3.3.3 Strain Tensor Elements in the Principle Coordinate System

Due to the cubic symmetry of Si and Ge there are only three non-zero components of the elastic stiffness tensor, namely $c_{11}$, $c_{12}$ and $c_{44}$ in the short-hand notation [50]. This fact allows to significantly simplify the calculations of $\varepsilon_{33}^{'}$ which are given below for the three substrate orientations $[001]$, $[110]$ and $[111]$. The calculations for an arbitrary substrate orientation are performed in the same manner. For these three substrate orientations the transformation operator takes the form:

$$\hat{U}_{(001)}=\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix},$$ (3.49)

$$\hat{U}_{(110)}=\begin{pmatrix}0 & -\frac{1}{\sqrt{2}} & \frac{1}... ...rt{2}}\\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ -1 & 0 & 0\end{pmatrix},$$ (3.50)

$$\hat{U}_{(111)}=\begin{pmatrix}\frac{1}{\sqrt{6}} & -\frac{1}{\sq... ... \frac{1}{\sqrt{3}}\\ \sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}}\end{pmatrix}.$$ (3.51)

Using (3.48) and (3.38) one obtains:

$$\varepsilon^{'(001)}_{33}=-\frac{2c_{12}}{c_{11}}\varepsilon_{\parallel},$$ (3.52)

$$\varepsilon^{'(110)}_{33}=-\frac{c_{11}+3c_{12}-2c_{44}}{c_{11}+c_{12}+2c_{44}}\varepsilon_{\parallel},$$ (3.53)

$$\varepsilon^{'(111)}_{33}=-\frac{2c_{11}+4c_{12}-4c_{44}}{c_{11}+2c_{12}+4c_{44}}\varepsilon_{\parallel}.$$ (3.54)

Now the transformation of the strain tensor according to (3.47) gives for the elements of the strain tensor in the principle coordinate system the following expressions.

$[001]$:

$$\varepsilon^{(001)}_{11}=\varepsilon^{(001)}_{22}=\varepsilon_{\parallel},$$

$$\varepsilon^{(001)}_{33}=-\frac{2c_{12}}{c_{11}}\varepsilon_{\parallel},$$ (3.55)

$$\varepsilon^{(001)}_{12}=\varepsilon^{(001)}_{13}=\varepsilon^{001}_{23}=0.$$

$[110]$:

$$\varepsilon^{(110)}_{11}=\varepsilon^{(110)}_{22}=\frac{2c_{44}-c_{12}}{c_{11}+c_{12}+2c_{44}}\varepsilon_{\parallel},$$

$$\varepsilon^{(110)}_{33}=\varepsilon_{\parallel},$$ (3.56)

$$\varepsilon^{(110)}_{12}=-\frac{c_{11}+2c_{12}}{c_{11}+c_{12}+2c_{44}}\varepsilon_{\parallel},$$

$$\varepsilon^{(110)}_{13}=\varepsilon^{(110)}_{23}=0.$$

$[111]$:

$$\varepsilon^{(111)}_{11}=\varepsilon^{(111)}_{22}=\varepsilon^{(111)}_{33}=\frac{4c_{44}}{c_{11}+2c_{12}+4c_{44}}\varepsilon_{\parallel},$$

$$\varepsilon^{(111)}_{12}=\varepsilon^{(111)}_{13}=\varepsilon^{(111)}_{23}=-\frac{c_{11}+2c_{12}}{c_{11}+2c_{12}+4c_{44}}\varepsilon_{\parallel}.$$ (3.57)

S. Smirnov: