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GATE 2023 PH – Question 22

Solid State Physics · Elements of crystallography · 1 mark · Multiple choice

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, a 1 and a 2 are two lattice vectors. Which one of the following is the area of the first Brillouin zone for this lattice?

Honeycomb lattice and primitive translation vectors.
  1. $8\pi^2/(3\sqrt3a^2)$
  2. $4\pi^2/(3\sqrt3a^2)$
  3. $8\pi^2/(\sqrt3a^2)$
  4. $4\pi^2/(\sqrt3a^2)$

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Correct answer: (C) $8\pi^2/(\sqrt3a^2)$

Explanation

For primitive translation length a the real-space cell area is $\sqrt3a^2/2$. The reciprocal-cell area is $(2\pi)^2/A=8\pi^2/(\sqrt3a^2)$.