Cathode edge displacement by voiding coupled with grain boundary grooving in bamboo like metallic interconnects by surface drift-diffusion under the capillary and electromigration forces

AbstractThe kinetics of cathode edge shrinkage and displacement (drift) coupled strongly with the grain boundary (GB) grooving is investigated using the novel mathematical model developed by Ogurtani, in sandwich type thin film bamboo lines. The computer simulations are performed under the constant current (CC) and the switch-over constant voltage (SOCV) operations. The cathode drift velocity and the cathode failure time show the existence of two distinct phases, depending upon the normalized electron wind intensity parameter χ; the capillary (χ⩽0.01) and the electromigration (EM) dominating regimes (χ>0.01), having current exponent n, equal to 0 and 1, respectively. Analysis of various experimental data on the cathode drift velocity results a consistent value for the surface drift-diffusion coefficient, 1.0×10-5exp(-1.00eV/kT)m2s-1, for copper interconnects exposed to some contaminations during the processing and testing stages. This is found to be an excellent agreement with the experimental values reported in the literature after applying the proper 1/kT correction on the apparent activation enthalpy associated with Nernst–Einstein mobility relationship. The complete cathode failure time (CCFT) due to the cathode area shrinkage by voiding is also formulated by inverse scaling and normalization procedures, which show exactly the same capillary and EM dominating regimes. This formula can be used to predict very accurate CCFT for metallic lines with bamboo-like, near-bamboo, and even with polycrystalline structures by proper calculation of the cathode-edge path length (CEPL) parameter, in terms of the actual line width, the thickness and the grain size

Surface morphological evolutions on single crystal films by strong anisotropic drift-diffusion under the capillary and electromigration forces

The morphological evolution of voids at the unpassivated surfaces and the sidewalls of the single crystal metallic films are investigated via computer simulations by using the novel mathematical model developed by Ogurtani relying on the fundamental postulates of irreversible thermodynamics. The effects of the drift-diffusion anisotropy on the development of the surface morphological scenarios are fully explored under the action of the electromigration (EM) and capillary forces (CF), utilizing numerous combination of the surface textures and the directions of the applied electric field. The interconnect failure time due to the EM induced wedge shape internal voids and the incubation time of the oscillatory surface waves, under the severe instability regimes, are deduced by the novel renormalization procedures applied on the outputs of the computer simulation experiments.Comment: 41 pages, 18 figures. related simulation movies utilizing numerous combination of the surface texture, see http://www.csl.mete.metu.edu.tr/aytac/thesis/movies/index.ht

Hillock formation by surface drift-diffusion driven by the gradient of elastic dipole interaction energy under compressive stresses in bi-crystalline thin films

We investigated surface drift diffusion induced grain boundary GB grooving and ridge hillock formation and growth, under the combined actions of the capillary forces and applied uniaxial compressive stresses, in bi-crystal thin films with dynamical computer simulations. In the present theory, the generalized driving force for the stress induced surface drift diffusion includes not only the usual gradient of the elastic strain energy density, but also the elastic dipole tensor interaction energy. During the morphological evolution of GB ridge formation and growth, triple junction TJ displacement and its velocity are continuously tracked down in order to resolve precisely the crossover time and depth at which velocity sign inversion takes place. An incubation time for the onset of the ridge growth stage coupled to the GB-TJ displacement velocity inversion is defined and its dependence on the stress is investigated. This analysis implies that the ridge growth stage is not controlled by Ziegler’s ‘maximum entropy production principle’ but rather Prigogine’s ‘minimum entropy production hypothesis’ for the stationary non-equilibrium states in complex systems, which are exposed to external applied body forces and surface traction

Thermal Grooving by Surface Diffusion: a Review of Classical Thermo-Kinetics Approach

I n polycrystalline materials wherever a grain boundary intersects a free surface and whenever the topographic variation associated with the atomic motion is favored by total free energy dissipation, the material surface grooves. In this review, we focused on the grain boundary grooving by surface diffusion which is an active mechanism at moderate temperatures and for grooves small in size. Starting with a description of the classical thermo-kinetics treatment of the process, we briefly reviewed Mullins’ very first modeling effort with a small slope assumption at the groove root and further considerations regarding finite slopes, different grain geometries, and anisotropic surface free energies. We concluded by giving examples of experimental observations in accord with theoretical calculation

Generic role of the anisotropic surface free energy on the morphological evolution in a strained-heteroepitaxial solid droplet on a rigid substrate

A systematic study based on the self-consistent dynamical simulations is presented for the spontaneous evolution of an isolated thin solid droplet on a rigid substrate, which is driven by the surface drift diffusion induced by the anisotropic capillary forces (surface stiffness) and mismatch stresses. In this work, we studied the affect of surface free energy anisotropies on the development kinetics of the 'Stranski-Krastanow' island type morphology. The anisotropic surface free energy and the surface stiffness were treated with well accepted trigonometric functions. Although, various tilt angles and anisotropy constants were considered during simulations, the main emphasis was given on the effect of rotational symmetries associated with the surface Helmholtz free energy topography in 2D space. Our computer simulations revealed the formation of an extremely thin wetting layer during the development of the bell-shaped Stranski-Krastanow island through the mass accumulation at the central region of the droplet via surface drift-diffusion. In the strong (anomalous) anisotropy constant domain, we demonstrated the existence of two distinct morphological modes: i) the complete stability of the initial Cosine-shaped droplet just above a certain anisotropy constant threshold level by spontaneous slight readjustments of the base and the height of the cluster; ii) the Frank-van der Merwe mode of thin film formation for very large values of the anisotropy constant by the spreading and coalescence of the droplets over the substrate surface. During the course of the simulations, we have continuously tracked both the morphology (i.e., the peak height, the extension of the wetting layer beyond the domain boundaries, and the triple junction contact angle) and energetic (the global Helmholtz free energy changes associated with the total strain and surface energy variations) of the system.Comment: anisotropic surface free energy, Quantum Dots, elastostatic load, morphological evolution, strained-heteroepitaxial soli

Unified theory of linear instability of anisotropic surfaces and interfaces under capillary, electrostatic, and elastostatic forces: The regrowth of epitaxial amorphous silicon

The first-order unified linear instability analysis (LISA) of the governing equation for the evolution of surfaces and interfaces under capillary, electromigration (EM), and elastostatic forces is developed. A formal treatment of the thermomigration (Soret effect) driven by the nonuniform temperature distribution caused by exothermic phase transformation (growth) at the surface and interfacial layers is presented and its apparent influence on the capillary force in connection with the stability is also established in a concise analytical form. This unified approach, which relies on a rigorous theory of irreversible thermodynamics of surfaces and interfaces, seriously considers the anisotropies associated with the generalized growth mobility, the interfacial specific Gibbs free energy (i.e., the surface stiffness), and the surface diffusivity in thin solid films. The singularity in the surface stiffness at the cusp regions of the Wulff construction of the surface Gibbs free energy is fully elaborated by using a modified cycloid-curtate function as a basis for generating the Dirac delta distribution, which shows an unusually strong anomalous effect on the surface morphological instability even in the absence of EM forces, as illustrated clearly by the graphical representation of the EM-induced instability threshold level as a function of tilt angle and wave number, in a three-dimensional plot for various intrinsic and normalized system parameters. In the development of LISA theory special attention is paid to the origin of the elastostatic forces, which include not only the elastic strain energy density, but also the elastic dipole tensor interaction between mobile atomic species and the applied stress field. The profound influence of the anomalous surface stiffness anisotropy on the surface morphological evolution under the applied stress system is demonstrated by three-dimensional computer graphics applied for copper and silicon thin single-crystal solid films having, respectively, sixfold {111}- and fourfold {100}-symmetric singular (vicinal) planes as the top surfaces, which reveal the fine features of the theory and give insight into some controversial issues related to LISA in the literature. This unified approach also considers the stress dependence of the generalized growth mobility and its profound influence on the stability of the interface displacement and roughening in thin solid films. As a special application of the theory, the effects of uniaxial and biaxial applied stresses on the recrystallization and the interfacial morphological evolution of amorphous Si deposited on silicon substrates are thoroughly analyzed and excellent quantitative agreement is found with the published experimental data in the literature

Thermal grain-boundary grooving in bicrystal thin solid films having strong anisotropic surface Gibbs free energy represented by the modified cycloid-curtate function

The variational non-equilibrium thermodynamic method is further extended to give full coverage for the tilted grain-boundary (GB) configuration with respect to the sidewalls of a bicrystal thin solid film having strong anisotropic specific surface Gibbs free energy associated with the singular directions (faceting, vicinal planes). A set of critical computer simulation experiments supported by the generalized longitudinal force diagrams is performed on the asymmetrically disposed (inclination) bicrystal thin metallic films having four- and six-fold anisotropic specific surface Gibbs free energy to demonstrate the various GB-groove root topologies caused by the grain-boundary grooving under the surface drift-diffusion driven by the capillarity forces (thermal grooving). In the computer simulations, the strongly anisotropic surface-specific Gibbs free energy associated with the cusp regions is represented by the modified cycloid-curtate function (MCCF) as a basis (generator) for the Dirac delta distribution function on the Wulff construction, which involves not only the Wulff surface roughness (WSR) parameter (anisotropy constant) but also the Wulff surface topography (WST) index (shape parameter) that may be used as a metric for the temperature roughening phenomenon. A special computer run is also designed using the realistic structural and physicochemical properties in order to simulate the thermal groove profiles of cube-textured pure nickel tape {Ni-99.99 wt%} annealed four hours in vacuum at 800 degrees C, and observed by the atomic force microcopy (AFM). The experimental line width fitting procedure applied to the simulation profile subjected to the self-similarity transformation, which resulted in almost perfect replication of the experimental digitized AFM photography has yielded, a mean surface (mass) diffusivity of nickel about 5.7 x 10(-13) m(2)/s (800 degrees C) which is in excellent quantitative agreement with the diffusivity relationship at T >= 1300 K reported in the literature on relatively contaminated surfaces, and obtained by high-precision profilometry measurements of the decay of capillary modes associated with the wide surface scratches. (C) 2009 Elsevier EN. All rights reserved

Dirichlet extremum problem associated with the asymmetric grain-boundary thermal grooving under the Dirac delta-type anisotropic surface stiffness in bicrystal thin solid films

A generalized Rayleigh-Ritz (RR) method combined with the Galerkin (RRG) functional space approximation is elaborated by using the extended and modified Laguerre functions manifold for the weak solution of the asymmetric grain-boundary thermal grooving problem with the Dirichlet boundary. This new hybrid RRG approach, which resembles the front-tracking method, reveals the fine features of the grain boundary groove-root topography (rough or faceted regions) more accurately than the previous approach under the severe nonanalyticity of the surface stiffness anisotropy, and showing almost excellent in accord with the experimental observations made by atomic force microscopy and scanning tunneling microscopy. The large deviations from Mullins' t(1/4) scaling law combined with the self-trapping (quasifaceting) are observed especially at low values of the normalized longitudinal mobilities, where the kinetics rather than the energetic considerations are found to be the dominating factor for the whole topographic appearances. For very high longitudinal mobilities, the smooth and symmetric groove profiles (no faceting) are found to be represented by the Mullins' function for the fourfold symmetry in the stationary state with great precision, if one modifies the rate parameter by the anisotropy constant and simultaneously utilizes the anisotropic complementary dihedral angle in the calculation of the slope parameter. A recently developed analytical theory fully supports this observation rigorously, and furnishes the quantitative determination of the threshold level of the anisotropy constant for the ridge formation, and as well as the penetration depth evaluation. (C) 2007 American Institute of Physics

The orientation dependent electromigration induced healing on the surface cracks and roughness caused by the uniaxial compressive stresses in single crystal metallic thin films

The first order unified linear instability analysis (ULISA) of the governing equation for evolutions of surfaces and interfaces under the capillary, electromigration, and elastostatic forces including the thermomigration (Soret effect) is developed very recently by the author. In the present application of the theory, the concurrent effects of uniaxial applied stresses and the electrostatic field on the sidewall morphological evolution of a single crystal thin metallic film are explored by dynamic computer simulations by taking the surface drift diffusion anisotropy fully into account. These computer experiments, which are supported by ULISA, clearly show that only the applied elastic compressive stresses are primary agents responsible for the morphological instability of the surface undulations through the elastic dipole tensor interactions but not the uniaxial tension loading in thin solid films. It is also demonstrated that these morphological instabilities manifested themselves as formations of the surface cracks and thus one may fully control the roughness. To do that, one needs to select crystal orientations properly with respect to the applied field so that a counteraction of the applied electrostatic fields (healing effect) is created above well defined threshold levels of electromigration. On the contrary to the healing effects, the improper selection of crystal orientations may drastically enhance the instability and eventually may cause catastrophic interconnect failure. At large normalized surface undulation amplitudes ((a) over bar >= 0.20), the drastic reductions in the decay rate constants (i.e., the strain relaxation rate) are detected in the nonlinear uniaxial tension regime compared to the ULISA theory regardless of the intensity of the normalized stress by analyzing the data obtained from the computer simulations. This situation is contrary to the results deduced from the low to moderate normalized amplitude ((a) over bar <= 0.10) measurements, where one finds that the decay rate constant closely obeys the prediction of the ULISA theory even for very high stress intensities

Mesoscopic irreversible thermodynamics of aging kinetics of alpha polypeptides [DNA] under various constraints: Special reference to the simple spring mechanics

The mesoscopic irreversible thermodynamic treatment of α-polypeptides and the helical polynucleotides (DNA) furnishes two sets of analytical expressions, which allow us not only to analyze the reversible force–extension experiments performed by atomic force microscopy (AFM) but also to predict the irreversible “aging” kinetics of the single-stranded and double-stranded polynucleotides (ssDNA and dsDNA) helical conformations exposed to aqueous solutions and applied static stress systems under the various constraints. The present physicochemical cage model emphasizes the fact that the global Helmholtz free energy of the helical conformation acts not only under the stored “intrinsic” unusual torsional and bending elastic energies inherited by the unfolded helical structure of the amino-acid (peptides) or the nucleic-acid (nucleotide) backbone but also reveals the importance of the interfacial Helmholtz free energy density associated with the interaction of the side-wall branches within the surrounding aqueous solutions. The analytical expression obtained for the unfolding force vs extension (FE) shows a strong non-linear elasticity behavior under the twist angle constraint when the interfacial Helmholtz energy term is incorporating into the scenario. This behavior is in excellent quantitative agreement with the AFM test results obtained by Idiris et al. (2000) on the poly-L-glutamic acid [Glu(n)-Cys] exposed to aqueous solutions, which show that acidity increases the degrees of helicity