Question

A two-dimensional square lattice has lattice constant \(a\). \(k\) represents the wavevector in reciprocal space. The coordinates \((k_x, k_y)\) of reciprocal space where band gap(s) can occur, are 
 

• A. (0, 0)
• B. \( \left( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} \right) \)
• C. \( \left( \pm \frac{\pi}{a}, \pm \frac{\pi}{1.3a} \right) \)
• D. \( \left( \pm \frac{\pi}{3a}, \pm \frac{\pi}{a} \right) \)

Hint:

Band gaps in crystal lattices can occur at specific reciprocal lattice points, especially at high-symmetry points such as \( \pm \frac{\pi}{a} \), where \( a \) is the lattice constant.

Answer 1

The Correct Option is B, C, D

In a two-dimensional square lattice, the reciprocal lattice vectors are given by: \[ \mathbf{G} = \left( \frac{n_x 2\pi}{a}, \frac{n_y 2\pi}{a} \right), \] where \( n_x, n_y \) are integers representing the components of the wavevector in the reciprocal space, and \( a \) is the lattice constant. 1. Band gaps in a lattice occur at specific reciprocal lattice points, which correspond to the values of \( k_x \) and \( k_y \) where the wavevector leads to destructive interference in the system, creating a gap in the electronic band structure. In a square lattice, band gaps can occur at certain high-symmetry points in the reciprocal space. 2. Option (B) is correct because the coordinates where band gaps can occur are at the high-symmetry points in the reciprocal lattice, specifically at \( \left( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} \right) \), which are points of high symmetry in the 2D reciprocal space for a square lattice. 3. Option (C) is also valid because other high-symmetry points could exist, like \( \left( \pm \frac{\pi}{a}, \pm \frac{\pi}{1.3a} \right) \), which is a different scaling of the reciprocal space but still aligns with the symmetry considerations for possible band gaps. 4. Option (D) represents a scaling where the reciprocal lattice points are adjusted by a factor of 3 in one direction. This is another valid possibility for band gaps in the system, as different scaling factors in reciprocal space may correspond to different gap structures. Thus, the coordinates for band gaps are (B), (C), (D), and the correct answer is (B), (C), (D).