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GATE 2021 PH – Question 20

Solid State Physics · Band theory of solids: nearly free electron model · 1 mark · Multiple select

Consider a two-dimensional square lattice with lattice constant $a$. Which of the following wave vectors can lie on Bragg planes where an electronic band gap may open? Select all that apply.

  1. $(0,0)$
  2. $(\pm\pi/a,\pm\pi/a)$
  3. $(\pm\pi/a,\pm\pi/(1.3a))$
  4. $(\pm\pi/(3a),\pm\pi/a)$

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Correct answer: (B) $(\pm\pi/a,\pm\pi/a)$; (C) $(\pm\pi/a,\pm\pi/(1.3a))$; (D) $(\pm\pi/(3a),\pm\pi/a)$

Explanation

At a Bragg plane, free-electron states $\mathbf k$ and $\mathbf k-\mathbf G$ are degenerate, so $2\mathbf k\cdot\mathbf G=G^2$. Taking $\mathbf G=(2\pi/a,0)$ gives $k_x=\pm\pi/a$, and taking $\mathbf G=(0,2\pi/a)$ gives $k_y=\pm\pi/a$. B and C satisfy the first condition and D the second. The origin does not satisfy a nonzero reciprocal-vector Bragg condition. Hence B, C and D.