Substrate curvature method

The ease with which the film mismatch stress can be estimated from the change in curvature of a relatively thick substrate subjected to thermal excursion has led to the widespread use of the substrate curvature method for the study of mechanical properties. As reviewed in Section 2.3, substrate 

Advantages in the use of substrate curvature method for the extraction of film properties can be summarized as follows. 

The curvature method also has some inherent limitations. Since the method entails determination of plastic response through the imposition of a temperature change or phase transformation, which is known to alter the plastic properties of metals, care should be exercised in the interpretation of strain relaxation phenomena from curvature measurements. There are also no clear means of isolating the individual contributions to overall curvature evolution seen experimentally from such factors as plastic yielding, strain hardening, diffusional creep, or microstructural changes, without recourse to other independent experimental and observational tools. 

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The thickness, density, and residual stress in sintered films can be determined by the methods described above. The X-ray diHfaction method of determining stress is generally more suitable for sintered films since it does not require the removal of the film, as do the substrate curvature methods. The surface roughness can be measured with a profilometer, whereas the adhesion of the film to the substrate can be determined by a pull test, in which a wire is bonded to the film and then pulled with the force needed to remove the film from the substrate. The amount of camber or warpage can also be determined with a profilometer. Grain and pore sizes can also be determined by the same techniques used for bulk ceramics. 

As noted in previous sections, the development of life prediction models for the reliability of patterned features such as periodic lines on substrates inevitably requires knowledge of intrinsic stress and mismatch stress generated during film growth, patterning, passivation and service. In this section, three prominent experimental methods for determining stress in thin films with patterned lines are considered the substrate curvature method, the x-ray diffraction method, and the micro-Raman spectroscopic method. The advantages and limitations of each of these techniques are also briefly addressed. 

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... 

Methods to measure changes in substrate curvature during stress evolution in a layered material can be broadly classified into the following... 

Many of the limitations of the above techniques are overcome by optical methods which offer the convenience, accuracy and flexibility to measure curvature through remote sensing capabilities. In this section, several different optical techniques for the measurement of substrate curvature are described, and their advantages and limitations are examined. 

Fig. 2.8. Schematic illustration of the scanning laser method for measuring substrate curvature.

The optical method for measuring substrate curvature is generally convenient for in-situ measurement of film deposition stress in CVD and MBE systems, provided that optical access is provided to the substrate and that the specimen is mounted in a manner which facilitates unconstrained curvature evolution in one direction. One of the most common optical methods of... 

In the discussion in Section 2.6.1 of substrate curvature in the range of behavior where the deformation is not axially symmetric, it was assumed a priori that the coordinate axes coincided with the axes of principal curvature. This assumption was incorporated in writing the transverse deflection w x, y) in the form given in (2.82). On the other hand, in determining curvature from measurements in this range of behavior, the directions of principal curvature are not known in advance. How is data to be interpreted in order to extract complete curvature information under the circumstances An approximation based on measurements obtained with the CGS method, as discussed in Section 2.3.4, is briefly considered here on the basis of a graphical construction known commonly as Mohr s circle. 

A single crystal alloy thin film with composition Sio.ssGeo.is is grown on an initially flat Si(OOl) substrate which is 0.5 mm thick. The lattice parameter of Si at room temperature is asi = 0.5431 nm, while that of Ge is ace = 0.5656 nm. Any dislocations formed as a consequence of epitaxial mismatch between the film and the substrate are known to be 60° dislocations with Burgers vectors in the family represented by (6.18). Assume that the biaxial moduli of Si(OOl) and Ge(OOl) crystals are Mgi(ooi) = 180.5 GPa and MGe(ooi) = 142 GPa, respectively, that the Poisson ratio of the film is i/f R 0.25, that Vq = 6/2, and that 6 w 0.4 nm. Curvature measurements are made continuously using the multibeam optical stress sensor method (see Section 2.3.2) so as to monitor the evolution of internal stress during film deposition. Estimate the substrate curvature at which misfit dislocations are first able to form at the interface between the film and the substrate. 

Continuum descriptions of the evolution of stress, substrate curvature and plastic deformation also provide a useful and broad framework with which experimental observations of strain relaxation in thin films on substrates can be assessed. As shown in later sections of this chapter, such analyses provide some tools with which the plastic deformation response can be experimentally determined by recourse to methods such as substrate curvature measurement, x-ray diffraction or indentation. 

It is evident from the discussion presented in Section 7.4 that observation of substrate curvature, interpreted as mean stress in a film bonded to an elastic substrate, during temperature cycling provide a convenient framework with which the inelastic strain relaxation characteristics of thin film materials can be inferred. Indeed, substrate curvature measurement made in the course of temperature cycling is among the most common experimental methods... 

The various curvature measurement methods described in Section 2.3 provide convenient tools with which the film stress can be assessed by using the approaches outlined in Chapters 2 and 3 and Section 7.4. However, these approaches are often predicated on assumed microstructural conditions which may differ markedly from real behavior in some material systems. Microstructural evolution and competition between strain relaxation mechanisms can influence the evolution of substrate curvature during thermal excursions and the manner in which such effects are strongly affected by the geometry and material properties of the particular thin film-substrate system being considered. The following points are discussed in order in the remainder of the section, primarily in the context of polycrystalline metal Aims ... 

Experimental techniques most commonly used to probe the plastic properties of thin film materials involve direct tensile loading of either a freestanding film or a film deposited onto a deformable substrate material, microbeam bending of films on substrates, substrate curvature measurement or instrumented depth-sensing nanoindentation. Sahent features of these methods, as well as specific examples of the adaptation of these methods for the study of mechanical properties in thin films, are briefly addressed in the following subsections. 

Fischer-Cripps AC (2000) A review of analysis methods for suh-micron indentation testing. Vacuum 58 569 Freund LB, Floro JA, 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... 

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