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]

[b]

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]

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

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

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.

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