# Honeycomb lattice Brillouin zone structure and direct lattice periodic boundary conditions

One way to construct the Brillouin zone of the Honeycomb lattice is by obtaining the standard Wigner-Seitz cell by constructing the perpendicular bisectors of the reciprocal lattice vectors and considering the minimum area enclosed by them. Now, if we impose periodic boundary conditions on the lattice, then only certain values of 'k' points are allowed and the number of such 'k' points should be equal to the number of lattice points (belonging to any one sublattice). How do we discretize 'k' points such that the honeycomb BZ is generated?

Another way gives us an alternative BZ which is a parallelogram. $$\vec{k}=\frac{m_{1}}{N} \vec{b_{1}}+\frac{m_{2}}{N} \vec{b_{2}}$$ where $$m_{1},m_{2}$$ are integers running from $$0$$ to $$N-1$$, $$N$$ being the number of lattice spacings in the direct lattice along the lattice vector directions and $$\vec{b_{1}},\vec{b_{2}}$$ are reciprocal lattice vectors. Is this BZ equivalent to the former one and if so how to prove it? Sure there areas are same, but can one to one correspondence of 'k' points be proved?

The discretization of $$\mathbf{k}$$ by periodic boundary conditions applied at the boundaries of a very large crystal is independent of the construction of the 1st Brillouin zone. In a two-dimensional material, if you consider a large rectangular piece of crystal with side lengths $$L_x$$ and $$L_y$$, then the spacing of discrete $$\mathbf{k}$$-values in $$x$$-direction is $$2\pi/L_x$$, and in $$y$$-direction it is $$2\pi/L_y$$, such that the total area $$A_k$$ taken up by a single discrete $$\mathbf{k}$$-value in reciprocal space is $$A_k = \frac{(2\pi)^2}{L_xL_y} = \frac{(2\pi)^2}{A},$$ where $$A=L_xL_y$$. In this sense, the discretized $$\mathbf{k}$$-points do not 'generate' the honeycomb BZ, as the way you obtain them does not refer to or depend on the symmetry of the crystal lattice that you consider. The periodic boundary condition merely provides you with the density of $$\mathbf{k}$$-points in reciprocal space.
The first Brillouin zone is a unique object by construction. It is the set of all points that are closer to the origin of reciprocal space (called the $$\Gamma$$-point) than to any other reciprocal lattice point. In other words, it is the primitive Wigner-Seitz-cell of the reciprocal lattice of the crystal under consideration. According to this definition, there is no alternative first BZ.