conceptual problem about the energy band along the High symmetry point in the Brillouin zone

Question

When we calculate the band structure of certain material, we only have to calculate the value along the high symmetry point which enclose the Irreducible Brillouin Zone. Why the information lie in the IBZ is enough for us to know the information of entire BZ ? How to find the IBZ given a reciprocal lattice vector in K-space without checking the table? Is there any proof ? Ive google it, but they just give you the table without telling you how to find out these High symmetry point.

Answer 1

I will try to outline this in simple terms. There are certainly more rigorous explanations.

The high-symmetry points in the Brillouin zone are those that remain physically identical when certain symmetry operations of the point-group are applied. Therefore, we notice that the first and higher derivatives of the dispersion relation have the same magnitude in directions that are mapped onto each other by these transformations.

For example, in a 2D material, the dispersion relation is $\epsilon\left(k_x,k_y\right)$ and we differentiate this with respect to a vector $\vec{k} = \vec{k}_x\sin{\alpha} + \vec{k}_y \cos{\alpha}$ for multiple $\alpha$. The magnitude and sign of the derivative are a periodic function whose period is defined by the symmetry of the high-symmetry point. Thus, extrema tend to be a high-symmetry points.