Elastic properties of polycrystalline silicon: experimental findings, effective estimates, and their relations

Abstract

Silicon has a large impact on today’s world economy, also known as Silicon Age. For instance, it is an extremely important material for renewable energy systems like photovoltaics. Thereby, the use of polycrystalline silicon has a very wide range of application. For a safe and economic operation with this material, the most accurate prediction or measurement of the elastic properties possible is of interest in the first place even if the focus is on the analysis of the inelastic behavior and related reliability and service life predictions. The problem of effective elastic parameters of polycrystals is also a question of material symmetry. The silicon single crystals obey cubic symmetry while for the aggregate, at random orientation of its constituents, isotropy results. We here give a synopsis on established analytical approaches used to predict effective values as well as a review on experimental outcomes at crystal and aggregate level. In context of present material, the methods are applied and effective properties are predicted analytically while results are compared in terms of the different approaches applied and the material data sets accessed. The results are also contrasted to the measured findings. The resulting deviations are discussed whereby the reasons for these discrepancies are identified. For the application of the effective properties in practicable calculations, this implies that special emphasis must be placed on the origin of these data. The results of mono- and polycrystal properties for both, experimental and analytical findings, are tabulated in clear and concise form, so that they are readily accessible to design engineers.

Size effects in numerical homogenization of polycrystalline silicon

“…To be specific, a certain part of the experimental results lie outside the theoretically admissible range, for isotropic and anisotropic aggregates. This problem is revealed and discussed by Aßmus and Altenbach [5]. In the present treatise, we want to address this phenomenon by means of computational investigations.…”

“… $$$$\begin{align} \lambda ^{\mathrm{c}}_{1}=C_{1111}+2C_{1122}&&\lambda ^{\mathrm{c}}_{2}=C_{1111}-C_{1122}&&\lambda ^{\mathrm{c}}_{3}=2C_{2323} \end{align}$$$$For silicon crystal aggregates, that is, silicon polycrystals, isotropy (superscript ○) is often assumed, at least in experiments, cf. [5]. In the isotropic case, the projector representation reduces to: $$$$\begin{align} \mbox{\boldmath $\mathbb {C}$}^{\circ }=\lambda ^{\circ }_{1} \mbox{\boldmath $\mathbb {P}$}^{\circ }_{1}+\lambda ^{\circ }_{2} \mbox{\boldmath $\mathbb {P}$}^{\circ }_{2} && \begin{aligned} \mbox{\boldmath $\mathbb {P}$}^{\circ }_{1}&=\mbox{\boldmath $\mathbb {P}$}^{\mathrm{c}}_{1}=\frac{1}{3}\mbox{\boldmath $I$}\otimes \mbox{\boldmath $I$}\\ \mbox{\boldmath $\mathbb {P}$}^{\circ }_{2}&=\mbox{\boldmath $\mathbb {I}$}^{\mathrm{sym}}- \mbox{\boldmath $\mathbb {P}$}^{\circ }_{1} \end{aligned} \end{align}$$$$The isotropic eigenvalues are related to the engineering parameters as follows.…”

“…For silicon crystal aggregates, that is, silicon polycrystals, isotropy (superscript •) is often assumed, at least in experiments, cf. [5]. In the isotropic case, the projector representation reduces to:…”

Size effects in the elastic properties of polycrystalline silicon

“…To be specific, a certain part of the experimental results lie outside the theoretically admissible range, for isotropic and anisotropic aggregates. This problem is revealed and discussed by Aßmus and Altenbach [12]. In the present treatise, we want to address this phenomenon by means of computational investigations.…”

“…For silicon crystal aggregates, that is, silicon polycrystals, isotropy (superscript $$\circ $$) is often assumed, at least in experiments, cf. [12]. In the isotropic case, the projector representation () reduces to $$\chi =2$$ [16].…”