Graphene

Graphene is a two-dimensional lattice which is built by a continuous arrangement of regular hexagons of sp² hybridized carbon atoms (see below). The distance between nearest neighbor atoms is 0.14 nm.

(a) What is the Bravais lattice of graphene?

(b) How many atoms are there in a primitive unit cell?

(c) The distance between the Bravais points is called the lattice constant, $a$. Express the primitive lattice vectors in terms of $a$. What is the lattice constant $a$ in Å?

(d) Draw the Wigner-Seitz primitive unit cell.

(e) The basis vectors describe the positions of the atoms in the unit cell. Write these basis vectors first in terms of the absolute positions (with $x$- and $y$- components and distances in Angstroms) and then in fractional coordinates. Fractional coordinates are usually given in terms of the conventional unit vectors. An atom at position $\vec{r}= f_1\vec{a}_1 + f_2\vec{a}_2$ has fractional coordinates $(f_1,f_2)$. For graphene, the conventional lattice vectors are the same as the primitive lattice vectors.

Solution