Stoneys formula

Freund, L.B., Floro, J.A., and Chason, E. (1999) Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations. Appl Phys. Lett., 74 (14), 1987. 

Zang, J., Liu, E Theory of bending of Si nanocantilevers induced by molecular absorption a modified Stoney formula for the cedibration of nanomechanochemical sensors. Nanotechnology 18,405501 (2007)... 

The expression for curvature in (2.7) is the famous Stoney formula relating curvature to stress in the film (Stoney 1909). Stoney s original analysis of the stress in a thin film deposited on a rectangular substrate was based on a uniaxial state of stress. Consequently, his expression for curvature did not involve use of the substrate biaxial modulus Mg. Consequently, (2.7) can be applied in situations in which mismatch derives from inelastic effects. However, the relationship (2.7) is based on Stoney s concept as outline in this section, and it has become known as the Stoney formula. It has the important property that the relationship between curvature k and membrane force / does not involve the properties of the film material. The elastic mismatch strain Cm corresponding to the stress <7 given in (2.8) is... 

Thus, the neutral plane always hes at a distance ghs from the midplane of the substrate and in the direction away from the film, no matter what the sign or magnitude of / might be. It is also noted that Khs 1, to the level of approximation assumed in deriving the Stoney formula (2.7). This implies that the differences among the curvatures of the midplane, neutral plane, top face and bottom face of the substrate are negligibly small. 

In the preceding section, an estimate was made of the curvature caused by the mismatch strain when a very thin film is bonded to the surface of a substrate. It was assumed that the change in film stress due to substrate deformation was negligible, and that the stiffness of the system depended only on the properties of the substrate. These assumptions led to the Stoney formula (2.7) relating the membrane force in the film to the curvature of the midplane of the substrate. The value of film thickness /if entered the derivation only peripherally. The issue is re-examined in this section for cases where the film thickness /if is not necessarily small compared to the substrate thickness /ig. 

The issue of film thickness effects on substrate curvature evolution is pursued now by recourse to the energy minimization method which was introduced in Section 2.1 for the derivation of the Stoney formula. All other features of the system introduced in that section are retained in this discussion, which follows the work of Freund et al. (1999). It is assumed that the film material carries an elastic mismatch strain in the form of an isotropic extension em (or contraction if Cm is negative) in the plane of the interface the physical origin of the mismatch strain is immaterial. The mismatch strain is spatially uniform throughout the film material. In this case, em is... 

Fig. 2.4. With reference to the expression (2.19) for substrate curvature for any values of hf/hs and M /Mg, the error in the Stoney formula for a system corresponding to parameter values in the region bounded by the two curves and the coordinate axes is less than 10%.

A second way to represent the influence of film thickness on substrate curvature is to establish the range of both parameters Mf/Ms and hf/hs for which the error incurred in using the Stoney formula is less than some prescribed value, say 10%. For example, the range of these parameters for which... 

The change in substrate curvature induced during film deposition or temperature excursion provides valuable insight into the evolution of mismatch stress in the thin film. As noted earlier, a particularly appealing feature of curvature measurement is that extraction of the membrane force / from substrate curvature by recourse to the Stoney formula (2.7) does not involve the material properties of the film, provided that the film is sufficiently thin compared to the substrate. Substrate curvature measurements also provide a means to assess the functional properties of thin films in photonic and microelectronic applications. For example, strain in films can modify the electronic transport characteristics of layered semiconductor systems through modification of the bandstructure of the material (Singh 1993). 

Since the original substrate is not generally flat, it is essential to measure the substrate radii of curvature pi and p2 before and after film deposition, respectively. Use of the Stoney formula (2.7) provides the average film stress in terms of the measured values of the radii of curvature of the substrate as... 

A linear regression analysis of the versus r data then provides an estimate of K from which the mean stress can be determined using the Stoney formula... 

A typical resolution of 15 km is achieved for the radius of curvature p, for an array consisting of four or five parallel laser beams. Prom the substrate curvature, the film stress is calculated using the Stoney formula as a function of film thickness, thereby providing a complete history of intrinsic stress evolution during the deposition of the film on a substrate. For a typical setup, the deposition of a 1 nm thick Sii a,Gej film with a stress of 50 MPa (i.e., a membrane force of 50 MPa-nm) on a 100-nm thick Si substrate is routinely detected (Floro et al. 1997). 

In this section, the effect on substrate curvature of the variation of mismatch strain and material properties through the thickness of layered films is analyzed. The derivation of the Stoney formula (2.7) in Section 2.1 refers only to the resultant membrane force in the film any through-the-thickness variation of mismatch strain in the film is considered only peripherally. Film thickness was taken into account explicitly in Section 2.2, but it was assumed there that mismatch strain and elastic properties of the material were uniform throughout the film. However, there are situations of practical significance for which this is not the case. Two of the most common cases are compositionally graded films in which the mismatch strain and the elastic properties vary continuously through the thickness of the film, and multi-layered films for which the mismatch strain and the elastic properties are discontinuous, but piecewise constant, from layer to layer throughout the thickness of the film. In both cases, the mismatch strain and the material properties are assumed to be uniform in the plane of the interface. With reference to the cylindrical r, 6,. z—coordinate system introduced in Section 2.1, the mismatch strain and film properties are now assumed to vary with z for fixed r and 6, but both are invariant with respect to r and 9, for fixed z. 

Fig. 2.14. Substrate curvature due to a mismatch strain which increases linearly through the film thickness for three values of stiffness ratio. The curvature is normalized by the value Kst based on the Stoney formula for this case.

It is possible to express the mismatch stress in term of mismatch strain for general anisotropy without reference to the substrate because the issue has been pursued under the same set of assumptions that underlie the derivation of the Stoney formula as outlined in Section 2.1. Recall the earlier assumptions that the film is very thin compared to the substrate and that the change in film strain associated with curvature of the substrate is small compared to the mismatch strain itself. Under these conditions, the film strain and film properties determine the stress which results in substrate curvature. The details of the resulting curvature do indeed depend on the properties of the substrate, and this issue will be taken up in Section 3.7. 

Certain key features of the analysis of curvature in Section 2.1 leading to the classic Stoney formula (2.7) when hi anisotropic stress film bonded to a substrate. Among these features is that it is only the resultant membrane force in the film prior to substrate deformation which contributes to substrate curvature. Any through-the-thickness variation of the film stress giving rise to this force... 

According to the Stoney formula (2.7), the substrate curvature is given in terms of this force, the biaxial modulus of the substrate and the thickness of the substrate. The biaxial modulus is approximately Mg = 76 x 10 N/m for polysihcon. The resulting estimate of radius of curvature is = 1.56m. 

The main calculation which must be done to estimate curvature in the presence of cracking is the determination of /, the force which plays the role here that served in the derivation of the Stoney formula. The... 

If both p and /if are very small compared to substrate thickness hg, then the configuration can be analyzed within the framework established in considering the Stoney formula (2.7). The main task here is to determine the resultant forces per unit length in coordinate directions that act on symmetry planes of the configuration with the substrate in its undeformed state these resultant forces are again denoted by fx and fy. The calculation can be pursued on the basis of elastic reciprocity (3.94), as in the case of a periodic array of cracks, and only a few steps in the procedure will be included. In the present case, the force resultants can be written as... 

Once the stress history is determined or, equivalently, once the dependence of stress on temperature of a prescribed temperature history is determined, the dependence of curvature on time or on temperature follows immediately from the Stoney formula (2.7) for thin films. Qualitatively, the stress versus temperature history depicted in Figure 7.16 captures the essential features of observations such as those reported by Doerner et al. (1986) and Shen and Suresh (1995a) for Al films on the Si substrates. Stress variations based on more complex constitutive behavior are considered in the following sections. 

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