6.2 Bulk Electron Mobility of Strained Si

In Figure 6.14 we compare simulation results for the electron mobility of strained Si obtained with FBMC and ABMC for the stress directions [100] and [110]. Mobility is plotted in three orthogonal directions, one being parallel and two being perpendicular to stress. Comparing simulation results from ABMC with FBMC allows: (i) a direct interpretation of the FBMC results, and (ii) the extraction of the limits of validity of the analytical band model.

Figure 6.14: Simulated bulk electron mobility of intrinsic Si as a function of strain for stress direction [100] (a) and [110] (b). Mobility from FBMC (solid lines) and ABMC (dotted lines) simulations is plotted along the stress direction and two orthogonal directions. Symbols indicate the change of mobility calculated using the piezoresistance coefficients [Smith54].

[a] [b]

In Figure 6.14a the simulation results from ABMC and FBMC for stress along [100] are compared and good agreement is observed. The resulting mobility is anisotropic in the (001) plane ( $\mu_{[100]}\neq \mu_{[010]})$, which is a result of the strain-induced $\Delta$-valley shifts. Mobility saturates at approximately 1% strain, regardless of the sign of strain. The saturated mobility values are larger for compressive strain, since for this type of strain four $\Delta$-valleys are being depopulated, while for tensile strain only two valleys are being depopulated. Thus, the larger amount of intervalley scattering and the larger transport mass of the two populated valley-pairs reduces the mobility enhancement for tensile stress.

In Figure 6.14b simulation results are shown for stress along [110]. At compressive stress (negative ${\ensuremath {\varepsilon _{xy}}}$), four valleys move down in energy, yielding a decreased mobility in the (001) plane and a mobility enhancement along [001]. However, if tensile stress is applied along [110], the mobilities along the three directions [110], $[1\bar{1}0]$, and [001] are different from each other with the largest mobility enhancement in [110] direction. Furthermore, no clear mobility saturation is observed as $\varepsilon_{xy}$ increases. The reason for the different mobility enhancement under [110] tensile stress compared to [100] stress is that for this particular stress direction the valley pair along [001] is primarily populated. As shown in Figure 6.10, this valley pair experiences a pronounced deformation as a function of shear strain $\varepsilon_{xy}$. In ABMC simulations this deformation was accounted for by using expressions (3.100), (3.94), and (3.99). It can be seen that the simulation results from ABMC qualitatively agree with those from FBMC up to 0.5% shear strain. At larger strain levels the band deformation is so pronounced that an energy band description in terms of an effective mass is no longer accurate, and FBMC have to be used even in the case of low-field transport to calculate the mobility. It is anticipated that a somewhat better agreement between ABMC and FBMC could be obtained, if the strain effect on the nonparabolicity coefficient $\alpha$ were included in the analytical band model.

Figure: (a) Constant-energy surfaces of the lowest Si conduction band under uniaxial tensile stress along [110] and projection on the (001) plane. (b) Simulated in-plane electron mobility for various stress levels. At 0.5 GPa simulation results are compared to the mobility calculated from the piezoresistance coefficients (open squares).

[a] [b]

Figure: (a) Constant energy surfaces of the lowest Si conduction band under uniaxial tensile stress along [001] and projection on the (110) plane. (b) Simulated in-plane electron mobility for various stress levels. At 0.2 GPa simulation results are compared to the mobility calculated from the piezoresistance coefficients (open squares).

[a] [b]

The simulated mobility enhancement for stress along [100] and [110] was compared with predictions from a model based on the linear piezoresistance coefficients [Smith54]. Good agreement is found for both stress directions at small stress ($<$ 0.2 GPa) where the model is valid (see Figure 6.14). Models solely based on strain-induced intervalley electron transfer [Herring56] fail to explain the origin of the non-vanishing shear piezoresistance coefficient of $\pi_{44}=$ -13.6e-11 Pa$^{-1}$ [Smith54]. Hence, these models are not capable of reproducing the anisotropy of electron mobility in uniaxially stressed channels with [110] channel direction [Maruyama90,Kanda91].

The anisotropy of the electron mobility in the (001)-plane arising from stress along the [110] direction is shown in Figure 6.15. It is a result of the stress-induced effective mass change only. In Figure 6.16 the anisotropic electron mobility resulting from stress along the [001] direction in the (110)-plane is plotted. The anisotropy of the electron mobility originates from a repopulation of the subband ladders induced by stress.

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