Substrate Of Finite Thickness Research Articles

Revised transmission line model for electromagnetic characterization of metasurfaces

We have revised the transmission line model with the purpose of simplifying the electromagnetic characterization of metasurfaces with both electric and magnetic responses. The revised model is an efficient tool for the case when a metasurface is located on top of a finite-thickness substrate or inside a finite-thickness slab. We have derived analytical expressions for the grid series impedance and shunt admittance of a substrated metasurface. The presented theory is illustrated by a numerical example of an array consisting of Mie-resonant silicon nanospheres. We have retrieved individual polarizabilities of a single sphere in a uniform host medium and compared them with the predictions of classical Mie theory. Further, we have studied a more practical case when the array of Si spheres is located on top of a finite-thickness glass substrate.

Compensating substrate-induced bianisotropy in optical metamaterials using ultrathin superstrate coatings

In this work, we propose an efficient approach to compensate for the commonly observed substrate-induced bianisotropy that occurs in on-wafer optical metamaterials at normal incidence. First, the consequence of placing a finite thickness substrate underneath a metamaterial is analyzed, indicating that the induced bianisotropy is a near-field effect. The properties of metamaterials sandwiched between an infinitely thick substrate and a finite-thickness superstrate with different permittivity and thickness values are then investigated. It is demonstrated from full-wave simulations that by adding an ultrathin superstrate with a judicious choice of its thickness and permittivity value, the substrate-induced bianisotropy of the system can be suppressed and even eliminated. In addition to the extracted nonlocal effective medium parameters, the induced electric and magnetic dipole moments calculated from the volumetric microscopic fields are also presented, validating that the magnetoelectric coupling compensation is a real physical phenomenon. This study will benefit future optical metamaterial design and implementation strategies as well as the corresponding fabrication and characterization methodologies.

Finite Substrate Effects on Critical Thickness in Epitaxial Systems

During the growth of an epitaxial overlayer on a thick substrate (GeSi on Si), an interfacial misfit dislocation becomes energetically favourable on exceeding the critical thickness. In substrates of finite thickness, the value of critical thickness is altered with respect to thick substrates. Thin substrates can bend and partially relax the coherency stresses, thus contributing to the altered value of the critical thickness. The current work aims at simulating the stress state of a growing finite epitaxial overlayer on a substrate of finite thickness, using finite element method. The numerical model is used to calculate the critical thickness for substrates with finite thickness. Eigenstrains will be imposed in selected regions in the domain towards this end. Size of the substrate for which it is not energetically favourable to accommodate a misfit dislocation is determined from the simulations (i.e. the system remains coherent for substrates below this thickness). Important effects arising in the transition regime of substrate thicknesses are also investigated.

Contractile Stresses in Cohesive Cell Layers on Finite-Thickness Substrates

Using a minimal model of cells or cohesive cell layers as continuum active elastic media, we examine the effect of substrate thickness and stiffness on traction forces exerted by strongly adhering cells. We obtain a simple expression for the length scale controlling the spatial variation of stresses in terms of cell and substrate parameters that describes the crossover between the thin and thick substrate limits. Our model is an important step towards a unified theoretical description of the dependence of traction forces on cell or colony size, acto-myosin contractility, substrate depth and stiffness, and strength of focal adhesions and makes experimentally testable predictions.

Directional and hemispherical mid-infrared transmittance through microscale slit arrays on a semi-transparent substrate at normal incidence

The directional and hemispherical transmittance through four microscale slit arrays on a semitransparent substrate were investigated at normal incidence. Materials and vertical dimensions of all arrays were the same, but lateral dimensions varied among the three simple slit arrays. The complex slit array had three identical slits and three lamellae of dissimilar widths in a profile period. The incidence covered a broad-band mid-infrared region (2.5–25μm) in which two linear polarizations were considered. Impacts of the structure profile, incidence polarization and wavelength, and optical constants of involved materials on spectra were studied numerically with programs based on the algorithm termed rigorous coupled-wave analysis (RCWA). Near-field patterns of electromagnetic fields that correlated with unique transmittance were also plotted to identify responsible physical mechanisms. In addition, the directional transmittance spectra of fabricated samples were measured with FT-IR spectrometry. Reasonable agreement between numerical and experimental results was obtained. Unique features in spectra were consistently found, regardless of the finite substrate thickness.

Wrinkling of a bilayer resting on a soft substrate under in-plane compression

The wrinkling problem of a bilayer film resting on a soft substrate under in-plane compression is investigated via theoretical analysis and numerical simulations. Such a system may undergo two different wrinkling modes, namely surface wrinkling of the top single layer on a composite substrate or bilayer wrinkling on a homogeneous substrate, depending on the geometric and material parameters. In the present paper, a theoretical solution is first derived for the former case, while a solution available in the literature is applied for the latter. The solution derived here can degenerate both to the classical solution for a stiff layer resting on a semi-infinite soft substrate and that for a stiff layer lying on a compliant substrate of finite thickness. A series of finite element simulations are performed to demonstrate the accuracy of the theoretical solution. In addition, it is found that, at a given compressive strain, a certain change in the elastic modulus of the intermediate layer may lead to a dramatic surface pattern transformation between the two wrinkling modes, indicating a novel approach to realise surface pattern switching.

The effect of a compliant substrate on the electromigration-driven surface morphological stabilization of an epitaxial thin film

We analyze the surface morphological stability of a coherently strained thin film that has been grown epitaxially on a compliant substrate of finite thickness and is subjected simultaneously to an external electric field, which drives surface electromigration. The compliant substrate has the ability to accommodate elastically some of the misfit strain that is developed in the epitaxial film due to the lattice mismatch between the film and substrate materials. We develop a three-dimensional model for the surface morphological evolution of the thin film and conduct a linear stability analysis for the morphological stability of the heteroepitaxial film’s planar state; of particular importance for the analysis is the elastostatic boundary-value problem for the heteroepitaxial film/substrate system. The analysis shows that surface electromigration due to a properly applied and sufficiently strong electric field can inhibit Stranski-Krastanow-type instabilities. Furthermore, we determine the critical electric-field strength as a function of material properties and heteroepitaxial system parameters, as well as the optimal direction of the electric field for the most efficient stabilization of the surface morphology. We find that using a compliant substrate reduces the critical strength of the externally applied electric field required for planar film surface stabilization by approximately two orders of magnitude compared to that needed to stabilize the planar surface of the same thin film when grown epitaxially on a practically infinite substrate. This critical electric-field strength also is found to be substantially lower than that required for planar film surface stabilization for the same film grown on an elastic substrate that is clamped to a holder and has thickness equal to the compliant substrate thickness. This critical strength requirement can be reduced further by decreasing the ratio of the film’s shear modulus with that of the substrate, and it can be minimized for an optimal value of the compliant substrate thickness. We conclude that surface electromigration can be used to control the onset of island formation on epitaxial film surfaces and that the required electric-field strength for such control can be optimized by efficient use of substrate engineering techniques.

Marangoni-induced deformation of evaporating liquid films on composite substrates

Marangoni convection plays an important role in hydrodynamics of evaporating liquid films and sessile drops. Evaporation of liquid films induces unsteady nonuniform temperature distribution across the liquid layer and in a substrate. If the substrate is composed of parts with different thermal properties, the interface temperature distribution becomes non-uniform, leading to appearance of Marangoni stresses, convective vortices, and film deformation. In this article, a model describing evaporation, Marangoni effect and interface dynamics of liquid films on composite substrates is developed. The film dynamics is described in the framework of long-wave theory. The unsteady heat conduction in the substrate is described using the Laplace transform method for semi-infinite substrates and using the separation of variables technique for substrates of finite thickness. The non-uniformity of substrate thermal properties has a pronounced effect on film dynamics.

Electromagnetic characterization of substrated metasurfaces

In this paper we present a macroscopic model of a metasurface—optically dense grids of resonant scatterers located on a refracting interface. Similar models were previously built for the case when the scatterers are non-resonant electric dipoles and for the case when there is no substrate. In the present case scatterers are resonant, can have magnetic response, and for substrates with large refraction index acquire also substrate-induced bianisotropic properties. Representing the homogenized response of a metasurface to incident electric and magnetic fields through surface susceptibilities we derive the reflection and transmission formulas for semi-infinite and finite-thickness substrates. We develop a robust algorithm of the retrieval of the characteristic surface parameters from the reflection and transmission coefficients measured or exactly calculated for three different incidence angles. With numerical examples we show that the retrieved susceptibilities of metasurfaces are independent on the incidence angle and can be really called characteristic parameters.

Optical switching in metal-slit arrays on nonlinear dielectric substrates

We propose a scheme for an optical limiter and switch of the transmitted light intensity in an array of subwavelength metallic slits placed on a nonlinear Kerr-type dielectric substrate of finite thickness, where the geometrical parameters are designed for operation at telecom wavelengths. Our approach is based on the abrupt changes of the output light intensity observed in these systems near transmission minima.

Analysis of current-driven surface morphological stabilization of a coherently strained epitaxial thin film on a finite-thickness deformable substrate

We report results for the surface morphological stability of a thin film that lies on a finite-thickness substrate and is subjected simultaneously to an external electric field. The film is grown epitaxially on the substrate and may undergo a Stranski–Krastanow instability due to its lattice mismatch with the substrate material. We develop a model for the surface morphological evolution of the thin film and conduct a linear stability analysis to examine the morphological stability of the epitaxial film’s planar surface state. Our analysis shows that surface electromigration due to a properly applied and sufficiently strong electric field can inhibit Stranski–Krastanow-type instabilities, which can be used to control the onset of island formation on the film surface. We find that using a finite-thickness substrate can have the beneficial effect of reducing the critical strength of the electric field required to stabilize the planar surface morphology of the epitaxial film with respect to the field strength required in the case of an infinitely thick substrate. We determine the critical electric-field strength as a function of material properties and heteroepitaxial system parameters, as well as the field’s optimal direction for the most efficient stabilization of the surface morphology. Detailed results are presented for the effects of the finite-thickness substrate on the stabilization of the film’s surface over a range of mechanical properties of the heteroepitaxial system’s constituents.

Crack propagating in functionally graded coating with arbitrarily distributed material properties bonded to homogeneous substrate

In this paper, a finite crack with constant length (Yoffe type crack) propagating in a functionally graded coating with spatially varying elastic properties bonded to a homogeneous substrate of finite thickness under anti-plane loading was studied. A multi-layered model is employed to model arbitrary variations of material properties based on two linearly-distributed material compliance parameters. The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the model. The numerical results show that the graded parameters, the thicknesses of the interfacial layer and the two homogeneous layers, the crack size and speed have significant effects on the dynamic fracture behavior.

Substrate Effect on the Optical Reflectance and Transmittance of Thin-Film Structures

A rigorous and consistent approach is demonstrated to develop a model of the 4M structure (the four-media structure of a film on a substrate of finite thickness). The general equations obtained for the reflectance and transmittance spectra of the 4M structure are simplified by employing a procedure of the so-called device averaging to reduce them to a succinct form convenient for processing of experimental spectra for the structures with a thick substrate. The newly derived equations are applied to two special cases: (i) an arbitrary film on highly absorbing substrates and (ii) a slightly absorbing film on transparent substrates. The reflectance and transmittance spectra represented in the simplified (with the device averaging) form have a practical application for determining the film thickness and optical constants from experimental spectra by using the known techniques.

Terahertz Intrinsic and Effective Surface Impedances of High-Temperature Superconducting Thin Films

Terahertz wave properties of high-temperature superconducting thin films are theoretically investigated based on the calculated surface impedances. The surface impedances for three model structures are considered in this work. We first treat the intrinsic bulk surface impedance for a superconductor occupying the half space. Second, the intrinsic film surface impedance of a superconducting film of finite thickness is calculated. Third, we calculate the effective surface impedance for a superconductor-dielectric layered structure, i.e., a superconducting film on the dielectric substrate of finite thickness. All calculations that will be made are based on the two-fluid model of superconductors together with the transmission line theory.

Magnetoelectric effects due to elastic coupling in ferroelectric/ferromagnetic multilayers

The critical behavior of a multilayered system of alternating ferroelectric and ferromagnetic thin films on a substrate of finite thickness is studied. A thermodynamic approach via the Ginzburg–Landau formulation is followed. Our interest is in the magnetoelectric effects exhibited in the critical behavior of the multilayer system, caused by the interlayer elastic interaction. The system is characterized by two bifurcation points. The system exhibits the strongest magnetoelectric effect near the second bifurcation.

Use of the optical admittance function and its WKB approximation to simulate and evaluate transmittance spectra of graded-index colloidal films

A method is presented for the simulation of transmittance spectra of graded-index dielectric layers deposited on a transparent substrate of finite thickness, and the simulation process is applied to films of silica and zinc oxide nanoparticles. The method is based on the optical admittance by solving its differential equation for an appropriate refractive index profile. When the refractive index changes with depth z according to the function N(z) = N(0)/(1+z/ζ)2, an exact solution exists which is equivalent to the WKB (geometric optical) approximate solution derived for the electromagnetic field in weakly inhomogeneous media. Using this solution allows us to characterize transparent graded-index films by two optical parameters—the mean refractive index and the grade of inhomogeneity, in addition to the film thickness through a fast curve fitting process.

Compact Spreading Resistance Model for Rectangular Contacts on Uniform and Epitaxial Substrates

We present a compact analytical spreading resistance model for substrate noise coupling analysis. The model can handle rectangular contacts on uniform substrates of finite thickness with a grounded backplane. In contrast to previously published compact models, the model does not require extraction of fitting parameters. The model is also scalable with the resistivity and thickness of the substrate, and with the contact size. The model is verified with extensive finite-element calculations, and the accuracy is shown to be good. We also show that the model can predict the spreading resistance on epitaxial substrates.

Dynamics of films and droplets spreading on porous substrates

This paper numerically investigates spreading and sorption of films and droplets on an unsaturated porous substrate, a mechanism taking place in paper coating and ink jet printing. The dynamics of the fluid on the substrate surface are described by lubrication approximation. Fluid flow within the porous substrate, caused by the droplet penetrating into it, is treated with the Richards’ equation, which models unsaturated flow in porous media and describes fluid transport in the porous layer after complete absorption of the film or droplet. Numerical results for sorption of an axisymmetric droplet into a substrate of finite thickness show that a sharp localization of the penetrating fluid, as required for fine printing resolution, can be achieved by a steep capillary sorption curve. The interaction of the fluid with the impermeable bottom of the layer may drastically delay droplet absorption and thus require matching layer thickness and droplet volume to avoid limiting printing process speed.

Natural convection air-cooling of a substrate-mounted protruding heat source in a stack of parallel boards

The problem numerically investigated is that of steady, two-dimensional laminar natural convection in a system of parallel vertical channels with a single protruding heat module mounted mid-height on a substrate of finite-thickness. The input heat flux at the base of the module is removed through the module to the flow by convection and through the substrate by conduction. The system simulates cooling passages in a stack of printed circuit boards (PCB) with heat modules attached on the surface of one board. Heat transfer from both sides of the boards is taken into account. The solution is computed simultaneously in the solid (module and substrate) and in the fluid regions taking into account the continuity of the temperature and heat flux at the solid–fluid interfaces. The objective of this work is to determine heat transfer and fluid flow characteristics. A parametric study was conducted by varying the thermal conductivity and the thickness of the substrate and the width of the module. A variation in the substrate conductivity considerably affects the module temperature while the flow structure is only slightly altered. It is demonstrated that streamwise conduction through the substrate is an important cooling mechanism and must be accounted for.

Influence of bond coat thickness on the cyclic rumpling of thermally grown oxides

Recent experimental measurements have revealed that the amplitudes of undulations in the thermally grown oxide (TGO) formed on a bond coat subject to cyclic thermal histories depend on the bond coat thickness, hbc, and exhibit a maximum when hbc≈100μm. The existing rumpling model does not predict the maximum. To account for this effect, the model has been extended to include finite substrate thickness. The embellished code predicts the maximum and demonstrates close correspondence between calculated and measured undulation amplitudes (provided that all of the strain misfits between the bond coat and substrate are included: thermal expansion, martensite transformation and swelling). The presence of the maximum is attributed to two opposing effects. When thin, the bond coat is unable to deform to the extent needed to accommodate the undulations in the TGO. Conversely, when it has finite thickness relative to the substrate, the strains induced in the substrate reduce the constraint imposed on the bond coat, again reducing its ability to accommodate the undulations.