On analytical estimates of the effective elastic properties of polycrystalline silicon

Abstract

Several analytical approaches can be utilized to estimate the elastic properties of polycrystalline silicon. In experimental studies, the notion of an macroscopically isotropic aggregate is introduced while the single crystals obey cubic symmetry. We here give a synopsis on analytical approaches to predict elastic properties and apply them to estimate effective parameters of polycrystalline silicon. Here, the predictions are based on the parameters associated with shear solely. The results are juxtaposed in terms of the approaches applied, while different measures are introduced for evaluation. In comparison with experimental findings, the geometric mean implies a reasonable estimation for the elastic properties of polycrystalline silicon.

Size effects in numerical homogenization of polycrystalline silicon

“…All three approaches result in subsequent formulations for the second eigenvalue. The unique estimate by Aleksandrov and Aizenberg was found reasonable for the elastic properties of polycrystalline silicon [15]. In summary, this results in the following analytical limits:…”

“…All three approaches result in subsequent formulations for the second eigenvalue. The unique estimate by Aleksandrov and Aizenberg was found reasonable for the elastic properties of polycrystalline silicon [15]. In summary, this results in the following analytical limits: Kröner 1977 [10] ‐ zeroth‐order bounds, $$\begin{aligned}[t] \text{anisotropic \ \ } &\lambda ^{-}_{2}=\min \lbrace \lambda ^{\mathrm{c}}_{2},\lambda ^{\mathrm{c}}_{3}\rbrace \ \ \ \ \ \lambda ^{+}_{2}=\max \lbrace \lambda ^{\mathrm{c}}_{2},\lambda ^{\mathrm{c}}_{3}\rbrace \end{aligned}$$ Voigt 1889 [16] ‐ uniform strain field ‐ upper first‐order bound, $$\begin{aligned}[t] \text{isotropic \ \ } &\lambda ^{\mathrm{V}}_{2}=\frac{2}{5}\lambda ^{\mathrm{c}}_{2}+\frac{3}{5}\lambda ^{\mathrm{c}}_{3} \end{aligned}$$ Reuss 1929 [17] ‐ uniform stress field ‐ lower first‐order bound, $$\begin{aligned}[t] \text{isotropic \ \ } &\lambda ^{\mathrm{R}}_{2}=(\frac{2}{5}\frac{1}{\lambda ^{\mathrm{c}}_{2}}+\frac{3}{5}\frac{1}{\lambda ^{\mathrm{c}}_{3}} ) ^{-1} \end{aligned}$$ Aleksandrov and Aizenberg 1966 [14] ‐ mathematical requirement, …”

“…The objective of this work is to investigate size [1] effects on the effective material stiffness because of a scale separation. The examination starts with bulk samples, and then a gradual transition to thin layers of the material is examined.…”

Size effects in the elastic properties of polycrystalline silicon

“…Note that the Voigt and Reuss bounds coincide with the so-called first-order bounds [28]. The unique estimate by Aleksandrov and Aizenberg was found reasonable for the elastic properties of polycrystalline silicon [29]. The factors and exponents used in Equations ( 16) arise from the share of the five-dimensional deviatoric space ND dev , where the averaging takes place.…”

“…$$${\lambda }_{2}^{{\rm{V}}}=\frac{2}{5}{\lambda }_{2}^{{\rm{c}}}+\frac{3}{5}{\lambda }_{3}^{{\rm{c}}}$$$ $$${\lambda }_{2}^{{\rm{R}}}={\left(\frac{2}{5}\frac{1}{{\lambda }_{2}^{{\rm{c}}}}+\frac{3}{5}\frac{1}{{\lambda }_{3}^{{\rm{c}}}}\right)}^{-1}$$$ $$${\lambda }_{2}^{{\rm{A}}}={\left({\lambda }_{2}^{{\rm{c}}}\right)}^{\frac{2}{5}}{\left({\lambda }_{3}^{{\rm{c}}}\right)}^{\frac{3}{5}}$$$ Note that the Voigt and Reuss bounds coincide with the so‐called first‐order bounds [28]. The unique estimate by Aleksandrov and Aizenberg was found reasonable for the elastic properties of polycrystalline silicon [29]. The factors and exponents used in Equations (16) arise from the share of the five‐dimensional deviatoric space $${ND}^{\text{dev}}$$, where the averaging takes place.…”