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Within the elastic limit of a material Hooke's law is a good approximation for relating stress and strain
The generalization of Hook's law for three-dimensional elastic bodies leads to
![$\displaystyle \sigma_{ij}=C_{ijkl}\,\varepsilon_{kl}\quad.$](img357.png) |
(3.10) |
The elastic stiffness tensor
is of fourth order and contains
(
) elements. Introducing additional symmetry considerations, the number of needed components can be reduced [162]. Cubic semiconductors like
,
or
are characterized by only three constants (
).
![$\displaystyle \left(\begin{array}{c} \sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz} \\...
...\ \varepsilon_{zz}\\ \gamma_{yz}\\ \gamma_{xz}\\ \gamma_{xy} \end{array}\right)$](img362.png) |
(3.11) |
If the strain, instead of stress, is the quantity of interest, it can be calculated by inversion of the elastic stiffness tensor (3.10)
![$\displaystyle \varepsilon_{ij}=S_{ijkl}\, \sigma_{kl}\quad,$](img363.png) |
(3.12) |
or in matrix form
![$\displaystyle \left(\begin{array}{c} \varepsilon_{xx}\\ \varepsilon_{yy}\\ \var...
...y}\\ \sigma_{zz} \\ \sigma_{yz}\\ \sigma_{xz} \\ \sigma_{xy} \end{array}\right)$](img364.png) |
(3.13) |
The stiffness constants are normally denoted as
, while the compliance constants are named as
. The compliance constants can be calculated from the stiffness constants with the following relations:
![\begin{displaymath}\begin{array}{cc} s_{11}=&\frac{c_{11}+c_{12}}{c_{11}^{2}+c_{...
...\quad,\mathrm{and}\\ s_{44}=&\frac{1}{c_{44}}\quad. \end{array}\end{displaymath}](img367.png) |
(3.14) |
Next: 3.4 Miller Index Notation
Up: 3. Strain and Semiconductor
Previous: 3.2 Stress
T. Windbacher: Engineering Gate Stacks for Field-Effect Transistors