Stress-induced curvature

In this paper, we report the evolution and quantification of the step-height-reduction-ratio, and subsequently extract the planarization distance for copper CMP for the first time. Conventional profilometry and the InSiteSOO photo-acoustic measurement tool were employed to quantify copper film thickness and topography. The Insite 300 operates upon the transient gradient technique and allows for noncontact, nondestructive metal film thickness measurement [5]. The utility of this metal thickness tool bypasses a number of issues which arise with conventional profilometry. The ability to accurately delineate metal feature edge positions and circumvent stress-induced curvature present in long profilometry scans is of principle importance. 

The final layer is a 0.5 pm thick gold metal layer on top of Poly2 for wires, bond pads, bimorphs, and potentially as an optically reflective surface. The gold is deposited on top of a thin (20 nm) chrome layer to promote adhesion. It is not possible to deposit gold on top of the Polyl layer. For a flat mirror to be formed, the stress-induced curvature from the metallization should be comprehended in the design [13]. 

V. A. Aksyuk, F. Pardo, and D.J. Bishop, Stress-induced curvature engineering in surface micromachined devices, SPIE 3680, pp. 984-993 (1999). 

T.G. Bifano, H.T. Johnson, P. Bierden, and R.K. Mali, Elimination of stress-induced curvature in thin-film structures, J. Microelectromechanical Systems 11(5), pp. 592-597 (2002). 

Figure 7.42 Almen test configuration to measure the specimen curvature, (a) Mounting block and (b) a thin foil before (1) and after grit blasting (2). The convex curvature will be reduced by tensile stresses induced during coating (3) (SAE Handbook, 1977).

A simple bend stress relaxation test can be used to compare the creep resistance of individual filaments [43]. A bending stress is applied with a graphite jig to the fiber which acquires a radius of curvature, R . The assembly is submitted to a heat treatment. After cooling, the graphite jig pieces are separated revealing a fiber with a creep induced curvature of radius R (with R > R ). The stress relaxation is quantified by a parameter m defined as m = 1 - (Ro/R). For given test conditions, creep resistant fibers are characterized by m-values close to 1 (i.e., R Ro) whereas creeping fibers display low m values (R Ro). 

The characteristic beam on elastic foundation stress distribution, as shown in Fig. 2 for the applied moment case, is frequently encountered in a wide range of adhesive bond situations. In addition to lap joints mentioned earlier, peel tests, moisture-induced stresses, and curvature mismatch situations all tend to exhibit this characteristic distribution. Clearly, the beam on elastic foundation has important qualitative and, in many cases, quantitative applicability to a host of adhesively bonded joints. 

Fig. 3.6. Schematic diagram of a thin film bonded to the snrface of a snbstrate. The film thickness hi is much less than the thickness hg of the substrate. Due to an elastic mismatch strain in the film, a membrane force per unit length must be imposed around the periphery of the film to balance the internal stress due to mismatch without deforming the substrate. Relaxation of this artificial membrane force resultant to render the edge free of applied moment induces curvature in the substrate.

The discussion up to this point has focused on the relationship between the curvature of an elastic substrate and the stress in a single layer or multilayer film in which the mismatch is invariant under any translation parallel to the interface. The films considered have also been continuous and of uniform thickness over the entire film-substrate interface. Within the range of small deflections, such an equi-biaxial film stress induces a spherical curvature in the substrate midplane, except very near the edge of the substrate. What is the deformation induced in the substrate if such a film does not have uniform thickness or if the mismatch stress varies with position along the interface This question is addressed in this section for the cases when the nonuniformity in mismatch stress or thickness varies periodically along the... 

The induced curvature changes in the coordinate directions due to these stress resultants are obtained by means of (3.73) as... 

The layer is flat for any uniform composition and it remains so if there is no lattice mismatch between constituents A and B. Assume that constituent B has an isotropic extensional mismatch strain ei with respect to constituent A. As a result, the midplane of the layer is curved, in general, for a nonuniform distribution From (2.58), it is known that the state of stress inducing this curvature is... 

Upon subsequent exposure to ambient environments in several humidity and temperature-controlled chambers these curvatures varied with time due to the two aforementioned contradictory time-dependent mechanisms. While the time-dependent moisture diffusion process serves as a stress-inducing mechanism, the time-dependent relaxation acts to reduce the level of those stresses. Note that relaxation depends on moisture content through the shift factor an m). 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

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