- 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

(3.1) |

We can define the following scalar quantities.

(3.2) |

The subscripts and in refer to the plane and the force direction, respectively. Similarly considering slices orthogonal to the and -directions, we obtain

(3.3) |

and

(3.4) |

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

(3.5) |

The condition of static equilibrium implies .

(3.6) |

Assuming to be a small displacement, a Taylor expansion about the point gives the absolute squared distance as

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

(3.8) | |

(3.9) |

(3.10) |

Here, denote the components of the strain tensor and are defined as

Assuming that the second order term in (3.11) can be neglected and the resulting tensor is

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

(3.13) |

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

(3.14) |

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

Here, 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 , and , 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

and the generalized Hook law in matrix form as

Si | ||||||

Ge |

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, .

The stiffness and compliance tensors are linked through the relation . Using this relation, the three independent compliance coefficients can be calculated as

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 , while the compliance coefficients are denoted by .

- represents a direction
- denotes equivalent directions
- represents a plane with the normal vector
- denotes equivalent planes

Here denotes the polar and 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

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

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

(3.25) |

with . Since epitaxial growth does not produce any in-plane shear strain in the interface coordinate system, we have

(3.26) |

The other three independent strain components, ( ) 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, . Therefore, using Hook's law stated in (3.15) we have

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

(3.28) |

which can be expressed in matrix form as

The matrix elements can be determined from the elastic stiffness tensor through the relation,

(3.30) |

where denotes the transformation matrix in (3.19). Once the matrix elements are known, (3.29) can be inverted to determine the ( ). Having determined the strain tensor in the interface coordinate system, the tensor can be transformed to the principal coordinate systemusing

Substrate Orientation | |||

0 | |||

0 | 0 | ||

0 | 0 |

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.

S. Dhar: Analytical Mobility Modeling for Strained Silicon-Based Devices