Effective Medium Approximation Model Research Articles

Spectroscopic ellipsometry study of Si nanocrystals embedded in a SiOx matrix: Modeling and optical characterization

In this work, we report on a spectroscopic ellipsometry study of thin films of silicon nanocrystals embedded in a SiOx matrix that is performed to better understand how substoichiometric SiOx species affect the optical properties of these systems. The silicon nanocrystals are fabricated in a SiOx matrix by thermal annealing of magnetron-sputtered thin films in a wide range of x values (0<x<2) in order to produce every possible substoichiometric SiOx species in various fractions. A generic optical model to describe the thin film content with various SiOx volumetric fractions is developed and used to analyze the data obtained from the spectroscopic ellipsometry. Tauc-Lorentz and Bruggeman effective medium approximation models are employed to determine the ellipsometric angles (Ψ and Δ) and the complex dielectric function. We show that the optical properties of the nc-Si embedded in a SiOx matrix are highly dependent on the value of x and on the volumetric fractions of the substoichiometric SiOx species present in the system.

Optical properties and laser damage threshold of HfO2–SiO2 mixed composite thin films

HfO(2)-SiO(2) mixed composite thin films have been deposited on fused silica substrate by co-evaporation of HfO(2) and SiO(2) through the reactive electron-beam evaporation technique. The composition-dependent refractive index and the absorption coefficient have been analyzed using different effective medium approximation (EMA) models in order to evaluate the suitability of these models for such mixed composite thin films. The discrepancies between experimentally determined and EMA-computed values are explained through microstructural and morphological evolutions observed in these mixed composite films. Finally, the dependence of the laser damage threshold as a function of silica content has been investigated, and the improved laser-induced damage threshold for films having more than 80% silica content has been explained through the defect-assisted multiphoton ionization process.

Assessment of effective-medium theories of ion-beam sputtered Nb&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;5&lt;/sub&gt;–SiO&lt;sub&gt;2&lt;/sub&gt; and ZrO&lt;sub&gt;2&lt;/sub&gt;–SiO&lt;sub&gt;2&lt;/sub&gt; mixtures

Single-layer mixture coatings of ZrO2–SiO2 and Nb2O5–SiO2 produced by the ion beam sputtering (IBS) deposition technique were investigated in detail. Effective medium approximation (EMA) models and two non-optical methods, namely Rutherford backscattering spectroscopy (RBS) and X-ray photoelectron spectroscopy (XPS) were applied for characterization of elemental composition of these films. The comparison of obtained results indicates discrepancies in atomic material concentrations. The reasons and potential sources of such discrepancies are discussed qualitatively and indicate limitations of optical models.

Assessment of effective-medium theories of ion-beam sputtered Nb&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;5&lt;/sub&gt;–SiO&lt;sub&gt;2&lt;/sub&gt; and ZrO&lt;sub&gt;2&lt;/sub&gt;–SiO&lt;sub&gt;2&lt;/sub&gt; mixtures

Single-layer mixture coatings of ZrO2–SiO2 and Nb2O5–SiO2 produced by the ion beam sputtering (IBS) deposition technique were investigated in detail. Effective medium approximation (EMA) models and two non-optical methods, namely Rutherford backscattering spectroscopy (RBS) and X-ray photoelectron spectroscopy (XPS) were applied for characterization of elemental composition of these films. The comparison of obtained results indicates discrepancies in atomic material concentrations. The reasons and potential sources of such discrepancies are discussed qualitatively and indicate limitations of optical models.

Importance of diffuse scattering phenomena in moth-eye arrays for broadband infrared applications

Moth-eye (ME) arrays with varying aspect ratios and profile heights were fabricated in Si using a general colloidal lithography and reactive ion etching technique. Antireflective (AR) properties of the arrays were rigorously assessed from the near to far infrared (λ=2-50 μm) using transmission and reflection measurements via dispersive and Fourier transform infrared spectroscopy and modeled using an effective medium approximation (EMA). Infrared transmission of low aspect ratio structures (~2) matched the EMA model, indicating that the most important factor for AR at higher wavelengths is structure height. High aspect ratio structures (>6) were highly transmissive (>90% of theoretical maximum) over a large bandwidth in the mid-infrared (20-50 μm). Specular reflectance, total transmission, and diffuse reflectance (DR) measurements indicate that ME structures do not reach the theoretical maximum at near-infrared wavelengths due to DR and forward scattering phenomena. Ultimately, correlating optical performance with feature geometry (pitch, profile, height, etc.) over multiple length scales allows intelligent design of ME structures for broadband applications.

Investigation of low field dielectric properties of anisotropic porous Pb(Zr,Ti)O3 ceramics: Experiment and modeling

Anisotropic porous Pb(Zr,Ti)O3 ceramics with various porosity degrees have been studied in order to determine the role of the pore shape and orientation on the low-field dielectric properties. Ceramic samples with formula Pb(Zr0.52Ti0.48)0.976Nb0.024O3 with different porosity degrees (dense, 10%, 20%, 40% vol.) have been prepared by solid state reaction. Taking into consideration the shape and orientation of the pore inclusions, the dielectric properties of porous ceramics have been described by using adapted mixing rules models. Rigorous bounds, derived on the basis on Variational Principle, were used to frame dielectric properties of porous composites. The finite element method (FEM) was additionally used to simulate the dielectric response of the porous composites under various applied fields. Among the few effective medium approximation models adapted for anisotropic oriented inclusions, the best results were obtained in case of needle-like shape inclusions (which do not correspond to the real shape of microstructure inclusions). The general case of Wiener bounds limited well the dielectric properties of anisotropic porous composites in case of parallel orientation. Among the theoretical approaches, FEM technique allowed to simulate the distribution of potential and electric field inside composites and provided a very good agreement between the computed permittivity values and experimental ones.

Thermal resistance and phonon scattering at the interface between carbon nanotube and amorphous polyethylene

We report a molecular dynamics study of heat conduction in carbon nanotube (CNT)/polyethylene (PE) composites. Particular attention was focused on two key interfacial effects: the thermal boundary resistance across the CNT/PE interface, and the reduction of CNT thermal conductivity due to boundary phonon scattering at the interface with amorphous PE matrix. The calculations reveal a relatively high thermal boundary resistance across the CNT/polyethylene interfaces (∼1×10−7m2KW−1) and a moderate but non-negligible reduction (∼28%) of CNT thermal conductivity. By a mode-dependent phonon transport analysis, the phonon boundary scattering rate was found to increase with decreasing frequency, which suggests a size effect in the thermal conductivity reduction. Finally, the likely impact of the thermal resistances on the effective thermal conductivity of a macroscopic CNT/PE composite is quantified based on an effective medium approximation model.

Si nanocrystal-based triple-layer anti-reflection coating for Si solar cells

A triple-layer anti-reflection coating (TL-ARC) with Si nanocrystals (nc-Si)-dielectric nanocomposite thin film structure is proposed for Si solar cells. The TL-ARC has a graded refractive index (RI) profile of high RI, medium RI, and low RI. Such RI profile is achieved with the structure consisting of a Si3N4 layer embedded with high concentration of nc-Si and another Si3N4 layer embedded with low concentration of nc-Si and a SiO2 layer. The design of the TL-ARC is carried out with the calculation of the effective indices of the high-RI and medium-RI layers with the Maxwell-Garnett effective medium approximation model. Due to the photoluminescence properties of nc-Si embedded in Si3N4 matrix, the TL-ARC has the inherent capability of down-converting ultraviolet photons to low-energy photons that are useful to Si solar cells. The deposition of the TL-ARC on Si solar cell is fabricated with plasma enhanced chemical vapor deposition in a single process step. The performance enhancement of Si solar cells by the TL-ARC has been demonstrated by experiments.

Novel Cu–Cr alloy matrix CNT composites with enhanced thermal conductivity

Carbon nanotubes (CNTs) are incorporated into the Cu–Cr matrix to fabricate bulk CNT/Cu–Cr composites by means of a powder metallurgy method, and their thermal conductivity behavior is investigated. It is found that the formation of Cr3C2 interfacial layer improves the interfacial bonding between CNTs and Cu–Cr matrix, producing a reduction of interfacial thermal resistance, and subsequently enhancing the thermal conductivity of the composites. The thermal conductivity of the composites increases by 12 % and 17 % with addition of 5 vol.% and 10 vol.% CNTs, respectively. The experimental results are also theoretically analyzed using an effective medium approximation (EMA) model, and it is found that the EMA model combined with a Debye model can provide a satisfactory agreement to the experimental data.

Charge Transport in ${\hbox{TiO}}_{2}$ Films With Complex Percolation Pathways Investigated by Time-Resolved Terahertz Spectroscopy

The depolarization fields play an important role in terahertz experiments on nanostructured samples with complex nanoparticle morphologies and percolation pathways. Namely, their effects can hide or distort peculiarities of nanoscopic charge transport in the spectra measured on these structures. We calculate the local fields for a large number of percolated and non-percolated two-dimensional model structures by numerical solving of Maxwell equations in the quasi-static limit. The results strongly suggest that in a broad family of structures a simple effective medium approximation model can be applied to characterize the effective response. The model consists in an equivalent circuit composed of a resistance accounting for the percolated chains with an additional parallel RC-branch describing the non-percolated part. The physical meaning of this model is discussed in the frame of the Bergman spectral representation of effective medium. We show a recipe for the retrieval of a response connected to the depolarization fields and to the nanoscale transport mechanisms from transient terahertz spectra. Finally, we use the model to interpret our THz photoconductivity spectra in various <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\hbox{TiO}}_{2}$</tex> </formula> films with nanofabricated percolation pathways.

Determination of bone mineral volume fraction using impedance analysis and Bruggeman model

Abstract Measurements by impedance spectroscopy and Bruggeman effective medium approximation model were employed in order to determine the mineral volume fraction of dry bone. This approach assumes that two or more phases are present into the composite: the matrix (environment) and the other ones are inclusion phases. A fragment of femur diaphysis dense bone from a young pig was investigated in its dehydrated state. Measuring the dielectric properties of bone and its main components (hydroxyapatite and collagen) and using the Bruggeman approach, the mineral volume filling factor was determined. The computed volume fraction of the mineral volume fraction was confirmed by a histogram test analysis based on the SEM microstructures. In spite of its simplicity, the method provides a good approximation for the bone mineral volume fraction. The method which uses impedance spectroscopy and EMA modeling can be further developed by considering the conductive components of the bone tissue as a non-invasive in situ impedance technique for bone composition evaluation and monitoring.

Model and simulation predictions of the thermal conductivity of compact random nanoparticle composites

The lattice thermal conductivity of compact random nanoparticle composites was investigated numerically as well as theoretically. By taking advantage of the three-dimensional Vorogoni diagrams, we were able to generate realistic numerical models of the composites. The phonon motion was then simulated in use of a Monte-Carlo simulator which employed an unstructured grid system (tetrahedron cells) and was developed under the single relaxation time approximation and the grey medium approximation. On the other hand, a new effective medium approximation (EMA) model was proposed in the present work to predict the thermal conductivity of nanoparticle composites which particles are all made of a same material type. This EMA model was next combined with the EMA model proposed by Liang and Ji (2000) for three bond percolation systems to predict the thermal conductivity of nanoparticle composites of two material types. It was found that with an effective particle diameter characterizing either the average particle volume or the average particle surface area, the model predictions agree excellently with the Monte-Carlo simulation results and match well with the measurements of nanostructured bulk alloys.

Temperature dependent optical properties of Si nanocrystals embedded in SiO2 matrix

The optical properties of Si nanocrystals (nc-Si) with different sizes (2∼5 nm) embedded in a SiO2 matrix synthesized by the SiOx/SiO2 superlattice approach were studied in a temperature range from room temperature to 600 K by spectroscopic ellipsometry. The Maxwell–Garnett effective medium approximation and Lorentz oscillator model were employed to extract the dielectric function of nc-Si. The results show that the temperature dependence of optical properties of nc-Si is sensitive to its size. An empirical expression of Varshni approximation was obtained to characterize the band-gap energy change of nc-Si at different temperatures.

Ellipsometry Study on Nanoparticles Grown by Atomic Layer Deposition

ABSTRACTSpectroscopic Ellipsometry (SE) was chosen to study thin film growth in atomic layer deposition (ALD). It was shown that Cauchy model had limitations in predicting the ultrathin film thickness at initial few deposition cycles, and the fitting results depend on wavelengths range greatly. Effective Medium Approximation (EMA) model is capable of predicting ultrathin film’s physical properties. Our experiments on Al2O3 growth give supporting evidence on the applicability of EMA model, where it is used to successfully explain the initial nucleation and island like growth. EMA model can be extended to be used for Palladium thin film, which can give reasonable thickness and void content.

Low field permittivity of ferroelectric-ferrite ceramic composites: Experiment and modeling

The dielectric properties of xNiFe2O4-(1−x)Pb0.988(Zr0.52Ti0.48)0.976Nb0.024O3 ceramic composites with different volume filling factors (x = 2, 5, 10, 20, 30, 40, 50, 60, 70 wt. %) prepared by sol-gel method have been investigated in the frequency range (106 ÷ 109) Hz. A reducing effective permittivity with increasing the ferrite amount was observed as result of the “sum property.” The experimental results have been compared with the effective permittivity values predicted by effective medium approximation (EMA) models and finite element method (FEM) calculations. The comparison was aimed to evaluate the appropriateness of the results of the two approaches in describing the effective dielectric properties of the composite with contrasting permittivities and to understand the role of microstructural characteristics on the dielectric response in a broad filling factor range. The best description of the experimental data in overall filling factor range is given by the results of FEM calculations which used realistic microstructures and among the EMA models, by the Lichtenecker's formula.

Modeling the thermal conductivity of graphene nanoplatelets reinforced composites

An analytical model is developed for the effective thermal conductivity of the composites with graphene nanoplates (GNPs) within the framework of differential-effective-medium (DEM) theory. Results of the present model are compared to an effective-medium-approximation (EMA) based model and available experimental results. Predictions on the effective thermal conductivity of GNP/PA-6 and GNP/epoxy composites are in good agreement with the experimental data. Moreover, the present model can well describe the large thermal conductivity enhancement in GNP composites and, in particular, the nonlinear dependence of effective thermal conductivity on the volume fractions of GNPs, which is superior to the EMA model.

Effective medium approximation model of sea foam layer microwave emissivity of a vertical profile

A sea foam layer produced by wave breaking consists of seawater-coated air bubbles, fluid water, and air. The non-uniformity and microstructure of the air–water mixture in a foam layer can cause some important effects on microwave or optical properties. Considering the vertical non-uniformity of the air–water volume, the air-volume fraction is derived as a function of the foam depth variable, coated air bubble velocity, and foam temperature using the gas convection–diffusion equation. For a vertical graded profile of the air volume fraction, we discuss the effects of the coated air bubble velocity parameter and the air–sea temperature difference on the air volume fraction. The results show that the coated air bubble velocity is a key parameter that widely modulates the air volume fraction. Furthermore, an effective medium approximation (EMA) of spherical shell microstructures is proposed to investigate the graded foam effective permittivity and the emissivity of the sea surface covered with the foam layer of the graded air volume fraction in a vertical profile, and good agreement is obtained on comparing the EMA results with the experimental data of foam layer microwave emissivities at frequencies 1.4, 10.8, and 36.5 GHz. In our EMA model, the depth average of graded foam permittivity is adopted. This model can produce reasonable results by only tuning the air bubble velocity. It indicates that the EMA model can combine the effects of graded foam permittivity and foam microstructures into the coated air bubble velocity parameter. Meanwhile, we have qualitatively discussed the influence of air–sea temperature difference on the brightness temperature of the foam layer. It is shown that a negative (or positive) air–sea temperature difference enhances (or reduces) the sea surface brightness temperature, and the brightness temperature difference increases with an increase in microwave frequency.

An investigation into the lattice thermal conductivity of random nanowire composites

The lattice thermal conductivity of compact random silicon and germanium nanowire composites was investigated by using a Monte-Carlo (MC) simulator, which was developed based on the gray medium approximation and unstructured grids. By defining the local equilibrium temperature as the one which preserves the local phonon energy in the interested heterogeneous subregion, we are able to obtain the effective thermal conductivity of random nanowire composites and explore the effects of the wire shape and the composition concentration on it. The results show that among the three kinds of wire shapes investigated – triangle, quadrangle, and voronoi, the random quadrilateral nanowire composites are the best thermal conductor under a fixed interface density or a fixed characteristic wire size and that the dependence of the effective thermal conductivity on the silicon volume concentration is approximately parabolic. The influences of these two factors are equally strong. Moreover, the parabolic dependence can be well explained by the effective medium approximation model for three bond percolation systems.

Preparation and photothermal characterization of nanocomposites based on high density polyethylene filled with expanded and unexpanded graphite: Particle size and shape effects

This work aimed at thermal transport characterization of high density polyethylene (HDPE) filled with 5 and 50 μm expanded graphite (EG) particles and with 0.4 μm unexpanded graphite (UG) particles. Sample platelets were produced by melt mixing followed by compression molding. Thermal conductivity k was determined by combining measurements of density, specific heat capacity and thermal diffusivity, the latter by modulated photothermal radiometry (PTR). Starting from an effective medium approximation model, we derived a linearized expression for the effective k of composites with low particle charge. It explains the unusually high experimental k values (up to four-fold increase) as the effect of strongly non-spherical EG particles (aspect ratio 1/p = 110–290). Larger particle sizes produce higher k enhancement, while the interfacial thermal resistance (Rbd = 2.1⋅10−7 m2⋅K/W) has an opposite effect. The same model is consistent with experimental k for low particle charge HDPE/UG composites. At higher particle charge the model fails due to particle interaction leading to validity break of the effective medium approximation.

The characterization of hydrogenated amorphous silicon and epitaxial silicon thin films grown on crystalline silicon substrates by using spectroscopic ellipsometry

Two series of silicon films on c-Si substrates with different thicknesses are deposited by radio frequency plasma enhanced chemical vapor deposition (rf-PECVD). The structures of samples are investigated by spectroscopic ellipsometry (SE) with fixed angle. The results show that the films are all amorphous for the first series of samples. In such a case an abrupt a-Si:H/c-Si heterojunction is formed which is beneficial for the passivation of the interface of HIT solar cell. For amorphous silicon films, the fitting result is acceptable by using Tauc-Lorentz Genosc model. For the second series, the films are of epitaxial Si at the initial deposition stages, the amorphous fraction increases with the increase of thickness. When the film thickness reaches a critical value of 46 nm, a transition to pure amorphous phase occurs. The epitaxial film shows excellent fitting by using the effective medium approximation (EMA) model under the assumption of the mixture of c-Si phases and a-Si:H. However, we obtain better fitting result by using a three-layer model, whose bulk layer is divided into EMA layer and a-Si layer after the transition to pure amorphous phase. This study indicates that the SE analysis, operated in different models, is effective to characterize different structures of silicon films on c-Si substrate.