3.5.2 $O_{h}$ symmetry

The point group $O_{h}$ contains 48 symmetry elements: listed in

• the unity operation,
• clockwise rotation of 180$^\circ$ about the principal axes ${\ensuremath{\mathitbf{e}}}_i$ (three operations),
• clockwise and counter-clockwise rotation of 120$^\circ$ about the four space diagonals (eight operations),
• clockwise rotation of 180$^\circ$ about the axes ${\ensuremath{\mathitbf{e}}}_p$ representing the diagonals on the faces of a unit cube (six operations),
• clockwise and counter-clockwise rotations of 90$^\circ$ about the principal axes ${\ensuremath{\mathitbf{e}}}_i$ (six operations).

In total these are 24 symmetry operations. Since the symmetry is preserved if any of these operations are followed by the inversion operation, there are 48 symmetry operations in total.

The 48 point symmetries are reflected in the energy bands and can be shown to leave the energy bands invariant under the following eight reflections

$$E_n(k_x,k_y,k_z) = E_{n}(\vert k_x\vert,\vert k_y\vert,\vert k_z\vert)$$ (3.37)

and six permutations

$$E_{n}(k_{x},k_{y},k_{z}) = E_{n}(k_{x},k_{z},k_{y}) = E_{n}(k_{y},k_{x},k_{z}) =$$

$$E_{n}(k_{y},k_{z},k_{x}) = E_{n}(k_{z},k_{x},k_{y}) = E_{n}(k_{z},k_{y},k_{x})\ .$$ (3.38)

A possible choice for the irreducible wedge of unstrained Si is depicted in Figure 3.5b. It should be noted that this specific choice is not unique, and other shapes for the irreducible wedge, reflecting the symmetries of $O_h$, can be found [Stanley98]. The volume of the specified irreducible wedge is $\Omega_\mathrm{BZ}/48$ in accordance with the relation (3.34).

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology