Question

Graphene is a two-dimensional material, in which carbon atoms are arranged in a honeycomb lattice with lattice constant $a$. As shown in the figure, $\vec{a_1}$ and $\vec{a_2}$ are two lattice vectors. Which one of the following is the area of the first Brillouin zone for this lattice?

• A. $\dfrac{8\pi^2}{3\sqrt{3}a^2}$
• B. $\dfrac{4\pi^2}{\sqrt{3}a^2}$
• C. $\dfrac{8\pi^2}{\sqrt{3}a^2}$
• D. $\dfrac{4\pi^2}{3\sqrt{3}a^2}$

Hint:

For a hexagonal lattice (graphene), the area of the first Brillouin zone is related to the lattice constant by $A_{\text{BZ}} = \dfrac{8\pi^2}{3\sqrt{3}a^2}$.

Answer 1

The Correct Option is A

- The area of the first Brillouin zone for a two-dimensional hexagonal lattice (graphene) is given by the formula: \[ A_{\text{BZ}} = \dfrac{8\pi^2}{3\sqrt{3}a^2}. \] This result comes from the reciprocal lattice vectors of the honeycomb structure.
Thus, the correct answer is (A).