3.3.2 Strain Resulting from Uniaxial Stress

This section describes the calculation of the strain tensor resulting from a uniaxial stress of magnitude $P$ along an arbitrary direction.

Analysis begins by adopting a coordinate system $(x',y',z')$ in which the $x'$ axis is parallel to the stress direction. This system is related to the coordinate system $(x,y,z)$ of the primary crystallographic axes of the semiconductor by a rotation $\ensuremath{{\underaccent{\bar}{U}}}$

$$\ensuremath{{\underaccent{\bar}{U}}}(\phi) = \begin{pmatrix}\cos ... ...\ \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta \\ \end{pmatrix},$$ (3.15)

where $\theta$ and $\phi$ are the polar and azimuthal angles of the stress direction relative to the crystallographic coordinate system. In the primed coordinate system, the stress tensor has only one non-zero component, ${\sigma_{xx}}' = P$. The stress tensor in the crystallographic system can be calculated from

$$\sigma_{ij} = U_{\alpha i} U_{\beta j} \sigma'_{\alpha \beta}\ .$$ (3.16)

If uniaxial stress is applied along one of the directions [100], [110], [111], and [120], the related stress tensors in the principal system become:

$$\ensuremath{{\underaccent{\bar}{\sigma}}}_{[100]} = \begin{pmatrix}P & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}\quad \ensuremath{{\underaccent{\bar}{\sigma}}}_{[110]} = \begin{pmatrix}P/2 & P/2 & 0 \\ P/2 & P/2 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$$

$$\ensuremath{{\underaccent{\bar}{\sigma}}}_{[111]} = \begin{pmatrix}P/3 & P/3 & P/3 \\ P/3 & P/3 & P/3 \\ P/3 & P/3 & P/3 \\ \end{pmatrix}\quad \ensuremath{{\underaccent{\bar}{\sigma}}}_{[120]} = \begin{pmatrix}P/5 & 2P/5 & 0 \\ 2P/5 & 4P/5 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}\quad$$ (3.17)

By inserting 3.17 in 3.14 the strain tensors can be calculated:

$$\ensuremath{{\underaccent{\bar}{\varepsilon}}}_{[100]} = P \begin{pmatrix}s_{11} & 0 & 0 \\ 0 & s_{12} & 0 \\ 0 & 0 & s_... .../2 & 0 \\ s_{44}/2 & s_{11}\!+\!s_{12} & 0 \\ 0 & 0 & 2 s_{12} \\ \end{pmatrix}$$

$$\ensuremath{{\underaccent{\bar}{\varepsilon}}}_{[111]} = \frac{P}{3} \begin{pmatrix}s_{11}\!+\!2s_{12} & s_{44}/2 & s_{4... ...& 0 \\ s_{44} & s_{12}\!+\!4s_{11} & 0 \\ 0 & 0 & 5 s_{12}\\ \end{pmatrix}\quad$$ (3.18)

In the same manner the strain tensor resulting from a uniaxial stress in general directions $[hk\ell]$ can be obtained by applying the proper coordinate transformation (3.15).

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