Single crystal silicon is aniostropic. The crystalline
directions of interest include the <100>, the <110>, and the
<111> crystal directions. Material properties in these crystalline
directions can be calculated from basic crystal properties, and results
of this analysis are shown in Appendix A. To simplify the initial design
process I assume that the silicon crystal can be considered isotropic.
Following the example of Spiering et al I choose a Young's Modulus of 150GPa,
and a Poisson's ratio of 0.17 for all calculations. It is the opinion of
these authors that these isotropic values best reflect the aniostropic
behavior of silicon in the <100> plane.

Young's Modulus |
150 GPa |

Poisson's Ratio |
0.17 |

Density |
2330 kg/m |

Thermal expansion coefficient of silicon.

The following table is extracted from data from Milek.

Temperature (K) |
100 |
200 |
400 |
1000 |

Linear Coefficient of Thermal Expansion
(10 |
-0.5 |
1.1 |
2.7 |
4.7 |

Fracture Strength of Silicon

Since silicon used is single crystal it is assumed for
all intents and purposes that the material does not yield until fracture
occurs. I assume that the design failure stress should be the fracture
strength of silicon. The fracture strength of silicon is given by Petersen
as being 7000 MPa. This extremely high failure stress is contradicted by
experience with anisotropically etched diaphragms where failures stresses
are estimated to be in the order of 300 MPa. Sooriakumar tracked this discrepancy
to the sharp corners introduced by aniostropic etching. Analysis of his
data shows stress concentration factors of up to 33 at the sharp corners
in aniostropically etched specimens. Rounding of the corners by isotropic
etching reduced stress concentration and increased failure load for the
specimens.

It is assumed in this design process that the fracture
stress of silicon is 7000 MPa, with stress concentration factors of 33
possible at sharp corners produced by aniostropic etching.

Fracture Toughness

Silicon is a brittle material. Failure usually occurs
along <111> cleavage planes. Analysis of failure in silicon can be helped
by the use of fracture mechanics models. Using these models requires knowing
the fracture toughness for the materials involved.

K_{1c} fracture toughness values are given for
different crystal directions

Silicon Direction |
K_{1c} (MPa m^{1/2 )} |

<111> |
0.83 to 0.95 |

<100> |
0.91 |

<110> |
0.94 |

Polycrystalline Silicon |
0.94 |

- Spiering, V.L., Bouwstra, S., Spiering, R.,
On chip decoupling zone for package-stress reduction.
*Sensors and Actuators*, A.39, 1993, 149-156. - Petersen, K.E., Silicon as a mechanical material,
*Proc. IEEE.*, Vol. 70, No. 5, 1982 - Sooriakumar, Chan, Savage and Fugate, A comparative study
of wet vs. dry isotropic etch to strengthen silicon micro-machined pressure
sensor,
*Electrochemical Soc. Proc.*, Vol. 95-27. - Ericson, F, et al., Hardness and fracture toughness of
semiconducting materials studied by indentation and erosion techniques,
*Materials Science and Engineering*, A 105/106 (1988) pp 131-141