3.5.1 Hierarchy of systems

The point group of the unstrained diamond structure is denoted by $O_h$ according to the Schönfließ notation, and contains 48 symmetry elements listed in Table 3.2 [Yu03]. The symmetry elements are given in terms of rotations and rotations followed by the inversion operation (I ${\ensuremath{\mathitbf{k}}} = -{\ensuremath{\mathitbf{k}}}$). The following notation for the point operations is chosen:

E	unity operation
n$_{k}^+$	clockwise rotation of angle $2\pi/n$ around axis ${\ensuremath{\mathitbf{e}}}_\mathrm{k}$
n$_{k}^-$	counter-clockwise rotation of angle $2\pi/n$ around axis ${\ensuremath{\mathitbf{e}}}_\mathrm{k}$
I	inversion
$\bar{n}_{k}^+$	clockwise rotation of angle $2\pi/n$ around axis ${\ensuremath{\mathitbf{e}}}_\mathrm{k}$ followed by inversion
$\bar{n}_{k}^-$	counter-clockwise rotation of angle $2\pi/n$ around axis ${\ensuremath{\mathitbf{e}}}_\mathrm{k}$ followed by inversion

The rotation axes ${\ensuremath{\mathitbf{e}}}_\mathrm{k}$ are grouped into five classes:

${\ensuremath{\mathitbf{e}}}_{i}$	$(1,0,0),\ (0,1,0),\ (0,0,1)$
	$(0,1,0),\ (\sqrt{3},-1,0),\ (-\sqrt{3},-1,0)$
${\ensuremath{\mathitbf{e}}}_{j}$	$(1,1,1),\ (-1,-1,1),\ (1,-1,-1),\ (-1,-1,-1)$
${\ensuremath{\mathitbf{e}}}_{p}$	$(1,1,0),\ (-1,1,0),\ (1,0,1),\ (0,1,1),\ (-1,0,1),\ (0,-1,1)$
${\ensuremath{\mathitbf{e}}}_{s}$	$(1,1,0),\ (-1,1,0)$

Table 3.2: Point group and symmetry elements of strained lattices that originate when stress is applied along various high symmetry directions to an initially cubic lattice $O_h$. The Schönfließ symbol is used to specify the point group. $\vert P(\Gamma )\vert$ denotes the number of elements of the point group.

Of all the point symmetry groups $P(\Gamma)$ of the crystal lattice, the group $O_h$ possesses the highest symmetry. The symmetry group $P(\Gamma')$ of the Bravais lattice of the strained crystal is a subgroup of the symmetry group $P(\Gamma')$ of the unstrained crystal and does not generally belong to the same crystal class as $O_h$. It contains only those symmetry elements which are preserved under strain. The effect of a homogeneous strain on the symmetry of the Bravais lattice depends on the specific form of applied strain.

By successively lowering the symmetry, one can go from $O_h$ to point group $S_2$ on two distinct paths [Bir74]:

$$O_{h} \rightarrow D_{4h} \rightarrow D_{2h} \rightarrow C_{2h} \rightarrow S_{2}$$ (3.35)

$$O_h \rightarrow D_{3d} \rightarrow C_{2h} \rightarrow S_{2}\ .$$ (3.36)

The symmetry elements of the five specified point groups $P(\Gamma')$, namely $D_{4h}, D_{3d}, D_{2h}, C_{2h}, S_{2}$, which are subgroups of $O_h$ are given in Table 3.2. The symmetry can be lowered by distorting the crystal by applying uniaxial stress. If, for example, stress is applied along a fourfold axis, the point group $O_h$ reduces to $D_{4h}$, which contains only $16$ symmetry elements. In Table 3.2 five directions of uniaxial stress are given that yield a direct $O_h \rightarrow P(\Gamma')$ symmetry reduction for any of the five subgroups of $O_h$.

From (3.34) it can be concluded that the higher the point symmetry of the crystal lattice, the smaller is the volume of the irreducible wedge. In the following, the symmetries of the band structure and the shape of the irreducible wedge in terms of the irreducible wedge of the unstrained crystal are given for the crystal systems $O_h, D_{4h}, D_{3d}, D_{2h}, C_{2h}$, and $S_{2}$.

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