3.5 Effect of Strain on Crystals with Diamond Structure

Application of strain to a crystal reduces its symmetry. The Bravais lattice basis vectors ${{\ensuremath{\mathitbf{a}}}_i}'$ of a crystal under homogeneous strain are obtained by deforming the vectors ${\ensuremath{\mathitbf{a}}}_i$ of the unstrained crystal

$${{\ensuremath{\mathitbf{a}}}_{i}}' = \left ( \mathbb{1} + \hat{\varepsilon} \right ) \cdot {\ensuremath{\mathitbf{a}}}_{i}\ ,$$ (3.30)

where $\hat{\varepsilon}$ denotes the strain tensor [Bir74]. Not only is the volume $\Omega'_0$ of the strained primitive unit cell modified

$$\Omega'_0 = \Omega_0(1 + {\ensuremath{\varepsilon_{xx}}} + {\ensuremath{\varepsilon_{yy}}} + {\ensuremath{\varepsilon_{zz}}})\ ,$$ (3.31)

but also the type of the Bravais lattice may change. The new Bravais lattice is then characterized by basis vectors which may be different from (3.30). However, the volume of the new primitive cell always differs from $\Omega_0$ by a factor proportional to the strain, given in (3.31) if the hydrostatic strain component, $\varepsilon_{xx}$ + $\varepsilon_{yy}$ + $\varepsilon_{zz}$, is nonzero. The Brillouin zone (BZ) of the strained crystal, being independent of the specific choice of basis vectors, can always be obtained by a suitable deformation of the BZ of the unstrained crystal [Bir74].

Generally, the strain tensor (3.5) determines only the deformation of the primitive cell as a whole, but the relative displacements of the atoms within the cell differ for the atoms forming the basis. This has no impact on the lattice symmetry but has to be accounted for in band structure calculations. In Section 3.8.2 it is shown how to take this internal displacement into account properly.

Points and lines of the BZ can be classified according to their symmetry. For each lattice there are symmetry operations that leave the lattice and hence the BZ invariant. These operations bring the BZ into coincidence with itself. In such an operation a specific vector $\mathitbf{k}$ is not necessarily projected onto itself or $\mathitbf{k}$ + $\mathitbf{G}$$_{lmn}$. For each vector $\mathitbf{k}$ there exists a set of symmetry operations $\alpha$ fulfilling

$$\alpha {\ensuremath{\mathitbf{k}}} = {\ensuremath{\mathitbf{k}}}\... ...hitbf{k}}} = {\ensuremath{\mathitbf{k}}} + {\ensuremath{\mathitbf{G}}}_{lmn}\ .$$ (3.32)

The number of symmetry operations $\alpha$ depends on the wave vector $\mathitbf{k}$ and is denoted as the symmetry group $P({\ensuremath{\mathitbf{k}}})$ of the respective $\mathitbf{k}$ vector

$$P({\ensuremath{\mathitbf{k}}}) = \{\alpha\, \vert\, \alpha {\ensu... ...tbf{k}}} = {\ensuremath{\mathitbf{k}}} + {\ensuremath{\mathitbf{G}}}_{lmn}\}\ .$$ (3.33)

Since the unity operation E always fulfills (3.33), the symmetry group $P({\ensuremath{\mathitbf{k}}})$ contains one element at least ( $\alpha=\mathrm{E}$). If the symmetry group of a given vector ${\ensuremath{\mathitbf{k}}}$ contains more elements than the symmetry group of neighboring points, this specific vector ${\ensuremath{\mathitbf{k}}}$ is referred to as symmetry point.

In Figure 3.5b the symmetry points (filled circles) and symmetry lines (open circles) of the fcc lattice are shown. Strictly, the given points $K$ and $U$ are not symmetry points according to the definition of (3.33), since they have the same symmetry as the points along the symmetry lines $\Sigma$ and $S$. Hence, they can be included in these symmetry lines.

Since the center point $\Gamma$ of the BZ is mapped onto itself at any point operation $\alpha$ of the crystal lattice, all symmetry operations of the lattice are included in the point group $P(\Gamma)$. Thus, this group determines the shape and volume of the irreducible wedge. The number of symmetry elements of $P(\Gamma)$ determines the volume of the irreducible wedge as [Nowotny98]

$$\Omega_\mathrm{irred} = \Omega_\mathrm{BZ}/\vert P(\Gamma)\vert\ ,$$ (3.34)

where $\vert P(\Gamma )\vert$ is the number of elements of the symmetry group $P(\Gamma)$.

Subsections

• 3.5.1 Hierarchy of systems
• 3.5.2 $O_{h}$ symmetry
• 3.5.3 $D_{4h}$ symmetry
• 3.5.4 $D_{3d}$ symmetry
• 3.5.5 $D_{2h}$ symmetry
• 3.5.6 $C_{2h}$ symmetry
• 3.5.7 $S_{\mathrm{2}}$ symmetry

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