Subsections

• 3.1.1 Stress
• 3.1.2 Strain
• 3.1.3 Stress-Strain Dependence
• 3.1.4 Miller Indices
• 3.1.5 Coordinate Transformation
• 3.1.6 Strain from Epitaxy

3.1 Theory of Elasticity

The property of solid materials to deform under the application of an external force and to regain their original shape after the force is removed is referred to as its elasticity. The external force applied on a specified area is known as stress, while the amount of deformation is called the strain. In this section, the theory of stress, strain and their interdependence is briefly discussed.

3.1.1 Stress

Imagine an object oriented in the cartesian coordinate system with a number of forces acting on it, such that the vector sum of all the forces is zero. Take a slice orthogonal to the $\mathrm{x}$-direction and define a small area on this slice as $\Delta A_x$. Let the total force acting on this small area be

$$\Delta {\mathbf{F}} = \Delta F_x \cdot {\mathbf{\hat{i}}} + \Delta F_y \cdot{\mathbf{\hat{j}}} + \Delta F_z \cdot{\mathbf{\hat{k}}}.$$ (3.1)

We can define the following scalar quantities.

$$T_{xx} = \lim_{\Delta {A_x} \rightarrow 0 } \frac{\Delta F_x}{\De... ...3mm} T_{xz} = \lim_{\Delta {A_x} \rightarrow 0 } \frac{\Delta F_z}{\Delta A_x}.$$ (3.2)

The subscripts $i$ and $j$ in $T_{ij}$ refer to the plane and the force direction, respectively. Similarly considering slices orthogonal to the $\mathrm{y}$ and $\mathrm{z}$-directions, we obtain

$$T_{yx} = \lim_{\Delta {A_y} \rightarrow 0 } \frac{\Delta F_x}{\De... ...{3mm} T_{yz} = \lim_{\Delta {A_y} \rightarrow 0 } \frac{\Delta F_z}{\Delta A_y}$$ (3.3)

and

$$T_{zx} = \lim_{\Delta {A_z} \rightarrow 0 } \frac{\Delta F_x}{\De... ...3mm} T_{zz} = \lim_{\Delta {A_z} \rightarrow 0 } \frac{\Delta F_z}{\Delta A_z}.$$ (3.4)

The scalar quantities can be arranged in a matrix form to yield the stress tensor

$$\ensuremath{{\underline{T}}} = \begin{pmatrix}\displaystyle T_{xx... ...& T_{xz} T_{yx} & T_{yy} & T_{yz} T_{zx} & T_{zy} & T_{zz} \end{pmatrix}.$$ (3.5)

The condition of static equilibrium implies $T_{ij} = T_{ji}$.

Figure 3.1: Components of the stress tensor.

3.1.2 Strain

A body under elastic deformation experiences an internal restoring force. The amount of deformation caused is called strain and the corresponding force, the stress. Consider a pair of points at locations ${\mathbf{x}}$ and ${\mathbf{x+dx}}$ which are deformed to locations ${\mathbf{x+u(x)}}$ and ${\mathbf{x+dx + u(x+dx)}}$. The absolute squared distance between the deformed points can be written as

$$\sum_i { \left({dx_i + u_i({\mathbf{x}}+{\mathbf{dx}})-u_i({\mathbf{x}})} \right)^2}.$$ (3.6)

Assuming ${\mathbf{dx}}$ to be a small displacement, a Taylor expansion about the point ${\mathbf{x}}$ gives the absolute squared distance as

$$\sum_i { \left({dx_i} + \sum_j{ \frac{\partial u_i}{\partial x_j ... ...{\frac{\partial u_i}{\partial x_j }dx_j\frac{\partial u_i}{\partial x_k} dx_k}.$$ (3.7)

Since the first term in (3.7) denotes the original squared distance between the points, the change in the squared distance becomes

$$D({\mathbf{dx}}) = \sum_{i,j} {dx_i \left(\frac{\partial u_i}{\pa... ...{\frac{\partial u_k}{\partial x_i }\frac{\partial u_k}{\partial x_j }dx_idx_j }$$ (3.8)

$$= \sum_{i,j} dx_i\left[ { \left(\frac{\partial u_i}{\partial x_j}... ...\frac{\partial u_k}{\partial x_i }\frac{\partial u_k}{\partial x_j}}\right]dx_j$$ (3.9)

$$= 2 \sum_{i,j} dx_i \varepsilon _{ij} dx_j.$$ (3.10)

Here, $\varepsilon _{ij}$ denote the components of the strain tensor and are defined as

$$\varepsilon_{ij} = \frac{1}{2}\left[\frac{\partial u_i}{\partial ... ... \frac{\partial u_k}{\partial x_i }\frac{\partial u_k}{\partial x_j} } \right].$$ (3.11)

Assuming that $$\frac{\partial u_k}{\partial x_i } \ll 1$$ the second order term in (3.11) can be neglected and the resulting tensor is

$$\varepsilon_{ij} = \frac{1}{2}\left[\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right].$$ (3.12)

Note that in literature, engineering shear strain components, $\gamma_{ij}$, are commonly used rather than the shear strain components described by (3.12). The relation is,

$$\gamma_{ij} = \varepsilon_{ij} + \varepsilon_{ji} = 2 \varepsilon_{ij}.$$ (3.13)

Arranging the strain components (3.12) in matrix form gives the strain tensor

$$\ensuremath{{\underline{\varepsilon }}} = \begin{pmatrix}\display... ...on _{zx} & \varepsilon _{zy} & \varepsilon _{zz} \end{pmatrix} \hspace*{1cm}$$ (3.14)

The sign convention adopted for stress is that tensile stress causes an expansion, whereas compressive stress causes a contraction.

3.1.3 Stress-Strain Dependence

The relation between stress and strain was first identified by Robert Hook [Hook78]. Hook's law of elasticity is an approximation which states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). The most general relationship between stress and strain can be mathematically written as

$$T_{ij} = C_{ijkl}\varepsilon_{ij}.$$ (3.15)

Here, $C_{ijkl}$ is a fourth order elastic stiffness tensor comprising 81 coefficients. However, depending on the symmetry of the crystal the number of coefficients can be reduced. For cubic crystals such as Si and Ge only three unique coefficients $c_{11}$, $c_{12}$ and $c_{44}$, exist. These coefficients are known as the stiffness constants. To simplify the notations, the stress and strain tensor can be written as vectors using the contracted notations

$$[Q_{xx},Q_{yy},Q_{zz},Q_{yz},Q_{xz},Q_{xy}] = [Q_1,Q_2,Q_3,Q_4,Q_... ...e{Q}}} = \ensuremath{{\underline{T}}}, \ensuremath{{\underline{\varepsilon }}},$$ (3.16)

and the generalized Hook law in matrix form as

$$\begin{pmatrix}T_1 T_2 T_3 T_4 T_5 T_6 \end{pma... ...silon_3 2\varepsilon_4 2\varepsilon_5 2\varepsilon_6 \end{pmatrix}.$$ (3.17)

	$c_{11}$	$c_{12}$	$c_{44}$	$s_{11}$	$s_{12}$	$s_{44}$
Si	$166.0$	$64.0$	$79.6$	$-2.13$	$7.67$	$12.6$
Ge	$126.0$	$44.0$	$67.7$	$-2.50$	$9.69$	$14.8$

Table 3.1: Elastic stiffness constants $c_{ij}$ in GPa [Levinstein99] and elastic compliance constants $s_{ij}$ in $10^{-12}$ m$^2$/N.

Of practical interest is the strain arising from a certain stress condition. The strain components can be obtained by inverting Hook's law and utilizing the compliance coefficients, $S_{ijkl}$.

$$\varepsilon_{ij} = S_{ijkl}T_{ij}.$$ (3.18)

The stiffness and compliance tensors are linked through the relation $\ensuremath{{\underline{S}}} = \ensuremath{{\underline{C}}}^{-1}$. Using this relation, the three independent compliance coefficients can be calculated as

$$s_{11} = \frac{c_{11}+c_{12}}{c_{11}^2 + c_{11} c_{12} - 2 c_{12}^2},$$

$$s_{12} = -\frac{c_{12}}{c_{11}^2 + c_{11} c_{12} - 2 c_{12}^2},$$

$$s_{44} = \frac{1}{c_{44}}.$$

The compliance coefficients for Si and Ge, together with the stiffness coefficients are listed in Table 3.1. It is interesting to note that traditionally the stiffness coefficients are denoted by $c_{ij}$, while the compliance coefficients are denoted by $s_{ij}$.

3.1.4 Miller Indices

Figure 3.2: Planes in the cubic system with the Miller indices marked.

The Miller indices, denoted as $h$, $k$ and $l$, are a symbolic vector representation for the orientation of atomic planes and directions in a crystal lattice. Defining three lattice vectors forming the lattice axes, any crystal plane would intersect the axes at three distinct points. The Miller indices are obtained by taking the reciprocal of the intercepted values. By convention, negative indices are written with a bar over the indices. In Fig. 3.2, three planes in the cubic system, along with their Miller indices are shown, where the following nomenclature is adopted [Davies98].

• $[hkl]$ represents a direction
• $\langle hkl \rangle\!$ denotes equivalent directions
• $(hkl)$ represents a plane with the normal vector $[hkl]$
• $\{ hkl \}$ denotes equivalent planes

3.1.5 Coordinate Transformation

Figure 3.3: Stress direction $[x',y',z']$ relative to the crystallographic coordinate system $[x,y,z]$.

It is often required to know the stress in the crystallographic coordinate system for a stress applied along a general direction. Consider a generalized direction $[x',y',z']$ in which the stress is applied. The stress in the crystallographic coordinate system $[x,y,z]$ can be calculated using the transformation matrix ${\underline{U}}$.

$$\ensuremath{{\underline{U}}}(\theta,\phi) = \begin{pmatrix}\cos \... ... 0 \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta \end{pmatrix}$$ (3.19)

Here $\theta$ denotes the polar and $\phi$ the azimuthal angle of the stress direction relative to the crystallographic coordinate system, as shown in Fig. 3.3. The stress in the crystallographic coordinate system is then given by

$$\ensuremath{{\underline{T}}}^{\ensuremath{{\mathrm{crys}}}} = \en... ...e{U}}} \cdot \ensuremath{{\underline{T}}} \cdot \ensuremath{{\underline{U}}}^T.$$ (3.20)

Applying a non-zero stress of magnitude ${\mathrm{P}}$ applied along the [100], [110] and [111] directions, the stress tensors in the principal coordinate system read, respectively

$$\ensuremath{{\underline{T}}}_{[100]} = \begin{pmatrix}P & 0 & 0 \... ...pmatrix}P/3 & P/3 & P/3 P/3 & P/3 & P/3 P/3 & P/3 & P/3 \end{pmatrix}.$$ (3.21)

From (3.18), the corresponding strain tensors can be determined.

$$\ensuremath{{\underline{\varepsilon}}}_{[100]} = \begin{pmatrix}s... ...\cdot P & 0 & 0 0 & s_{12}\cdot P & 0 0 & 0 & s_{12}\cdot P \end{pmatrix}$$ (3.22)

$$\ensuremath{{\underline{\varepsilon}}}_{[110]} = \begin{pmatrix}(... ...dot P/4 & (s_{11}+s_{12})\cdot P/2 & 0 0 & 0 & s_{12} \cdot P, \end{pmatrix}$$ (3.23)

$$\ensuremath{{\underline{\varepsilon}}}_{[111]} = \begin{pmatrix}(... ... s_{44} \cdot P/6 & s_{44} \cdot P/6 & (s_{11}+2s_{12}) \cdot P/3 \end{pmatrix}$$ (3.24)

3.1.6 Strain from Epitaxy

For the case of epitaxially grown Si on SiGe, the in-plane strain $\varepsilon _{\parallel}$ is given by (2.1). Consider an interface (primed) coordinate system, the $\ensuremath{{\mathrm{z}}}$-axis of which is perpendicular to the Si/SiGe interface. The strain tensor in this coordinate system can be written as

$$\ensuremath{{\underline{\varepsilon '}}} = \begin{pmatrix}\displa... ..._{31}' & \varepsilon _{32}' & \varepsilon _{33}' \end{pmatrix} \hspace*{1cm}$$ (3.25)

with $\varepsilon _{11}'=\varepsilon _{22}' = \varepsilon _{\parallel}$. Since epitaxial growth does not produce any in-plane shear strain in the interface coordinate system, we have

$$\varepsilon '_{12}= \varepsilon '_{21} = 0.$$ (3.26)

The other three independent strain components, ( $\varepsilon '_{13},\varepsilon '_{23},\varepsilon '_{33}$) can be determined as described below [Hinckley90].

Since the strain is applied uniformly to the Si layer, all external stress components in the vertical direction vanish, $T_{13} = T_{23} = T_{33} = 0$. Therefore, using Hook's law stated in (3.15) we have

$$C_{\alpha\beta ij}'\varepsilon '{ij} = 0 \hspace*{2cm} (\alpha,\beta)=(1,3),(2,3),(3,3).$$ (3.27)

where summation over repeated indices is implied. Expanding (3.27) gives

$$C_{\alpha\beta 33}'\varepsilon '_{33} + 2C_{\alpha\beta 23}'\vare... ...'_{31} = - (C_{\alpha\beta 11}' + C_{\alpha\beta 22}')\varepsilon '_{\parallel}$$ (3.28)

which can be expressed in matrix form as

$$\begin{pmatrix}\displaystyle C_{3333}' & C_{3323}' & C_{3331}' \\... ...le C_{2311}' + C_{2322}' \displaystyle C_{3111}' + C_{3122}'. \end{pmatrix}.$$ (3.29)

The $C_{\gamma\delta kl}'$ matrix elements can be determined from the elastic stiffness tensor $C_{\gamma\delta kl}$ through the relation,

$$C_{\gamma\delta kl}' = U_{\alpha\gamma} U_{\beta\delta} U_{ik} U_{jl} C_{\alpha\beta ij}.$$ (3.30)

where $\ensuremath{{\underline{U}}}$ denotes the transformation matrix in (3.19). Once the matrix elements are known, (3.29) can be inverted to determine the ( $\varepsilon '_{13},\varepsilon '_{23},\varepsilon '_{33}$). Having determined the strain tensor in the interface coordinate system, the tensor can be transformed to the principal coordinate systemusing

$$\varepsilon _{\alpha\beta} = U_{\alpha i} U_{\beta j}\varepsilon _{ij}.$$ (3.31)

Substrate Orientation
	$[100]$	$[110]$	$[111]$
$\varepsilon _{11}$	$$\varepsilon _{\parallel}$$	$\varepsilon _{\parallel}\displaystyle \frac{2c_{44}-c_{12}}{c_{11}+c_{12}+2c_{44}}$	$\varepsilon _{\parallel}\displaystyle \frac{4c_{44}}{c_{11}+2c_{12}+4c_{44}}$
$\varepsilon _{22}$	$$\varepsilon _{\parallel}$$	$\varepsilon _{\parallel}\displaystyle \frac{2c_{44}-c_{12}}{c_{11}+c_{12}+2c_{44}}$	$\varepsilon _{\parallel}\displaystyle \frac{4c_{44}}{c_{11}+2c_{12}+4c_{44}}$
$\varepsilon _{33}$	$$\frac{-2c_{12}}{c_{11}}\varepsilon _{\parallel}$$	$\varepsilon _{\parallel}$	$\varepsilon _{\parallel}\displaystyle \frac{4c_{44}}{c_{11}+2c_{12}+4c_{44}}$
$\varepsilon _{12}$	0	$-\varepsilon _{\parallel}\displaystyle \frac{c_{11}+2c_{12}}{c_{11}+c_{12}+2c_{44}}$	$-\varepsilon _{\parallel}\displaystyle \frac{c_{11}+2c_{12}}{c_{11}+2c_{12}+4c_{44}}$
$\varepsilon _{13}$	0	0	$-\varepsilon _{\parallel}\displaystyle \frac{c_{11}+2c_{12}}{c_{11}+2c_{12}+4c_{44}}$
$\varepsilon _{23}$	0	0	$-\varepsilon _{\parallel}\displaystyle \frac{c_{11}+2c_{12}}{c_{11}+2c_{12}+4c_{44}}$

Table 3.2: Components of the strain tensor for different orientations of the SiGe substrate.

Equations (3.29) and (3.31) can be solved to obtain the strain tensor for epitaxial growth of Si on an arbitrarily oriented SiGe substrate. Table 3.2 lists the expressions for the strain tensor components for the high-symmetry (001), (111) and (110) oriented SiGe substrates.