My goal is to analytically derive the first Brillouin zone of the honeycomb lattice. Geometrically it's clear how to do this by just finding the space on the lattice nearest to a particular point of the lattice; this is just another hexagon, so the first Brillouin zone should be a hexagon in $k$-space.

Here's my attempt so far. Let the nearest neighbor spacing be $a$, and choose lattice vectors $\vec{a}_{1} = a\sqrt{3}(-1/2,\sqrt{3}/2)$ and $\vec{a}_{2} = a\sqrt{3}(1/2, \sqrt{3}/2)$. I assume that a translation by $N_{1}\vec{a}_{1}$ or $N_{2}\vec{a}_{2}$ is a symmetry of the lattice, i.e., we have periodic boundary conditions and these translations take us around the torus. This seems to suggest the constraints

$$\exp\left[iN_{1}\vec{a}_{1}\cdot\vec{k}\right] = \exp\left[iN_{2}\vec{a}_{2}\cdot\vec{k}\right] = 1$$

After some massaging, this gives the equations

$$k_{x} = \frac{2\pi}{a\sqrt{3}}\left(\frac{q}{N_{2}}-\frac{m}{N_{1}}\right), k_{y}=\frac{2\pi}{3a}\left(\frac{q}{N_{2}}+\frac{m}{N_{1}}\right)$$

for integers $q,m$ that have to be chosen to restrict the momenta to the first Brillouin zone. However I can't see how some choice of these integers traces out a hexagon. Is this reasoning somehow flawed, or is there perhaps a way to extract the Brillouin zone from this result?



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