I am quite confused by the Brillouin zones. I know there is a dispersion relation $E=E(k)$ for the first Brillouin zone. But is this dispersion relation periodic across different Brillouin zones? Thinking in one way that larger $k$ should give rise to larger energy tends me to think that in higher-order Brillouin zones the energy is larger, but the Bloch theorem seems to hint a periodic structure. I also found references that seemed to give contradictory answers. For example, page 13 in this document seems to indicate energy is larger in higher-order zones, but page 4 in this other document seems to indicate energy is periodic across different Brillouin zones. I need some really good explanations.


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Basically, there are several ways to describe how your energy eigenvalues depend on the wavevector, in so-called zone-scheme representations:

  • An extended zone-scheme, like the one displayed on page 13 of your first reference and Fig. 2a of the second one, is useful to compare the energy dispersion of a crystalline electron compared to the parabolic one of a free electron.
  • However, since all lattice points are equivalent by translational symmetry, the choice of the origin in momentum space is completely arbitrary. As a consequence, in a repeated zone-scheme, the same dispersion relation is repeated all over $k$-space, at each reciprocal lattice point. An example of a repeated zone-scheme is Fig. 2b in your second reference. This scheme is useful to visualize the periodicity of the dispersion relation in momentum space $E(k)=E(k+G)$.
  • Still, the repeated zone-scheme is clearly redundant, as all contributions from all reciprocal lattice points to the first Brillouin zone, i.e. in $-\frac{\pi}{a}\leq k \leq\frac{\pi}{a}$, correspond to all contributions from all reciprocal lattice points to each Brillouin zone. So we are free to pick one Brillouin zone and we choose to restrict to the first. This results in a reduced zone-scheme, as displayed in Fig. 2c in your second reference.

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