Showing papers by "Zhigang Suo published in 1994"

A new procedure for measuring the decohesion energy for thin ductile films on substrates

Abstract: A novel testing technique has been developed capable of measuring the interfacial fracture resistance, Γi, of thin ductile films on substrates. In this technique, the thin film on the substrate is stressed by depositing onto the film a second superlayer of material, having a large intrinsic stress, such as Cr. Subsequent processing defines a precrack at the interface between the film and the substrate. The strain energy available for driving the debond crack is modulated by varying the thickness of the Cr superlayer. Spontaneous decohesion occurs for superlayers exceeding a critical thickness. The latter is used to obtain Γi from elasticity solutions for residually stressed thin films. The technique has been demonstrated for Cu thin films on silica substrates.

Cracking in ceramic actuators caused by electrostriction

Abstract: Many perosvskite-type ceramics deform appreciably under electric fields; they make good actuators which deliver motions upon receiving electrical signals. High electric fields are usually applied to induce large strains. Fracture has been observed in the actuators under electrical loading. In this theoretical study, the phenomenon is examined on the basis of electrostriction and fracture mechanics. Attention is focused on a crack emanating from an internal electrode or a conducting damage path. At the edge of the conducting path, the electric field is intense and nonuniform, inducing incompatible electrostrictive strains. Consequently, a stress field is set up in the ceramic, localized around the edge of the conducting path. The condition for the stress to extend a crack is estimated by two models, using either quadratic or step-like electrostriction law. It is found that, under a given electric field, cracking is suppressed in a multilayer actuator if the ceramic layers are sufficiently thin.

Electromigration instability: Transgranular slits in interconnects

Abstract: Evidence has recently accumulated that an interconnect under intense electric current can fail by a transgranular slit. A rounded void first forms, enlarges, and drifts. When the void becomes sufficiently large, a narrow slit emerges at the expense of the void, running across the linewidth. In this letter we describe a physical mechanism that explains this instability. Both electric current and surface energy drive atoms to diffuse on the void surface, but in the opposite directions. The slit emerges if the electric current prevails. An approximate analysis shows how the slit selects its width and velocity.

Cleavage due to dislocation confinement in layered materials

Abstract: The effects of dislocation confinement on fracture behavior in laminates consisting of alternating submicron ductile and brittle layers are studied. When the ductile layer thickness is below the micron level, dislocations must be treated individually. Dislocations emitted from the crack tip have two effects: they blunt the crack and thereby reduce the tensile stress at the crack tip ; and pile up against an interface and send a back stress to the crack tip to hinder further dislocation emission. Consequently, an equilibrium number of dislocations exists at a given load level. We estimate this number by considering the stability conditions for dislocations threading in the ductile layer, and dislocation pile-up is treated as an equivalent superdislocation. Furthermore, the competition between further dislocation emission and cleavage at the blunted crack tip is considered. Our result shows that because of the confinement, as the applied load increases, the tensile stress at the blunted crack tip also increases. Cleavage occurs when the tensile stress at the crack tip reaches the theoretical strength. Given a sufficiently thin constraining layer, cleavage can even occur in ductile metals such as copper and aluminum. The implications of this model for several material systems are discussed.

Diffusive void bifurcation in stressed solid

Abstract: Interconnects are susceptible to solid diffusion under residual stress, electric current, and elevated temperature. As atoms diffuse, voids nucleate, drift, and enlarge. At some point, the voids of rounded shape can collapse to narrow slits and sever the interconnects. The fatal slits are often found to be transgranular, i.e., each slit cuts across a single grain. They have raised many concerns, but the underlying mechanism has remained unclear. It is proposed that a void changes shape due to surface diffusion under the combined action of surface energy, elastic energy, and electric current. The void will be rounded if surface energy prevails, but will collapse to a slit if the elastic energy or the electric current prevails. A cylindrical void in an infinite crystal under biaxial stresses, but under no electric current, is analyzed. Four things are done, as follows: (1) A suitable thermodynamic potential is minimized and maximized to select, among a family of ellipses, equilibrium void shapes. The bifurcation diagram consists of a subcritical pitchfork and two Griffith cracks. (2) A void under biased stresses is analyzed to illustrate the effect of imperfections. (3) Exact initial bifurcation modes are determined. The critical loads for the successive modes are closely separated, indicating that the shape evolution will be sensitive to initial imperfections. (4) A variational principle for shape evolution under stress, current and surface energy is identified. Stress‐induced evolution time is estimated by using this variational principle.

Emergence of cracks by mass transport in elastic crystals stressed at high temperatures

Abstract: Single crystals are used under high temperatures and high stresses in hostile environments (usually gases) A void produced in the fabrication process can change shape and volume, as atoms migrate under various thermodynamic forces A small void under low stress remains rounded in shape, but a large void under high stress evolves to a crack The material fractures catastrophically when the crack becomes sufficiently large In this article three kinetic processes are analyzed : diffusion along the void surface, diffusion in a low melting point second phase inside the void, and surface reaction with the gases An approximate evolution path is simulated, with the void evolving as a sequence of spheroids, from a sphere to a pennyshaped crack The free energy is calculated as a functional of void shape, from which the instability conditions are determined The evolution rate is calculated by using variational principles derived from the balance of the reduction in the free energy and the dissipation in the kinetic processes Crystalline anisotropy and surface heterogeneity can be readily incorporated in this energetic framework Comparisons are made with experimental strength data for sapphire fibers measured at various strain rates

Cavity and dislocation instability due to electric current

Abstract: Alectric current drives atomic diffusion in aluminum lines. The phenomenon, known as electromigration, has been a persistent problem in integrated circuits, as the current density increases and the linewidth shrinks for every new generation of the technology. As atoms diffuse, cavities nucleate, enlarge and migrate. Evidence has accumulated that an aluminum line often fails when a rounded cavity collapses into a narrow slit across the width of the line. The fatal slits are often found to be transgranular, i.e. each slit cuts across a single grain. We investigate theoretically this instability phenomenon. Because the shape changes in a relatively short time below half the melting temperature of aluminum, atoms are assumed to diffuse only along the cavity surface, under the combined action of the electron wind and surface tension. When subjected to a small electric field, a circular cavity migrates as atoms diffuse from one portion of the cavity surface to another ; the cavity remains circular, so that capillarity does not drive diffusion. However, the cavity buckles above a critical electric field, elongating either along or normal to the electric field. The surface tension now drives the atomic diffusion to restore the circular shape. The instability sets in if the electric field prevails over the surface tension. The general problem is formulated in terms of nonlinear differential equations, and the pre-bifurcation, bifurcation and post-bifurcation solutions are discussed. In addition, we analyse an analogous phenomenon: a prismatic dislocation loop climbs due to core diffusion driven simultaneously by the electron wind and line tension. The climb relocates mass at relatively low temperatures and thereby degrades the aluminum lines.

On the tensile properties of a fiber reinforced titanium matrix composite. II: Influence of notches and holes

Abstract: The effects of holes and notches on the ultimate tensile strength of a unidirectionally reinforced titanium matrix composite have been examined. During tensile loading, a narrow plastic strip forms ahead of the notch or hole prior to fracture, similar to that observed in thin sheets of ductile metals. Examination of the fibers following dissolution of the matrix indicates that essentially all the fibers within such a strip are broken prior to catastrophic fracture of the composite. The trends in notch-strength have been rationalized using a fracture mechanics-based model, treating the plastic strip as a bridged crack. The observations suggest that the bridging traction law appropriate to this class of composite is comprised of two parts. In the first, the majority of fibers are unbroken and the bridging stress corresponds to the unnotched tensile strength of the composite; in the second, the fibers are broken and the bridging stress is governed by the yield stress of the matrix, with some contribution derived from fiber pullout. This behavior has been modeled by a two-level rectilinear bridging law. The parameters characterizing the bridging law have been measured and used to predict the notch strength of the composite. A variation on this scheme in which the fracture resistance is characterized by an intrinsic toughness in combination with a rectilinear bridging traction law has also been considered and found to be consistent with the predictions based on the two-level traction law.

Notch-sensitivity and shear bands in brittle matrix composites

Abstract: Matrix cracking, fiber breaking and interface sliding cause nonlinear deformation in fiber-reinforced brittle matrix composites. When a notched sample is loaded in tension, the nonlinear deformation usually localizes around the notch, spreads the stress in the ligament more evenly, and thereby leads to a higher fracture load. We simulate the interplay of two deformation mechanisms: a tensile band ahead of, and shear bands perpendicular to, a notch. The shear deformation evens out the stress distribution in the tensile band, and the strength of the tensile band sets the extent of the shear deformation. Each band is simulated by a traction-deformation law. The work of fracture is computed from a small-scale inelastic problem, and the fracture loads of notched samples from a large-scale inelastic problem. Several important conclusions emerge from the simulation. First, weak shear bands can substantially increase the work of fracture. Second, the fracture loads of notched samples are well correlated with the unnotched strength, work of fracture and notch size, by a formula independent of the shear band description. The results of the simulation are used to explain the available experimental data, and to suggest an evaluation procedure for notch-sensitivity.

Electromigration-induced dislocation climb and multiplication in conducting lines

Abstract: Electromigration along dislocation cores is considered as a mass-transport mechanism, in conducting lines of bamboo-like grains, below one half of the melting temperature. Given that the dislocation density in annealed metals may be insufficient to account for the observed mass-transport rate, this paper focused on electromigration-induced dislocation motion and multiplication. A prismatic loop climbs like a rigid ring, as electromigration relocates atoms along the core, from one portion of the loop to the other. Each loop is therefore a mass carrier: a vacancy loop migrates towards the cathode and an interstitial loop towards the anode. Furthermore, a thread of an edge dislocation multiplies prismatic loops under a sufficiently high electric field. A bamboo grain-boundary catches loops on one side and emits on the other. Available empirical facts are discussed according to this picture, including lifetime, linewidth, stress gradient and alloying.

Mechanics of some degradation mechanisms in ferroelectric ceramic actuators

Abstract: Several degradation mechanisms in ferroelectric ceramics are analyzed in this article. A ferroelectric crystal under cyclic electric field fatigues by forming a-domain bands. An energy based model is proposed, indicating that these bands are retarded by the electric field, but driven by the shear stress resolved onto the bands. The stress has been attributed to the misfit strain near the 180 degree(s) domain wall and the edge of the electrodes. A second problem is related to fracture of piezoelectric ceramics. A double-cantilever beam subjected to combined electrical and mechanical loadings is analyzed using finite elements. Also analyzed is electrode debonding, which is shown to decrease capacitance.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Dislocation Climb in the Electron Wind

Abstract: When a current flows in a conductor, the electron wind causes atoms to diffuse. This paper considers the consequences of such diffusion along dislocation cores. A dislocation climbing in a crystal is viewed as a non-equilibrium thermodynamic system to define the force that drives self-diffusion along the core. Not only is a dislocation a mass-transport pipe, it also climbs and generates more dislocations—all in the electron wind. A prismatic loop moves like a rigid disk, as atoms electro-migrate along the core from one edge of the loop to the other. Each loop is therefore a mass carrier responding to an electric current. Interstitial and vacancy loops can be simultaneously generated and subsequently climb in the opposite directions. The process transports mass in single crystal or bamboo-like interconnects at moderate temperatures.

Transgranular Slits in Aluminum Interconnects Caused by Thermal Stress and Electric Current

Abstract: An aluminum interconnect can fail by a transgranular slit. We show how a rounded void collapses to such a slit, as atoms diffuse on the void surface, driven by electric current, thermal stress, and surface energy. We use a variational principle to simulate the shape evolution, and identify the critical conditions to trigger the shape instability.

Precipitate Drifting and Coarsening Caused by Interface Electromigration

Abstract: A mechanism is proposed to explain electromigration-enhanced precipitate coarsening in Al-Cu alloy interconnects. The interface between the a-phase matrix and a θ-phase Al2C11 precipitate is incoherent, along which both Al and Cu atoms diffuse under an applied electric field. Depending on the relative mobility of Al and Cu, the diffusion causes the precipitate to migrate towards either the positive or the negative electrode. The velocity of a spherical precipitate is proportional to the electric field and the mobilities and inversely proportional to its radius. A critical electric field or precipitate radius exists, above which the precipitate can penetrate a grain boundary. Consequently, the precipitates agglomerate by the synergism between the surface tension-induced ripening and the current-induced migration. The resulting particles are distantly separated, depleting Cu atoms from the rest of the interconnect. The mechanism appears to limit the lifetime of interconnects having bamboo-like grains, tested below 300°C, less than half of the melting temperature of Al.