3.1 Strain

First a basic set of expressions to describe strain in semiconductors is introduced [161,162,163]. The orthonormal vector set $\vec{x},\,\vec{y},\, \mathrm{and}\,\vec{z}$ is the basis of the unstrained solid. $\vec{x}',\,\vec{y}',\, \mathrm{and}\,\vec{z}'$ represent the distorted vectors under a uniform deformation,

$$\begin{array}{ccc} \vec{x}'&=&(1+\epsilon_{xx})\,\vec{x}+\eps... ..._{zy}\,\vec{y}+(1+\epsilon_{zz})\,\vec{z},\nonumber \end{array}$$ (3.1)

where $\epsilon_{i j}$ are the deformation coefficients of the system.

Assuming a uniform deformation, a point located at $\vec{r}=x\,\vec{x}+y\,\vec{y}+z\,\vec{z}$ will be shifted to $\vec{r}'=x\,\vec{x}'+y\,\vec{y}'+z\,\vec{z}'$, which leads to the displacement $\vec{R}$ defined as

$$\begin{array}{ccl} \vec{R}\equiv\vec{r}'-\vec{r}&=&x\,(\vec{x... ..._{zx}\,x+\epsilon_{zy}\,y+\epsilon_{zz}\,z)\vec{z}. \end{array}$$ (3.2)

Further generalization leads to a description for non-uniform deformation, introducing a position dependent vector function $\vec{u}(\vec{r})$,

$$\vec{R}(\vec{r})= u_{x}(\vec{r})\,\vec{x}+ u_{y}(\vec{r})\,\vec{y}+u_{z}(\vec{r})\,\vec{z}$$ (3.3)

Restricting to small displacements from $\vec{r}$, the displacement function $\vec{u}(\vec{r})$ can be developed into a Taylor series and truncated after the linear term at $\vec{R}\left(\vec{0}\right)=\vec{0}$, leading to a relation between the local displacement tensor and the displacement function,

$$\begin{array}{ccccc} \epsilon_{xx}&=&\frac{\partial u_{x}}{\p... ...silon_{zz}=\frac{\partial u_{z}}{\partial z} \end{array}\qquad.$$

Therefore, the displacement can be expressed as $\varepsilon_{ij}=\frac{\partial u_{j}}{\partial r_{i}}$. Frequently, the displacements $\varepsilon_{i j}$ are expressed via the strain tensor $\epsilon_{i j}$ to describe the deformation of a body in three dimensions. In the limit of small deformations, the strain tensor is known as the Green tensor or Cauchy's infinitesimal strain tensor,

$$\varepsilon_{ij}=\frac{\epsilon_{ij}+\epsilon_{ji}}{2}\quad.$$ (3.4)

The relative length change in the $\vec{x}_{i}$ direction is described by the diagonal coefficients $\varepsilon_{ii}$, while the off-diagonal elements $\varepsilon_{ij}\: (i\neq j)$ denote the angular distortions by shear strains.

Also very common are the engineering strains $e_{ij}$, which are linked with the strain tensor as follows:

$$\left(\begin{array}{ccc} e_{xx}&e_{xy}&e_{xz}\\ e_{yx}&e_{yy}&e_{... ...2\varepsilon_{zx}&2\varepsilon_{zy}&\varepsilon_{zz}\\ \end{array}\right)\quad.$$ (3.5)

The Voigt notation uses the six indipendent components of the strain tensor in a more compact vector form

$$\left(\varepsilon_{xx}, \varepsilon_{yy},\varepsilon_{zz},\gamma_... ...\gamma_{xy}\right)=\left(e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}\right) \quad.$$ (3.6)