Numerical calculations in literature exhibit a considerable quantitative
difference in the saturation value of the in-plane electron mobility in
biaxially-strained Si layers. Enhancement factors ranging from 56 %
[Bufler97] up to 180% [Yamada94] have been simulated while
measurements indicate a value of around 97% [Ismail93]. We have adopted
a somewhat conservative value of 70% for the enhancement factor, which lies
in-between other reported values [Fischetti96] [Vogelsang93].
Type of scattering
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To achieve such values, it was required to adjust the coupling constants of the -type and -type phonons. The reported values of the coupling constants for inter-valley scattering in Si are shown in Table 5.1. Since at high strain values the -type phonon scattering is completely suppressed in the valleys, the only way to obtain an increase in the saturation mobility is by reducing the -type coupling constants. This adjustment however would result in an increased value of the unstrained electron mobility. To restore the latter, it was required to increase the -type phonon coupling constant. The coupling constants of the -type phonons were reduced by a factor of 1.06 and that for -type phonons increased by a factor of 1.16, as compared to the original values proposed by Jacoboni [Jacoboni83]. The only parameter of the inter-valley scattering model (4.67) is the phonon energy. A value of has been assumed. While the model based on piezoresistance coefficients assumes a linear relationship between mobility enhancement and strain, with the proposed model the mobility enhancement saturates at large strain values as observed experimentally.
The electron mobility components for different orientations of the underlying SiGe layer is obtained by transformation of the mobility tensor from the principle coordinate system to the interface coordinate system, using the transformation matrix (3.19). Fig. 5.1 shows the electron lattice mobility components obtained using (4.72) for substrate orientations (001) and (110), respectively. For substrate orientation (001) the two in-plane components of the electron mobility are equal with the maximum mobility saturating at a value above 2400 cmVs at about 30% Ge content. For (110) orientation (Fig. 5.1b), the Monte Carlo simulation results demonstrate that the in-plane component ( ) of the electron mobility is equal to the perpendicular component (). This feature is also reproduced by the analytical model.
For the (123) oriented SiGe substrate three distinct components of the mobility can be seen in Fig. 5.2. The analytical model shows excellent agreement with the Monte Carlo simulation results for a large range of Ge content in the SiGe layer. The deviation for very large strain levels (Ge content 0.85) is due to the fact that the proposed model does not consider the population of the L valleys.
The dependence of the electron mobility in strained Si on the in-plane angle can be obtained by taking the projection of the mobility tensor in the direction of the in-plane vector .
The variation of the in-plane masses parallel ( ) and perpendicular ( ) to the stress, and the longitudinal () masses for the -valleys as a function of the strain as calculated using (3.73) to (3.75) is shown in Fig. 5.4b. The values of the parameters and were chosen as 7.0 eV and 0.53 eV, respectively. It can be seen that there is a significant change in for increasing strain along .
In Fig. 5.5 the anisotropy of the mobility is compared for and stress directions with and without the change in the effective masses. It can be clearly seen that cannot be neglected for uniaxial stress. For uniaxial compression, on the other hand, there is a negligible change in the effective masses of the lowered -valleys.
Fig. 5.9 shows the analytically calculated electron mobility components for uniaxial compressive and tensile stresses, respectively. Also shown for comparison are the Monte Carlo simulation results (symbols). Fig. 5.9a indicates that compressive stress increases the electron mobility in the plane. The mobility saturates at 6900 cm/Vs for stress values greater than GPa. This mobility improvement is nearly 2.8 times the strain enhanced bulk mobility in Si. On the contrary, applying uniaxial tensile stress along  results in larger out-of-plane mobilities, as shown in Fig. 5.9b.
The temperature dependence of the electron mobility is shown in Fig. 5.10a. The value of the parameter in (4.87) has been extracted from the Monte Carlo data and the values of and in (4.88) were chosen as 1.79 and , respectively. Fig. 5.10b shows a comparison of the mobility components in the plane as a function of compressive stress along at T = 200K, as obtained from Monte Carlo simulations and the analytical model. Very good agreement is observed.