Tarik Omer Ogurtani

Mesoscopic nonequilibrium thermodynamics of solid surfaces and interfaces with triple junction singularities under the capillary and electromigration forces in anisotropic three-dimensional space

A theory of irreversible thermodynamics of curved surfaces and interfaces with triple junction singularities is elaborated to give a full consideration of the effects of the specific surface Gibbs free energy anisotropy in addition to the diffusional anisotropy, on the morphological evolution of surfaces and interfaces in crystalline solids. To entangle this intricate problem, the internal entropy production associated with arbitrary virtual displacements of triple junction and ordinary points on the interfacial layers, embedded in a multicomponent, multiphase, anisotropic composite continuum system, is formulated by adapting a mesoscopic description of the orientation dependence of the chemical potentials in terms of the rotational degree of freedom of individual microelements. The rate of local internal entropy production resulted generalized forces and conjugated fluxes not only for the grain boundary triple junction transversal and longitudinal movements, but also for the ordinary points. The natural combination of the mesoscopic approach coupled with the rigorous theory of irreversible thermodynamics developed previously by the global entropy production hypothesis yields a well-posed, nonlinear, moving free-boundary value problem in two-dimensional (2D) space, as a unified theory. The results obtained for 2D space are generalized into the three-dimensional continuum by utilizing the invariant properties of the vector operators in connection with the descriptions of curved surfaces in differential geometry. This mathematical model after normalization and scaling procedures may be easily adapted for computer simulation studies without introducing any additional phenomenological system parameters (the generalized mobilities), other than the enlarged concept of the surface stiffness.

Computer simulation of void growth dynamics under the action of electromigration and capillary forces in narrow thin interconnects

A comprehensive picture of void dynamics in connection with the critical morphological evolution has been developed in order to understand the conditions under which premature failure of metallic thin interconnects occurs. Our mathematical model on the mass flow and accumulation on void surfaces, under the action of applied electrostatic and elastostatic force fields, and capillary effects, follows an irreversible but discrete thermodynamic formulation of interphases and surfaces. This formalism also takes into account in a natural way the mass transfer process (the void growth), between bulk phase and the void region in multi-component systems, in terms of the normalized local values of Gibbs free energy of transformation with respect to the specific surface Gibbs free energy, in addition to the contribution due to local curvature of the advancing reaction front, rather rigorously.

Grain boundary grooving and cathode voiding in bamboo-like metallic interconnects by surface drift diffusion under the capillary and electromigration forces

The process of grain boundary (GB) grooving and cathode voiding in sandwich type thin film bamboo lines are simulated by introducing a mathematical model, which flows from the fundamental postulates of irreversible thermodynamics. In the absence of the electric field, the computer studies on the triple junction kinetics show that it obeys the first order reaction kinetics at early transient stage, which is followed by the familiar time law as t¯1∕4, at the steady state regime. The applied electric field (EF) in constant current experiments modifies this time law drastically above the well-defined electron wind intensity (EWI) threshold, and puts an upper limit for the groove depth, which decreases monotonically with EWI. Below the threshold level, the capillary regime predominates, and EF has little effect on the general kinetics of GB grooving, other than the linear increase in total elapsed time with EWI. An analytical formula for the cathode failure time in constant voltage test is obtained in terms of...

Electromigration-induced void grain-boundary interactions: The mean time to failure for copper interconnects with bamboo and near-bamboo structures

A well-posed moving boundary-value problem, describing the dynamics of curved interfaces and surfaces associated with voids and/or cracks that are interacting with grain boundaries, is obtained. Extensive computer simulations are performed for void configuration evolution during intergranular motion, under the actions of capillary and electromigration forces in thin-film metallic interconnects with bamboo structures. The analysis of experimental data, utilizing the mean time to failure formulas derived in this paper, gives consistent values for the interface diffusion coefficients and enthalpies of voids. 5.85×10−5exp(−0.95eV∕kT)m2s−1 is the value obtained for voids that form in the interior of the copper interconnects avoiding any surface contamination. 1.80×10−4exp(−1.20eV∕kT)m2s−1 is obtained for those voids that nucleate either at triple junctions or at the grain-boundary technical surface intersections (grain-boundary groove), where the chemical impurities such as Si, O, S, and even C are segregated ...

Unified theory of kink dragging with a special reference to the flow stress and the Snoek–Köster relaxation in anisotropic body‐centered‐cubic metals with heavy interstitials

The drag force acting on a kink moving rigidly and uniformly along the dislocation line and in the atmosphere of mobile octahedral (heavy) interstitials is investigated numerically in anisotropic body‐centered‐cubic metals. It is shown by extensive computer modeling experiments that the drag force versus normalized kink velocity plot goes through a maximum which is composed of three subpeaks, one acoustic and two optical in character. The acoustic part and the optical mode‐2 are strongly and selectively associated with the isotropic and the pure shear parts of the elastic dipole tensor of the interstitial species, respectively. An extremely accurate and the concise analytical expression is deduced for the linear viscosity regime which exhibits the behavior of Navier–Stokes fluids, which can be used directly in Snoek–Koster relaxation. The flow stress theory of Seeger for the ultrahigh‐purity body‐centered‐cubic metals is extended and coupled with the present treatment of the kink dragging which yields an ...

Kinetics of hopping of octahedral interstitials in arbitrary time dependent and inhomogeneous field: Body‐centered cubic lattice

Atomic movements of interstitials (octahedral sites) species in an arbitrary time and space dependent field have been formulated in terms of a set of coupled linear differential equations for a body‐centered cubic (bcc) lattice. The decoupled form of these equations has been solved exactly using the discrete Fourier k space transformation supplemented by a Laplace transform with respect to time. An explicit and compact expression for the Fourier transform of the partial concentration of interstitials is obtained for fields which have absolutely convergent norms in the time domain. Some special cases of general physical interest such as simple harmonic or static inhomogeneous fields are also treated. For the latter situations, a steady‐state power dissipation and the thermodynamic equilibrium partition of defects over energetically distinguishable sites are uniquely determined, respectively. The inverse relaxation time spectrum of octahedral‐interstitials which has three distinct branches, one acoustic and...

Nonlinear theory of power dissipation due to the motion of heavy interstitials in a fluctuating inhomogeneous field with a strong bias: Special reference to the Snoek–Köster relaxation

Atomic movements of heavy interstitials (octahedral sites) in an arbitrary time‐dependent and inhomogeneous field with a strong static bias have been described by a system of nonlinear autonomous first‐order differential equations for a body‐centered‐cubic lattice. The linearized form of these equations in the vicinity of the thermodynamic equilibrium state (equilibrium with respect to the strong and inhomogeneous static bias field) has been solved exactly, using the discrete Fourier k‐space transform supplemented by a Laplace transform with respect to time. The power dissipation associated with the hopping motion of interstitial species in the presence of either a simple harmonic vibration or of propagating acoustic waves in the media is also determined. The implications of the nonlinear theory for the effective activation enthalpy of the Snoek–Koster relaxation are also worked out for highly localized or delocalized static interaction fields, respectively.

Internal friction and viscosity associated with mobile interstitials in the presence of a kink harmonically or uniformly moving in anisotropic body‐centered cubic metals

The power dissipation (internal friction) due to mobile octahedral interstitials in the stress field of a harmonically oscillating or uniformly moving rigid kink of a dislocation line is derived for anisotropic body‐centered ‐cubic metals using the discrete Fourier k‐space transformation of the elastic Green’s function in connection with the general linear‐response theory of inhomogeneous interactive fields. The viscosity and the drag force acting on the kink moving rigidly and uniformly along the dislocation line are formulated which show strong velocity and the temperature dependent behavior. A discrete Debye relaxation spectrum, the first term of which is identical in form to the ordinary Snoek peak, is found to represent the anelastic behavior associated with the kink enhancement. The inverse relaxation time spectrum of octahedral interstitials which has three distrinct branches, one acoustic and two optical in the first Brillouin zone, is also presented and discussed.

The kinetics of hopping motion of interstitials with chemical reactions in arbitrary time dependent inhomogeneous interactive fields

Atomic hopping motions of interstitial species with chemical reactions in an arbitrary time‐ and space‐dependent field have been formulated in terms of a linear but coupled set of differential equations for a general lattice structure. The decoupled form of these equations has been solved exactly using the discrete Fourier k‐space transformation supplemented by a Laplace transformation with respect to time. An explicit and compact expression for the Fourier transform of the partial concentration of interstitials is obtained for interactive fields which have absolutely convergent norms in the time domain. Some special cases of general interest such as the simple harmonic, almost periodic, or static inhomogeneous fields are also treated extensively. The power dissipation and the storage terms associated with interstitial hopping motion in harmonic or almost periodic fields are formulated rigorously. The effect of static inhomogeneous bias field due to other imperfections on the relaxation time spectrum is c...

Nonlinear theory of the dislocation-enhanced Snoek effect and its connection with the geometric and/or thermal kink oscillations on nonscrew dislocations in body-centered-cubic metals

The dislocation‐enhanced Snoek effect, which is induced by the forced vibrations of the interacting kinks (geometric and/or thermal) on nonscrew dislocations in the atmosphere of mobile interstitials, is formulated by taking into account the nonlinear character of the atomic hopping motion of interstitials, as well as the existence of the strong and inhomogeneous static interaction field associated with the kinked dislocation lines. The desired explicit and transparent expression for the internal friction behavior of defects is also worked out, which reveals clearly the effect of the site saturation on the relaxation strength as well as on the activation enthalpies, especially for the case of highly localized binding interaction (hydrostatic) between interstitial foreign atoms and the dislocation lines.