3.3.1 Strain Tensor in the Interface Coordinate System

The energy splitting and the hydrostatic shift of the mean energy depend on the orientation of the applied forces.

The interface coordinate system is specified as a system with its $z$-axis perpendicular to the hetero-interface. The form of the strain tensor $\hat{\boldsymbol{\varepsilon}}^{'}$ in this coordinate system can be found as follows.

The condition of biaxial dilatation or contraction gives:

$$\varepsilon^{'}_{11}=\varepsilon^{'}_{22}=\varepsilon_{\parallel},$$ (3.32)

where $\varepsilon_{\parallel}$ is the in-plane strain given as the relative lattice mismatch:

$$\varepsilon_{\parallel}=\frac{a_{s}-a_{l}}{a_{l}}.$$ (3.33)

Here $a_{l}$ is the lattice constant of the layer and $a_{s}$ that of the substrate. The substrate is assumed to be thick enough to remain unstrained. Further, the condition of vanishing in-plane shear implies:

$$\varepsilon^{'}_{12}=0.$$ (3.34)

It is also assumed that there is no any film distortion which means the following conditions:

$$\varepsilon^{'}_{13}=\varepsilon^{'}_{23}=0.$$ (3.35)

This is justified for the case of substrates with high rotational symmetry. In other cases it is relatively weak for SiGe structures.

Thus under these conditions the strain tensor for the SiGe active layer is diagonal in the interface coordinate system. The two diagonal elements are known to be equal to $\varepsilon_{\parallel}$. To determine the third diagonal element Hooke's law is applied. It linearly relates the components of the stress and the strain tensors $\sigma_{ik}$ and $\varepsilon_{jl}$:

$$\sigma^{'}_{\alpha\beta}=c^{'}_{\alpha\beta ij}\varepsilon^{'}_{ij},$$ (3.36)

where $c_{ijkl}$ is a tensor of rank four called the elastic stiffness tensor. As the only external stress is in-plane, the out-of-plane component vanishes

$$\sigma^{'}_{33}=0$$ (3.37)

which gives for the third diagonal component of the strain tensor:

$$\varepsilon^{'}_{33}=-\frac{c^{'}_{3311}+c^{'}_{3322}}{c^{'}_{3333}}\varepsilon_{\parallel}.$$ (3.38)

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