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6.1.1 Strain-Induced Shift of the Conduction Band Minimum

In Figure 6.1 the calculated band edge energies of biaxially strained Si and Ge are compared to those of Rieger and Vogl [Rieger93]. Good agreement can be observed for both strained Si for (001)-oriented Si$ _{1-y}$Ge$ _y$ buffers and strained Ge for (001)-oriented Si$ _{1-y}$Ge$ _y$ buffers for the whole range of substrate mole-fractions $ y$.

Figure 6.2: Calculated band edge energies of the $ \Delta $-valleys and $ L$-valleys of strained Si on (a) (110)-oriented and (b) (111)-oriented Si$ _{1-y}$Ge$ _y$ relaxed buffer with respect to valence band edge.
[a]\includegraphics[width=8.cm]{xcrv-scipts/epmSubOr110_bw.eps} [b]\includegraphics[width=8.cm]{xcrv-scipts/epmSubOr111_bw.eps}

In Figure 6.2 the band edge energies of the conduction band valleys of Si grown on Si$ _{1-y}$Ge$ _y$ buffers with orientation (110) and (111) are presented. For (111)-oriented buffers the $ \Delta $-valleys do not split in agreement to the theoretical model (3.47). For (110)-oriented buffer the splitting between the $ \Delta $-valleys of strained Si is very small. From a deformation potential theory which neglects the lifting of the degeneracy of the two lowest conduction bands at the $ X$ points the splitting cannot be reproduced. Using the deformation potential theory the valley shift between the $ \Delta _{[001]}$-valley pair and the $ \Delta _{[010]}$-valley pair increases linearly with strain [Singh93]

$\displaystyle \delta E_0^{\Delta_{[001]}} - \delta E_0^{\Delta_{[100]}} = \Xi_u^\Delta {\ensuremath{\varepsilon_{\vert\vert}}} \tilde{c}\ ,$ (6.1)

where $ {\ensuremath{\varepsilon_{\vert\vert}}}=(a_\mathrm{sige} - a_\mathrm{si})/a_\mathrm{si}$, $ \tilde{c} =
(c_{11} + 2c_{12})/(c_{11} + c_{12} + 2c_{44})$, and $ a_\mathrm{sige}$ is the lattice constant of the Si$ _{1-y}$Ge$ _y$ buffer.

When taking into account the additional valley shift of the $ \Delta _{[001]}$-valley due to shear strain, the total valley shift is obtained from

$\displaystyle \delta E_0^{\Delta_{[001]}} + \delta E_1^{\Delta_{[001]}} - \delt...
...math{\varepsilon_{\vert\vert}}} \tilde{c} \frac{\Delta}{4} \kappa^2 \right )\ .$ (6.2)

Figure 6.3 shows that the model (6.2) agrees much better with the results of EPM calculations.

Figure 6.3: Valley splitting of the $ \Delta _{[001]}$ and $ \Delta _{[100]}$ valleys of biaxially strained Si on a (110) Si$ _{1-y}$Ge$ _y$ buffer.
\includegraphics[width=11.cm]{xcrv-scipts/DeltaValleySplitEpi110_bw.eps}

The shifts of the band edge of the $ \Delta $-valleys and the $ L$-valleys are calculated for uniaxial stress along four different directions. Figure 6.4 shows that the valley shifts are linear for stress up to 2 GPa, and that the largest splitting among the $ \Delta $-valleys is obtained for the stress direction $ \langle
100 \rangle$. Since the conduction band edges are given with respect to edge of the top valence band, which is either the heavy-hole or the light-hole band depending on the sign of stress, the slope of the valley splitting at 0 GPa changes.

Figure 6.4: Calculated energies of the $ \Delta $-valleys and $ L$-valleys of Si with respect to the valence band edge as a function of uniaxial stress for various stress directions.
\resizebox{77mm}{!}{\includegraphics{xcrv-scipts/epmStress100_bw.eps}} \resizebox{77mm}{!}{\includegraphics{xcrv-scipts/epmStress110_bw.eps}}
\resizebox{77mm}{!}{\includegraphics{xcrv-scipts/epmStress111_bw.eps}} \resizebox{77mm}{!}{\includegraphics{xcrv-scipts/epmStress120_bw.eps}}


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