The Young's modulus (E) of a material is a key parameter for mechanical engineering design. Silicon, the most common single material used in microelectromechanical systems (MEMS), is an anisotropic crystalline material whose material properties depend on orientation relative to the crystal lattice. This fact means that the correct value of E for analyzing two different designs in silicon may differ by up to 45%. However, perhaps, because of the perceived complexity of the subject, many researchers oversimplify silicon elastic behavior and use inaccurate values for design and analysis. This paper presents the best known elasticity data for silicon, both in depth and in a summary form, so that it may be readily accessible to MEMS designers.

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What is the Young's Modulus of Silicon?

The lattice parameter of high‐purity silicon is measured as a function of temperature between 300 and 1500 K, and the linear thermal expansion coefficient is accurately determined. Precise measurements are made by the high‐temperature attachment for Bond’s x‐ray method to a few parts per million. It is found that the temperature dependence of the linear thermal expansion coefficient α(t) is empirically given by α(t)=(3.725{1−exp[−5.88×10−3{(t−124)} +5.548×10−4t)×10−6 (K−1), where t is the absolute temperature ranging from 120 to 1500 K. It is shown that the lattice parameter in the above temperature range can be calculated using α(t) and the lattice parameter at 273.2 K (0.5430741 nm). Measured values of the lattice parameter and the thermal expansion coefficient for high‐purity float‐zoned (100 kΩ cm) and Czochralski‐grown (30 Ω cm) single crystals are uniformly distributed within ±1×10−5 nm and ±2×10−7 K−1 with respect to the values obtained from the above empirical formula.

Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K

Studies of the thermal expansion of solids in the temperature range 1–100 K are reviewed. Effects due to thermal vibrations of the lattice, electrons, magnetic interactions and defects are fully discussed. The survey of experimental data is intended to be comprehensive. From the large volume of theoretical work in recent years the authors have aimed to select for discussion the most significant contributions.

Thermal expansion of solids at low temperatures

The success of first-principles electronic-structure calculation for predictive modeling in chemistry, solid-state physics, and materials science is constrained by the limitations on simulated length scales and timescales due to the computational cost and its scaling. Techniques based on machine-learning ideas for interpolating the Born-Oppenheimer potential energy surface without explicitly describing electrons have recently shown great promise, but accurately and efficiently fitting the physically relevant space of configurations remains a challenging goal. Here, we present a Gaussian approximation potential for silicon that achieves this milestone, accurately reproducing density-functional-theory reference results for a wide range of observable properties, including crystal, liquid, and amorphous bulk phases, as well as point, line, and plane defects. We demonstrate that this new potential enables calculations such as finite-temperature phase-boundary lines, self-diffusivity in the liquid, formation of the amorphous by slow quench, and dynamic brittle fracture, all of which are very expensive with a first-principles method. We show that the uncertainty quantification inherent to the Gaussian process regression framework gives a qualitative estimate of the potential’s accuracy for a given atomic configuration. The success of this model shows that it is indeed possible to create a useful machine-learning-based interatomic potential that comprehensively describes a material on the atomic scale and serves as a template for the development of such models in the future.

https://link.aps.org/pdf/10.1103/PhysRevX.8.041048

Machine Learning a General-Purpose Interatomic Potential for Silicon

The thermo-optic coefficient dn/dT of silicon was measured at 1550 nm in the wide temperature range from 5 K to 300 K. For this purpose an interferometric measurement scheme was applied using the silicon sample as a Fabry-Perot etalon. The high resolution of this setup revealed a thermo-optic coefficient as low as 10−8 K−1 at 5 K. The presented results show an excellent agreement with former measurements above 30 K including a value of dn/dT=1.8×10−4 K−1 at 300 K.

/pdf/thermo-optic-coefficient-of-silicon-at-1550-nm-and-cryogenic-3kjdvnxfog.pdf

Thermo-optic coefficient of silicon at 1550 nm and cryogenic temperatures

Linear thermal expansion measurements have been carried out from 6 to 340 K on a high‐purity silicon sample using a linear absolute capacitance dilatometer. The accuracy of the measurements varies from ±0.01×10−8 K−1 at the lowest temperatures to ±0.1×10−8 K−1 or 0.1%, whichever is greater, near room temperature, and is sufficient to establish silicon as a thermal expansion standard for these temperatures. The agreement with previous data is satisfactory at low temperatures and excellent above room temperature where laser‐interferometry data of comparable accuracy exist. Thermal expansions calculated from ultrasonic and heat‐capacity data are preferred below 13 K where experimental problems occurred.

Linear thermal expansion measurements on silicon from 6 to 340 K

Linear-thermal-expansivity data are given to 1.5 K for the two different amorphous solids, fused silica (amorphous Si${\mathrm{O}}_{2}$) and PMMA [poly(methylmethacrylate)]. Both solids exhibit a negative expansivity contribution at low temperatures which is linear in the absolute temperature and which corresponds to a common Gr\"uneisen parameter, ${\ensuremath{\gamma}}_{1}=\ensuremath{-}16$. A calculation is given for the corresponding expansivity which would be expected for a continuous distribution in energy of two-level states using both symmetric and asymmetric tunneling models. Inconsistencies between the postulates which must be made to explain the present results and those required to explain other low-temperature properties of amorphous solids lend support to the conclusion from other experiments that the tunneling model does not explain satisfactorily the low-temperature properties of amorphous solids.

K. G. Lyon

Papers

Linear thermal expansion measurements on silicon from 6 to 340 K

Tunneling models and the experimental thermal expansivities of fused silica and poly(methylmethacrylate) (PMMA) below 4 K