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?
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?
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For a hexagonal lattice (graphene), the area of the first Brillouin zone is related to the lattice constant by $A_{\text{BZ}} = \dfrac{8\pi^2}{3\sqrt{3}a^2}$.
Updated On: Aug 30, 2025
- $\dfrac{8\pi^2}{3\sqrt{3}a^2}$
- $\dfrac{4\pi^2}{\sqrt{3}a^2}$
- $\dfrac{8\pi^2}{\sqrt{3}a^2}$
- $\dfrac{4\pi^2}{3\sqrt{3}a^2}$
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The Correct Option is A
Solution and Explanation
- 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).
Thus, the correct answer is (A).
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