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## 4.1.2 Small Strain Values

The valley splitting was shown to be linear in strain for small shear strain values and to depend strongly on the film thickness . To support these findings we reformulated (4.14) for the sum and the difference of and . First we introduce the transformation rules for and as, (4.17)

or (4.18)

We only show the derivation for (4.14a), due to the similarity with (4.14b). Using the above given transformation and rewriting (4.14a) to separate and leads to the following expression. (4.19)

Further simplification steps result in: (4.20)

Now we re-express as function of (Appendix B.) resulting in (4.21)

The derivation of the fraction containing and can be found in Appendix B.. (4.22)

For zero stress the ratio on the right hand side of (4.22) is equal to zero, and the standard quantization condition is recovered. Due to the plus/minus sign in the right-hand side of (4.22), the equation splits into two non-equivalent branches for and non-parabolic bands. (4.22) is nonlinear and can be solved only numerically. However, for small the solution can be thought in the form , where is small. Substituting into the right-hand side of (4.22) and solving the equation with respect to , we obtain for the valley splitting: (4.23)

In accordance with earlier publications [182,183,184,185], the valley splitting is inversely proportional to the third power of and the third power of the film thickness . The value of the valley splitting oscillates with film thickness, in accordance with [183,184,185]. In contrast to previous works, the subband splitting is proportional to the gap at the X-point, and not at the -point. Since the parameter , which determines non-parabolicity, depends strongly on shear strain, the application of uniaxial  stress to  ultra-thin Si film generates a valley splitting proportional to strain.    Next: 4.1.3 High Values of Up: 4.1 Unprimed Subbands Previous: 4.1.1 Scaled Energy Dispersion

T. Windbacher: Engineering Gate Stacks for Field-Effect Transistors