Physical Modeling of Electron Mobility Enhancement for Arbitrarily Strained Silicon

The band structure (BS) of Si with arbitrary stress/strain conditions has been calculated using the empirical non-local pseudopotential method (EPM). It is shown that the change of the effective masses cannot be neglected for general stress conditions and how this effect together with the strain induced splitting of the conduction bands can be used to optimize electron mobility enhancement Δμn. The new findings have been incorporated into an existing low-field mobility model. BAND STRUCTURE CALCULATIONS Strain effects on the electronic BS of Si are often described using deformation potential theory, which allows the determination of the strain induced splitting of the conduction bands. However, experiments [1] have shown that the mobility enhancement Δμn cannot solely be attributed to this effect, and a recent study has shown that a stress along the 〈110〉 direction leads to a change of the effective mass Δm∗ [2]. In this work, the effect of general strain conditions on the BS is studied by means of EPM calculations. The number of symmetry elements P (Γ) at the center of the Brillouin zone of the strained lattice determines the volume of the irreducible wedge via Ωirred = ΩBZ/P (Γ). For stress along 〈100〉, 〈111〉, and 〈110〉, as shown in Fig. 1, P (Γ) is 16, 12, and 8, respectively, while for stress along other directions the lattice is invariant only to inversion, thus P (Γ) = 2. MOBILITY ENHANCEMENT Bulk mobility was calculated by means of Monte Carlo simulations [3]. In Fig. 2 the strain induced valley splitting is shown for biaxially strained and uniaxially stressed Si (along 〈110〉 and 〈120〉). It can be seen that biaxial tension is more effective in splitting the conduction bands than uniaxial tension in 〈110〉 and 〈120〉. Note that for 〈120〉 stress the conduction bands split into three two-fold degenerate pairs. The in-plane effective masses of the lowest valley were extracted from EPM calculations. Fig. 3 shows how uniaxial tensile stress along 〈110〉 yields a pronouncedΔmt, whereas the effect is smaller for stress along 〈120〉 stress and negligible for biaxial tensile strain. The direction of stress in turn leads to a pronounced anisotropy of the mobility in the transport plane. In Fig. 4 the anisotropy of Δμn is compared for different stress directions. It can be clearly seen that Δmt cannot be neglected for 〈110〉 uniaxial stress. The two beneficial effects on the mobility arising from Δmt and the valley splitting can be combined to yield the highest mobility enhancement in a system with in plane biaxial tension and uniaxial stress along 〈110〉. In Fig. 5 the in-plane mobilities parallel and perpendicular to [110] are shown. The physically based low-field mobility model in [4] was extended to take into account the stress induced effective mass change of the lowest Δ2 valley. For 〈110〉 stress the effective mass tensor becomes non-diagonal, with mc/2(m t,|| +m −1 t,⊥) in the diagonal and mc/2(m t,|| − m −1 t,⊥) in the offdiagonal. Good agreement between Monte Carlo simulation and the analytical model is observed (Fig. 5). This work has been partly supported by the Austrian Science Fund (FWF), project 17285-N02. REFERENCES [1] K. Uchida et al., IEDM, pp. 229–232 (2004). [2] K. Uchida et al., IEDM, pp. 135–138 (2005). [3] Institute for Microelectronics, VMC 1.0 User’s Guide, TU Wien, http://www.iue.tuwien.ac.at/software/vmc (2004). [4] S. Dhar et al., TED, vol. 52, pp. 527–533 (2005). Fig. 1. Irreducible wedge for stress applied in [100], [111], and [110] direction. 0 0.2 0.4 0.6 0.8 1 strain [%] 0 50 100 150 200 en er gy [m eV ] biaxial strain uniaxial <110> stress uniaxial <120> stress