Maxwell-Garnett Approximation Research Articles

Contribution of Embedded Bromine Molecules to the Dielectric Properties of Fluorinated Graphite Intercalation Compounds

Here, we study the dielectric properties of fluorinated graphites C2Fx with different fluorine contents (x ≈ 1.05, 0.85, and 0.60) and embedded bromine molecules by impedance spectroscopy. An analysis of the results using the Maxwell-Garnett approximation made it possible to estimate the contributions of the fluorinated graphite matrix and bromine guests to the dielectric permittivity of the compound. The permittivity of the C2Fx matrix increases with decreasing fluorine content and does not depend on temperature. The change in the permittivity upon cooling/heating of the samples is provided by the polarization of bromine molecules Br2. The significant temperature-dependent dielectric response of C2F0.60 is associated with the formation of bromine and polybromide ions, such as Br2– and Br3–, in the interlayer space of the matrix. The decrease in the permittivity at low temperature is explained by the freezing of the ion mobility.

Optical Parameters of Atomically Heterogeneous Systems Created by Plasma Based Low Energy Ion Beams: Wavelength Dependence and Effective Medium Model

The article presents the irradiation effects of low energy (∼0.5 keV) inert gaseous Argon ion beams on optical constants [real (n) and imaginary (k) parts of the refractive index], dielectric constants, skin depth, and optical conductivity of copper (Cu), silver (Ag), and aluminum (Al) metallic thin films (MTF). The optical constants of pristine MTF are obtained by employing the universal Kramers-Kronig (KK) technique. The reflectivity of pristine MTF measured using UV-VIS-NIR spectrophotometry is used as an input parameter in the KK technique to determine the optical constants as a function of energy [or wavelength (λ)] of incident light ranging between ∼1–4.96 eV (or 250–1,200 nm). For the irradiated MTF, the optical constants including the skin depth (δ = λ/2πk), optical conductivity (σ = nkc/λ), and dielectric constants (ϵ1 = n2 − k2 and ϵ2 = 2nk) with varying ion fluence have been investigated by implementing the Maxwell-Garnett (MG) approximation, used to determine the effective dielectric constants of a random mixture of two different mediums. Additionally, n and k obtained from MG approximation have been compared with those obtained using the pseudo- Brewster angle technique for four different laser wavelengths (405, 532, 632.8 and 670 nm) and are found to be in good agreement with each other. It is observed that the optical constants and optical conductivity of the MTF decrease with increase in ion beam fluence, while the skin depth increases. Besides the optical constants, the behavior of skin depth, dielectric constants, and optical conductivity of the irradiated MTF with varying fluence are discussed in this article.

Rietveld refinement analysis and influence of individual phases on the magnetoelectrically coupled (1-x)BSZT+ xNCZGF multiferroic composites

The magnetoelectric voltage coefficient at room-temperature is realized in various (1-x)BSZT+ xNCZGF composites by studying refined crystal structures, surface morphology, magnetic, electric and ferroeletric properties. The ferroelectric (BSZT) and ferrite (NCZGF) phases of the composites are prepared using standard solid-state reaction technique. Rietveld refinement confirmed that the BSZT phase forms perovskite structure with P4mm space group and NCZGF phase forms spinel structure with Fd3¯m space group. From FESEM surface morphology it was observed that the average grain diameter ranges 0.5–2.21 µm. EDX analysis confirms the formation of respective phases and composites. Initial permeability increases from 55 to 130 with increasing NCZGF content which is consistent with the value calculated theoretically using different models. The saturation magnetization also increases from 12 to 37 emu/g with increasing NCZGF content, consistent with the calculated value using law of approach to saturation technique. The dielectric constant of the composites is found to decrease with increasing NCZGF content which is in line with the Maxwell-Garnett, Bruggeman, Looyenga and Lichteneker theoretical approximations. Frequency dependence of the dielectric constant exhibits dispersion, which can be modeled using a modified Debye function that considers the probability of several ions contributing to the relaxation. The ac conductivity follows the Jonscher’s power law, and the conduction mechanism is attributed to the small polaron hopping. Complex impedance is evaluated using the Cole-Cole plot ensures the dominance of both grain and grain boundary resistances. The P-E hysteresis loops ensure the typical ferroelectric nature at room-temperature for x ≤ 0.2. The simultaneous M-H and P-E hysteresis loops confirm the magnetoelectric coupling between ferrite and ferroelectric phases. Furthermore, the maximum magnetoelectric voltage coefficient for 0.9BSZT+ 0.1NCZGF composite is found to be ~194 mV cm−1 Oe−1 that can be used in multifunctional devices.

Reflection of Light from Gold and Silver Plasmon Films: A Comparison of Experiment and Theory

The spectra of light reflection by plasmonic metal films deposited in vacuum on dielectric substrates and subsequently annealed at high temperatures are investigated. To explain the optical properties of plasmonic films, a theoretical model of a plane-layered metal-dielectric medium is used. One of the layers of such a structure has the dielectric constant of the metal under study, and the second layer has the dielectric constant calculated within the Maxwell-Garnett approximation for a heterogeneous medium consisting of a suspension of metallic nanoellipsoids in air. The introduction of this layer makes it possible to qualitatively take into account the inhomogeneities of the plasmon film surface. It is shown that the developed model allows a number of features in the experimental reflection spectra to be explained.

Evolution of Shape and Volume Fraction of Superconducting Domains with Temperature and Anion Disorder in (TMTSF)2ClO4

In highly anisotropic organic superconductor (TMTSF)2ClO4, superconducting (SC) phase coexists with metallic and spin-density wave phases in the form of domains. Using the Maxwell-Garnett approximation (MGA), we calculate the volume ratio and estimate the shape of these embedded SC domains from resistivity data at various temperature and anion disorder, controlled by the cooling rate or annealing time of (TMTSF)2ClO4 samples. We found that the variation of cooling rate and of annealing time affect differently the shape of SC domains. In all cases the SC domains have oblate shape, being the shortest along the interlayer z-axis. This contradicts the widely assumed filamentary superconductivity along the z-axis, used to explain the anisotropic superconductivity onset. We show that anisotropic resistivity drop at the SC onset can be described by the analytical MGA theory with anisotropic background resistance, while the anisotropic Tc can be explained by considering a finite size and flat shape of the samples. Due to a flat/needle sample shape, the probability of percolation via SC domains is the highest along the shortest sample dimension (z-axis), and the lowest along the sample length (x-axis). Our theory can be applied to other heterogeneous superconductors, where the size d of SC domains is much larger than the SC coherence length ξ, e.g., cuprates, iron-based or organic superconductors. It is also applicable when the spin/charge-density wave domains are embedded inside a metallic background, or vice versa.

A Scaled Effective Medium Theory for Calculation of Resonant and Relaxation Frequency in Magnetic Composites

Scaled Effective Medium Theory (ScEMT) is applied to predict magnetic resonant and relaxation frequency in polymer-magnetic particle composites with favorable comparison to measured data. A single scaling function is identified that uses magnetic particulate resonant and relaxation frequencies, or magnetization and anisotropy field, volume fraction and DC susceptibility as predicted by ScEMT. Previous publications demonstrated that ScEMT improved the prediction of DC susceptibility as compared to classical models. Maxwell-Garnett (MGT) and Coherent Model Approximation (CMA) serve as theoretical baselines for comparison. However, both require separate scaling functions in their prediction of resonant and relaxation. Measured data are presented that suggest a single scaling function of ScEMT is sufficient to calculate both parameters. The paper emphasizes the application of the models and shows a wide range of particulate chemistries. ScEMT calculates susceptibility, resonant and relaxation frequency that agree with measurement. The paper concludes by predicting dispersive permeability that represents improvement over both CMA and MGT models. Future studies will address formulation of an EMT model (s) that describe mixtures of hard and soft magnetic materials mixed in a polymer composite. EMT for composite dielectric properties will be expanded to address the chaining. Early results of that effort will be reported in a separate paper.

Отражение света от плазмонных пленок золота и серебра: сравнение эксперимента и теории

The spectra of light reflection by plasmonic metal films deposited in vacuum on dielectric substrates and subsequently annealed at high temperatures are investigated. To explain the optical properties of plasmonic films, a theoretical model of a flat-layered metal-dielectric medium is used. One of the layers of such a structure has the dielectric constant of the metal under study, and the second layer has the dielectric constant calculated within the Maxwell-Garnett approximation for a heterogeneous medium consisting of a suspension of metallic nanoellipsoids in air. The introduction of this layer makes it possible to qualitatively take into account the inhomogeneities of the plasmonic film surface. It is shown that the developed model makes it possible to explain a number of features in the experimental reflection spectra

Polaritonic cylinders as multifunctional metamaterials: Single scattering and effective medium description

Polaritonic materials, owing to a strong phonon-polariton resonance in the THz and far-infrared parts of the electromagnetic spectrum, offer both high-index dielectric and metallic response in this regime. This complex response makes them suitable candidates for the design of metamaterial-related phenomena and applications. Here we show that one type of polaritonic-material-based structures that are particularly suitable for the achievement of a wide range of metamaterial properties are systems of polaritonic rods. To study the interplay between the material and the structural resonances in such systems we employ as model systems rods of LiF and SiC and we calculate first the scattering properties of a single rod, identifying and discussing the behavior of the different resonances for different rod diameters. To analyze the response of ensembles of polaritonic rods we employ an effective medium approach based on the Coherent Potential Approximation (CPA), which is shown to be superior to the simple Maxwell-Garnett approximation for polaritonic and high-index dielectric metamaterials. Calculating and analyzing the CPA effective parameters, we found that our systems exhibit a large variety of interesting metamaterial properties, including hyperbolic dispersion, epsilon-near-zero and negative refractive index response. This rich response, achievable in almost any system of polaritonic rods, is highly engineerable by properly selecting the radius and the filling ratio of the rods, making polaritonic rod systems an ideal platform for demonstration of multifunctional metamaterials.

Effects of Density and Diameter on Surface Optical Phonon Modes in GaAs Nanowire Bundles.

We report the systematic investigation of the surface optical phonon modes in Au-catalyzed GaAs nanowires grown on an Au pre-patterned GaAs(111)B substrate using μ-Raman spectroscopy. We employed electron-beam dose rate as a control parameter during the substrate patterning step for adjusting the nanowire base diameter and coverage, which are independent from the nanowire growth conditions. We have experimentally studied the effect of the fill factor and average diameter on the surface optical phonon modes and explained the red-shift and broadening of the surface optical phonon frequencies by employing the dielectric continuum model. The surface optical phonon mode shift is exhibited to be sensitive to fill factor, rather than base diameter. The decrease in the average diameter from 280 nm to 180 nm results in the asymmetric broadening and red-shift of the surface optical phonon frequency (~1.83 cm-1) but the theoretical calculation from the isolated single nanowire-based dielectric continuum model cannot solely explain the behaviors of the surface optical phonon mode. In contrast, the change in the fill factor from 0.01 to 0.83 results in a shift of the surface optical phonon frequency (~6.5 cm-1) from the GaAs bulk value. The red-shift and asymmetric broadening of the surface optical phonons, in an agreement with the Maxwell-Garnett approximation, are consequences of dipolar interaction of randomly aligned neighboring nanowires and the polar nature of GaAs nanowire bundles. This work suggests the pre-patterning parameter dependent surface optical phonon characteristics of GaAs nanowire bundles which are of great importance in the nondestructive characterization of low-dimensional opto-electronic materials and devices.

Prediction of effective thermal conductivity of multicomponent tribocomposites taking into account contact thermal resistance between inclusions and matrix

A method is proposed for predicting the effective thermal conductivity of a matrix composite with several types of spherical inclusions with contact thermal resistance at the boundary of the matrix and inclusions. The method is based on a generalized effective-field approximation for an inhomogeneous medium with inclusions with a shell. Model calculations were performed for a matrix tribocomposite with two types of inclusions. Keywords: effective thermal conductivity, contact thermal resistance, composite material, matrix, inclusion with a shell, Maxwell—Garnett approximation, generalized effective-field approximation. alekc_sap@mail.ru

Dielectric tensor of a rectangular arrangement of Ag nanoparticles in anisotropic [formula omitted]: Analysis of the negative epsilon conditions

In the present paper we report the optical response of an ordered array of silver (Ag) nanoparticles (NPs) using the uniaxial LiNbO3 as a model case. We investigate the anisotropic effects on the effective dielectric tensor taking into account the charges interaction of particles. The Ag NPs dielectric function is described through a modified Drude model whereas the LiNbO3 dielectric functions are deduced from their experimentally established Sellmeier equations. The effective dielectric tensor components of the ensemble aggregates of Ag NPs and uniaxial LiNbO3 crystal are treated through the extended Maxwell–Garnett approximation. Following the asymmetric behaviour of uniaxial crystals, Ag NPs are sited in the ordinary plane of the crystal, giving rise to different responses in the x, y and z directions of applied electric field. Real and imaginary parts of the effective dielectric tensor components of the aggregate ensemble are investigated in terms of different structural parameters, such as the interparticle spacings, the NPs filling factor and sizes of spherically embedded nanoparticles. We demonstrate that an adequate choice of structural parameters such as NPs sizes, interparticle separation gaps or the filling factor can determine the optical properties of layered Ag–embedded uniaxial crystals such as LiNbO3. We show that the negative epsilon (NE) condition is satisfied from a critical size of Ag NPs when the filling factor and the interparticles distances (a and b) have particular values. This condition defines an interval of energies, called NE range, which clearly depend on values of structural parameters defined in the model. This NE range shows some type of bandwidth anisotropy when it is compared among the x-, y-, z- and xy-components of the effective dielectric tensor. We analyze some anisotropic features such as the bandwidth and shift resonance energies in the real and imaginary parts of the dielectric tensor when structural parameters change.

The IR plasmonic properties of sub-wavelength ITO rod arrays predicted by anisotropic effective medium theory

Simple three-layer Fresnel equations combined with Maxwell–Garnett approximation were applied to study the IR plasmonic properties of indium-tin-oxide (ITO) nanorods. By treating the anisotropic nanorod layer as a layer with an effective dielectric constant, and using anisotropic effective medium theory, we were able to accurately predict the surface plasmon resonance behavior of ITO nanorods with different nanorod length, spacing, and tilt angle. This model allows a fast and computationally inexpensive calculation to predict the plasmonic properties of arrayed nanorods.

Simulation of magneto-optical properties of nanocomposites (CoFeZr)x(Al2O3)1-x

The magnetooptical spectra of the transverse Kerr effect of (CoFeZr)x(Al2O3)1-x nanocomposites are calculated within the framework of the symmetrized Maxwell-Garnett (SMG) approximation. The quasi-classical size effect and the size distribution of the granules were taken into account. The calculation results for the spectral range 0.5-3.5 eV and different concentrations of the magnetic component are in semi-quantitative agreement with the experimental data. A possible reason for the existing discrepancy is the difference between the optical and magneto-optical parameters of ferromagnetic granules from those corresponding for bulk samples.

Nonlinear thermal conductivity of periodic composites

Composites have been widely used to realize various functions in thermal metamaterials, thus becoming important to predict heat transport properties according to geometric structures and component materials. Based on a first-principles approach, namely, the Rayleigh method, here we develop an analytical way to calculate temperature-dependent (i.e., nonlinear) thermal conductivities of a composite with circular inclusions arranged in a periodic rectangular array. We focus on both weak and strong nonlinearity. As a result, we find that the temperature-dependence (nonlinearity) coefficient of the whole periodic composite can be larger than that of the nonlinear component inside this composite. Simulation results from finite element analysis show that the Rayleigh method can be also more accurate than the Maxwell-Garnett or Bruggeman effective medium approximations. As a model application, we further tailor the nonlinearity to design a thermal diode, for which heat flux along one direction is much larger than that along the opposite. This work provides a different theory for handling periodic structure with thermally responsive thermal conductivities, and it could be useful for designing thermal metamaterials with diverse properties including rectification.

Symmetrized Maxwell–Garnett Approximation as an Effective Method for Studying Nanocomposites

The symmetrized Maxwell-Garnett (SMG) approximation is considered as the most optimal method of an effective medium for the description of nanocomposite structures. This approximation takes into account the microstructure of the sample, which makes it possible to calculate the metal-dielectric system. Thus, SMG describes with good accuracy the structure of the nanocomposite. Besides, this approximation is applicable for granular alloys consisting of metal components. As a result, this technique can be considered as a universal approximation to describe a wide class of nanostructured materials. At the same time, this article discusses various methods of effective environment. In these methods, the metal component of nanocomposites and the dielectric matrix are replaced by an effective medium with effective permittivity εeff. It is necessary that the particles (granules) in such structures be small in comparison with the wavelength of electromagnetic radiation incident on the sample. Based on this, the spectral dependences of the transverse Kerr effect (TKE) in magnetic nanocomposites were calculated with (CoFeZr)(Al2O3) structure as an example at different concentrations of the magnetic component. The simulation was carried out at small and large concentrations (below and above the percolation threshold). The spectral dependences were obtained taking into account the form factor of nanoparticles and the quasi-classical size effect. Besides, the authors note and discuss in this paper the contribution of various mechanisms that affect the type of spectra of the transverse Kerr effect. Using the symmetrized Maxwell-Garnett approximation, the effective values of the granule size of the nanocomposites under study were found, and the tensor of effective dielectric permittivity (TEDP) was calculated. The obtained TEDP values allowed to simulate the spectral dependences of the magneto-optical transverse Kerr effect. The authors discuss and draw conclusions about the features of the obtained spectral dependences in both the visible and infrared regions of the spectrum. In addition, the practical and fundamental importance of the obtained results is noted. The importance of effective medium methods for the study of optical, transport and magneto-optical properties of magnetic nanocomposites is shown.

Observation of plasmon-phonons in a metamaterial superconductor using inelastic neutron scattering

Metamaterial approach is capable of drastically increasing the critical temperature, Tc, of composite metal-dielectric superconductors. Tripling of Tc was observed in bulk Al-Al2O3 core-shell metamaterials. A theoretical model based on the Maxwell-Garnett approximation provides a microscopic explanation of this effect in terms of electron-electron pairing mediated by a hybrid plasmon-phonon excitation. We report the first observation of this excitation in Al-Al2O3 core-shell metamaterials using inelastic neutron scattering. This result provides support for this novel mechanism of superconductivity in metamaterials and explains the 50 year old mystery of enhanced Tc in granular aluminium films.

The Transverse Kerr Effect, Modeled in (CoFeZr)x(Al2O3)(1 −х) Nanocomposites

Spectral dependences of the transverse Kerr effect (TKE) are studied in (CoFeZr)x(Al2O3)(1 − х) magnetic nanocomposites. TKE spectra are modeled for an unannealed composite using the Maxwell–Garnett approximation (MGA) in an effective medium. TKE calculations are made by considering the particle sizes of the nanocomposite (the quasi-classical size effect). The experimental data are found to be consistent with the theoretical TKE spectra.

A shape model of internally mixed soot particles derived from artificial surface tension

Abstract. To retrieve the physical properties of aerosols from multi-channel ground-based and satellite measurements, we developed a shape model of coated soot particles and created a dataset of their optical properties. Bare soot particles were assumed to have an aggregate shape, and two types of aggregates with different size–shape dependences were modeled using a polyhedral Voronoi structure. To simulate the detailed shape properties of mixtures of soot aggregates and adhered water-soluble substances, we propose a simple model of surface tension derived from the artificial surface potential. The light-scattering properties of the modeled particles with different volume fractions of water-soluble material were calculated using the finite-difference time-domain method and discrete-dipole approximation. The results of the single-scattering albedo and asymmetry factors were compared to those of conventional internally mixed spheres (i.e., effective medium spheres based on the Maxwell-Garnett approximation and simple core-shell spheres). In addition, the lidar backscattering properties (i.e., lidar ratios and linear depolarization ratios) of the modeled soot particles were investigated. For internally mixed soot particles, the lidar backscattering properties were sensitive to the shape of the soot particles and the volume mixing ratio of the assumed water-soluble components. However, the average optical properties of biomass smoke, which have been reported from in situ field and laboratory measurements, were difficult to explain based on the individually modeled particle. Nonetheless, our shape model and its calculated optical properties are expected to be useful as an alternative model for biomass smoke particles in advanced remote sensing via multi-channel radiometer and lidar measurements.

Absorption of electromagnetic radiation by a nanoparticle in a nanocomposite: Going beyond the Maxwell-Garnett approximation

A model has been proposed for calculating the polarizability of a metal nanoparticle in a composite consisting of a dielectric matrix and monodisperse nanoparticles with more exact allowance for the influence of neighboring particles going beyond the Maxwell-Garnett approximation. The main attention has been focused on the imaginary component of the polarizability, which determines electromagnetic radiation absorption by a nanoparticle in the dipole approximation. It has been shown that the absorption may be noticeably enhanced or diminished depending on the incident radiation frequency.

Effect of porosity on the electrical conductivity of LAMOX materials

Porous La2Mo1.5W0.5O9 (LMW05) samples with different levels of densification were obtained by varying the sintering temperature in the range of 750 to 1100°C. Impedance spectroscopy was used to evaluate the effect of porosity on the electrical response of LMW05-type compounds. The bulk conductivity was found to depend on the microstructure and the pore volume fraction. Effective medium theories (Bruggeman and Maxwell–Garnett approximations) were used to predict the macroscopic properties of the heterogeneous medium. The Bruggeman approximation proved to be better suited to predicting the bulk electrical response of porous LMW05 samples in the pore volume fraction range of 0 to 0.22. Compared with other ceramic materials, LMW05 was found to have a low blocking factor (αR). This characteristic of LMW05 could be related to the microstructure of ceramic materials. Two porous YSZ model samples with the same porosity but different pore morphologies were used to explain why microstructural defects (MD) have a small impact on the electrical properties of LMW05.