1 Introduction

1.1 Motivation

Polycrystalline silicon offers advantageous properties for a broad range of technical applications based on their semiconductor specificity. It is an important material in high-technology semiconductor devices such as microprocessors, integrated circuits, and solar cells. Production capacities of silicon are currently far more than 500,000 metric tons per year worldwide [1] which are offered at costs just above 30 U.S. dollars per kg [2]. There are various processes for producing polycrystalline silicon while the Siemens process (chemical vapor deposition) is still the prevailing technology [3]. Here, differences in resulting aggregates certainly lie in the degree of purity.

In addition to the semiconductor properties, which are in the foreground, the mechanical properties of this material are of great interest. Silicon has the diamond cubic crystal structure featuring an face-centered cubic (fcc) lattice. So, on micro scale, the single crystals obey cubic symmetry, i.e. their material properties depend on the orientation relative to the crystal lattice. While elastic properties of monocrystalline silicon are well studied, the properties of polycrystalline silicon are often still in the dark [4,5,6]. Here the variation in elastic properties is mainly influenced by the features crystal size distribution, crystal orientation distribution, crystal interactions, and the influence of the distribution functions of crystal interfaces and crystal edges [7]. Astonishingly, these polycrystals properties have not yet been studied to any considerable extent in the required depth, at least not for silicon. Thus, the elastic properties of polycrystalline silicon are typically assumed to be orientation independent, which can be substantiated by a series of experiments [8]. Nevertheless, the experimental findings are subject to a large dispersion which leaves open the reliability of these data.

On the other hand, there is also the possibility of theoretical predictions of these properties. Theses predictions are realized by the calculation of the constitutive relationship between macroscopic field quantities—here stress and strain—as a function of material structure and material properties at the micro scale. Thus, the bulk properties of polycrystalline materials are appropriate averages of the properties of their microconstituents.

It is worth examining theoretical and experimental results more closely since effective characteristics are not only important in the rigidity rating of silicon structures. Effective elastic properties also play a major role in the evaluation of strength properties, c.f. [9,10,11]. It should thus be clarified in which bounds the elastic properties of anisotropic and isotropic silicon polycrystals aggregates are enclosed. In addition, the specification of unique estimates is of interest. Such estimates must withstand the comparison with experimental results, and vice versa.

1.2 Outline

The manuscript is organized in five sections. In Sect. 2 we introduce a constitutive relationship that is appropriate for silicon. As silicon is a brittle material, it is reasonable to reduce our concern to linear elasticity. We distinguish between monocrystalline and polycrystalline silicon on the basis of the different crystal systems [12, 13]. For the sake of clarity and comprehensibility in context of the representation of constitutive measures, we make use of the projector representation of stiffnesses popularized by Rychlewski [14]. In contrast to classically applied engineering parameters like Young’s modulus and Poisson’s ratio which arise as natural parameters in different physical experiments, this representation is based on mathematical properties of the constitutive tensor. Since homogenization is a purely mathematical procedure, the eigenvalues are more suitable since they will help us to provide a concise presentation in the analytical determination of effective properties of the silicon polycrystal. Nevertheless, we use the natural parameters to determine and visualize the directionality of mechanical properties. In the process, extremes of these parameters are discussed as well.

In Sect. 3, experimental findings on mono- and polycrystalline silicon are compiled on the basis of a literature review. In addition to the description of basic determination methods, techniques are shown that significantly reduce the experimental effort, e.g. in the determination of aggregate parameters. Since the physical experiments are based on the determination of natural parameters, all findings are converted to eigenvalues. The results of present examination are compared, significant parameters are determined and their scatter is discussed. However, in the context of experimental results, we refrain from detailed explanations of experimental procedures, set-up and conduct of corresponding investigations and refer to pertinent publications.

Section 4 is dedicated to the analytical estimation of effective elastic properties of polycrystalline silicon. The representation is restricted explicitly to the problem of an isotropic aggregate of cubic single crystals. We refrain from detailed derivations of the respective theoretical approaches and concentrate on application and evaluation on the basis of the present problem. Fully analytic forms of the different constitutive tensors for the effective material have been worked out in the respective treatises cited. However, in what follows we have brought the original descriptions in a form which is consistent with the notation used in this treatise.

In Sect. 5, the methods for the prediction of the effective aggregates properties are applied. The problem is to calculate linear elastic properties of a polycrystalline composite of cubic phases basing on single crystal data summarized in Sect. 3. The analytical results are compared with and evaluated against each other. This leads to bandwidths that can be specified for isotropic and generally anisotropic silicon aggregates. Most suitable approaches to estimate effective properties are identified. Furthermore, the analytically results for polycrystalline silicon are compared to the results of the literature review from Sect. 3 and discussed. The conformity and divergence between theoretical prediction and experimentally determined quantity is illustrated by means of suitable measures. Finally, possible reasons for the discrepancies that occurred are discussed and their validity is verified.

The paper concludes with a summary of the key findings while indications of possible further work are given.

1.3 Notation

A direct notation is preferred for the mathematical representations in present treatise. First-order tensors are denoted as bold minuscules (e.g. \({{\varvec{a}}}\)), second-order tensors as bold majuscules (e.g. \({{\varvec{A}}}\)), and fourth-order tensors as black-board bold majuscules (e.g. \({{\varvec{\mathbb A}}}\)). The dyadic product and single scalar contractions are denoted like \(({{\varvec{a}}}\otimes {{\varvec{b}}}\otimes {{\varvec{c}}})\,{\varvec{:}}\,({{\varvec{d}}}\otimes {{\varvec{e}}})=({{\varvec{b}}}\,{\varvec{\cdot }}\,{{\varvec{d}}})({{\varvec{c}}}\,{\varvec{\cdot }}\,{{\varvec{e}}}){{\varvec{a}}}\). The superscript index \(\top \) denotes the transpose of a tensor which is defined via \({{\varvec{a}}}\,{\varvec{\cdot }}\,{{\varvec{B}}}^\top \,{\varvec{\cdot }}\,{{\varvec{c}}}={{\varvec{c}}}\,{\varvec{\cdot }}\,{{\varvec{B}}}\,{\varvec{\cdot }}\,{{\varvec{a}}}\) or \({{\varvec{A}}}\,{\varvec{:}}\,{{\varvec{\mathbb B}}}^\top \,{\varvec{:}}\,{{\varvec{C}}}={{\varvec{C}}}\,{\varvec{:}}\,{{\varvec{\mathbb B}}}\,{\varvec{:}}\,{{\varvec{A}}}\). The components of a tensor are given with respect to the orthonormalized bases \({{\varvec{e}}}_i\). We make use of Einstein’s summation convention with implicit summation from 1 to 3 that appear pairwise in a product, e.g., the second-order identity tensor is given by \({{\varvec{I}}}={{\varvec{e}}}_i\otimes {{\varvec{e}}}_i\). The fourth-order identity is given by \({{\varvec{\mathbb I}}}={{\varvec{e}}}_i\otimes {{\varvec{e}}}_j\otimes {{\varvec{e}}}_i\otimes {{\varvec{e}}}_j\). We furthermore require the transposer \({{\varvec{\mathbb T}}}={{\varvec{e}}}_i\otimes {{\varvec{e}}}_j\otimes {{\varvec{e}}}_j\otimes {{\varvec{e}}}_i\) to assemble the symmetric part of fourth-order identity, e.g. \(2{{\varvec{\mathbb I}}}^{\textrm{sym}}={{\varvec{\mathbb I}}}+{{\varvec{\mathbb T}}}\). The transposer maps any second-order tensor in to its transposed, \({{\varvec{\mathbb T}}}\,{\varvec{:}}\,{{\varvec{A}}}={{\varvec{A}}}^\top \). The superscript index \(-1\) denotes the inverse of a tensor, e.g., \({{\varvec{A}}}\,{\varvec{\cdot }}\,{{\varvec{A}}}^{-1}={{\varvec{I}}}\) holds for second-order tensors and \({{\varvec{\mathbb A}}}\,{\varvec{:}}\,{{\varvec{\mathbb A}}}^{-1}={{\varvec{\mathbb I}}}^{\textrm{sym}}\) for fourth-order tensors. Orthogonal tensor have the property \({{\varvec{A}}}^{-1}={{\varvec{A}}}^\top \). Further we use the Rayleigh product which is denoted by \({{\varvec{A}}}\star {{\varvec{\mathbb B}}}^{0}={{\varvec{\mathbb B}}}\). It maps all basis vectors of \({{\varvec{\mathbb B}}}^0\) simultaneously without changing its components. We also apply the Frobenius norm of fourth-order tensors which is defined via \(\vert \vert {{\varvec{\mathbb A}}}\vert \vert =\sqrt{{{\varvec{\mathbb A}}}\,{\varvec{:}}\,\,{\varvec{:}}\,{{\varvec{\mathbb A}}}}\).

2 Constitutive relations on crystal and aggregate level

2.1 Constitutive law

Pure elastic behavior in context of small deformations is well represented by Hooke’s law. This law is given as a linear mapping of the strains \({{\varvec{E}}}\) onto the stresses \({{\varvec{T}}}\).

$$\begin{aligned} {{\varvec{T}}}={{\varvec{\mathbb C}}}\,{\varvec{:}}\,{{\varvec{E}}} \end{aligned}$$
(1)

Herein \({{\varvec{\mathbb C}}}\) is the fourth-order elasticity tensor while \({{\varvec{E}}}\) and \({{\varvec{T}}}\) are second-order tensors for the linearized strains and the Cauchy stresses. The elasticity tensor exhibits the major symmetry as well as the left and right subsymmetry, shown below for arbitrary  \({{\varvec{A}}}\),  \({{\varvec{B}}}\).

$$\begin{aligned} {{\varvec{A}}}\,{\varvec{:}}\,{{\varvec{\mathbb C}}}\,{\varvec{:}}\,{{\varvec{B}}}={{\varvec{B}}}\,{\varvec{:}}\,{{\varvec{\mathbb C}}}\,{\varvec{:}}\,{{\varvec{A}}} \qquad {{\varvec{A}}}\,{\varvec{:}}\,{{\varvec{\mathbb C}}}\,{\varvec{:}}\,{{\varvec{B}}}={{\varvec{A}}}\,{\varvec{:}}\,{{\varvec{\mathbb C}}}\,{\varvec{:}}\,{{\varvec{B}}}^\top ={{\varvec{A}}}^\top \,{\varvec{:}}\,{{\varvec{\mathbb C}}}\,{\varvec{:}}\,{{\varvec{B}}} \end{aligned}$$
(2)

Due to these symmetry properties, we can find a unique representation of \({{\varvec{\mathbb C}}}\) by the aid of the spectral decomposition [14, 15].

$$\begin{aligned} {{\varvec{\mathbb C}}}=\sum _{\alpha =1}^{n}\lambda _\alpha {{\varvec{\mathbb P}}}_\alpha \end{aligned}$$
(3)

Herein n denotes the maximum number of distinct \(\alpha \). In general \(2\le n\le N\) holds, where \(N=6\) indicates the maximum number of pairwise orthogonal, one-dimensional subspaces [16]. Furthermore, \(\lambda _\alpha >0\) are the real eigenvalues and \({{\varvec{\mathbb P}}}_\alpha \) are eigenprojectors of \({{\varvec{\mathbb C}}}\). The eigenprojectors obey the following projector rules.

$$\begin{aligned}&{{\varvec{\mathbb P}}}_\alpha \;\,{\varvec{:}}\,{{\varvec{\mathbb P}}}_\alpha ={{\varvec{\mathbb P}}}_\alpha \qquad \qquad \text {idempotence} \end{aligned}$$
(4a)
$$\begin{aligned}&{{\varvec{\mathbb P}}}_\alpha \;\,{\varvec{:}}\,{{\varvec{\mathbb P}}}_\beta ={{\varvec{\mathbb O}}}\qquad \qquad \text {orthogonality}\end{aligned}$$
(4b)
$$\begin{aligned}&\textstyle \sum _{\alpha =1}^{n}{{\varvec{\mathbb P}}}_\alpha ={{\varvec{\mathbb I}}}^{\textrm{sym}}\qquad \text {completeness} \end{aligned}$$
(4c)

Herein, \({{\varvec{\mathbb O}}}\) is the fourth-order zero tensor and \({{\varvec{\mathbb I}}}^{\textrm{sym}}\) is the fourth-order identity on symmetric second-order tensors. The inverse of the elasticity tensor is determined by the inversion of the eigenvalues.

$$\begin{aligned} {{\varvec{\mathbb C}}}^{-1}=\sum _{\alpha =1}^{n}\frac{1}{\lambda _\alpha }{{\varvec{\mathbb P}}}_\alpha \end{aligned}$$
(5)

2.2 Single crystal elasticity

Silicon monocrystals obey cubic symmetry. With respect to Eq. (5) \(n=3\) holds. The distinct cubic eigenvalues are

$$\begin{aligned} \lambda _1^{\textrm{c}}=C_{1111}+2C_{1122}, \qquad \lambda _2^{\textrm{c}}=C_{1111}-C_{1122}, \qquad \text {and}\quad \lambda _3^{\textrm{c}}=2C_{2323}\,. \end{aligned}$$
(6)

Herein \(C_{ijkl}\) are the lattice parameters. Furthermore, \(\lambda _1^{\textrm{c}}\) is element of a one-dimensional dilatoric space and the two others belong to the five-dimensional deviatoric space which features two subspaces. Here, \(\lambda _2^{\textrm{c}}\) is element of a two-dimensional deviatoric subspace and \(\lambda _3^{\textrm{c}}\) is of a three-dimensional deviatoric subspace [16]. For silicon crystals \(\lambda ^{\textrm{c}}_2<\lambda ^{\textrm{c}}_3\) holds. The cubic eigenprojectors [14] are

$$\begin{aligned} {{\varvec{\mathbb P}}}_1^{\textrm{c}}=\frac{1}{3}\, {{\varvec{I}}}\otimes {{\varvec{I}}}\,,\qquad {{\varvec{\mathbb P}}}_2^{\textrm{c}}= {{\varvec{\mathbb D}}}-{{\varvec{\mathbb P}}}_1^{\textrm{c}}\,,\qquad {{\varvec{\mathbb P}}}_3^{\textrm{c}}={{\varvec{\mathbb I}}}^{\textrm{sym}}- {{\varvec{\mathbb D}}}\,, \end{aligned}$$
(7)

while

$$\begin{aligned} {{\varvec{\mathbb D}}}=\sum \limits _{i=1}^3 {{\varvec{g}}}_i \otimes {{\varvec{g}}}_i\otimes {{\varvec{g}}}_i\otimes {{\varvec{g}}}_i={{\varvec{Q}}}\star {{\varvec{\mathbb D}}}^0 \quad \text {with}\quad {{\varvec{\mathbb D}}}^0=\sum \limits _{i=1}^3 {{\varvec{e}}}_i \otimes {{\varvec{e}}}_i\otimes {{\varvec{e}}}_i\otimes {{\varvec{e}}}_i\;, \end{aligned}$$
(8)

is the anisotropic portion and \({{\varvec{I}}}={{\varvec{e}}}_i\otimes {{\varvec{e}}}_i\) is the identity on first-order tensors. Herein we make use of the lattice vectors \({{\varvec{g}}}_i\) of the single crystal and the fixed sample basis \({{\varvec{e}}}_i\). These orthonormal vectors are related via the versor \({{\varvec{Q}}}={{\varvec{g}}}_i\otimes {{\varvec{e}}}_i=Q_{ij}{{\varvec{e}}}_i\otimes {{\varvec{e}}}_j\) that maps the sample basis onto the local lattice vectors.

2.3 Aggregate elasticity

For an isotropic approximation \(n=2\) [14] holds. We here make use of the superscript index \(\circ \) for the description of isotropic parameters. The distinct isotropic eigenvalues are given as

$$\begin{aligned} \lambda _1^\circ =3K=\frac{Y}{1-2\nu }\quad \text {and}\quad \lambda _2^\circ =2G=\frac{Y}{1+\nu }\,. \end{aligned}$$
(9)

Herein, \(\lambda _1^\circ \) is element of a one-dimensional dilatoric space, while \(\lambda _2^\circ \) is element of a five-dimensional deviatoric space. The variables K and G are known as bulk and shear modulus, while Y and \(\nu \) are known as Young’s modulus and Poisson’s ratio. The associated isotropic projectors are as follows [14].

$$\begin{aligned} {{\varvec{\mathbb P}}}_1^\circ ={{\varvec{\mathbb P}}}_1^{\textrm{c}} \qquad {{\varvec{\mathbb P}}}_2^\circ ={{\varvec{\mathbb I}}}^{\textrm{sym}}-{{\varvec{\mathbb P}}}_1^\circ \end{aligned}$$
(10)

2.4 Graphical representation and comparison

To demonstrate directional dependencies for mono- and the polycrystal, we here make use of three-dimensional elasticity bodies. Again, the difference in the constitutive measures introduced in preceding section is shown as follows

$$\begin{aligned} {{\varvec{\mathbb C}}}^{\textrm{c}}= \lambda _1^{\textrm{c}}{{\varvec{\mathbb P}}}_1^{\textrm{c}}+ \lambda _2^{\textrm{c}}{{\varvec{\mathbb P}}}_2^{\textrm{c}}+ \lambda _3^{\textrm{c}}{{\varvec{\mathbb P}}}_3^{\textrm{c}} \qquad {{\varvec{\mathbb C}}}^\circ = \lambda _1^\circ {{\varvec{\mathbb P}}}_1^\circ + \lambda _2^\circ {{\varvec{\mathbb P}}}_2^\circ \;, \end{aligned}$$
(11)

where the eigenvalues are determined by a projection of stiffness tensor, i.e.

$$\begin{aligned} \lambda _\alpha ^{\square }={{\varvec{\mathbb C}}}^{\square }\,{\varvec{:}}\,\,{\varvec{:}}\,\frac{{{\varvec{\mathbb P}}}^{\square }_\alpha }{\vert \vert {{\varvec{\mathbb P}}}^{\square }_\alpha \vert \vert ^2}\quad \forall \,\alpha \in \{1,\dots ,n\}\wedge \square \in \{\textrm{c},\circ \}\;, \end{aligned}$$
(12)

with \(\vert \vert {{\varvec{\mathbb P}}}_\alpha \vert \vert ^2={{\varvec{\mathbb P}}}_\alpha \,{\varvec{:}}\,\,{\varvec{:}}\,{{\varvec{\mathbb P}}}_\alpha \) being the dimension of the respective subspace , while holds. There are only two characteristic parameters that allow a unique three-dimensional representation of the directional properties of the elastic behavior. These parameters are the Young’s modulus Y and the bulk modulus K. Following [14], we can determine these parameters on the basis of an stiffness tensor \({{\varvec{\mathbb C}}}\) of arbitrary material symmetry in dependence of a direction \({{\varvec{d}}}\), cf. [17].

$$\begin{aligned} Y({{\varvec{d}}})&=\big [\left( {{\varvec{d}}}\otimes {{\varvec{d}}}\right) \quad \,{\varvec{:}}\,\;{{\varvec{\mathbb C}}}^{-1}\,{\varvec{:}}\,\left( {{\varvec{d}}}\otimes {{\varvec{d}}}\right) \big ]^{-1} \end{aligned}$$
(13)
$$\begin{aligned} K({{\varvec{d}}})&=\big [\quad 3\,{{\varvec{I}}}\qquad \,{\varvec{:}}\,\;{{\varvec{\mathbb C}}}^{-1}\,{\varvec{:}}\,\left( {{\varvec{d}}}\otimes {{\varvec{d}}}\right) \big ]^{-1} \end{aligned}$$
(14)

Herein, the direction is parameterized in spherical coordinates with respect to a unit sphere, i.e. by \({{\varvec{d}}}=\cos (\varphi )\sin (\vartheta ){{\varvec{e}}}_1+\sin (\varphi )\sin (\vartheta ){{\varvec{e}}}_2+\cos (\vartheta ){{\varvec{e}}}_3\) with \(\Vert {{\varvec{d}}}\Vert =1\). The measures \(\varphi \in \;]0,2\pi ]\) and \(\vartheta \in [0,\pi [\) are the azimuth and the polar angle. By the aid of the computational implementation given in [17],Footnote 1 we can determine the graphical representations given in Fig. 1. There, characteristics of cubic (monocrystal) and isotropic (polycrystal) behavior are juxtaposed whereby we make use of exemplary material parameters for both, crystal and aggregate elasticity as given in [18]. To relate the measures used for visualization to the eigenvalues used in Eq. (11), we can rewrite the elasticity tensors whereby we distinguish the cubic and the isotropic case by means of the index.

$$\begin{aligned} {{\varvec{\mathbb C}}}^{\textrm{c}}= 3K^{\textrm{c}}{{\varvec{\mathbb P}}}_1^{\textrm{c}}+ 2G_1^{\textrm{c}}{{\varvec{\mathbb P}}}_2^{\textrm{c}}+ 2G_2^{\textrm{c}}{{\varvec{\mathbb P}}}_3^{\textrm{c}} \qquad {{\varvec{\mathbb C}}}^\circ = 3K^\circ {{\varvec{\mathbb P}}}_1^\circ + 2G^\circ {{\varvec{\mathbb P}}}_2^\circ \end{aligned}$$
(15)

It is evident that the bulk modulus is not subjected to a directional dependency on both, crystal and aggregate level, i.e. \(K({{\varvec{d}}})=K\). In contrast, Young’s modulus is only direction-independent at aggregate level. A pronounced dependence with the following characteristics is evident.

$$\begin{aligned} Y({{\varvec{d}}})= {\left\{ \begin{array}{ll} \min &{}\text {if}\;{{\varvec{d}}}={{\varvec{g}}}_i \;\Rightarrow Y_{\min }\\ \max &{}\text {if}\;{{\varvec{d}}}={{\varvec{d}}}_S\Rightarrow Y_{\max } \end{array}\right. } \end{aligned}$$
(16)
Fig. 1
figure 1

Directional dependencies of Young’s and bulk modulus for mono- (top) and polycrystalline (bottom) silicon

Full size image

Herein, \({{\varvec{d}}}_S\) is the vector of the space diagonal of the cubic primitive cell. The four space diagonal vectors are defined as follows.

$$\begin{aligned} {{\varvec{d}}}_{S}=\frac{({{\varvec{g}}}_1\pm {{\varvec{g}}}_2\pm {{\varvec{g}}}_2)}{\sqrt{3}} \end{aligned}$$
(17)

Herein again, \({{\varvec{g}}}_i\) are the principal crystallographic axes. In materials science, the so-called Miller indices [uvw] and (hkl) are used for the lattice-related characterization of directions and planes [19]. The indices are always integer and relative prim. Negative values are marked by overlining. All crystallographically equivalent directions are bracketed in angles, e.g. \(\langle 100\rangle \). Crystallographically equivalent planes are written in curly brackets, e.g. \(\{111\}\). Crystallographically equivalent are all directions that can be transformed into each other by symmetry transformations of the crystal, cf. [20], e.g. all three edges of the cubic unit cell, the six face diagonals or the four space diagonals [21]. Thus, the principal crystallographic axes can also be specified as \({{\varvec{g}}}_i\,\hat{=}\, \langle 100\rangle \) when using Miller notation. However, by the aid of Eq. (16) we can compute Young’s moduli in extremal directions  \({{\varvec{d}}}\). In dependence of Si monocrystal input (6), the Young’s modulus is varying between

$$\begin{aligned} Y({{\varvec{g_i}}})=\frac{3\lambda _1^{\textrm{c}}\lambda _2^{\textrm{c}}}{2\lambda _1^{\textrm{c}}+\lambda _2^{\textrm{c}}} \end{aligned}$$
(18)

and

$$\begin{aligned} Y({{\varvec{d}}}_S)=\frac{3\lambda _1^{\textrm{c}}\lambda _3^{\textrm{c}}}{2\lambda _1^{\textrm{c}}+\lambda _3^{\textrm{c}}}\,, \end{aligned}$$
(19)

while Young’s modulus along the face diagonals of the cubic primitive cells is determined by

$$\begin{aligned} Y({{{\varvec{d}}}_F})=\dfrac{6\lambda _1^{\textrm{c}}\lambda _2^{\textrm{c}} \lambda _3^{\textrm{c}}}{3\lambda _1^{\textrm{c}}\lambda _2^{\textrm{c}} +\lambda _1^{\textrm{c}}\lambda _3^{\textrm{c}}+2\lambda _2^{\textrm{c}}\lambda _3^{\textrm{c}}} \end{aligned}$$
(20)

with \({{\varvec{d}}}_F=\frac{{{\varvec{g}}}_i \pm {{\varvec{g}}}_j}{\sqrt{2}} \;\forall \,i< j\) being a face diagonal of the cubic primitive cell, i.e. results refer to \(\langle 110\rangle \). Concluding, in the case of an isotropic material ( \(\lambda _2=\lambda _3\)) as introduced in Sect. 2.3, \(Y_{\min }=Y_{\max }=Y({{\varvec{d}}})=Y\) holds. We can then state Young’s modulus in terms of the isotropic eigenvalues.

$$\begin{aligned} Y= \frac{3\lambda _1^{\circ }\lambda _2^{\circ }}{2\lambda _1^{\circ }+\lambda _2^{\circ }} \end{aligned}$$
(21)

The bulk modulus is defined as the relative change in the volume of a body induced by a change in pressure acting uniformly over the bodies surface. For the cubic and the isotropic case this measure is determined identically.

$$\begin{aligned} K({{\varvec{d}}})=K=\frac{1}{3}\lambda ^{\textrm{c}}_1=\frac{1}{3}\lambda ^{\circ }_1 \end{aligned}$$
(22)

Obviously, the anisotropic behavior of the cubic single crystal is associated with a directional dependency of shear modulus and Poisson’s ratio. Thus, we give expressions for these measures, too. Following [14], we can determine these parameters as follows.

$$\begin{aligned} G({{\varvec{d}}},{{\varvec{n}}})&=\big [\left( {{\varvec{d}}}\otimes {{\varvec{n}}}+{{\varvec{n}}} \otimes {{\varvec{d}}}\right) \,{\varvec{:}}\,\;{{\varvec{\mathbb C}}}^{-1}\,{\varvec{:}}\,\left( {{\varvec{d}}} \otimes {{\varvec{n}}}+{{\varvec{n}}}\otimes {{\varvec{d}}}\right) \big ]^{-1} \end{aligned}$$
(23)
$$\begin{aligned} \nu ({{\varvec{d}}},{{\varvec{n}}})&=-Y({{\varvec{d}}})\quad \left( {{\varvec{d}}}\otimes {{\varvec{d}}}\right) \quad \,{\varvec{:}}\,\;{{\varvec{\mathbb C}}}^{-1}\,{\varvec{:}}\,\left( {{\varvec{n}}}\otimes {{\varvec{n}}}\right) \end{aligned}$$
(24)

Both quantities are not defined on a unit sphere, but are specified on a plane by direction \({{\varvec{d}}}\) and normal \({{\varvec{n}}}\), i.e. \({{\varvec{d}}}\) and \({{\varvec{n}}}\) are mutually orthogonal vectors. There is therefore an infinite number of planes onto which these two parameters can be projected. Nevertheless, we can determine some specific values for the parameter conjugated to the Young’s modulus, i.e. for Poisson’s ratio. To be exact, Poisson’s ratio is the ratio of the lateral contraction along direction \({{\varvec{d}}}\) to the longitudinal extension along direction \({{\varvec{n}}}\), perpendicular to \({{\varvec{d}}}\). Usually, Poisson’s ratio is determined for any given \({{\varvec{d}}}\) and \({{\varvec{n}}}\), cf. [22]. In case of cubic symmetry it is possible to specify bounds of this parameter, cf. [23]. Upper and lower bounds are defined as follows [24].

$$\begin{aligned} \nu ({{\varvec{d}}},{{\varvec{n}}})= {\left\{ \begin{array}{ll} \min &{}\text {if}\;{{\varvec{d}}}\,\hat{=}\,\langle 110\rangle \wedge {{\varvec{n}}}\,\hat{=}\,\langle 1{\overline{1}}0\rangle \quad \Rightarrow \nu _{\min }\\ \max &{}\text {if}\;{{\varvec{d}}}\,\hat{=}\,\langle 110\rangle \wedge {{\varvec{n}}}\,\hat{=}\,\langle 001\rangle \quad \Rightarrow \nu _{\max } \end{array}\right. } \end{aligned}$$
(25)

These extremes are for a stretch along the face diagonal of the cubic cell and a resulting lateral strain measured along a perpendicular face-diagonal or a perpendicular cube-axis direction, respectively. By exploiting the properties of the orthogonal vectors \({{\varvec{d}}}\) and \({{\varvec{n}}}\) in Eq. (24), we can determine the extremes of Poisson’s ratio as follows.

$$\begin{aligned} \nu _{\min }&=\frac{\lambda ^{\textrm{c}}_1(3\lambda ^{\textrm{c}}_2 -\lambda ^{\textrm{c}}_3)-2\lambda ^{\textrm{c}}_2\lambda ^{\textrm{c}}_3}{\lambda ^{\textrm{c}}_1(3\lambda ^{\textrm{c}}_2+\lambda ^{\textrm{c}}_3) +2\lambda ^{\textrm{c}}_2\lambda ^{\textrm{c}}_3} \end{aligned}$$
(26a)
$$\begin{aligned} \nu _{\max }&=\frac{(\lambda ^{\textrm{c}}_1-\lambda ^{\textrm{c}}_2)\lambda ^{\textrm{c}}_3}{\lambda ^{\textrm{c}}_1(\lambda ^{\textrm{c}}_2+\lambda ^{\textrm{c}}_3) +\lambda ^{\textrm{c}}_2\lambda ^{\textrm{c}}_3} \end{aligned}$$
(26b)

In the isotropic case, \(\nu _{\min }=\nu _{\max }=\nu ({{\varvec{d}}},{{\varvec{n}}})=\nu \) holds. Poisson’s ratio is then given as follows in terms of the isotropic eigenvalues.

$$\begin{aligned} \nu =\frac{\lambda _1^{\circ }-\lambda _2^{\circ }}{2\lambda _1^{\circ }+\lambda _2^{\circ }} \end{aligned}$$
(27)

To be complete, we supplement the extremal values of the shear modulus in cubic symmetry. The shear modulus generally is defined as the ratio of shear stress to the shear strain. This of course refers to the shear plane and the shear direction. Upper and lower bounds are as follows.

$$\begin{aligned} G({{\varvec{d}}},{{\varvec{n}}})= {\left\{ \begin{array}{ll} \min &{}\text {if}\;{{\varvec{d}}}\,\hat{=}\,\langle 110\rangle \wedge {{\varvec{n}}}\,\hat{=}\,\langle 1{\overline{1}}0\rangle \quad \Rightarrow G_{\min }\\ \max &{}\text {if}\;{{\varvec{d}}}\,\hat{=}\,\langle 110\rangle \wedge {{\varvec{n}}}\,\hat{=}\,\langle 0kl\rangle \quad \Rightarrow G_{\max } \end{array}\right. } \end{aligned}$$
(28)

Even for the shear modulus of the cubic system, we can specify extremes which are determined as follows.

$$\begin{aligned} G_{\min }&= \frac{1}{2}\lambda _2^{\textrm{c}} \end{aligned}$$
(29a)
$$\begin{aligned} G_{\max }&=\frac{1}{2}\lambda _3^{\textrm{c}} \end{aligned}$$
(29b)

In the case of isotropy, \(G_{\min }=G_{\max }=G({{\varvec{d}}},{{\varvec{n}}})=G\) holds. The isotropic shear modulus is defined as below.

$$\begin{aligned} G=\frac{1}{2}\lambda ^\circ _2 \end{aligned}$$
(30)

3 Experimental findings of silicon’s mechanical properties

3.1 General remarks

The parameters of Hooke’s law are usually determined in static or dynamic tests. Here, tension-compression, torsional or bending and vibration or wave propagation tests are preferably used. In the isotropic case, two elastic constants have to be determined. Here, Obermeier [25] and Sharpe, Yuan, Vaidyanathan, and Edwards [26] reviewed different test methods for mechanical properties of silicon aggregates. The mechanical tests are more complicated if the material is cubic. For example, Hall [27] considered sound-velocity measurements by a pulse-echo technique whereby elastic parameters were computed based on the density of silicon which was determined by Hall-effect measurements. However, for what follows, we restrict our review to material parameters determined at room temperature and ambient pressure. For dependencies of elastic properties deviating from these conditions we refer to Lyon, Salinger, Swenson, and White [28] or Roberts [29].

3.2 Monocrystal properties

As was introduced in Sect. 2.2, the elastic behavior of a cubic silicon monocrystals is determined by three independent material parameters. The first known publication of elasticity parameters for a silicon monocrystal was authored by Mason in 1958 [30]. Further basic data were provided by McSkimin and Andreatch [31]. Parameters with a slightly better accuracy were proposed by Hall [27]. This essential data is summarized in Table 1. The difference inbetween the data sets presented there does not seem to be of significance, at least at this point. However, it is worth to mention that the elastic properties of a semiconductor material may depend upon the electron concentration [32]. Einspruch and Csavinszky [33] measured the effect of doping of silicon samples, which had a direct impact on \(\lambda _2^{\textrm{c}}\). At room temperature, however, only deviations in the order of magnitude of the fluctuations of the parameters listed in Table 1 were found.

Table 1 Eigenvalues of an Si monocrystal at room temperature
Full size table
Fig. 2
figure 2

Parameter space of experimentally determined eigenvalues and their scattering for silicon monocrystals

Full size image

In context of cubic symmetry, the Zener ratio [34], i.e.

(31)

is usually applied to quantify the degree of anisotropy. This ratio may be interpreted as the ratio of the shear moduli, since \(\lambda _3^{\textrm{c}}\) is associated with the shear modulus at shearing in \(\langle 100\rangle \)-direction on a \(\{100\}\)-plane, and \(\lambda _2^{\textrm{c}}\) is associated with the shear modulus at shearing in \(\langle 110\rangle \)-direction on a \(\{110\}\)-plane. The Zener ratio also reveals the isotropy condition \(\lambda _2^{\textrm{c}}=\lambda _3^{\textrm{c}}\) by

(32)

If the crystal is isotropic, this measure turns 1. All results deviating from 1 indicate anisotropy. Examples include copper and aluminium with and , respectively [35, 36]. For present data sets of silicon we obtain while the scattering of the input has only a minor influence which is reflected in the fourth decimal place. In comparison to other cubic materials, monocrystalline silicon has a moderate degree of anisotropy.

We can quantify the scattering of the eigenvalues as follows whereby the subscript index \(\gamma \) is used for any eigenvalue assemblage of the three parameter sets given in Table 1.

(33)

The variety of eigenvalues of the monocrystal are shown in Fig. 2. on the left-hand side. The colors used thereby are associated to the colors used in Table 1. The white marks on the grey colored bounding planes ( \(\lambda _{\alpha }^{\textrm{c}}\)- \(\lambda _{\beta }^{\textrm{c}}\)-planes for \(\alpha \ne \beta \) while \(\alpha ,\beta \in \{1,2,3\}\) holds) serve to illustrate the scattering.

To make the scatter of the input data a little more tangible, we use the representative directions of the Young’s modulus body, cf. Fig. 1 top left. By the aid of Eqs. (18), (19), and (20) we have determined specific values for the Young’s modulus based on the different inputs. The results are given in Table 2. To make it clear again, the Young’s modulus of silicon is bounded by the extremal values located at \(\langle 100\rangle \) and \(\langle 111\rangle \) crystal direction. It is worth to note that the Young’s modulus along \(\langle 111\rangle \) is almost \(45\%\) larger than that along \(\langle 100\rangle \), i.e. , cf. Eqs. (16), (18), (19). Obviously, these specific values differ only slightly with regard to the varying input data. To be complete, we list extremes of shear modulus and Poisson’ ratio in Table 3. It is evident that considerably bigger fractional variations arise for \(\nu ({{\varvec{d}}},{{\varvec{n}}})\) in comparison to \(Y({{\varvec{d}}})\), i.e. minimum Poisson’s ratio is slightly above \(17\%\) of maximum Poisson’s ratio for all thee entries. For shear modulus, the minima are around \(70\%\) of the maxima.

Table 2 Values for Young’s modulus in dependence of characteristic directions \({{\varvec{d}}}\) of the Si monocrystal
Full size table
Table 3 Extremes for Poisson’s ratio and shear modulus of the cubic Si monocrystal
Full size table

3.3 Polycrystal properties

The elastic behavior of the silicon polycrystal is usually assumed to be direction-independent. As introduced in Sect. 2.3, such isotropic aggregate elasticity is sufficiently described by two independent material parameters. These can be usually determined by standard materials engineering tests, e.g., tensile tests. Sharpe [8] summarized different methods for tests of micro-electro-mechanical systems (MEMS) in detail, where polycrystalline silicon is widely used. These tests are:

  • tensile tests at thin films [37,38,39,40,41]

  • flexure-beam micromirror device tests [42]

  • in-plane bending tests of thin beams [43]

  • bulge tests at thin films [44, 45]

  • resonance tests at thin films [46]

In the referenced sources, the focus is on determining Young’s modulus. Complete data sets for the material behavior of polycrystalline silicon are very scarce in the literature. For the most part, as already mentioned, only Young’s modulus is addressed. From the introductory fundamentals, it can be concluded that isotropic elastic material behavior requires a conjugate parameter. In Sect. 2.3, this is given by several pairings, i.e. via the eigenvalues \(\lambda _1^\circ \) and \(\lambda _2^\circ \), via bulk K and shear modulus G, as well as via Young’s modulus Y and Poisson’s ratio \(\nu \). Typically in experiments, the latter parameters Y and \(\nu \) are measured, e.g. in a tensile test. Unfortunately, in the literature often only Young’s modulus is specified. The reason for this is that Poisson’s ratio is difficult to capture metrologically. Hence, engineers use a trick related to the bulk modulus of mono- and polycrystal. Due to the isotropic nature of volumetric changes, the bulk modulus is the same for cubic monocrystal and isotropic aggregate, i.e. \(K^{\textrm{c}}=K^\circ =K\), and therefore \(\lambda _1^{\textrm{c}}=\lambda _1^\circ =\lambda _1\) holds. By the knowledge of the bulk modulus from monocrystal data and the measured Young’s modulus of the polycrystal, one may deduce an estimation for the Poisson’s ratio of the isotropic aggregate.

$$\begin{aligned} \nu =\frac{3K-Y^{\delta }}{6K}=\frac{\lambda _1-Y^{\delta }}{2\lambda _1} \end{aligned}$$
(34)

Hereby, \(\delta \) is a designator for experimentally determined parameters. It is worth mentioning, that in the isotropic case, Poisson’s ratio is bounded by \(\nu \in \;]-1,0.5[\). This results from the demand of positive-definiteness of the (symmetric) stiffness tensor, i.e. all eigenvalues are positive. By implication, however, this also entails the range for the Young’s modulus to be determined experimentally is bounded due to \(Y=\lambda _1(1-2\nu )\), i.e. \(Y^{\delta }\in \;]0,3\lambda _1[\). However, silicon is non-auxetic, i.e. \(\nu >0\) applies for any combination of \({{\varvec{d}}}\) and \({{\varvec{n}}}\) ensured by for the isotropic aggregate [47]. Here, \(\nu \in \;]0,0.5[\) holds, so that the range of the expected Young’s modulus is restricted to

$$\begin{aligned} \lambda _1>Y^{\delta }>3\lambda _1\,. \end{aligned}$$
(35)

Since in the literature, \(\nu \approx 0.22\) is reported occasionally for polycrystalline silicon, c.f. [40], one would expect a Young’s modulus around \(165\,\textrm{GPa}\) for such an aggregate when considering the reformulation of Eq. (34). However, we apply this operation vice versa to determine Poisson’s ratio, while for a particular value of \(\lambda _1\) in Eq. (34) we make use of the mean value by utilizing the N parameters \(\gamma \) of \(\lambda _1^{\textrm{c}}\) given in Table 1, i.e.

$$\begin{aligned} {\lambda }_1^\delta = \frac{1}{N}\sum _\gamma ^{N} (\lambda _1^{\textrm{c}})_\gamma ^{}\;. \end{aligned}$$
(36)

Thus, this appropriate substitution allows us to generate complete data sets from Young’s module measurements alone. The sources referred to at the outset of present section lead to parameters that are summarized in Table 4. At first glance we can note relatively large variations in the obviously most fundamental mechanical property, the Young’s modulus. In what follows, we apply quantities converted into the eigenvalues that allows us to build the subsequent stiffnesses based the experimental data given on Table 4.

(37)

while the eigenvalues of the experimental findings are related via Eq. (9). We can now introduce the isotropic eigenvalue ratio of the experimentally determined stiffnesses, i.e.

(38)

This parameter is related to the bulk-to-shear-modulus ratio , also known as Pugh’s ratio [48], i.e. . This key figure is often used as fingerprint for the material behavior, i.e. ductility or brittleness. Usually, for a solid, for small values one expects a brittle manner but larger values indicate a ductile manner. The ductile-brittle transition (db)—for isotropic materials—is specified with [49]. In present case, unsurprisingly points to brittle behavior, i.e. linear elastic until fracture. Thus, at room temperature, the fracture behavior of silicon is purely brittle, that neither a plastics zone nor dislocation generation are observed during the initiation and progression of cracking. However, we are not focusing on any inelastic processes here, but the present eigenvalue ratio confirms the choice of the constitutive law in Sect. 2.1. At a glance, the mean value of the experimental findings is \(\textrm{Mean}(Y^{\delta })=166.2\,\textrm{GPa}\) with a standard deviation \(\sqrt{\textrm{Var}(Y^{\delta })}=4.04\,\textrm{GPa}\). Compared with the estimation by Eq. (34), i.e. ca. \(165\,\textrm{GPa}\), the experimental findings are found in the expected range, at least on average. However, it is worth mentioning that the scatter in the experimental results reported in Table 4 is up to \(\pm 5\%\) depending on the test method. To put it in figures, considering a mean of \(165\,\textrm{GPa}\), \(Y\in [156.75,173.25]\,\textrm{GPa}\) results at worst.

Table 4 Ten experimentally determined datasets for an Si polycrystal
Full size table
Fig. 3
figure 3

Parameter space of experimentally determined isotropic eigenvalues and scattering for Si polycrsytals

Full size image

4 Estimation of cubic crystal aggregates properties

4.1 Analytical homogenization

The prediction of crystal aggregates elastic properties is a long standing problem, cf. [50, 51]. Besides computational homogenization methods, there are analytical methods that have found widespread application. We here summarize several such approaches appropriate for the determination of effective isotropic elastic properties of cubic silicon aggregates. These approaches typically result in bounds and estimates of the polycrystals effective behavior. In the course of the specification of constitutive measures given in Sect. 2, subsequent description is based on operations with cubic eigenvalues. Again, due to the isotropic nature of volumetric changes, the bulk modulus is not subject to any change for mono- and polycrystal. Thus, for all subsequent procedures, \(\lambda _1^\circ =\lambda _1^{\textrm{c}}=\lambda _1^{}\) holds. For a concise description, we restrict ourselves to the presentation for the determination of the second eigenvalue, cf. Eq. (12) for \(\alpha =2\), of an isotropic effective stiffness, i.e.

$$\begin{aligned} \lambda _2^{\circ }={{\varvec{\mathbb C}}}^{\circ }\,{\varvec{:}}\,\,{\varvec{:}}\,\frac{{{\varvec{\mathbb P}}}_2^{\circ }}{\vert \vert {{\varvec{\mathbb P}}}_2^{\circ }\vert \vert ^2}={{\varvec{\mathbb C}}}^\circ \,{\varvec{:}}\, \,{\varvec{:}}\,\frac{1}{5}\,{{\varvec{\mathbb P}}}_2^{\circ }\;, \end{aligned}$$
(39)

which results from the tensorial operations of the respective approach.

4.2 Bounds

In engineering science, bounds have initially been proven particularly useful. These imply ranges for the elasticity parameters. Compared to computational homogenization procedures, bounds provide quick and simple estimates for the effective behavior of an aggregate. Moreover, the knowledge of the aggregates geometrical features is not available in most experimental investigations. Even if an accurate determination of the micro-geometry of the aggregate is possible, obtaining this information and its numerical parameterization can be time-consuming what is often also associated with a high computational burden, cf. e.g. [52, 53] or [54]. Thereby, usually, cross-sectional images are taken, which provide only limited information, cf. [55, 56]. Another aspect is the value of bounds in the optimization of mechanical properties by specifically modifying micro-structural properties of artificially constructed aggregates which can later be implemented in manufacturing processes, cf. [57].

However, the range of experimental results can be narrowed by means of bounds, at least the validity of basic assumptions can be verified. Hereby, bounds of different order are introduced, where the order is associated with the probabilistic description of microstructures, cf. [58]. Low-order bounds naturally give only rough information of the effective properties. High-order bounds give access to a better understanding of the range in which the effective properties are assured to be contained. In what follows, we confine our considerations to zeroth-, first-, and second-order bounds.

Zeroth-order bounds  Kröner [59] introduced bounds independently of the microstructural configuration of the aggregate by using information of the material constituents of the heterogeneous material. So, the determination of these bounds is here based on the monocrystals material parameters solely.

$$\begin{aligned}&\lambda _2^{\textrm{K}+}=\text {max}( \lambda _2^{\textrm{c}}, \lambda _3^{\textrm{c}}) \end{aligned}$$
(40a)
$$\begin{aligned}&\lambda _2^{\textrm{K}-}=\text {min}( \lambda _2^{\textrm{c}}, \lambda _3^{\textrm{c}}) \end{aligned}$$
(40b)

Since in the case of silicon \(\lambda _2^{\textrm{c}}<\lambda _3^{\textrm{c}}\) holds, \(\lambda _2^{\textrm{K}+}=\lambda _3^{\textrm{c}}\) and \(\lambda _2^{\textrm{K}-}=\lambda _2^{\textrm{c}}\) results. These bounds usually span a huge range for the domain of effective parameters since they do not incorporate any information of the microstructure statistics. However, the zeroth-order bounds are generally isotropic [60]. Even anisotropic behaviour of the aggregate takes place within these bounds.

First-order bounds  First-order bounds are associated with the names Voigt and Reuss. They are applicable if information on volume fractions of the microstructure of the aggregate is available. Then, they simply result from the volume average of the stiffness and compliance. Thereby, Voigt [61] assumed that all crystals in a polycrystal have a uniform strain field. This results in a weighted sum of second and third cubic eigenvalues.

$$\begin{aligned} \lambda _2^{\textrm{V}}=\frac{2}{5}\lambda _2^{\textrm{c}} +\frac{3}{5}\lambda _3^{\textrm{c}} \end{aligned}$$
(41a)

Contrary, Reuss [62] assumed that all crystals in a polycrystal obey a uniform stress state. This results in a weighted sum of the reciprocals of the eigenvalues.

$$\begin{aligned} \lambda _2^{\textrm{R}}=\frac{5\lambda _2^{\textrm{c}}\lambda _3^{\textrm{c}}}{2\lambda _3^{\textrm{c}}+3\lambda _2^{\textrm{c}}} \end{aligned}$$
(41b)

Voigt and Reuss bounds are very popular nowadays. Often, they are a good approximation for the effective stiffness since these bounds seem close to each other. This immediately implies that they narrow the domain of effective values compared to zero-order bounds.

Second-order bounds  A further delimitation of the effective behaviour is possible via second-order bounds. Hashin and Shtrikman [63] proposed a variational principle to get tighter bounds. In contrast to the procedure of the bounds previously introduced, Hashin and Shtrikman do not relate stress and strain fields, but quantities that represent a deviation from a reference solution. Thus, errors due to certain assumptions in the method influence the solution far less. A detailed derivation is given in [63,64,65,66,67,68] while the approach explicitly for the present case of cubic crystals in an isotropic aggregate is sketched in [69]. Upper and lower bound of such second-order bounds in the case of \(\lambda _2^{\textrm{c}}<\lambda _3^{\textrm{c}}\) are given as follows.

$$\begin{aligned} \lambda ^{\textrm{HS}+}_2&= \lambda _3^{\textrm{c}} +2\left[ \frac{5}{\lambda _2^{\textrm{c}}-\lambda _3^{\textrm{c}}} +\frac{6(\lambda _1^{\textrm{c}} +3\lambda _3^{\textrm{c}})}{5\lambda _3^{\textrm{c}}(\lambda _1^{\textrm{c}} +2\lambda _3^{\textrm{c}})} \right] ^{-1} \end{aligned}$$
(42a)
$$\begin{aligned} \lambda ^{\textrm{HS}-}_2&= \lambda _2^{\textrm{c}} +3\left[ \frac{5}{\lambda _3^{\textrm{c}}-\lambda _2^{\textrm{c}}} +\frac{4(\lambda _1^{\textrm{c}} +3\lambda _2^{\textrm{c}})}{5\lambda _2^{\textrm{c}} (\lambda _1^{\textrm{c}}+2\lambda _2^{\textrm{c}})} \right] ^{-1} \end{aligned}$$
(42b)

4.3 Unique estimates

While the above approaches serve to enclose the effective properties, there are also approaches for determining unique effective estimates. In the sequel, some well-known approaches are compiled, which have proven successful in predicting effective material parameters.

Aleksandrov and Aizenberg [70] proposed that the natural logarithm of the overall stiffness is equal to the average of the logarithms of the monocrystals stiffness. This results in an averaging scheme around the condition of the commutation of inverse and averaging oprations.

$$\begin{aligned} \lambda _2^{\textrm{A}}=(\lambda _2^{\textrm{c}})_{}^{\frac{2}{5}} (\lambda _3^{\textrm{c}})_{}^{\frac{3}{5}} \end{aligned}$$
(43)

Hill [71] proposed two mean values of the bounds suggested by Voigt and Reuss. These are the arithmetic mean

$$\begin{aligned} \lambda _2^{\textrm{H}+}=\frac{1}{2}\left( \lambda _2^{\textrm{V}} +\lambda _2^{\textrm{R}}\right) \, \end{aligned}$$
(44)

and the geometric mean

$$\begin{aligned} \lambda _2^{\textrm{H}-}=\left( \lambda _2^{\textrm{V}} \lambda _2^{\textrm{R}}\right) ^{\frac{1}{2}}\,. \end{aligned}$$
(45)

Allthough the arithmetic mean has found a much higher spread in application, cf. [18], we will nevertheless incorporate the geometric mean of first-order bounds, too. However, it is worth to mention that both effective estimates by Hill are purely empirical, i.e., there is no physical reasoning.

Huang [72] introduced an effective estimate by the aid of the perturbation method. For an isotropic aggregate of cubic crystals, this results in subsequent expression.

$$\begin{aligned} \lambda _2^{\textrm{Hg}}=\frac{2}{5}\lambda _2^{\textrm{c}} +\frac{3}{5}\lambda _3^{\textrm{c}}-\frac{12}{25} \frac{(\lambda _2^{\textrm{c}}-\lambda _3^{\textrm{c}})^2 (5\lambda _1^{\textrm{c}}+6\lambda _2^{\textrm{c}} +9\lambda _3^{\textrm{c}})}{(2\lambda _2^{\textrm{c}} +3\lambda _3^{\textrm{c}})(5\lambda _1^{\textrm{c}} +4\lambda _2^{\textrm{c}}+6\lambda _3^{\textrm{c}})} \end{aligned}$$
(46)

Obviously, this estimate is located below the first-order upper bound since the first term in Eq. (46) is identified as the Voigt estimate.

Fokin [73, 74] developed a method called singular approximation approach. This approach has been formulated for aggregates of cubic crystals by Matthies and Humbert [75]. In the case of an isotropic self-consistent solution, cf. [50], this approximation reduces as follows.

$$\begin{aligned} \lambda _2^{\textrm{SA}}=\frac{2\lambda _2^{\textrm{c}}p_2 +3\lambda _3^{\textrm{c}}p_3}{2p_2+3p_3} \end{aligned}$$
(47)

where the parameters \(p_\chi \) are determined as follows.

$$\begin{aligned} p_\chi = \left[ 1+\frac{2}{5(\lambda _2^{\mathrm c})^{\frac{2}{5}}(\lambda _3^{\mathrm c})^{\frac{3}{5}}}\frac{\lambda _1+3(\lambda _2^{\mathrm c})^{\frac{2}{5}}(\lambda _3^{\mathrm c})^{\frac{3}{5}}}{\lambda _1+2(\lambda _2^{\mathrm c})^{\frac{2}{5}}(\lambda _3^{\mathrm c})^{\frac{3}{5}}} \left( \lambda _\chi -(\lambda _2^{\mathrm c})^{\frac{2}{5}}(\lambda _3^{\mathrm c})^{\frac{3}{5}}\right) \right] ^{-1}{} & {} \forall \;\chi \in \{2,3\} \end{aligned}$$
(48)

We herein may identify terms defined by the geometric mean of Aleksandrov and Aizenberg. Detailed derivations can be found in [76].

5 Results and discussion

5.1 Analytical results

Based on aforementioned formulae of Sect. 4, it is possible to determine datasets of isotropic effective aggregate behavior and their bounds based on the monocrystals cubic input data given in Table 1. This results in isotropic effective aggregate stiffnesses for the respective homogenization approach presented in Sects. 4.2 and 4.3.

$$\begin{aligned} {{\varvec{\mathbb C}}}^{\square } = \lambda _1^{}{{\varvec{\mathbb P}}}_1 +\lambda _2^{\square }{{\varvec{\mathbb P}}}_2^\circ{} & {} \forall \;{\square }\in \{\text {K}{+}, \text {K}{-}, \text {V}, \text {R}, \text {A}, \text {H}{+}, \text {H}{-}, \text {HS}{+}, \text {HS}{-}, \text {Hg}, \text {SA}\} \end{aligned}$$
(49)

However, we tabulate computed effective values of the elastic behavior of the silicon aggregate in Table 5. These effective outcomes based on all input given in Table 1 will be utilized for evaluation in subsequent sections. These results are also visualized in Fig. 4. Since Kröner bounds do not contribute significantly in narrowing the isotropic effective mechanical properties, they are not included in the graphical representation. However, the effective estimates presented are also converted to engineering parameters which are presented in Table 6 of App. A.1.

Table 5 Computed bounds and effective eigenvalues for an macroscopically isotropic Si aggregate
Full size table
Fig. 4
figure 4

Parameter space of analytically determined isotropic eigenvalues of the effective medium and scattering in homogenization procedure due to different inputs

Full size image

Generally, we can identify the following relations which hold true for all silicon monocrystals inputs:

  • As mentioned, the zeroth-order bounds by Kröner [Eqs. (40a) and (40b)] give only a rough delimitation of the domain of possible effective results.

  • Approaches based on Voigt and Reuss [Eqs. (41a) and (41b)] lead to realistic outer bounds. All other estimates are within these extremes.

  • Hashin–Shtrikman bounds considerably limit the range of feasible estimates compared to Voigt-Reuss bounds.

  • Estimates by Hill [Eqs. (44) and (45)], Aleksandrov and Aizenberg [Eq. (43)], Huang [Eq. (46)], and the singular approximation approach Eq. (47) lie in between the Hashin–Shtrikman bounds.

  • Hill’s estimates generate the smallest values within the Hashin–Shtrikman bounds.

  • Huang’s estimate generate the highest values within the Hashin–Shtrikman bounds.

  • The estimate proposed by Aleksandrov and Aizenberg is located above Hill’s estimates but below the singular approximation approach.

In summary, the zeroth- (Kröner), the first- (Voigt and Reuss) and second-order bounds (Hashin-Strikman) enclose the effective material behavior of the polycrystal. The unique estimates determined are all within the range of the second-order bounds. Thus, subsequent relation holds for present material.

$$\begin{aligned} \lambda _2^{\mathrm {K+}}>\lambda _2^{\textrm{V}}> \lambda _2^{\mathrm {HS+}}> \lambda _2^{\textrm{Hg}}>\lambda _2^{\textrm{SA}}>\lambda _2^{\textrm{A}}> \lambda _2^{\mathrm {H+}}>\lambda _2^{\mathrm {H-}}> \lambda _2^{\mathrm {HS-}}>\lambda _2^{\textrm{R}}>\lambda _2^{\mathrm {K-}} \end{aligned}$$

This equivalently leads to the following bounds hierarchy with concomitant account taken of the unique estimates.

$$\begin{aligned} {{\varvec{\mathbb C}}}^{\mathrm {K+}}>{{\varvec{\mathbb C}}}^{\textrm{V}}>{{\varvec{\mathbb C}}}^{\mathrm {HS+}}> {{\varvec{\mathbb C}}}^{\textrm{Hg}}>{{\varvec{\mathbb C}}}^{\textrm{SA}}>{{\varvec{\mathbb C}}}^{\textrm{A}}>{{\varvec{\mathbb C}}}^{\mathrm {H+}}> {{\varvec{\mathbb C}}}^{\mathrm {H-}}>{{\varvec{\mathbb C}}}^{\mathrm {HS-}}>{{\varvec{\mathbb C}}}^{\textrm{R}}>{{\varvec{\mathbb C}}}^{\mathrm {K-}} \end{aligned}$$

However, it is worth mentioning that the global bounds of the cubic monocrystal, i.e., \(Y_{\max }\) and \(Y_{\min }\) of Table 2, correspond to the zeroth-order bounds \({{\varvec{\mathbb C}}}^{\mathrm {K+}}\) and \({{\varvec{\mathbb C}}}^{\mathrm {K+}}\). This is particularly evident in the sectional view of the plots of Young’s modulus contrasted in Fig. 5, left-hand side (The grey line represents the bisector of the first quadrant in the 1–2-plane). There, the Young’s modulus bodies of \({{\varvec{\mathbb C}}}^{\square }\,\forall \;\square \in \{\mathrm {K+,K-,c}\}\) are opposed. The minima and maxima of the Young’s modulus body of the silicon monocrystal are the points of contact to the zeroth-order upper and lower bound of the aggregates elasticity. The red points indicate contact with the upper bound while the blue ones do that for the lower bound. On the right hand-side of Fig. 5, the bounds of different order considerd here are compared. It turns out that by increasing the order of approximation, a substantial containment of the effective material behaviour can be achieved, at least with respect to Young’s modulus. In figures, this means a reduction of bandwidth,

  • \(\Delta Y^{\textrm{K}}=57.67\,\textrm{GPa}\) (zeroth order),

  • \(\Delta Y^{\textrm{VR}}=6.3\,\textrm{GPa}\) (first order), and

  • \(\Delta Y^{\textrm{HS}} =0.57\,\textrm{GPa}\) (second order)

with respect to input of Hall [27]. Focusing on the relationship of the unique estimates within the second-order bounds, it must be noted due to the narrow bandwidth of \(0.57\,\textrm{GPa}\), it is difficult to judge the predictive quality of the respective approaches that lead to unique results.

Fig. 5
figure 5

Visual comparison of calculated bounds (data used here is restricted to input of [27] for the sake of clarity)

Full size image

5.2 Scattering due to variance of input data

It is possible to judge the effective properties range of the aggregate’s behavior while there are different appropriate scalar measures for the size of the domain spanned by corresponding bounds [75, 77, 78]. We here make use of the Frobenius norm of the difference between respective upper and lower bounds [79]. In terms of the stiffnesses determined in Sect. 4.2, such distances are defined as follows for zeroth-order bounds

(50)

first-order bounds

(51)

as well as second-order bounds

(52)

Obviously all distances are related to increments of the degree of anisotropy , cf. Equation (32), where \({{\mathcal {O}}}\in \{0,1,2\}\) is the order of the bounds considered. The results shown in Fig. 6 (left-hand side) reveal that in present case distances decrease with increasing order. When comparing the distances with regard to the different inputs, we find only very small and almost negligible changes. Contrary, as was to be expected, the improvement of the distance between upper and lower bounds in the degree of approximation order is clearly evident. The first-order bounds show the largest distances. For present material \(D^\textrm{VR}\approx 11\cdot 10^{-2} D^\textrm{K}\) and \(D^\textrm{HS}\approx 8.9\cdot 10^{-2} D^\textrm{VR}\) results. With each increase in the order of approximation, a reduction of the distances by approximately one order of magnitude can be noted. Specifically this means that for present material the Hashin–Shtrikman bounds are tighter by a factor of around 11 compared to Voigt-Reuss bounds, and by a factor of around 100 compared to Kröner bounds.

Fig. 6
figure 6

Distances of classical bounds, shear modulus ratio of estimates to Voigt bound, and isotropic eigenvalue ratio

Full size image

5.3 Dispersion due to the choice of homogenization approach

As mentioned in Sect. 4, \({{\varvec{\mathbb C}}}^\textrm{V}\) represents the upper bound of the isotropic elastic behavior. In the sequel we will hence normalize the homogenization results to this measure. More precisely, we reduce this normalization to the second isotropic eigenvalue  \(\lambda _2^\textrm{V}\). This restriction is possible since the first eigenvalues are all identical, at least in context of the analytical procedure, cf. Table 5.

To judge the dispersion of the results due to the homogenization approach, we make use of the shear modulus ratio, i.e. we normalize above mentioned equations by the shear modulus predicted by the Voigt bound.

(53)

Clearly, applying this scheme to the first eigenvalue, i.e. determining the bulk modulus ratio, will theoretically lead to

(54)

for any homogenization approach. However, results of Eq. (53) are depicted in Fig. 6 (center). It is shown that all results of the analytically determined shear modulus in relation to the Voigt bound are within a range from around . This narrow range of dispersion is not least due to the low degree of anisotropy of the silicon single crystals. All shear modulus ratios within Hashin-Shtrikman bounds are closely spaced by . All results of do not differ with respect to the input variables of the monocrystal, at least not in a perceptible range.

When applying the normalization in Eq. (53), all results depend on Zener ratio and the eigenvalue ratio solely. Since the Zener ratio is almost identical for all inputs, cf. Table 1, we reduce further evaluations to the eigenvalue ratio as already introduced in Eq. (38). Due to different analytical approaches, we introduce the eigenvalue ratio in subsequent manner.

(55)

Results of this ratio are depicted in Fig. 6 on the right-hand side. The eigenvalue ratio of all analytical results between Voigt and Reuss estimates is bounded as while results based on Aleksandrov and Aizenberg [70] lead to . Obviously, the theoretical estimates result in a tighter range of the eigenvalue ratio than it is the case with the experimental findings, cf. Table 4 ( ), which is related to the much larger scatter of the different tests, at least with respect to the second eigenvalue.

Obviously, while comparing computed effective parameters for an isotropic silicon aggregate, we can generally determine three simple but significant parameters. Those are based on the purposeful approaches by

  • Voigt [61], i.e. the arithmetic mean representing the upper bound,

  • Reuss [62], i.e. the harmonic mean representing the lower bound, and

  • Aleksandrov and Aizenberg [70], i.e. the geometric mean obviously representing a good approximation for a unique estimate.

These cases define the two reasonable extremes while the latter point is a good approximation for a unique estimate due to the central placement within the narrow second-order bounds, in which all unique estimates are located [80]. The straightforward mathematical handling of all three approaches is also a reason for this choice. Hence, for comparison of the analytical with the experimental findings, we restrict further examinations to these key approaches indexed by \(\{\textrm{V,A,R}\}\).

5.4 Comparison with aggregates experimental findings

The compiled experimental data and the calculated parameters of polycrystalline silicon are juxtaposed in Fig. 7. Unsurprisingly, the experimental findings for the first eigenvalue \(\lambda _1^\delta \) lie in between the theoretical values bounded by the red data points (McSkimin and Andreatch [31]) and the yellow data points (Hall [27]) while both represent minimal and maximal values in present considerations. Again, the determination of this first eigenvalue is in present work based on theoretical considerations. We additionally make use of the arithmetic mean (36) of monocrystal data. In contrast, the experimental findings of the second eigenvalue \(\lambda _2^\delta \) partly exceed the upper bound represented by Voigt’s approach. The second eigenvalues of [43]( ), [37]( ), [40]( ), [43]( ), [43]( ), and [43]( ) are up to \(8.5\,\textrm{GPa}\) above Voigt bound \(\lambda _2^{\mathrm V}\). From a theoretical standpoint, such a transgression is impossible in terms of an isotropic aggregate. The second eigenvalues of an isotropic aggregate must be within the Voigt-Reuss bounds. Nonetheless, we continue to incorporate these data and investigate the deviation of the analytical results from the experimental findings. We therefore compare the stiffnesses of Eqs. (37) and (49). Usually, an error function e is introduced to compare the calculated stiffnesses ( \(\square \)) of data sets \(\gamma \) with experimental polycrystals data ( \(\delta \)).

(56)
Fig. 7
figure 7

Comparison of calculated and experimental results and their deviations in terms of error function and computation- to-experiment ratio

Full size image

The results of this error are depicted in Fig. 7 (center). The plot confirms that is closest to the Voigt bound, is closest to the Reuss bound, and , , are closest to the estimate according to Aleksandrov and Aizenberg. Experimental results show the strongest deviation to all three estimates. However, since the error measure e only gives an absolute value, no assessment on the direction of the prediction can be made, i.e. whether over- or underestimation is present. Since in present study such an error measure is dependent on the two isotropic eigenvalues only, we additionally can introduce the computation-to-experiment ratio .

(57)

From a theoretical standpoint, all aggregate stiffnesses \({{\varvec{\mathbb C}}}\) of the effective estimates differ by the second eigenvalue solely. Thus, we can reduce our concern to the ratio . For results with applies that experimental findings exceed the analytical predictions, while vice versa, for results with analytical predictions exceed experimental results. The results for this measure are depicted in Fig. 7 at the right-hand side. Obviously, experimental findings , , and fit quite well in the predictions with the geometric mean. In contrast, and deviate the most from the predictions. Considering the approach of Aleksandrov and Aizenberg [70] as a good prediction for the effective behavior, the experimental findings vary approximately between \(-2\%\) and \(+8\%\) of .

5.5 Discussion

Obviously, there are discrepancies between experimental results and analytically estimates. Quite concretely, some of experimental results lie outside the admissible range predicted for an isotropic bulk. In this section we want to discuss which influences may have disturbed the experimental findings.

One reason may, that one has to account for temperature influences at experiments, i.e., the stiffness measured is a function of the temperature, \({{\varvec{\mathbb C}}}(\theta )\). E.g., Michalicek, Séné, and Bright [42] report a fairly linear dependency of Young’s modulus Y and absolute temperature \(\theta \) in form of the following relationship,

(58)

where Z is the slope and \(Y_0\) is the inhomogeneity. Referring to room temperature which we have assumed in present work, \(Y(273.15\,\textrm{K})=163.58\,\textrm{GPa}\) results, which is a pretty good finding in comparison with the analytical approximations. In contrast, the experimentally determined maximum of  \(174\,\textrm{GPa}\) from [43] in this context would mean a temperature below absolute zero, to be exact \(-49.83\,\textrm{K}\). A temperature below \(0\,\textrm{K}\) is impossible [81]. Such large deviations of the experimentally determined data for the Young’s modulus can therefore not be justified by temperature effects.

The experimental data summarized in Sect. 3.3 was, to a large extent, determined on thin layers or thin films. Here, size effects can have an influence on the mechanical behavior, even in the elastic range [82]. Generally, size effects may result in an increase or decrease in the quantities representing the mechanical properties. [83]. Such size effects were experimentally determined by, e.g., Mizubayashi, Yoshihara, and Okuda [84]. There, for example, a decrease of Young’s modulus for aluminum films with a thickness below \(100\,\textrm{nm}\) is reported. Besides, such size effects are also known for foams or bones, cf. [85, 86]. Nowadays, generalized continuum theories which incorporate an inner length scale are fashionable to explain such size effects, cf. [87, 88]. In contrast, Knowles [89] showed that such effects can also be described by means of plane state conditions, which especially appear during bending of relatively wide beams, in which the transverse bending is restricted within the central portion of the beam, as described by [90]. This is based on the premise of Altenbach and Eremeyev [91] who theoretically proved such size effects on plate structures due to thickness dependence.

When measuring on such thin structures, it is convenient to invoke the concepts of plane state conditions, i.e. the plane stress or the plane strain state. Theses states are related to individual assumptions on stresses and strains which lead to alterations in the components of the reduced constitutive tensor [92]. Such procedure leads to reduced constitutive laws compared to Eq. (1). We hence mark these new quantities in question with a triangle, i.e. \({{\varvec{T}}}^\triangle ={{\varvec{\mathbb C}}}^\triangle \,{\varvec{:}}\,{{\varvec{E}}}^\triangle \). The relations of elastic parameters between the three-dimensional state as assumed in Sect. 2 and the plane states mentioned was derived by Nordmann, Aßmus, Glüge, and Altenbach [93] in a lengthly and detailed procedure. Amongst others, they have determined the isotropic reduced stiffnesses for plane stress state (superscript index  \(\triangle =\mathrm {p\sigma }\)) and plane strain state (superscript index  \(\triangle =\mathrm {p\varepsilon }\)) which reads as follows.

$$\begin{aligned} {{\varvec{\mathbb C}}}^\triangle =\sum _{\alpha =1}^{2}\lambda _\alpha ^\triangle {{\varvec{\mathbb P}}}_\alpha ^\triangle \quad \text {with}\quad {{\varvec{\mathbb P}}}_1^\triangle =\frac{1}{2}{{\varvec{I}}}^\triangle \otimes {{\varvec{I}}}^\triangle \; \text {and}\;{{\varvec{\mathbb P}}}_2^\triangle =[{{\varvec{\mathbb I}}}^\triangle ]^{\textrm{sym}}-{{\varvec{\mathbb P}}}_1^\triangle \end{aligned}$$
(59)

Herein, we have introduced as the symmetric part of the fourth-order identity and \({{\varvec{I}}}^\triangle ={{\varvec{e}}}_I\otimes {{\varvec{e}}}_I={{\varvec{I}}}-{{\varvec{e}}}_3\otimes {{\varvec{e}}}_3\) as the second-order identity, both defined on reduced, two-dimensional bodies, cf. [94, 95]. In the process, the fourth-order identity \({{\varvec{\mathbb I}}}^\triangle ={{\varvec{e}}}_I\otimes {{\varvec{e}}}_J\otimes {{\varvec{e}}}_I\otimes {{\varvec{e}}}_J\) and the transposer \({{\varvec{\mathbb T}}}^\triangle ={{\varvec{e}}}_I\otimes {{\varvec{e}}}_J\otimes {{\varvec{e}}}_J\otimes {{\varvec{e}}}_I\) are reformulated, whereby majuscule indices, i.e. I or J, are running from 1 to 2 only. Furthermore, \({{\varvec{\mathbb P}}}_\alpha ^\triangle ={{\varvec{\mathbb P}}}_\alpha ^{\textrm{p}\sigma }={{\varvec{\mathbb P}}}_\alpha ^{\textrm{p}\varepsilon }\) are the isotropic projectors of the plane states, i.e. \({{\varvec{\mathbb P}}}_1^\triangle \) maps into a one-dimensional dilatoric space and \({{\varvec{\mathbb P}}}_2^\triangle \) maps into a two-dimensional deviatoric space. The isotropic eigenvalues can be determined as follows.

$$\begin{aligned} \lambda _1^{\triangle }={{\varvec{\mathbb C}}}^{\triangle }\,{\varvec{:}}\,\,{\varvec{:}}\,\frac{{{\varvec{\mathbb P}}}_1^{\triangle }}{\vert \vert {{\varvec{\mathbb P}}}_1^{\triangle } \vert \vert ^2}={{\varvec{\mathbb C}}}^\triangle \,{\varvec{:}}\,\,{\varvec{:}}\,\,{{\varvec{\mathbb P}}}_1^{\triangle }\;, \quad \lambda _2^{\triangle }={{\varvec{\mathbb C}}}^{\triangle }\,{\varvec{:}}\,\,{\varvec{:}}\,\frac{{{\varvec{\mathbb P}}}_2^{\triangle }}{\vert \vert {{\varvec{\mathbb P}}}_2^{\triangle } \vert \vert ^2}={{\varvec{\mathbb C}}}^\triangle \,{\varvec{:}}\,\,{\varvec{:}}\,\frac{1}{2}\,{{\varvec{\mathbb P}}}_2^{\triangle }\;, \end{aligned}$$
(60)

According to further executions in [93] we can state the reduced isotropic stiffnesses in terms of eigenvalues for both plane state conditions as follows.

$$\begin{aligned} \lambda _1^{\mathrm {p\sigma }}=\frac{3\lambda _1^\circ \lambda _2^\circ }{\,\lambda _1^\circ +2\lambda _2^\circ } \qquad \lambda _1^{\mathrm {p\varepsilon }}=\frac{1}{3}[2\lambda _1^\circ +\lambda _2^\circ ] \qquad \lambda _2^{\triangle }=\lambda _2^\circ \quad \forall \;\triangle \in \{\textrm{p}\sigma ,\textrm{p}\varepsilon \} \end{aligned}$$
(61)

Obviously, the second eigenvalue is identical for both plane states and, both coincide with the second eigenvalue of the three-dimensional state. This is since the second eigenvalue—associated with the shear modulus—is already a two-dimensional measure. Thus, only the first eigenvalue of plane state stiffnesses-associated with the bulk modulus—has to be recalculated. We can state the following relations for the first eigenvalues.

$$\begin{aligned} \lambda _1^{\mathrm {p\sigma }}\le \lambda _1^{\circ } \qquad \frac{1}{\lambda _1^{\mathrm {p\varepsilon }}}\le \frac{1}{\lambda _1^{\circ }} \end{aligned}$$
(62)

This implies that the stiffness is decreased in plane stress state and increased in plane strain state. For a plane strain state, this reflects the reaction stresses due to the kinematic constraint of plane deformations. Hence, for present considerations, the plane strain condition is of interest. The eigenvalues of the plane strain state stiffness \({{\varvec{\mathbb C}}}^{\mathrm {p\varepsilon }}\) are given as follows.

$$\begin{aligned} \lambda _1^{\mathrm {p\varepsilon }}=2K^{\mathrm {p\varepsilon }} \qquad \lambda _2^{\mathrm {p\varepsilon }}=2G^{\mathrm {p\varepsilon }} \end{aligned}$$
(63)

Here, in fact, \(K^{\mathrm {p\varepsilon }}\) and \(G^{\mathrm {p\varepsilon }}\) are the bulk and shear moduli of plane strain state by which the constitutive tensor (59) is determined completely for the case \(\Delta ={\mathrm {p\varepsilon }}\). This implies, that Young’s modulus and Poisson’s ratio need a redefinition, i.e. \(Y^{\mathrm {p\varepsilon }}\) and \(\nu ^{\mathrm {p\varepsilon }}\)

As already reported, the experiments are focused on the determination of Young’s modulus. We will therefore recalculate the Young’s moduli determined experimentally by transforming them into plane strain state conditions whereby we simultaneously transform these engineering parameters \(Y^{\mathrm {p\varepsilon }}\) and \(\nu ^{\mathrm {p\varepsilon }}\) into eigenvalues \(\lambda _\alpha ^\circ \). This is done by the set of equations given in Table 7 of Sect. A.2. However, we restrict our comparison to the arithmetic, geometric, and harmonic means determined analytically. Thus we can compare these the transformed data of experimental findings given in Fig. 3 directly. This juxtaposition is given in Fig. 8. Even when the relations \(K^{\textrm{p}\varepsilon }>K\) and \(G^{\textrm{p}\varepsilon }=G\) are obvious due to the linear dependence to the eigenvalues, cf. Equation(61), we can render the following relations, at least in the case of a positive eigenvalue ratio, cf. Eq. (38).

  • \(Y^{\textrm{p}\varepsilon }>Y\)

  • \(\nu ^{\textrm{p}\varepsilon }>\nu \)

Further comparisons, e.g. related to the stiffness tensors \({{\varvec{\mathbb C}}}\) and \({{\varvec{\mathbb C}}}^{\textrm{p}\varepsilon }\), would be vague at this point, as these two tensors refer to different dimensions [93].

We now consider the experimental findings in question, i.e. [43]( ), [37]( ), [40]( ), [43]( ), [43]( ), and [43]( ), to be results of an investigation under plane strain conditions. The outcome of all theses experiments is the Young’s modulus tabulated in Table 4, while in terms of the plane stress state we now designate them as . Another known measure whose validity remains is the bulk modulus K where we follow the procedure of Eq. (36) to determine a unique value as given in Table 4. By the aid of the relations between different engineering measures in different states given in Tables 8 and 9, on may derive subsequent expression for the relation between the original bulk modulus K, plane stress Young’s modulus \(Y^{\textrm{p}\varepsilon }\), and the Young’s modulus of the bulk Y.

$$\begin{aligned} Y_\delta = \frac{3 Y^{\textrm{p}\varepsilon }_\delta K \left[ 9 K - 2 q^{}_\delta + 2 Y^{\textrm{p}\varepsilon }_\delta \right] }{\left[ 6K + Y^{\textrm{p}\varepsilon }_\delta - q^{}_\delta \right] ^2} \qquad q^{}_\delta =\sqrt{{Y^{\textrm{p}\varepsilon }_\delta \left( 9 K + Y^{\textrm{p}\varepsilon }_\delta \right) }} \end{aligned}$$
(64)
Fig. 8
figure 8

Experimentally determined elasticity parameters after applying the correction for plane strain state measurements, compared with analytically determined estimates (theoretical results are here restricted to input based on [27])

Full size image

The results of this rectification are visualized in Fig. 8 on the left-hand side while the respective associated \(\nu \) is determined on the basis of Eq. (34). It can be seen that the corrected parameter sets (green) are much closer to the untreated experimental results (black), compared to Fig. 7 (left-hand side). Based on the engineering parameters, it is easy to transform these results to the space of isotropic eigenvalues. These recomputed eigenvalues are juxtaposed in Fig. 8 on the right-hand side. One recognizes immediately, that the experimental findings of sources , , and now fit into the Voigt-Reuss bounds determined analytically. The results of sources , , and still lie outside above Voigt bound. Considering the Voigt-Reuss bounds made visible in Fig. 8 (left-hand side), it is recognizable that the parameters are partly only just beyond the permissible range. Therefore, and could be assumed to be within a tolerance due to measurement uncertainty. This does not apply to data set .

To be just, these converted results should be considered with a certain degree of caution. The material parameters of the two plane states (designated with \(\textrm{p}\varepsilon \) or \(\textrm{p}\sigma \)) are based on the projection of the elasticity tensor onto a plane. Unlike transformations, projections are non-invertible. Thus, the determination of plane material parameters is justified, while the re-transformation to parameters of the bulk is not, at least not generally.

In case of experimental results outside the admissible range, the reasons already referred to in [18] and [69], i.e., an anisotropic aggregate behavior, must be considered.

The anisotropic behavior of aggregates is described through a texture, i.e. the presence of preferred orientations of the single crystals in a polycrystal. This naturally yields the question of the samples texture symmetry since the effective stiffness \(\overline{{{\varvec{\mathbb C}}}}\) then incorporates an isotropic and an anisotropic part, cf. [96].

$$\begin{aligned} \overline{{{\varvec{\mathbb C}}}}={{\varvec{\mathbb C}}}^\circ +{{\varvec{\mathbb C}}}'\qquad {{\varvec{\mathbb C}}}^\circ =(\overline{{{\varvec{\mathbb C}}}} \,{\varvec{:}}\,\,{\varvec{:}}\,{{\varvec{\mathbb P}}}_1){{\varvec{\mathbb P}}}_1+\frac{1}{5}(\overline{{{\varvec{\mathbb C}}}} \,{\varvec{:}}\,\,{\varvec{:}}\,{{\varvec{\mathbb P}}}_2^\circ ){{\varvec{\mathbb P}}}_2^\circ \end{aligned}$$
(65)

The tensor \({{\varvec{\mathbb C}}}'\) constitutes the purely anisotropic part of the effective stiffness while \({{\varvec{\mathbb C}}}'={{\varvec{\mathbb O}}}\) represents the isotropy condition in terms of crystal orientations, i.e. the absence of texture. Since at this juncture, to the best of the authors knowledge, tangible information on textures in polycrystalline silicon cannot be found in the literature, this approach initially leads only to vague estimations. However, even in the case of a textured aggregate, effective stiffnesses must lie within the zeroth-order bounds, i.e.

$$\begin{aligned} \overline{{{\varvec{\mathbb C}}}}\in [{{\varvec{\mathbb C}}}^{\textrm{K}+},{{\varvec{\mathbb C}}}^{\textrm{K}-}]\;, \end{aligned}$$
(66)

that means for example within the extremes of the single crystals Young’s modulus, cf. Eq. (16) and Table 2. These bounds represent two extreme cases which result from \(\langle 100\rangle \) or \(\langle 111\rangle \) texture. However, for the reviewed literature, all experimental findings lie in between these extremes, i.e. \( {{\varvec{\mathbb C}}}^{\delta }\in \;]{{\varvec{\mathbb C}}}^{\textrm{K}+},{{\varvec{\mathbb C}}}^{\textrm{K}-} [ \). Therefore, it is worthwhile to pursue the approach to determine the effective anisotropic behavior by at least investigating the arithmetic, harmonic, and geometric means of textured silicon aggregates, which considerably narrow the effective properties compared to zeroth-order bounds determined here, cf. Fig. 5. Nevertheless, the aggregates elastic anisotropy in a Frobenian sense, i.e. , should not be very strong what is attributed to the relatively moderate degree of anisotropy of the silicon monocrystals, cf. of Eq. (32) and Table 1.

6 Summary and concluding remarks

6.1 Summary

We have presented in this paper a general overview of the elastic properties of polycrystalline silicon. This survey has three aspects: Firstly, the literature on experimental findings is thoroughly reviewed. Secondly, theoretical approaches to determine the effective behavior analytically are compiled. Thirdly, experimental findings and theoretical results are compared and their relations are revealed.

Due to the brittle characteristics of silicon, we can assume purely linear material behavior, i.e. geometrical and physical linearity. While delineating the theory of the elastic behavior, both, for the single crystal and the polycrystal, we assume the aggregates isotropy which seems quite a good approximation when following the tracts on experimental investigations of polycrystalline silicon. For the single crystal, we indicate extremes due to the special anisotropic behavior. We furthermore clearly address the differences between crystal and aggregate level and introduce visualizations that underline these features.

In sequence, we present a thorough review of the literature on experimental findings. First, we rely on analyses of the individual crystal. It turns out that the first analyses of the elasticity parameters of monocrystalline silicon date back about 60 years. Although there have been a few subsequent studies since then, the differences in their results are almost negligible. Furthermore, the degree of anisotropy of monocrystalline silicon is comparably low. Nonetheless, we analyze the differences of classical elasticity parameters caused by anisotropy. The scattering in the cubic properties is given by \(\Delta \lambda ^{\textrm{c}}_1=150\,\textrm{MPa}\) and \(\Delta \lambda ^{\textrm{c}}_2=\Delta \lambda ^{\textrm{c}}_3=100\,\textrm{MPa}\).

The scattering in experimental findings for polycrystalline silicon is considerably larger compared to that for monocrystalline silicon. They can be expressed by the second eigenvalue solely, where \(\Delta \lambda ^{\delta }_2=14.21\,\textrm{GPa}\) results. This corresponds to a variance of the Young’s modulus similar in magnitude while the mean value is \(\textrm{Mean}(Y^{\delta })=166.2\,\textrm{GPa}\).

Several, now classical analytical approaches for the prediction of isotropic elastic properties are then introduced and has been applied to estimate the behavior of the silicon aggregate. The deviatoric portions of the single crystal elasticity tensor are used to form the aggregate effective parameters. This first specifies different order bounds of the elastic behavior while we also introduce approaches which result in the prediction of a unique effective value. The determination of effective parameters is given in terms of the eigenvalues of the stiffness tensor. The reasons are obvious: eigenvalues are advantageous in mathematical handling and are justified physically clearly. Hence, the effective parameters are presented by simple algebraic relations. The high degree of lucidity and clarity in present descriptions is solely based on the projector representation of the stiffness. We give exact solutions for all approaches, while different measures are introduced to describe distances between bounds to judge homogenization approaches and influences due to scattering in the data input. Such an in-depth analysis is motivated by some divergences of experimental findings compared to theoretically possible ranges. An attempt was made to evaluate these deviations by means of error measures. Just for comparison to the experimental findings, \(Y^{\textrm{A}}\approx 162.69\,\textrm{GPa}\) results for the geometric mean which at first glance reflects the basic trend of underestimation by the analytical predictions.

6.2 Conclusion and prospects

The validity of analytical and experimental results are judged by a combination of rationality and empiricism, where physical reasonability comes to the fore. Although the experimental data confirm the magnitude of the effective material parameters which is predicted by the analytical estimates, there are some discrepancies as the nature of the experimental samples are not sufficiently described. It is shown that certain parts of the experimental findings lie outside Voigt-Reuss bounds. Thereby, all outliers lie above the expected maximum value, i.e. the Voigt bound. We then make use of significant properties of the bounds. While one is usually interested in the smallest possible bounds of the effective material behavior, we have also shown the benefits of very coarse bounds, here the zeroth-order bounds. These are generally isotropic, but also delimit a potentially anisotropic material behavior of the aggregate.

Various possible reasons for the experimental deviations are considered and their influence on the mechanical behavior is investigated. Temperature effects could be excluded. Due to the experiments usually conducted on thin layers, the consideration of the plane stress or plane strain conditions is reasonable as much larger values of the Young’s modulus—the relevant measure in these experiments—must be expected there. The extent of the influence of plane state conditions remains open for the moment, while plane strain state could be identified as the decisive condition. Individual values of the experimental outliers could be mapped into the admissible range for an effective isotropic bulk, i.e. within the first-order bounds, by means of a correction procedure. However, this did not apply to all experimental data. Here it has to be noted that the measurements on thin layers may also induce anisotropy. Since it is proven for all experimental data that they lie within the zeroth-order bounds, i.e., in the bounds in which the effective parameters of any polycrystal aggregate can be found, anisotropic material behavior must be presupposed for the alleged outliers.

In contrast to several specifications of silicon aggregate’s elasticity given in the literature, the present paper for the first time defines a range in which experimental findings must be located. The suggested analytical approaches can be applied without any further restriction to polycrystalline silicon in a physically linear context. The simple bounds introduced represent the enclosures for the real material behavior. It has been shown that by the increase of the approximation order of the bounds, the domain of effective parameters could be narrowed considerably. For isotropic aggregates, second-order bounds turned out to be sufficiently accurate since all unique estimates are very close in between these bounds. Obviously, the approach by Aleksandrov and Aizenberg represents a unique estimate which is very well suited to predict effective isotropic properties due to the close distance between relevant bounds and due to its simple determination.

Silicon is by far not as well studied as, e.g., steel, at least when taking account of the distribution of composition, size, and orientation of its micro-constituents. A common source of anisotropy in a polycrystal is the microscopic variation in orientation of its crystals. However, in the case of aggregate anisotropy, it is necessary to investigate whether the approximations to isotropic elastic behavior introduced here analytically are reasonable, i.e., the deviation of isotropic to anisotropic prediction is acceptable. Otherwise, investigations by the aid of the finite element method become necessary. Such analyses appear to be appropriate only if data on the distribution functions of the grains, grain edges and grain interfaces are available. For present problem one can furthermore suspect a size effect or a scaling problem also since the experimental findings were determined on dimensionally reduced structures, i.e. beams or plates, while the analytical methods to determine effective properties assume a bulk material. Such problems are stenciled in [97] for microstructured materials. Hence, this is an area for continuing research. Both perspectives, anisotropy and size effects, will be explored in future work.

Table 6 Computed parameters of the effective medium for an macroscopically isotropic Si aggregate in terms of Y and  \(\nu \)
Full size table

However, due to the extensive survey on silicon in general, the analysis and the evaluation presented here, this paper can be valuable for a very broad audience. Thus, we hope to have sufficiently answered not only the question “What is the Young’s modulus of silicon?”, c.f. [90, 98], but also gave an comprehensive insight into the behavior of different material parameters of mono- and polycrystalline silicon.