3.6.1 Strain-Induced Conduction Band Splitting

In this section we give expressions for the strain-induced energy shifts of the nondegenerate energy levels of the conduction band edges of the cubic crystal class $O_h$.

At the $\Gamma$ and $L$ point, and along the $\Delta$ symmetry line, the deformation potential operators $\mathcal{D}^{\alpha\beta}$ are scalars and given by one ore two independent constants. Neglecting the strain-induced splitting of the degenerate conduction bands $\Delta _1$ and $\Delta _{2'}$ at the $X$ point for the moment, the energy shifts of the conduction band edge of valleys along the $\langle 100 \rangle$ and $\langle111\rangle$ direction can be calculated from two independent deformation potential constants [Balslev66]

$$\delta E^{v_{i}}_0 = \ensuremath {\Xi_\mathrm{d}}^{v}\, (\ensuremath{{\underaccent{\bar}{\varepsilon}}}) + \ensuremath {\... ...\ensuremath{{\underaccent{\bar}{\varepsilon}}} {\ensuremath{\mathitbf{a}}}_i\ .$$ (3.47)

Here, $\ensuremath {\Xi_\mathrm{d}}^{v}$ denotes the dilatation- and $\ensuremath {\Xi_\mathrm{u}}^{v}$ the uniaxial deformation potential constant for a valley of type $v = \Delta,L$, and ${\ensuremath{\mathitbf{a}}}_i$ is a unit vector parallel to the ${\ensuremath{\mathitbf{k}}}$ vector of valley $i$. The valley shift of the $\Gamma_2'$ conduction band minimum can be obtained from a single deformation potential constant

$$\delta E^{\Gamma}_0 = \ensuremath {\Xi_\mathrm{d}}^{\Gamma}\, (\ensuremath{{\underaccent{\bar}{\varepsilon}}})\ .$$ (3.48)

The valley splitting from uniaxial stress along any direction can be obtained from the strain tensor using the relations above. The strain tensors resulting from uniaxial stress are discussed in Section 3.3.2. The analytical expressions for the energy shifts of the conduction band valleys for three stress directions [100], [110], and [111] are given in in Table 3.4.

Table 3.4: Strain-induced energy shifts of the conduction band valleys of cubic semiconductors when uniaxial stress $P$ is applied along three high symmetry directions. The energy shifts are divided by $P$.

stress direction	valley	valley direction	$\delta E / P$
$[100]$	$\Delta$	$[100]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Delta}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{\Delta}} s_{11}$
	$\Delta$	$[010][001]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Delta}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{\Delta}} s_{12}$
	$L$	$[111][11\bar{1}][1\bar{1}1][\bar{1}11]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{L}}(s_{11} + 2 s_{12})+\ensuremath {\Xi_\mathrm{u}}^{\mathrm{L}}/3(s_{11} + 2 s_{12})$
	$\Gamma$	$[000]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Gamma}}(s_{11}+2s_{12})$
$[110]$	$\Delta$	$[100][010]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Delta}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{\Delta}}/2(s_{11}+s_{12})$
	$\Delta$	$[001]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Delta}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{\Delta}} s_{12}$
	$L$	$[111][11\bar{1}]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{L}}(s_{11} + 2 s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{L}}/3(s_{11} + 2 s_{12} + s_{44})$
	$L$	$[\bar{1}11][1\bar{1}1]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{L}}(s_{11} + 2 s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{L}}/3(s_{11} + 2 s_{12} - s_{44})$
	$\Gamma$	$[000]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Gamma}}(s_{11}+2s_{12})$
$[111]$	$\Delta$	$[100][010][001]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Delta}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{\Delta}}/3(s_{11}+2s_{12})$
	$L$	$[111]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{L}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{L}}/3(s_{11}+2s_{12}+2 s_{44})$
	$L$	$[\bar{1}11][1\bar{1}1][11\bar{1}]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{L}}(s_{11}+2s_{12}) + \ensuremath {\Xi_\mathrm{u}}^{\mathrm{L}}/3(s_{11}+2s_{12}-2/3 s_{44})$
	$\Gamma$	$[000]$	$\ensuremath {\Xi_\mathrm{d}}^{\mathrm{\Gamma}}(s_{11}+2s_{12})$

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