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 [181]. To support these findings we reformulated (4.14) for the sum and the difference of $X_{1}$ and $X_{2}$. First we introduce the transformation rules for $y_{n}$ and $\bar{y}_{n}$ as,

$$y_{n} = \frac{X_{1}-X_{2}}{2} \quad\mathrm{and}\quad \bar{y}_{n} = \frac{X_{1}+X_{2}}{2}\quad\:$$ (4.17)

or

$$(y_{n}+\bar{y}_{n})^{2}= X_{1}^{2} \quad\mathrm{and}\quad (y_{n}-\bar{y}_{n})^{2}= X_{2}^{2}\enspace.$$ (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 $y_{n}$ and $\bar{y}_{n}$ leads to the following expression.

$$\sin(y_{n} k_{0} t) + \sin(\bar{y}_{n} k_{0} t)= \frac{c(X_{2})}{c(X_{1})} \cdot \left(-\sin(y_{n} k_{0} t) + \sin(\bar{y}_{n} k_{0} t)\right)$$ (4.19)

Further simplification steps result in:

$$\sin(y_{n} k_{0} t) = \frac{c(X_{2})-c(X_{1})}{c(X_{2})+c(X_{1})}\, \sin(\bar{y}_{n} k_{0} t)$$ (4.20)

Now we re-express $\bar{y}_{n}$ as function of $y_{n}$ (Appendix B.) resulting in

$$\bar{y}_{n}^{2} = \frac{1-y_{n}^{2}-\zeta^{2}}{1-y_{n}^{2}}\enspace.$$ (4.21)

The derivation of the fraction containing $c(X_{1})$ and $c(X_{2})$ can be found in Appendix B..

$$\sin(y_{n} k_{0} t)=\pm \frac{\zeta\, y_{n} \sin\left(\sqrt{\frac... ...^{2}}{1-y_{n}^{2}}} k_{0}t\right)}{\sqrt{(1-y_{n}^{2})(1-\zeta^{2}-y_{n}^{2})}}$$ (4.22)

For zero stress the ratio on the right hand side of (4.22) is equal to zero, and the standard quantization condition $q_{n}=\pi n/k_{0}t$ 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 $\zeta\neq0$ and non-parabolic bands. (4.22) is nonlinear and can be solved only numerically. However, for small $\zeta$ the solution can be thought in the form $y_{n}=q_{n}\pm\zeta$, where $\zeta$ is small. Substituting $y_{n}=q_{n}$ into the right-hand side of (4.22) and solving the equation with respect to $\zeta$, we obtain for the valley splitting:

$$\Delta E_{n}=4 \left(\frac{\pi n}{k_{0} t}\right)^{2} \frac{\Xi_{u'} \varepsilon_{xy}}{k_{0} t}\, \frac{\sin(k_{0} t)}{\vert 1-q_{n}^{2}\vert}$$ (4.23)

In accordance with earlier publications [182,183,184,185], the valley splitting is inversely proportional to the third power of $k_{0}$ and the third power of the film thickness $t$. 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 $\delta$ at the X-point, and not at the $\Gamma$-point. Since the parameter $\zeta$, which determines non-parabolicity, depends strongly on shear strain, the application of uniaxial [110] stress to [001] ultra-thin Si film generates a valley splitting proportional to strain.