6.2 Geometric Properties of the Irreducible Wedge

Due to additional symmetries of the lattice structure of relaxed silicon the energy bands are invariant under some particular spatial reflections and permutations. Therefore the energy information is redundant in the full Brillouin zone and can be reduced to an irreducible wedge containing only $1/48$ th. The first irreducible wedge has six vertices

$$\Gamma = \frac{2\pi}{a_0} \begin{pmatrix}0 \ 0 \ 0 \end{pmatrix} , \quad L = \frac{2\pi}{a_0} \begin{pmatrix}1/2 \ 1/2 \ 1/2 \end{pmatrix} , \quad K = \frac{2\pi}{a_0} \begin{pmatrix}3/4 \ 3/4 \ 0 \end{pmatrix},$$

$$X = \frac{2\pi}{a_0} \begin{pmatrix}1 \ 0 \ 0 \end{pmatrix} , \quad W = \frac{2\pi}{a_0} \begin{pmatrix}1 \ 1/2 \ 0 \end{pmatrix} , \quad U = \frac{2\pi}{a_0} \begin{pmatrix}1 \ 1/4 \ 1/4 \end{pmatrix},$$ (6.7)

and five faces

$$k_x + k_y + k_z = \frac{3}{2} \frac{2\pi}{a_0}, \qquad k_x = \frac{2\pi}{a_0},$$ (6.8)

$$k_x = k_y, \qquad k_y = k_z, \qquad k_z = 0.$$ (6.9)

The first two planes (6.8) are part of the Brillouin zone boundary, whereas the last three ones (6.9) are inside of the Brillouin zone. Figure 6.3 gives a sketch of the irreducible wedge placed within the first octant of the first Brillouin zone.

Figure 6.3: First Brillouin zone of the reciprocal lattice with emphasis on the first octant which carries the first irreducible wedge.

In the following the tessellation of the first octant of the Brillouin zone and the irreducible wedge with emphasis on the band structure of silicon are under consideration. Different approximations for the silicon band structure are presented, where one of the models is used for the construction of a Brillouin zone mesh, shown in the next section.