3.3.3 Strain Resulting from Epitaxy

In techniques using global strain, biaxial strain is induced in the Si layer by epitaxial growth on a relaxed virtual substrate with different lattice constant. The strain tensor depends on the mismatch of the lattice constants and on the substrate orientation. The strain in the plane of the hetero-interface can be determined from the lattice mismatch

$${\ensuremath{\varepsilon_{\vert\vert}}} = \frac{a_s-a_0}{a_0}\ ,$$ (3.19)

where $a_0 = 5.431$Åis the lattice constant of Si [Levinshtein99] and $a_l$ that of the substrate layer.

The strain tensor for arbitrary substrate orientations can be calculated according to [Hinckley90] and is given here for some frequently used substrate orientations:

$$\ensuremath{{\underaccent{\bar}{\varepsilon}}}_{(001)} = {\ensuremath{\varepsilon_{\vert\vert}}} \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \displaystyle -\frac{2 c_{12}}{c_{11}} \\ \end{pmatrix}$$

$$\ensuremath{{\underaccent{\bar}{\varepsilon}}}_{(110)} = {\ensuremath{\varepsilon_{\vert\vert}}} \begin{pmatrix}\display... ... \frac{2c_{44}-c_{12}}{c_{11}+c_{12}+2c_{44}} & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$

$$\ensuremath{{\underaccent{\bar}{\varepsilon}}}_{(111)} = {\ensuremath{\varepsilon_{\vert\vert}}} \begin{pmatrix}\display... ...c_{44}} & \displaystyle \frac{4c_{44}}{c_{11}+2c_{12}+4c_{44}} \\ \end{pmatrix}$$ (3.20)

Here, the elastic stiffness constants, given in Table 3.1, were used. The strain tensor contains shear components when the Si layer is grown on a (110) and (111) oriented substrate. These components can take large values. For example, the magnitude of the shear component $\varepsilon_{xy}$ is larger that $3 \%$, when growing Si on (110) oriented Ge.

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