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3.5.4 $ D_{3d}$ symmetry

Figure 3.9: Irreducible wedge of the first BZ of a diamond structure stressed along direction $ [111]$.
\includegraphics[scale=1.0, clip]{inkscape/bz111_colored.eps}

Under uniaxial stress along the threefold symmetry axes $ {\ensuremath{\mathitbf{e}}}_j$ representing the cube diagonals, the cube of the $ O_h$ lattice becomes a rhombohedron with an arbitrary angle between the edges. The cubic fcc lattice is transformed to a primitive rhombohedral lattice ( $ \Gamma'=\Gamma_{rh}$) belonging to the crystal class $ D_{3d}$ of the trigonal (= rhombohedral) system [Bir74].

Apart from the trivial unity transformation and inversion, the symmetry operations of $ D_{3d}$ include threefold rotations about the direction of stress and twofold rotations around three twofold axes perpendicular to the threefold axis. All twelve symmetry operations and the resulting invariances of the energy band structure are listed in Table 3.3. Note that in this table the Miller index notation of the cubic lattice is chosen to specify the directions of the axes of rotation, whereas the directions used in Table 3.2 are given using the Bravais basis vectors of the trigonal lattice:

$\displaystyle {\ensuremath{\mathitbf{a}}}_1 = \begin{pmatrix}\frac{a}{2\sqrt{3}... ...}}}_3 = \begin{pmatrix}\frac{-a}{\sqrt{3}} \\ 0 \\ \frac{c}{3} \end{pmatrix}\ .$ (3.42)

These basis vectors are equivalent to those of the fcc lattice when $ a=(\sqrt{3}a_0)/2$ and $ c=(3a_0)/2$.

From (3.18) it can be seen that the strain tensor resulting from stress along $ \langle111\rangle$, which yields a $ O_h \rightarrow D_{3d}$ symmetry reduction, contains equal off-diagonal components $ {\ensuremath{\varepsilon_{12}}} = {\ensuremath{\varepsilon_{13}}} = {\ensuremath{\varepsilon_{23}}}$ and equal diagonal components $ {\ensuremath{\varepsilon_{11}}} = {\ensuremath{\varepsilon_{22}}} = {\ensuremath{\varepsilon_{33}}}$. Note that biaxial strain in the $ \{111\}$ plane also yields $ {\ensuremath{\varepsilon_{12}}} = {\ensuremath{\varepsilon_{13}}} = {\ensuremath{\varepsilon_{23}}}$, and $ {\ensuremath{\varepsilon_{11}}} = {\ensuremath{\varepsilon_{22}}} = {\ensuremath{\varepsilon_{33}}}$, hence the same symmetry reduction occurs.


Table 3.3: Symmetry operations leaving the band structure invariant in the $ D_{3d}$ class. The direction of the rotation axes are given using the Miller index notation of the cubic lattice.
\begin{table}\centering \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccc}... ...{2}_{[0\bar{1}1]}$\ & $E_{n}(k_{x},k_{z},k_{y})$\ \\ \end{tabular*} \end{table}


The twelve involved symmetry operations give rise to a volume of the irreducible wedge of $ \Omega_{\mathrm{BZ}}/12$, which can be mapped onto four irreducible wedges of the unstrained crystal as shown in Figure 3.9. The four wedges can be transformed into the first wedge by the symmetry operations

$\displaystyle \ensuremath{{\underaccent{\bar}{T}}}_1 = \begin{pmatrix}1 & 0 & 0... ...{\bar}{T}}}_4 = \begin{pmatrix}1 & 0 & 0\\ 0 & 0 & -1\\ 0 &-1 & 0\end{pmatrix}.$ (3.43)


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