Construction of first Brillouin zone

Question

For a square lattice with lattice translation vector, $T=a\hat{x}+a\hat{y}$, we find that the reciprocal lattice vectors are given by, $$A=\frac{2\pi}{a}\hat{x},$$ and, $$B=\frac{2\pi}{a}\hat{y},$$ such that the Brillouin zone ranges from $-\frac{\pi}{a}\to\frac{\pi}{a}$. That is it ranges from $-\frac{|A|}{2}\to\frac{|A|}{2}$, where $|A|$ is the magnitude of $A$, and where $|A|=|B|$.

Now consider a hexagonal lattice with receiprocal lattice vectors, $$A=2\pi\hat{x}+\frac{2\pi}{\sqrt{3}}\hat{y},$$ and, $$B=\frac{4\pi}{\sqrt{3}},$$ both with magnitude, $|A|=|B|=\frac{4\pi}{\sqrt{3}}$. Does the first Brillouin zone run from, $=\frac{2\pi}{\sqrt{3}}\to\frac{2\pi}{\sqrt{3}}$? If this is the case, can we say that the first Brillouin zone is constructed similarly to the Wigner-Seitz unit cell in that it bisects lines to closest neighbors in reciprocal space to form a closed cell?

Answer 1

For the First case where you have a square lattice, the first Brillouin zone is a square that lies between $\frac{-|A|}{2} \rightarrow \frac{|A|}{2}$ in the $\hat{x}$ direction and $\frac{-|B|}{2} \rightarrow \frac{|B|}{2}$ in the $\hat{y}$ direction. Assuming that you chose a lattice point as your origin in the k-space. (Here is a reference for how to draw it https://www.doitpoms.ac.uk/tlplib/brillouin_zones/printall.php)

For the Second case choose an origin, using your reciprocal lattice vectors construct the overall lattice structure. Then you will see that A Hexagonal lattice in the position space (let's say with a side length of 1) is a hexagonal lattice in the momentum space that is oriented by $90^\circ$ (with a side length of $\frac{2 \pi}{\sqrt{3}}$. Then you construct the Brillouin zones by drawing the bisectors (Bragg Plane's) between consecutive neighbouring points.

Here's an image I have taken from ( Carvalho, Alexandre F.. “Simultaneous synthesis of diamond on graphene for electronic application.” (2015) ). Their lattice vectors are shifted by $\frac{2 \pi}{\sqrt{3}} \hat{y}$ compared to yours but it gives the same result.