Finite Film Thickness Research Articles

Role of residual stress on crack penetration and deflection at a bimaterial interface in a 4-point bend test

The four-point bend (4PB) test has emerged as a method of choice in the semiconductor industry for obtaining bimaterial interface adhesion data. When measuring the interface adhesion using 4PB test, it is essential to obtain a crack through the interface of interest. In our previous work the effect of finite flaw size, boundary conditions and film thickness on the crack deflection or penetration at a bimaterial interface was studied, and it was determined that prior work on semi-infinite media cannot be directly used in the case of 4PB test. In this paper, we briefly discuss the origins and development of residual stress, and will expand our previous work to include the role these residual stresses play on the competition between deflection and penetration energy release rates of a bimaterial interface and the extent of which the previous assumption of two semi-infinite media can be accepted.

Optical properties of a nearly ferroelectric superconductor

Optical properties of a nearly ferroelectric superconductor in the dielectric response are theoretically investigated based on the electrodynamics of the nearly ferroelectric superconductors. The optical reflectance and transmittance of TE and TM waves are calculated and analyzed for two model structures, a superconductor occupying the half-space and a superconducting film of finite thickness. At the oblique incidence to a free-space-superconductor interface, it is found that the Brewster angle of a TM wave is close to the grazing angle of incidence. In the transmittance spectrum for a superconducting film, a sharp comblike distribution of the resonance peaks is found. These peaks are not regularly spaced in the frequency domain. The number of peaks increases substantially with an increase in the thickness of the film. In addition, the effect of the oblique incidence on this anomalous comblike distribution in a TM wave is stronger than that of a TE wave.

Distribution of the magnetic field induced by a current passing through slabs in the superconducting and normal states

The magnetic field induced by a current passing through a long slab is calculated. For the uniformly distributed current, simple formulas for the field components inside and outside the slab are derived. For the nonuniformly distributed current, the magnetic field is numerically calculated for a thin wide superconducting film (the thickness of the film is on the order of the magnetic field penetration depth). It is shown that the current-induced field distribution depends on the form of the exponential dependence of the current density. Experimental distributions are compared with results of a theory where a finite-thickness film is considered as two-dimensional (indefinitely thin). Comparison is also made between experimental and calculated values of the magnetic fields. For a number of points on a plane parallel to the slab and on a plane passing through its center, experimental data are compared with the results obtained by mere summation of partial contributions to the field.

Bending Elasticity of Charged Surfactant Layers: The Effect of Layer Thickness

The bending properties of charged one-component surfactant films of finite thickness have been theoretically investigated. It is demonstrated that finite thickness effects are of crucial importance for layers formed by an ionic surfactant with a flexible hydrophobic tail, whereas the influence on layers formed by a surfactant with a rigid tail is less pronounced. As a matter of fact, in the former case, the spontaneous curvature and mean and Gaussian bending constants all become significantly modified as compared to an infinitely thin surface and assume identical values as if the surfactant layer were bent at constant layer thickness. As a result, the spontaneous curvature is found to decrease, whereas the magnitudes of the mean and Gaussian bending constants both increase with increasing layer thickness as well as with increasing hydrophobic-hydrophilic interfacial tension. All of these trends are consistent with experimental observations. In addition, it is demonstrated that separating the hydrophilic-hydrophobic interface and the surface of charge a certain distance from each other tends to increase the spontaneous curvature and the mean bending constant, whereas the Gaussian bending constant becomes increasingly negative. It is also found that the work of bending a bilayer into a geometrically closed vesicle is substantially raised to large positive values for surfactants with flexible aliphatic chains, whereas the corresponding quantity is negative for surfactants with rigid tails, indicating that stable bilayer structures may only be formed by the former surfactant. Furthermore, each of the bending elasticity constants for monolayers formed by a double-chain ionic surfactant are found to assume lower values as compared with layers formed by the corresponding single-chain surfactant.

Wetting transitions in a binary thin-film

The effect of finite thickness of a binary thin-film system exhibiting a consolute critical point on the wetting behavior at the planar film-substrate interface was examined by accounting for the short-range chemical interaction between the film and substrate. Owing to the limited amount of mass available within the film, the first-order adsorption transition obtained from the one-dimensional equilibrium analyses has a physical meaning only when the film is initially uniform in composition. The line of first-order adsorption transition is strongly affected by the finite film thickness, with the composition line shifting inward towards the bulk miscibility gap as the film thickness descreases. However, when the film is initially in a two-phase mixture state, two-dimension simulations showed that the wetting transition from partial to perfect wetting is hardly influenced by the film thickness. The results of the present study suggest that the wetting behavior of a thin film at a given film composition and temperature depends critically on the initial state of the film.

The effect of finite film thickness on the crystallization kinetics of amorphous film and microstructure of crystallized film

A model for crystallization kinetics of thin amorphous film is developed by extension the Kolmogorov–Johnson–Mehl–Avrami (KJMA) model to take into account a finite film thickness. Two model versions, volume induced crystallization (VIC) and surface induced one (SIC), are explored. Finite film thickness effects lead to important consequences in the VIC: the crystallization profile reaches maximum in film middle, the crystallites population in a film is always higher than in a bulk material, the thinner film the slower it crystallizes and a spatially inhomogeneous structure with a fine-grained subsurface layer is formed. A VIC-kinetics follows a generalized KJMA equation with parameters depending on a film thickness and a range of the KJMA model validity is found. A SIC-kinetics obeys 2D KJMA equation in sufficiently thin film and is almost linear in a thick film. The model is extended to the case of non-constant growth rate and crystallization of subsurface film layer, and is shown to be consistent with experimental data.

Adsorption of Helium on Au Nanowires

The growth of helium films on Au nanowires is investigated within finite range density functional theory. It is shown that as in previous studies of helium adsorption on spherical surfaces, submonolayer condensation and stable layering transitions take place, and that a growth instability appears above a finite film thickness. The competition between unstable film growth on a single tube and stable capillary condensation in an array of nanowires is examined in a simplified mean field-like model. Results are displayed to illustrate adsorption and its stable and unstable patterns outside solid nanowires, cavities in bulk gold and in the gap between a cavity and an inner concentric cylinder.

Numerical Simulation of Electron Transport Through Constriction in a Metallic Thin Film

Electron transport through a constriction in a thin metallic film of finite thickness is simulated by numerically solving the Boltzmann transport equation (BTE) within the relaxation time approximation under linear response conditions. Such a structure closely represents vias connecting metal levels of different widths. Predicted reduction in effective electrical conductance due to electron surface scattering, which is significant when the dimensions are of the order of carrier mean free path, is compared with that for a constriction between semi-infinite spaces available in the literature. A simple expression for the size effect on conductance is fit to the simulated results applicable for constriction in a finite size thin film. The results could enable better estimate of effective resistance of next generation on-chip interconnections.

Waves induced by instability in falling films of finite thickness

Nonlinear gravity-capillary waves in falling viscous films of finite thickness are here investigated. Mathematical modelling of this problem is based on an evolution system with a single similarity parameter δ , which depends on the mean film thickness, gravity and liquid properties. To investigate spatially periodic structures, we apply the Galerkin procedure to obtain a dissipative dynamical system which contains disturbance wavenumber in addition to the parameter δ . Two-parametric manifold of its limit cycles is numerically constructed. Attracting invariant solutions, limit cycles and invariant torii, developing from different initial data are revealed. Parallel with the analysis of solutions to the dynamical system, the corresponding periodic waves in physical space are analysed and their mechanical meaning is discussed. The dominating waves that correspond to the attracting limit cycles have successfully been compared with experimental data.

Drying of poloxamer hydrogel films

The drying of hydrogel films formed by Poloxamer 407 poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) amphiphilic block copolymer was investigated at various air relative humidity (RH) conditions ranging from 11 to 97%. Initially, the amount of water lost increased linearly with the drying time. After this linear region (stage I), a nonlinear behavior was observed (stage II). The drying rate increased with decreasing RH, thus greatly shortening the drying time. A decrease of the film thickness also shortened the drying time; however, the drying mechanism did not change. Three models for one-dimensional water diffusion were used to fit the experimental results at different RH conditions and film thicknesses. Model 1 assumes semi-infinite medium and constant diffusion coefficient, and fits very well the data in stage I of the drying process. The fitted water diffusion coefficient (D) is 5 × 10−10 m2/s, whereas the effects of the RH are captured by a proportionality constant (α) that appears in the boundary condition. Model 2 considers a finite (constant) film thickness and captures the experimental observations over the whole drying period for the same D and α as in Model 1. The analytical solutions available for Models 1 and 2, used together with the experimentally derived model parameters D and α, allow for easy estimation of drying time and water loss from Poloxamer hydrogel films of various compositions and thicknesses and at different relative humidities. Numerical solutions for water diffusion under conditions of decreasing film thickness and diffusion coefficient being a function of concentration are also presented (Model 3). It becomes apparent from the fit of the data to the different models that the drying rate is more sensitive to the boundary condition at the film–air interface (represented by α) than to the diffusion in the film. It is notable that the α values obtained from the fits of the Poloxamer hydrogel drying rate are comparable to those obtained from drying of water films under the same experimental conditions. © 2004 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 93:1454–1470, 2004

Finite size effects in phase transformation kinetics in thin films and surface layers

In studies of phase transformation kinetics in thin films, e.g. crystallization of amorphous films, until recent time is widely used familiar Kolmogorov–Johnson–Mehl–Avrami (KJMA) statistical model of crystallization despite it is applicable only to an infinite medium. In this paper a model of transformation kinetics in thin films based on a concept of the survival probability for randomly chosen point during transformation process is presented. Two model versions: volume induced transformation (VIT) when the second-phase grains nucleate over a whole film volume and surface induced transformation (SIT) when they form on an interface with two nucleation mode: instantaneous nucleation at transformation onset and continuous one during all the process are studied. At VIT-process due to the finite film thickness effects the transformation profile has a maximum in a film middle, whereas that of the grains population reaches a minimum inhere, the grains density is always higher than in a volume material, and the thinner film the slower it transforms. The transformation kinetics in a thin film obeys a generalized KJMA equation with parameters depending on a film thickness and in limiting cases of extremely thin and thick film it reduces to classical KJMA equation for 2D- and 3D-system, respectively.

An analytical model for ultrathin films with spatially varying magnetic anisotropies

The continuum theory of micromagnetism is used to study ultrathin films with magnetic anisotropy varying laterally on the nanometer scale. A series of infinitely long stripes with alternating in- and out-of-plane uniaxial anisotropy is considered as a model for patterned ferromagnetic films. The analytical solution based on an effective anisotropy description of the dipolar energy is given as a function of the material parameters (exchange and anisotropy constants) and the width of the stripes. States of uniform and non-uniform magnetization are obtained. For film of finite thickness the exact value of the dipolar energy is calculated numerically and the numerical solution is compared with the analytical one. We find that the correction to the dipolar energy is negligible for nickel, while for cobalt and iron it has to be considered as a function of the anisotropies, the width of the stripes and the thickness of the film.

Sulfur diffusion in cadmium telluride thin films: Part 1: the grain boundary diffusion coefficient

Previous data on sulfur diffusion in single crystal CdTe has been expanded to include the more practical example of diffusion in polycrystalline CdTe thin films. Samples of vacuum evaporated CdTe were sealed, with a source of sulfur vapour, in evacuated silica ampoules and annealed at temperatures of either 372, 450 or 550 °C for times up to 28 days. Sulfur diffusion profiles were measured by secondary ion mass spectrometry. A new method to correct for finite film thickness was used to determine the grain boundary diffusion coefficients. These show that films grown at high temperatures, such as those produced by close-space sublimation, may possess sulfur enriched grain boundaries that extend through the complete thickness of the CdTe film. However, extensive decoration of the grain boundary by hexagonal CdS x Te 1− x is believed unlikely.

Derivation of expressions for the spontaneous curvature, mean and Gaussian bending constants of thermodynamically open surfactant monolayers and bilayers

We have derived expressions for the spontaneous curvature H0, the mean and Gaussian bending constants, kc and k̄c, respectively, for a surfactant film of finite thickness that is open in a thermodynamic sense. Geometrical packing constraints are taken into account and give rise to explicit large and important contributions to kc, k̄c, and kcH0. From its contribution to the latter quantity we may deduce that surfactant aggregates (micelles, vesicles, microemulsion droplets) are expected to dramatically increase their size with increasing surfactant tail length. Moreover, the coupling between free energy contributions related to surfactant head group and tail with geometrical packing constraints give rise to dominant terms on the form 2ξpH0, where ξp is the thickness of a planar film, in the expressions for kc. In the case of repulsive head group effects that favor a large spontaneous curvature, such as electrostatics, these terms raise kc and thus increase the rigidity of the film. Due to the constraint of constant free monomer chemical potentials, the composition of the film becomes a function of curvature. As a result, the ability of a surfactant film to have different surfactant compositions in differently curved parts (e.g., inner and outer layer of a vesicle, central parts and end caps of rod or threadlike micelles, etc.) may considerably reduce kc, whereas k̄c and kcH0 are mainly unaffected by mixing. The magnitude of the reduction of kc of a binary surfactant film increases with increasing asymmetry between the two surfactants with respect to charge number, head group size, and tail volume.

Fluorescence of a dye-doped cholesteric liquid crystal film in the region of the stop band: theory and experiment

Due to the sinusoidal modulation of the dielectric properties along the helical axis, cholesteric liquid crystals exhibit a photonic stop band for circularly polarized light, which strongly affects the emission of fluorescent guest molecules. In this paper, we discuss the resulting changes in the emission spectrum. In an analytical treatment, we first derive the photonic densities of states of the two normal light modes for propagation parallel to the helical axis, taking into account multiple reflections due to the finite film thickness. Then we discuss the influence of the degree of order of the dye's transition dipole moment on the emission characteristics. Finally, we present experimental results, which show excellent quantitative agreement with our theoretical description.

Self-consistent study of localization in thin films

We present an investigation of the Anderson localization of electronic states due to disorder in films of finite thickness. We first present numerical results that give some evidence for a transition from localized to delocalized states as the thickness of the film is increased, for disorder strengths below the localization threshold in three dimensions. We then examine the situation analytically using an extension of the self-consistent diagrammatic method developed by Vollhardt and Wolfle for noninteracting electrons.

Impurity segregation in thin films: effect of lateral interaction and site competition

A theoretical model describing both segregation kinetics and equilibrium, taking into account site competition and lateral interaction, for the cases of a semi-infinite medium and films of finite thickness has been developed. It has been shown that limiting the number of impurity atoms (i.e. film thickness) affects the segregation process similar to the effect of temperature increase. The level of equilibrium segregation decreases with film thinning for a given bulk concentration and temperature. A criterion for classification into “thick” and “thin” films, depending on impurity segregation activity and temperature, has been formulated. At low temperatures, the critical thickness is mainly determined by the d/ c 0 δ ratio and does not show any pronounced temperature dependence. At high temperatures, the dimensional range widens with elevation of temperature due to the strong (exponential) temperature dependence of equilibrium segregation.

Strong pinning by correlated and point disorder in YBa 2Cu 3O 7− δ thin films

We present a detailed study of the critical current density of YBa 2Cu 3O 7− δ thin films grown by pulsed laser deposition. Among the salient features are the total absence of any traits characteristic for pinning by extended linear defects, and an unexpected increase of the critical current density with film thickness. It turns out that critical current the dependence of critical current density on thickness is correlated with the field dependence of the same quantity; thicknesses of the both are very well described by extending to finite film thickness the Ovchinnikov–Ivlev theory of strong pinning [Phys. Rev. B 43 (1991) 8024].

General expression for the angular dependence of the critical field in layered superconductors

Starting from the Ginzburg-Landau equation we derive an analytical expression for the angular and temperature dependence of the critical field in layered superconducting materials. This formula is shown to reproduce as the limiting cases the well known results of Tinkham for the isolated film of finite thickness and of Lawrence-Doniach for the stacked array of films of zero thickness with its particular anisotropic Ginzburg-Landau limit. Since this general formula is very simple, we suggest that it should be used in all cases where the dimensional crossover of the critical field is studied.

The transient diffusion of hydrogen in finite metal films and spheres containing trapping sites

Finite difference methods have been used to calculate the transient and steady-state diffusion behavior of H-atoms permeating films of finite thickness and outgassing from uniformly charged spheres. The relationships between the lattice diffusivity obtained from defect-free materials and the diffusivities obtained from samples containing vacancy-related trapping sites and metastable (dislocation-related) traps have been calculated.