: the green atom sits atop the lattice point, and the red atom is defined in this case to be 'd' to the left of the green atom. Now, you may travel from lattice point to lattice point and experience the exact same nearest neighbors.

Note: you will often hear a different way of describing these non-bravais lattices. You can describe graphene as being two triangle lattices with a one-atom basis, offset from one another. Similarly, the diamond cubic structure may be described as two simple cubic lattices offset from one another. These are geometrically equivalent to the lattice+basis=crystal model.

Answer 2

The unit cell in articles is a primitive cell, and the two points are not two lattices - instead, they are two sublattices of a primitive cell, since they cannot get to each other from translation over a unit vector.