Simple Effective Medium Approximation Research Articles

Doping effects on optical and conductive properties of titanium dioxide in terahertz range

Terahertz time domain spectroscopy (THz-TDS) has been used to investigate the effect of bismuth doping on the refractive index, absorption coefficient, dielectric constant, and the conductivity of TiO2. Simple effective-medium approximation along with THz-TDS has been utilized for the extraction of these parameters from the composite samples. These values are increasing with dopant concentration. Moreover, the fitting of conductivity with Drude-smith model revealed that the charge carriers are localized and back scattering is the major conductivity mechanism.

A simple model to predict effective conductivity of multicomponent matrix-based composite materials with high volume concentration of particles

For high-contrast-component-property composites, most popular effective medium approximations that are based on the dilute analytical solution for an ellipsoidal inclusion embedded in an infinite matrix diverge significantly from each other and from experimental and numerical data. This paper discusses the challenges in this research area and proposes some homogenization strategies to solve the problems. In this work, a simple effective medium approximation is constructed to predict the effective conductivity of multicomponent matrix-based composites containing high concentrations of particles in d-dimensional space (d=2,3). The maximum volume fraction that depends on particle size distribution and the geometric approximation are accounted for in the micromechanical model to obtain new closed-form solutions. A flexible version of the model containing a free parameter is proposed to make the predictions more accurate at very high-volume fraction of inclusions. Applications are illustrated by comparing the theoretical predictions with the available experimental data or finite element simulations for various types of material to show the agreeable results close to the maximal packing points of the interactions.

Simple effective medium approximation with Rayleigh scattering.

We present a simple analytical effective medium approximation (EMA) that can account for finite-sized and interacting inclusions in the Rayleigh scattering regime. The polarizability of an individual inclusion is calculated using the Mie solution and net polarizability is formed, leading to effective parameters. The Mie coefficients are expanded into polynomials and curtailed to the sixth order, which is accurate in the Rayleigh regime. Results obtained using the developed models are compared with experimental results and existing EMAs. The analytical formula outperforms other EMAs for large inclusion sizes, high filling factors, and high relative refractive indices.

High-Order Hilbert Curves: Fractal Structures with Isotropic, Tailorable Optical Properties

© 2015 American Chemical Society. Fractals are promising candidates as nonperiodic, nonresonant structures exhibiting a homogeneous, isotropic, and frequency-independent effective optical response. We present a comprehensive optical investigation of a metallic Hilbert curve of fractal order N = 9 in the visible and near-infrared spectral range. Our experiments show that high-order fractal nanostructures exhibit a nearly frequency independent reflectance and an in-plane isotropic optical response. The response can be simulated in the framework of a simple effective medium approximation model with a limited number of parameters. It is shown that high-order Hilbert structures can be considered as a transparent in-plane metal, the dielectric function of which is modified by the filling factor f, hence creating a tunable conductive effective metal with tailorable plasma frequency and variable reflectance without going through an insulator-to-metal transition.

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.

Structure of plasmonic aerogel and the breakdown of the effective medium approximation

A method for making aerogel doped with gold nanoparticles (GNPs) produces a composite material with a well-defined localized surface plasmon resonance peak at 520 nm. The width of the extinction feature indicates the GNPs are well dispersed in the aerogel, making it suited to optical study. A simple effective medium approximation cannot explain the peak extinction wavelengths. The plasmonic field extends on a scale where aerogel cannot be considered isotropic, so a new model is required: a 5 nm glass coating on the GNPs models the extinction spectrum of the composite material, with air (aerogel), methanol (alcogel), or toluene filling the pores.

Second Critical Point in First Order Metal-Insulator Transitions

For first order metal-insulator transitions we show that, together with the dc conductance zero, there is a second critical point where the dielectric constant becomes zero and further turns negative. At this point the metallic reflectivity sharply increases. The two points can be separated by a phase separation state in a 3D disordered system but may tend to merge in 2D. For illustration we evaluate the dielectric function in a simple effective medium approximation and show that at the second point it turns negative. We reproduce the experimental data on a typical Mott insulator such as MnO, demonstrating the presence of the two points clearly. We discuss other experiments for studies of the phase separation state and a similar phase separation in superconductors with insulating inclusions.

Local field distribution near corrugated interfaces: Green function formalism versus effective medium theory

The recent Green function formalism (GFF) has been used to study the local field distribution near a periodic interface separating two homogeneous media of different dielectric constants. In the GFF, the integral equations can be solved conveniently because of the existence of an analytic expression for the kernel (Greenian). However, due to a severe singularity in the Greenian, the formalism was formerly applied to compute the electric fields away from the interface region. In this work, we have succeeded in extending the GFF to compute the electric field inside the interface region by taking advantage of a sum rule. To our surprise, the strengths of the electric fields are quite similar in both media across the interface, despite of the large difference in dielectric constants. Moreover, we propose a simple effective medium approximation (EMA) to compute the electric field inside the interface region. We show that the EMA can indeed give an excellent description of the electric field, except near a surface plasmon resonance.

Anisotropic Dielectric Function in Polar Nanoregions of Relaxor Ferroelectrics

The Letter suggests treating the infrared reflectivity spectra of single crystal perovskite relaxors as fine-grained ferroelectric ceramics: locally frozen polarization makes the dielectric function strongly anisotropic in the phonon frequency range and the random orientation of the polarization at nanoscopic scale requires taking into account the inhomogeneous depolarization field. Employing a simple effective medium approximation (the Bruggeman symmetrical formula) turns out to be sufficient for reproducing all principal features of room temperature reflectivity of Pb(Mg(1/3)Nb(2/3))O3. One of the reflectivity bands is identified as a geometrical resonance entirely related to the nanoscale polarization inhomogeneity. The approach provides a general guide for systematic determination of the polar mode frequencies split by the inhomogeneous polarization at the nanometer scale.

Modeling of magneto-optical properties of periodic nanostructures

Optical and magneto-optical (MO) response from nanometer-size grating is calculated using improved Fourier modal method. The MO grating is modeled by an anisotropic MO layer of effective medium. The complex refractive indices and MO parameter of the effective layer are compared with simple effective medium approximation. Dependence of the optical and MO parameters on the filling factor of the grating is presented. MO grating with period in the sub-wavelength scale could be used as an effective medium with desired optical and MO properties.

Direct Determination of Effective Interfacial Optical Constants by Nonlinear Optical Null Ellipsometry of Chiral Films

Nonlinear optical null ellipsometry (NONE) measurements of chiral interfaces allowed direct experimental measurement of the linear interfacial optical constants in surface second harmonic generation (SHG) measurements. Since phase information is retained in NONE measurements, the real and imaginary components of the interfacial refractive index (n and k, respectively) were uniquely obtained from the measured chiral chi((2)) tensor elements of a fluorescein-labeled bovine serum albumin film. The sensitivity of the calculated chi((2)) tensor elements on the assumed values of the interfacial optical constants allowed measurements of n and k to four significant figures with no additional adjustable parameters and independent of molecular symmetry. The optical constants measured by SHG agreed within a relative error of 0.8% with values predicted independently using a simple effective medium approximation, also with no adjustable parameters. Additionally, those same optical constants produced relationships between the achiral chi((2)) tensor elements in excellent agreement with predictions for systems exhibiting weak orientational order. This study suggests that the far-field intensity and polarization state of the nonlinear optical beam may be largely independent of the near-field optical constants within the interfacial layer in the limit of a film thickness much less than the wavelength of light.

Weighted effective-medium approximations for elastic quasisymmetric completely random composites

Abstract The envelopes of the realizable effective-medium hierachical models contain a major central part in the middle between the available bounds on the elastic moduli of isotropic quasisymmetric inhomogeneous materials and hence are expected to cover most (if not all) the property range for the realistic quasisymmetric completely random composites, the constituents of which are combined randomly (i.e. in the utmost uncorrelated manner). A simple effective-medium approximation scheme, based on a weighted sum of the extreme models and containing a few geometric weight parameters, is proposed to describe the effective curves for the mixtures. The geometric parameters can be evaluated from the respective dilute suspension results, or from measurements of the effective properties at some proportions of the components of particular composites.

Renormalized effective-medium theory for weakly nonlinear composites.

The effective response is calculated in random nonlinear conductance networks consisting of two kinds of conductors obeying a current-voltage ($I\ensuremath{-}V$) characteristic of the form $I={\ensuremath{\sigma}}_{i}V+{\ensuremath{\chi}}_{i}{V}^{3}$, ($i=1, 2$). The effective linear response ${\ensuremath{\sigma}}_{e}$ and nonlinear response ${\ensuremath{\chi}}_{e}$ are calculated in a simple effective-medium approximation (EMA); the results are compared with numerical simulations in nonlinear conductance networks. It is found that while the simple EMA fits the simulation data of ${\ensuremath{\sigma}}_{e}$ reasonably well over the entire range of volume fraction, strong deviations are obtained for ${\ensuremath{\chi}}_{e}$. In an attempt to improve the EMA, we invoke a real-space renormalization-group (RG) analysis to rescale the network by small-cell transformations and bring it farther away from percolation, where EMA is applicable. The renormalized EMA (REMA) formalism, which couples RG and EMA, is able to extend the range of validity of EMA substantially. Results from the REMA are compared with numerical simulations. We find a remarkable improvement over the simple EMA in the entire range of volume fraction, especially at high contrast between the components.