# The Biaxial Moduli of Cubic Materials Subjected to an Equi-biaxial Elastic Strain

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## Abstract

Formulae for the biaxial moduli along the directions of principal stress for \((\mathit{hkl})\) interfaces of cubic materials are given for situations in which there is equi-biaxial strain within the plane. These formulae are relevant in the consideration of the deposition of thin films on single crystal substrates such as silicon. Within a particular \((\mathit{hkl})\), the directions defining these principal biaxial moduli are shown to be those along which there are the extreme values of the shear modulus and Poisson’s ratio. Conditions for stationary values of the biaxial moduli are also derived, from which the conditions for the global extrema of the biaxial moduli are established.

## Keywords

Anisotropy Cubic materials Elasticity Biaxial moduli Tensor algebra## Mathematics Subject Classification (2000)

74E10 74E15 74B05## 1 Introduction

In contrast to mathematical expressions and their graphical representations for Young’s modulus, Poisson’s ratio and the shear modulus, as a function of crystal orientation for cubic crystals [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], expressions in the literature for the orientation dependence of the biaxial modulus of cubic materials subjected to an equi-biaxial elastic strain are limited to circumstances where this strain is in a {001}, {111} or {011} plane [17, 18, 19, 20, 21, 22, 23]. In almost all studies in which biaxial moduli are of interest, the material can reasonably be assumed to be isotropic, e.g., if the material is a glass or a fully annealed polycrystalline metal or ceramic. In such circumstances, it is straightforward to show that if there is zero stress perpendicular to the plane on which there is equi-biaxial strain (the Kirchhoff hypothesis of thin plate theory [20]), the single-valued biaxial modulus is \(E/(1- \nu)\), where \(E\) is Young’s modulus and \(\nu\) is Poisson’s ratio [19, 20, 22].

However, as Janssen et al. note [22], biaxial moduli are important in the deposition of thin films on single crystal substrates, such as in the semiconductor industry where single crystal silicon is a common substrate, and where a state of equi-biaxial strain in the film and substrate has been introduced during deposition and/or by a change in temperature. For example, a polycrystalline metal film might be deposited on the silicon substrate at a high temperature, after which the thin film/substrate assembly is cooled to room temperature. This equi-biaxial strain introduces a state of biaxial stress in the thin film and curvature in the thin film/substrate assembly [19, 20, 22] which can be measured experimentally. This physical situation of an equi-biaxial strain in the material of interest and a zero normal stress also applies to the situation where a single crystal thin film with a \((\mathit{hkl})\) orientation parallel to the substrate and with a cubic crystal structure is deposited onto an isotropic substrate. Janssen et al. comment on the need to use correct mathematical expressions when deducing film stress from wafer curvature measurements and show how biaxial moduli of (001) and (111) silicon substrates can be derived [22].

The purpose of the work reported here is the generation of analytical expressions and their graphical representations for the biaxial moduli of cubic materials relevant for thin film/substrate assemblies in which there is an equi-biaxial elastic strain, and where the substrate is a single crystal, cut so that the interface between the thin film and the substrate is a general \((\mathit{hkl})\) plane, rather than a plane of high symmetry such as (001) or (111). For such a situation, in which the film could be a polycrystalline film isotropic in the plane of the film, the substrate will have induced in it curvature associated with two principal radii of curvature orthogonal to one another. These principal radii of curvature are directly related to the two orthogonal principal biaxial moduli in the \((\mathit{hkl})\) plane. For the high symmetry situations of (001) and (111) cubic substrates, the two principal biaxial moduli are identical and the substrate appears to be transversely isotropic, so that the curvature induced by an equi-biaxial strain is radially symmetric [22].

The paper is organised as follows. A statement of the problem of determining principal biaxial moduli for a plane of a general anisotropic material of undeclared symmetry subjected to in-plane biaxial strain is given in Sect. 2. The process of how the equations generated for a general anisotropic material of undeclared symmetry reduce for equi-biaxial strain when the material is cubic is then presented in Sect. 3. Computations of the principal biaxial moduli for various \((\mathit{hkl})\) for cubic materials and their extrema as a function of \((\mathit{hkl})\) are presented in Sect. 4. Finally, in Sects. 5 and 6, remarks are made on the magnitude of shear stresses out of the plane on which there is an equi-biaxial strain and conclusions drawn on the practical consequences for cubic materials of the results established in Sect. 4.

## 2 General Tensor Transformation Relations for Biaxial Strains in Thin Plates

For a general anisotropic material, and for a general \((\mathit{hkl})\), the number of independent components of the stiffness tensor makes evaluation of the principal biaxial moduli computationally challenging, even for this special case of equi-biaxial strain. However, for cubic materials, the effect of crystal symmetry simplifies the problem considerably, as is shown in the next section.

## 3 Equations Defining the Biaxial Moduli for Cubic Materials when There Is an Equi-biaxial Strain

This is the same condition as Eq. (19) of Knowles and Howie [16] defining the axes of the sextic equation describing the variation of the shear modulus as a function of shear direction of a particular plane of shear \((\mathit{hkl})\); Knowles and Howie show how this equation can be solved to determine these axes for a general \((\mathit{hkl})\). It is evident from Eqs. (8) and (21) that the condition expressed by Eq. (21) is equivalent to the statement that \(C'_{1233} = C'_{3312} = 0\). It is also evident that equivalent descriptions of this condition can be derived by permuting 1, 2, 3 and 3 in Eq. (8), so that, for example, the condition \(C'_{1332} = 0\) is also defined by Eq. (21).

It is further shown in Appendix A that the condition described by Eq. (21) is also equivalent to the condition for the extreme values of shear modulus and Poisson’s ratio for a particular \((\mathit{hkl})\) in a cubic material specified by Norris [12]. Hence, we have the interesting result that, on a particular \((\mathit{hkl})\) for a cubic material used as a substrate, the biaxial moduli defining the principal radii of curvature in an isotropic thin film/substrate assembly in which a state of equi-biaxial strain has been induced in both the thin film and the substrate are parallel to extrema in both the shear modulus and Poisson’s ratio on that plane.

## 4 Computation of Principal Biaxial Moduli for Various \((\mathit{hkl})\)

We are now in a position to consider both special and general results from this analysis.

### 4.1 (001) Interface Orientation

### 4.2 (111) Interface Orientation

### 4.3 (011) Interface Orientation

### 4.4 \((0\mathit{kl})\) Interface Orientation

It is evident that the disparity between the values of \(M [100]\) and \(M [0l\bar{k}]\) is greatest when (\(0\mathit{kl}\)) is (011). Also noteworthy is the fact that for \(A > 1\), both \(M [100]\) and \(M [0l\bar{k}]\) increase from minima at (001) and (010) to a maximum at (011). For \(A < 1\), these trends in \(M [100]\) and \(M [0l\bar{k}]\) are reversed.

### 4.5 \((\mathit{hhl})\) Interface Orientation

For the three materials for which \(A > 1\), the maximum in \(M [\mathit{ll}\overline{2h}]\) occurs around a value of \(\theta\) of 46–47°, while \(M [1\bar{1}0]\) increases monotonically from its value at (001) to its value at (110). For Nb, for which \(A < 1\), there is a minimum in \(M [\mathit{ll}\overline{2h}]\) at a slightly higher value of \(\theta\) of ≈ 49°, while for \(M [1\bar{1}0]\), its value appears to plateau above \(\theta \approx 63^{\circ}\).

However, for Nb where \(A < 1\), \(b > 0\) and \(c < 0\), one of the roots of Eq. (39) lies within the range \(0 \leq \cos^{2}\theta \leq\) 1, giving a minimum of 136.9 GPa at a value of \(\theta = 70.86^{\circ}\), very close to (221) at \(\theta = 70.54^{\circ}\). By comparison, the value of \(M [1\bar{1}0]\) at \(\theta = 63^{\circ}\) is 138.2 GPa and that at \(\theta = 90^{\circ}\), when the interface plane is (110), is 138.4 GPa.

It is evident that the results for the principal values of \(M\) for (*hll*) planes on the boundary of the standard 001–011–111 stereographic triangle are equivalent by symmetry to those principal values for \((\mathit{hhl})\) for \(54.74^{\circ} \leq \theta \leq 90^{\circ}\): the relevant values of the biaxial moduli along the directions of principal stress are then \(M[01\bar{1}]\) and \(M [\overline{2l}\mathit{hh}]\). Thus, for example, for (122), these values can be obtained from Fig. 2 for Si, Cu, \(\beta\)-brass and Nb by reading the results for \(M [1\bar{1}0]\) and \(M [\mathit{ll}\overline{2h}]\) for \((\mathit{hhl})\) at 70.53°.

### 4.6 \((\mathit{hkl})\) Interface Orientation

Direction cosines defining the directions of principal stress \(\eta_{1}\) and \(\eta_{2}\) for various \((\mathit{hkl})\) within the standard 001–011–111 stereographic triangle subjected to an equi-biaxial strain. The planes (011), (156), (134), (123), (235) and (112) all lie on the same great circle whose pole is [\(1 1\bar{1}\)] and share the same value of \(Q\) defined in Eq. (11) of 0.5

| \(\eta_{1}\) | \(\eta_{2}\) |
---|---|---|

(159) | [\(\overline{0.0622}\ \ \overline{0.8695}\ \ 0.4900\)] | [\(0.9934\ \ \overline{0.1015}\ \ \overline{0.0540}\)] |

(125) | [\(\overline{0.2294}\ \ \overline{0.8870}\ \ 0.4007\)] | [\(0.9561\ \ 0\overline{.2826}\ \ \overline{0.0782}\)] |

(234) | [\(\overline{0.1763}\ \ \overline{0.7431}\ \ 0.6455\)] | [\(0.9116\ \ \overline{0.3707}\ \ \overline{0.1778}\)] |

(245) | [\(\overline{0.0884}\ \ \overline{0.7603}\ \ 0.6436\)] | [\(0.9504\ \ \overline{0.2577}\ \ \overline{0.1740}\)] |

(367) | [\(\overline{0.0643}\ \ \overline{0.7439}\ \ 0.6652\)] | [\(0.9487\ \ \overline{0.2522}\ \ \overline{0.1904}\)] |

(156) | [\(\overline{0.0243}\ \ \overline{0.7660}\ \ 0.6424\)] | [\(0.9916\ \ \overline{0.1001}\ \ \overline{0.0818}\)] |

(134) | [\(\overline{0.0639}\ \ \overline{0.7906}\ \ 0.6090\)] | [\(0.9785\ \ \overline{0.1696}\ \ \overline{0.1174}\)] |

(123) | [\(\overline{0.1410}\ \ \overline{0.8014}\ \ 0.5813\)] | [\(0.9532\ \ \overline{0.2684}\ \ \overline{0.1388}\)] |

(235) | [\(\overline{0.2569}\ \ \overline{0.7799}\ \ 0.5707\)] | [\(0.9104\ \ \overline{0.3935}\ \ \overline{0.1280}\)] |

Principal values of \(M\) in GPa for various \((\mathit{hkl})\) within the standard 001–011–111 stereographic triangle and at its corners and on its edges for planes parallel to \((\mathit{hkl})\) subjected to an equi-biaxial strain. Directions \(\eta_{1}\) and \(\eta_{2}\) are specified in Table 1 for \((\mathit{hkl})\) within this stereographic triangle. For \((\mathit{hkl})\) at the corners and on the edge of this stereographic triangle, \(\eta_{1}\) and \(\eta_{2}\) and the values of \(M\) are those determined by the relevant equations in Sects. 4.1–4.5. (001) and (111) are isotropic in their biaxial modulus and so for these planes a single value of \(M\) is given

Material | Si | Cu | | Nb | ||||
---|---|---|---|---|---|---|---|---|

| \(M (\eta_{1})\) | \(M (\eta_{2})\) | \(M (\eta_{1})\) | \(M (\eta_{2})\) | \(M (\eta_{1})\) | \(M (\eta_{2})\) | \(M (\eta_{1})\) | \(M (\eta_{2})\) |

(001) | 180.3 | 114.8 | 52.4 | 227.6 | ||||

(111) | 229.1 | 261.0 | 254.1 | 143.6 | ||||

(159) | 225.5 | 194.5 | 244.0 | 172.9 | 239.0 | 143.7 | 164.5 | 202.8 |

(125) | 212.8 | 194.8 | 211.1 | 169.2 | 196.0 | 138.4 | 183.8 | 205.0 |

(234) | 233.6 | 214.0 | 268.6 | 225.5 | 265.7 | 210.9 | 142.5 | 169.3 |

(245) | 235.8 | 208.9 | 272.9 | 213.3 | 271.8 | 195.6 | 140.9 | 177.4 |

(367) | 236.6 | 210.0 | 275.1 | 216.3 | 274.2 | 199.3 | 138.8 | 175.2 |

(122) | 237.1 | 212.1 | 276.6 | 221.6 | 275.5 | 205.8 | 136.9 | 171.4 |

(011) | 240.4 | 197.0 | 281.5 | 184.6 | 284.6 | 159.1 | 138.4 | 195.4 |

(156) | 238.3 | 199.1 | 276.9 | 189.2 | 278.6 | 165.0 | 141.1 | 192.7 |

(134) | 235.6 | 201.8 | 270.8 | 195.3 | 270.7 | 172.9 | 144.7 | 189.1 |

(123) | 231.9 | 205.5 | 262.5 | 203.6 | 260.0 | 183.6 | 149.6 | 184.3 |

(112) | 225.9 | 211.5 | 249.2 | 216.9 | 242.8 | 200.9 | 157.4 | 176.4 |

The data in Table 2 confirm that the differences between the principal values of \(M\) for a particular \((\mathit{hkl})\) are greatest when \((\mathit{hkl}) = \{011\}\). In this context, the data for the planes (011), (156), (134), (123), (235) and (112) are relevant, because for these planes \(Q = a_{31}^{4} + a_{32}^{4} + a_{33}^{4}\) is constant, and so the denominators of Eqs. (10) and (14) are fixed while their numerators vary.

### 4.7 Global Extrema of \(M\) and \(\bar{M}\)

For the three materials discussed in Sects. 4.4–4.6 for which \(A > 1\), the global maxima in \(M\) occur for {011} planes in \(\langle 01\bar{1}\rangle\) directions, and the global minima occur for {001} planes. For \(A < 1\), the global maxima in \(M\) occur for {001} planes, whereas the global minima occur for \(\{\mathit{hhl}\}\) planes in \(\langle 1\bar{1}0\rangle\) directions. If Eq. (42) is not satisfied, these \(\{\mathit{hhl}\}\) will be {110}. If Eq. (42) is satisfied, as for Nb, these planes are very close to being {122}. Moving away from (122) towards (125) on the great circle whose pole is \(2\bar{1}0\), i.e., through (367), (245) and (123), is consistent with the statement that the global minima in \(M\) occur for \(\{\mathit{hhl} \} \approx \{122\}\) for Nb (Table 2). For KCl, for which Eq. (42) is also satisfied, the \(\{\mathit{hhl}\}\) planes with the global minima for \(M\) are close to being {133}.

These results can be confirmed using the formalism established by Norris [11] when determining the conditions for the global extrema of Poisson’s ratio for cubic materials, suitably adapted for biaxial moduli. The formalism, which makes use of expressions for stationary values of engineering moduli in general in anisotropic materials, is used in Appendix B to validate the results quoted in this Section for stationary conditions and global extrema.

It is apparent that in the formalism of Brańka et al., the three materials with \(A < 1\) for which there is a global minimum in \(M\) along [\(1\bar{1}0\)] on \((\mathit{hhl})\) interfaces away from (110) are at the positions of (0.924, 0.673), (0.781, 0.490) and (0.307, 0.456) within (\(X,Y\)) space for KCl, PbS and Nb respectively. These regions in (\(X,Y\)) space are some way from the very limited domain in (\(X,Y\)) space near \(X = 0\) and \(Y = 0\), i.e., for materials for which \(A \gg 1\), where there are the most extreme maxima and minima in Poisson’s ratio, as documented by Brańka et al.

More interestingly, (\(X,Y\)) space can be used to explore the possibility for \(A > 1\), or equivalently, \(Y < 0.25\), that the local maxima in \(M\) along [\(\mathit{ll}\overline{2h}\)] on \((\mathit{hhl})\) interfaces away from (110) can be greater than the value of \(M\) along [\(1\bar{1}0\)] on (110). It is evident from Fig. 2 that, for Si, Cu and \(\beta\)-brass these local maxima have values which are a significant fraction of \(M [1\bar{1}0]\) on (110): 0.965, 0.947 and 0.919 for Si, Cu and \(\beta\)-brass respectively, the (\(X,Y\)) values of which are (0.520, 0.160), (0.171, 0.078) and (0.084, 0.029) respectively.

_{3})

_{6}(NO

_{3})

_{2}[33]. However, the criterion \(12c_{44} < 0.29 (c_{11} + 2c_{12})\) is not met in any of these materials. It is most close to being met in nickel hexamine nitrate at −34°C, for which \(c_{11} = 9.275~\mbox{GPa}\), \(c_{12} = 8.77~\mbox{GPa}\) and \(c_{44} = 0.699~\mbox{GPa}\) [33]; were \(0.253~\mbox{GPa} < c_{44} < 0.648~\mbox{GPa}\) in this \(Fm\bar{3}m\) material, Eq. (48) would be satisfied.

By comparison with the global extrema for \(M\), global extrema for \(\bar{M}\) are easier to specify. \(Q\) has limiting values of 1 at {001} orientations and \(1/3\) at {111} orientations. An examination of Eq. (17) shows that for \(A > 1\), \(\bar{M}\) has maxima at {111} orientations and minima at {001} orientations; for \(A < 1\), \(\bar{M}\) has maxima at {001} orientations and minima at {111} orientations.

## 5 Stresses \(\sigma '_{13}\) and \(\sigma '_{23}\)

So far in this analysis we have only considered the stresses \(\sigma '_{11}\), \(\sigma '_{22}\) and \(\sigma '_{12}\) within \((\mathit{hkl})\) and the stress \(\sigma '_{33}\) normal to \((\mathit{hkl})\) arising from an equi-biaxial strain within \((\mathit{hkl})\). The constraint that \(\sigma '_{33} = 0\) enables a relationship between the principal strains \(\varepsilon '_{11} = \varepsilon '_{22} = \varepsilon\) and \(\varepsilon '_{33}\) to be established. We have yet to examine the stresses \(\sigma '_{13}\) and \(\sigma '_{23}\).

For (\(0\mathit{kl}\)) planes, with \(k\), \(l\) both ≠ 0, using Eq. (30) we can choose \(1'\) to be [100] and \(2'\) to be [\(0\ a_{33}\ \overline{a_{32}}\)]. While \(\sigma '_{13} = 0\), \(\sigma '_{23} = 0\) only when \(a_{32} = a_{33}\), i.e., only for (011). For all other (\(0\mathit{kl}\)) planes apart from the special cases of (010) and (001), \(\sigma '_{23} \ne 0\). For \((\mathit{hhl})\) planes \(\sigma '_{13} = 0\) for \(1'\) parallel to [\(1 \bar{1} 0\)], but \(\sigma '_{23} = 0\) along [\(\mathit{ll}\overline{2h}\)] only if \(l = 0\), \(h = 0\) or \(h^{2} = l^{2}\), e.g., for (001), (111) and (110).

Hence, for a general \((\mathit{hkl})\), both \(\sigma '_{13} \ne 0\) and \(\sigma '_{23} \ne 0\). These stresses can be argued to be small relative to \(\sigma '_{11}\) and \(\sigma '_{22}\), even for highly anisotropic cubic materials, since they both depend on \(H\) modulated by a direction cosine expression which, from the form of this expression, is small. Even so, it is important to recognise that, for interface planes other than {001}, {111} and {011}, shear stresses out of the plane will arise naturally in cubic materials as a consequence of a state of equi-biaxial strain within the plane and a zero normal stress to the plane. It also follows that any calculation of elastic strain energy per unit volume for a \((\mathit{hkl})\) oriented cubic single crystal subjected to an equi-biaxial strain must include these terms.

## 6 Discussion and Conclusions

The main results of this paper are Eqs. (10) and (21). Equation (10) specifies the biaxial modulus along a particular direction of choice in a \((\mathit{hkl})\) plane of a cubic crystal subjected to an equi-biaxial strain in the plane, with the additional constraint that the stress normal to \((\mathit{hkl})\) is zero. In this context, \((\mathit{hkl})\) could either be a plane of a cubic crystal substrate or the orientation of planes parallel to the substrate of a single cubic crystal deposited on a suitable substrate, such as a glass slide which would be isotropic, or another cubic crystal with which it has a cube-cube orientation relationship. For two orthogonal directions within a general \((\mathit{hkl})\) plane, the biaxial moduli will be parallel to the directions of principal stress within the \((\mathit{hkl})\) plane. These directions of principal stress are specified by Eq. (21) and are parallel to extrema in both the shear modulus and Poisson’s ratio on that plane. They are necessarily extrema of the biaxial moduli on that plane simply because they are principal stresses.

Examination of the biaxial moduli as a function of \((\mathit{hkl})\) shows that for cubic crystals for which the anisotropy ratio \(A > 1\), the minimum value of the biaxial modulus occurs when \((\mathit{hkl})\) is {001}, in which case the biaxial modulus is isotropic in the plane. Apart from specific unusual sets of circumstances discussed in Sect. 4.7, the maximum value of the biaxial modulus occurs on {011} along \(\langle 01\bar{1}\rangle\). The maximum difference between the principal biaxial moduli on \((\mathit{hkl})\) occurs on {011} for all cubic crystals for which \(A > 1\).

For \(A < 1\), the formal minimum value of the biaxial modulus occurs when \((\mathit{hkl})\) is a plane of the form \(\{\mathit{hhl}\}\) where \(h\) and \(l\) are determined by Eqs. (39), (41) and (42) and the direction is \(\langle 1\bar{1}0\rangle\). However, in practice, the value of the \(\langle 1\bar{1}0\rangle\) biaxial modulus on the (110) plane for a material with \(A < 1\) will have a value sufficiently close to any formal minimum away from (110) that it can be regarded from an engineering point of view as being a plane with the minimum value of the biaxial modulus, as the calculations for Nb, KCl and PbS have shown. The maximum value of the biaxial modulus occurs on {001}, in which case the biaxial modulus is isotropic in the plane. As for materials with \(A > 1\), the maximum difference between the principal biaxial moduli on \((\mathit{hkl})\) occurs on {011}.

The most obvious context in which to use equations for biaxial moduli is in the deposition of thin films on substrates [19, 20, 22]. While most single crystal silicon substrates used for thin film deposition are (001) and (111), a number of other possible substrate orientations are now offered by manufacturers such as (011), (112) and even (531) (e.g., [34, 35]). For (011) silicon substrates, the elastic biaxial moduli differ by 22 % along [\(01 \bar{1}\)] and [100], and so any computations of elastic responses of microelectromechanical systems must take this into account, as Hopcroft et al. have recently discussed [23].

A second practical application is in the evaluation of residual stress levels in relatively thick coatings deposited on substrates, where the thickness of the coating is not necessarily small in comparison with the thickness of the substrate [20, 36]. Thus, for example, in the deposition of plasma electrolytic oxidation coatings on metallic substrates, the coatings can be ∼100 μm thick and the substrates 300–500 μm thick [36]. In such a situation, the misfit strains generate significant levels of stress in both the coating and the substrate. Although it is usual for such substrates to be polycrystalline, and therefore isotropic in their elastic response, such substrates could in principle be \((\mathit{hkl})\) planes of cubic single crystals.

## Notes

### Acknowledgements

I would like to thank Prof. Andrew N. Norris of the Department of Mechanical and Aerospace Engineering at Rutgers University, Piscataway, New Jersey for drawing my attention to Refs. [11] and [12] of this paper. I would also like to thank Dr Philip R. Howie of the Department of Materials Science and Metallurgy at the University of Cambridge for constructive comments on this paper and Prof. T. William Clyne of the Department of Materials Science and Metallurgy at the University of Cambridge for stimulating discussions on the mechanics of coating/substrate systems, and specifically on the creation of curvature arising from misfit strains in such systems. Finally, I would like to acknowledge an anonymous referee of the original manuscript for drawing my attention to the paper of Brańka et al., Ref. [15], in which the (\(X, Y\)) plane discussed in Sect. 4.7 is introduced for the representation of elastic moduli.

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